Variance Premium and Implied Volatility in a Low-Liquidity Option Market

We propose an implied volatility index for Brazil that we name “IVol-BR”. The index is based on daily market prices of options over Ibovespa—an option market with relatively low liquidity and few option strikes. Our methodology combines standard international methodology used in high-liquidity markets with adjustments that take into account the low liquidity in Brazilian option markets. We do a number of empirical tests to validate the IVol-BR. First, we show that the IVol-BR has significant predictive power over future volatility of equity returns not contained in traditional volatility forecasting variables. Second, we decompose the squared IVol-BR into (i) the expected variance of stock returns and (ii) the equity variance premium. This decomposition is of interest since the equity variance premium directly relates to the representative investor risk aversion. Finally, assuming Bollerslev, Tauchen, & Zhou (2009) functional form, we produce a time-varying risk aversion measure for the Brazilian investor. We empirically show that risk aversion is positively related to expected returns, as theory suggests.

We propose an implied volatility index for Brazil that we name "IVol-BR". The index is based on daily market prices of options over Ibovespa-an option market with relatively low liquidity and few option strikes. Our methodology combines standard international methodology used in high-liquidity markets with adjustments that take into account the low liquidity in Brazilian option markets. We do a number of empirical tests to validate the IVol-BR. First, we show that the IVol-BR has significant predictive power over future volatility of equity returns not contained in traditional volatility forecasting variables. Second, we decompose the squared IVol-BR into (i) the expected variance of stock returns and (ii) the equity variance premium. This decomposition is of interest since the equity variance premium directly relates to the representative investor risk aversion. Finally, assuming Bollerslev, Tauchen, & Zhou (2009) functional form, we produce a time-varying risk aversion measure for the Brazilian investor. We empirically show that risk aversion is positively related to expected returns, as theory suggests.

INTRODUCTION
This is the first article to propose an implied volatility index for the stock market in Brazil. 1 We call our implied volatility index "IVol-BR". The methodology to compute the IVol-BR combines state-of-theart international methodology used in the US with adjustments we propose that take into account the low liquidity in Brazilian option market. The average daily volume traded in Ibovespa Index options is US$20 million 2 and, as consequence, few option strikes are traded. The methodology we propose can be applied to other low-liquidity markets. This is the first contribution of the paper.
The IVol-BR has good empirical properties. First, by regressing future realized volatility on the IVol-BR and a number of traditional volatility forecasting variables, we show that the IVol-BR does contain information about future volatility. Second, we decompose the squared IVol-BR into (i) the expected variance of stock returns and (ii) the equity variance premium (the difference between the squared IVol-BR and the expected variance). This decomposition is of interest since the equity variance premium directly relates to the representative investor's risk aversion. Then, we use such a decomposition to pin down a time-varying risk aversion measure for the Brazilian market. Finally, based on the results using a sample from 2011 to 2015, we show that both the risk aversion measure and the variance premium are good predictors of future stock returns in Brazil. 3 This is the second contribution of this paper.
An implied volatility index is useful for both researchers and practitioners. It is commonly referred to as the "fear gauge" of financial markets. The best known example is the VIX, first introduced by the Chicago Board of Options Exchange (CBOE) in 1993. 4 The squared implied volatility of a stock market reflects the dynamics of two very important variables. The first relates to the level, or quantity, of risk that the representative investor faces: the expected future variance of the market portfolio. The second relates to the price of such risk, the risk aversion of this investor.
The economic intuition for this is the following. Since options' payoffs are asymmetric, the value of any option is increasing in the expected variance of the underlying asset. Because of that, options are often used as a protection against changes in variance. Since the typical risk-averse investor dislikes variance, options are traded with a premium because of their insurance value. As a consequence, the squared implied volatility (which is directly computed from option prices) also has a premium with respect to the (empirical) expected variance: the first should always be higher than the second. This is the so-called "variance premium". The more the investor dislikes variance, the more she is willing to pay for the insurance that options provide. Therefore, the higher the risk aversion, the higher the variance premium (see, for instance, Bollerslev et al., 2009 andBollerslev, Gibson, &Zhou, 2011).
We decompose the squared IVol-BR (hereinafter "IVar") into (i) the expected variance conditional on the information set at time t and (ii) the variance premium at time t . We estimate component (i) by searching for the best forecasting model for future variance, based on Bekaert & Hoerova (2014). Following the recent literature on variance forecasting (Chen & Ghysels, 2011;Corsi, 2009;and Corsi & Renò, 2012), we use high-frequency data for this task. Then, we compute the variance premium, i.e. component (ii), as the difference between the implied variance and the estimated expected variance. Finally, we use a closed-form equation for the variance premium, based on Bollerslev et al. (2009), which is an increasing function of the risk aversion coefficient of the representative investor, to pin down a time-varying risk aversion measure of the representative investor in the Brazilian market.
Our paper relates to Bekaert & Hoerova (2014) that shows the variance premium calculated with an econometric model for expected variance has a higher predicting power over future returns than the variance premium calculated under the assumption that the expected variance follows a random walk process, as in Bollerslev et al. (2009). In particular, we confirm Bekaert & Hoerova (2014) finding that the predicting properties of the variance premium can be significant as early as a month in advance rather than only at the quarterly frequency found by Bollerslev et al. (2009). However, we go beyond Bekaert & Hoerova (2014) and directly relate the risk aversion series obtained from the variance premium with future returns. We show the risk-aversion series is a strong predictor of future returns with a slightly superior fit than the variance premium.
The US evidence on the predicting properties of the variance premium, first shown by Bollerslev et al. (2009), has recently been extended to international developed markets. Bollerslev, Marrone, Xu, & Zhou (2014) finds that the variance premium is a strong predictor of stock returns in France, Germany, Switzerland, The Netherlands, Belgium, the UK and, marginally, in Japan. Our results confirm Bollerslev et al. (2014) findings for the Brazilian market. To the best of our knowledge, our study is the first to show this for an emerging economy.
The paper is divided as follows. Section 2 presents the methodology to compute the IVol-BR. Section 3 decomposes the squared IVol-BR into expected variance and variance premium, computes a timevarying risk-aversion measure of the representative investor in the Brazilian market, and documents the predictive power of both the variance premium and the risk-aversion measure over future stock returns. Section 4 concludes.

IMPLIED VOLATILITY INDEX FOR THE BRAZILIAN STOCK MARKET
The methodology we use to compute an index that reflects the implied volatility in the Brazilian stock market (called IVol-BR) combines the one used in the calculation of the "new" VIX (described in Carr & Wu, 2009) with some adjustments we propose, which take into account local aspects of the Brazilian stock market-mainly, a relatively low liquidity in the options over the Brazilian main stock index (Ibovespa) and, consequentially, a low number of option strikes.
Options over Ibovespa expire on even months: February, April, etc. Because of that, we compute the IVol-BR in order to reflect the implied volatility with a 2-month ahead horizon. It is calculated as a weighted average of two volatility vertices: the "near-term" and "next-term" implied volatilities of options over the Ibovespa spot. On a given day t , the near-term refers to the closest expiration date of the options over Ibovespa, while the next-term refers to the expiration date immediately following the near-term. For instance, on any day in January 2015, the near-term refers to the options that expire in February 2015, while the next-term refers to the options that expires in April 2015. 5 On Table 1 we show the quarterly daily averages of the number and financial volume of options contracts traded.
The formula for the square of the near and next-term implied volatilities combines the one used in the calculation of the "new" VIX (described in Carr & Wu, 2009) with a new adjustment factor needed to deal with the low liquidity in Brazil: where • k = 1 if the formula uses the near-term options and k = 2 is the formula uses the next-term options-that is, σ 2 1 (t ) and σ 2 2 (t ) are, respectively, the squared near-and the next-term implied volatilities on day t ; 6 • F ( t,T k ) is the settlement price on day t of the Ibovespa futures contract which expires on day T k (T 1 is the near-term expiration date and T 2 is the next-term expiration date); • K 0 is the option strike value which is closest to F ( t,T k ) ; • K i is the strike of the i -th out-of-the-money option: a call if K i > K 0 , a put if K i < K 0 and both if K i = K 0 .
• r t is risk-free rate from day t to day T k , obtained from the daily settlement price of the futures interbank rate (DI); is the market price on day t of option with strike K i ; • j is a new adjustment factor that can take the values 0, 1 or 2-in the methodology described in Carr & Wu (2009) j is always equal to 1 (we explain this adjustment below).
After calculating both the near-and next-term implied volatilities using equation (1), we then aggregate these into a weighted average that corresponds to the 2-month (42 business days) implied volatility, as follows: where IVol t is the IVol-BR in percentage points and annualized at time t ; N T 1 is the number of minutes from 5 pm of day t until 5 pm of the near-term expiration date T 1 ; N T 2 is the number of minutes from 5 pm of day t until 5 pm of the next-term expiration date T 2 ; N 42 is the number of minutes in 42 business days (42 × 1440); and N 252 is the number of minutes in 252 business days (252 × 1440). 7 5 All options expire on the Wednesday closest to the 15th day of the expiration month. 6 The rollover of maturities occurs when the near-term options expire. We tested rolling-over 2, 3, 4, and 5 days prior to the near-term expiration to avoid microstructure effects, but the results do not change. 7 On days when the weight of the second term of equation (2) is negative, we do not use the next-term volatility, i.e., the IVol-Br index equals the near-term volatility. The methodology presented above departs from the standard one described in Carr & Wu (2009) for three reasons which are related to the relatively low liquidity and low number of strikes traded in the options market in Brazil: 1) We introduce the adjustment factor j in equation (1) to account for the following: (i) there are days when only a call or a put at K 0 is traded- Carr & Wu (2006) have always both a call and a put-; moreover, (ii) we have to define K 0 as the option strike value which is closest to F ( t,T k ) and, because of that, we may have either K 0 > F (t,T k ) or K 0 < F ( t,T k ) -Carr & Wu (2006) define K 0 as the option strike value immediately below F ( t,T k ) . Depending on the situation we face regarding (i) and (ii), the value of j is set to 0, 1 2, as follows (an explanation about this can be found in the Appendix): 2) We widen the time frame of options prices to the interval [3 pm, 6 pm]. For each strike, we use the last deal in this interval to synchronize the option price with the settlement price of the Ibovespa futures; 3) We only calculate σ 2 1 (t ) and σ 2 2 (t ) if, for each vertex, there are at least 2 trades involving OTM call options at different strikes and 2 trades involving OTM put options also at different strikes-this is done in order to avoid errors associated with lack of liquidity in the options market. If on a given day only one volatility vertex can be calculated, we suppose that the volatility surface is flat and the IVol-Br is set equal to the computed vertex. If both near-and next-term volatilities cannot be calculated, we report the index for that day as missing.
The volatility index calculated according to equations (1) and (2) could be biased because it considers only traded options at a finite and often small number of strikes. To assess the possible loss in the accuracy of the integral calculated with a small number of points, we refine the grid of options via a linear interpolation using 2,000 points of the volatility smile that can be obtained from the traded options (based on the procedure suggested by Carr & Wu, 2009). 9 The results did not change.
The IVol-BR series, computed according to the methodology described above, is available for download at the webpage of the Brazilian Center for Research in Financial Economics of the University of Sao Paulo (NEFIN). 10 Figure 1 plots the IVol-Br for the period between August 2011 and February 2015, comprising 804 daily observations.  The "coarse" volatility smile for both near and next-term is retrieved from the options market values and the Black-76 formula. We then refine the grid of strike prices K i using the implied volatilities and implied deltas of the options with the formula: where w = 1 for calls and w = −1 for puts; N −1 (·) is the inverse of the standard normal cumulative density function. To simplify the process of retrieving K i , we transform all traded options in calls (via put-call parity) and create a smile in the (∆ call , σ ) space. We then generate 2,000 points by linearly interpolating this smile considering two intervals: (i) the interval [ ∆ max ; ∆ min ] of deltas of the traded options; (ii) the interval [99; 1] of deltas. 10 http://www.nefin.com.br An implied volatility index should reflect the dynamics of (i) the level, or quantity, of risk that investors face-the expected future volatility-and (ii) the price of such risk-the risk-aversion of investors. Given that, the IVol-BR should be higher in periods of distress. As expected, as Figure 1 shows, the series spikes around events that caused financial distress in Brazil, such as the Euro Area debt crisis (2011), the meltdown of oil company OGX (2012), the Brazilian protests of 2013, the second election of Mrs. Rousseff (2014) and the corruption and financial crisis in Petrobras (2015).
It is also interesting to compare the IVol-BR with the VXEWZ, the CBOE's index that tracks the implied volatility of a dollar-denominated index (EWZ) of the Brazilian stock market. Figure 2 shows the evolution of both series.
As Figure 2 presents, the VXEWZ is often higher than IVol-BR. This happens because the VXEWZ, which is constructed using options over the EWZ index (that tracks the level in dollars of the Brazilian stock market), embeds directly the exchange rate volatility. In turn, the IVol-BR is constructed using options over the Ibovespa itself and, hence, reflects only the stock market volatility. Thus the IVol-BR is better suited to describe the implied volatility of the Brazilian stock market for local investors or foreign investors that have hedged away the currency risk. During the period depicted in Figure 2 there were important changes in the exchange rate volatility that directly impacted the VXEWZ but not the IVol-BR.  Implied Volatility (% per year) 0 1 j u l 2 0 1 1 0 1 j a n 2 0 1 2 0 1 j u l 2 0 1 2 0 1 j a n 2 0 1 3 0 1 j u l 2 0 1 3 0 1 j a n 2 0 1 4 0 1 j u l 2 0 1 4 0 1 j a n 2 0 1 5 IVol−BR VXEWZ

Comparison of IVol-BR with ATM volatility
In this section we compare the IVol-BR to the Black-Scholes implied volatility calculated using only atthe-money options; we call this volatility index "IVol-ATM." To compute it, we first calculate the Black-Scholes put and call implied ATM-volatilities from the two strikes immediately below and above the current futures' settlement price. Then, we linearly interpolate the put and call implied volatilities to obtain the forward implied volatility. This procedure is done to both near-and next-term options. Finally, we standardize the near-and next-term forward implied volatilities to have a basis period of two months (42 business days) to obtain the Ivol-ATM. Figure 3 shows the daily time-series of the Ivol-BR and the Ivol-ATM from 2011 to 2015.
In section 3.1 we decompose the implied variance, calculated as the IVol-BR squared, into (i) the expected variance of stock returns and (ii) the equity variance premium. To do this, we first estimate a model that represents the conditional expectation of investors of future variance. Then, by calculating the difference between the implied variance and the estimated expected variance, we arrive at a daily measure of the variance premium. In section 3.2, from the volatility premium, we produce a time-varying risk aversion measure for the Brazilian investor from the variance premium. In section 3.3 we show that the variance premium and the risk aversion measure are able to predict future stock returns as theory suggests: when variance premium (risk-aversion) is higher, expected returns are higher.

Decomposing implied variance into expected variance and variance premium
To decompose implied variance into expected variance and a premium, we first search for the model that best forecasts variance. Because the implied variance, calculated by squaring the IVol-BR, reflects markets expectations for the two-months ahead, the measure of expected variance of interest is also over the same two-month period.
The variance of returns is a latent, unobservable variable. Fortunately, we can obtain a good estimator of the variance of returns from high frequency data and use the estimated time-series, the so-called realized variance, as the dependent variable of our forecasting model. Formally, the realized variance over a two-month period at day t is calculated by summing squared 5-minute returns over the last 42 trading days: where ∆ = 5/425 is the 5-minute fraction of a full trading day with 7 hours including the opening observation; ⌊·⌉ is the operator that approximates to the closest integer; and r i = 100× is a 5-minute log-return in percentage form on the Ibovespa index, except when i refers to the first price of the day, in which case r i corresponds to the opening/close log-return. Following the recent literature on variance forecasting (Chen & Ghysels, 2011, Corsi, 2009, Corsi & Renò, 2012, and Bekaert & Hoerova, 2014, we construct several explanatory variables (predictors) from a 5-minute returns data set. 13 First, we include in the set of explanatory variables lags of realized variance at heterogeneous frequencies to account for the clustering feature of stock returns variance. In the spirit of Corsi's HAR model (Corsi, 2009), lags of bimonthly, monthly, weekly and daily realized variances are included: RV (42) t , RV (21) t , RV (5) t and RV (1) t . Formally: for each k = 42, 21, 5, 1.
One important feature of variance is the asymmetric response to positive and negative returns, commonly referred to as leverage effect. To take this into account, Corsi & Renò (2012) suggests including lags of the following "leverage" explanatory variables: with k = 42, 21, 5, 1. For a convenient interpretation of the estimated parameters, we take the absolute value of the cumulative negative returns. 13 We thank BM&FBovespa for providing the intraday data set.
Finally, we follow Bekaert & Hoerova (2014) and include the lagged implied variance as explanatory variable. Importantly, as will be shown, this variable contains information about future realized variance that is not contained in lagged realized variance and other measures based on observed stock returns.
To find the best forecasting model, we apply the General-to-Specific (GETS) selection method proposed by David Hendry (see for instance Hendry, Castle, & Shephard, 2009). The starting model, also called GUM or General Unrestricted Model, includes all the variables described above plus a constant: To avoid multicolinearity, the lagged realized variance measures were excluded from the initial set of explanatory variables since by construction they are approximately equal to RV (k ) t ≈ C (k ) t + J (k ) t . However, in a robustness exercise below, we include these variables in other forecasting models.
Following an iterative process, the method searches for variables that improve the fit of the model but penalizes for variables with statistically insignificant parameters. The regressions are based on daily observations. Table 3 shows the estimates of the final model-the GETS model. Eight variables plus a constant remain in the GETS model: IVar t , C (42) t and Lev (5) t . Importantly, the coefficient on the implied variance is positive (0.152) and highly significant. This indicates that, as expected, IVar does contain relevant information about future variance, even after controlling for traditional variance forecasting variables. From the GETS model, we calculate a time-series of expected variance. We name the difference between implied variance and this time-series of expected variance as the variance premium: where ] is the GETS model expected variance computed using information up until day t ; the subscript t + 42 emphasizes the fact that it is the expected variance over the same horizon as the implied variance, IVar t . Figure 4 shows both series and Figure 5 shows the variance premium. We observe that the premium varies considerably. The 3-month moving average shown in Figure 5 suggests that the average premium varies and remains high for several months.  Table 3. Both series are in percentage points and annualized. 0 500 1000 1500 2000 0 1 j a n 2 0 1 2 0 1 j a n 2 0 1 3 0 1 j a n 2 0 1 4 0 1 j a n 2 0 1 5 data Implied Variance Expected Variance Figure 5. THE VARIANCE PREMIUM This figure shows the weekly time-series of the variance premium calculated by the difference of the implied variance and expected variance as predicted by the GETS model shown on Table 3, and its three month moving average.
−50 0 50 100 0 1 j a n 2 0 1 2 0 1 j a n 2 0 1 3 0 1 j a n 2 0 1 4 0 1 j a n 2 0 1 5 data Variance Premium 3−Month Avg. Premium Table 3. GENERAL-TO-SPECIFIC BEST MODEL The table shows the estimates of the best variance forecasting model following the General-to-Specific selection method. The starting model, also called GUM or General Unrestricted Model, comprises of all independent variables. The standard errors reported in parenthesis are robust to heteroskedasticity. Regressions are based on daily observations. The corresponding p -values are denoted by * if p < 0.10, ** if p < 0.05 and *** if p < 0.01.

The variance risk premium and the risk aversion coefficient
An implied variance index reflects the dynamics of two very important variables. The first relates to the level, or quantity, of risk that investors face: the expected future variance of the market portfolio, estimated above. The second relates to the price of such risk: the risk aversion of the representative investor.
Since options' payoffs are asymmetric, the value of any option (call or put) is increasing in the expected variance of the underlying asset. Because of that, options are often used as a protection against changes in expected variance. Since the typical risk-averse investor dislikes variance, options are traded with a premium because of such an insurance value. As a direct consequence, the implied variance (IVar, the IVol-BR squared), which is computed directly from options prices, also has a premium with respect to the expected variance. That is, the more risk-averse the investor is, the more she is willing to pay for the insurance that options provide, i.e., the higher the variance premium.
In order to make this connection between risk aversion and variance premium more precise, we need to impose some economic structure. To do this, we use Bollerslev et al. (2009) economic model, which is an extension of the long-run risk model of Bansal & Yaron (2004). We assume that the following closed-form equation for the variance premium holds for each t : 14 where ψ is the coefficient of elasticity of intertemporal substitution, γ t is the time-varying risk aversion coefficient, q the volatility of the volatility, and ρ σ is the auto-regressive parameter in the volatility of consumption.
Using the estimated weekly series for the variance premium computed above and usual parameter calibration, 15 we pin down a time-series for the time varying risk aversion coefficient of the representative investor in Brazil. 16 The resulting series is plotted in Figure 6. The smallest value for γ t is 1 on August 22, 2014; and the highest value is 57 on February 13, 2015. The average risk aversion level is 26. Such values are consistent with the results in Zhou (2009)-an average risk aversion higher than 10 is needed to match the empirical moments of the variance premium (see his Table 8). 0 1 j a n 2 0 1 2 0 1 j a n 2 0 1 3 0 1 j a n 2 0 1 4 0 1 j a n 2 0 1 5 data Risk Aversion 3−Month Avg. Risk Aversion Using different methodology and data sets, other papers find (unconditional) lower estimates of risk aversion for the Brazilian representative investor. For instance, Fajardo, Ornelas, & Farias (2012) using data on currency options for the Brazilian Real from 1999 to 2011, estimate the implied riskneutral density and, by comparing it to the objective density, find a coefficient of relative risk aversion of around 2.7. Similarly, Issler & Piqueira (2000) using data on aggregate consumption from 1975 to 1994, 14 We use their simpler equation, where they assume a constant volatility of volatility (the process q is constant at all t ). 15 We set ψ = 1.5, q = 10 −6 , κ 1 = 0.9 and ρ σ = 0.978 following the calibration in Bansal & Yaron (2004) and Bollerslev et al. (2009). 16 Equation (4) is quadratic on the risk-aversion coefficient γ t . In order to avoid complex roots, we shift the variance premium upward so that the minimum variance premium corresponds to the minimum value of γ t = 1. document a coefficient of relative risk aversion lower than five. In the next section we assess to which extent the lower risk aversion found in other studies can be attributed to differences in calibration.
3.2.1. Sensitivity of risk-aversion to calibration parameters A number of papers estimate the elasticity of intertemporal substitution for the Brazilian representative investor. Issler & Piqueira (2000) find a relatively low ψ of 0.29 using annual consumption data from 1975 to 1994-less than half of the ψ they find for the US (0.72) for the same period and using the same methodology. A low value of ψ for the Brazilian economy is also reported by Havranek, Horvath, Irsova, & Rusnak (2015); the authors find that households in developing countries and countries with low stock market participation, such as Brazil, substitute a lower fraction of consumption intertemporally in response to changes in expected asset returns, with implied values for ψ often below one. On the other hand, using a quarterly data set of 1975 to 2000, Araújo (2005) finds a much higher range of possible ψ 's, with the minimum value of ψ being equal to 2.5. Given that there is no consensus on which is the ψ that best characterize the behavior of the representative investor in Brazil, we assess the sensitivity of our results to the choice of ψ by comparing the estimates of γ t for three different values of ψ : 0.50, 1.50, and 3.00. Another parameter that was used in the calibration that can vary across countries is κ 1 . The parameter κ 1 is equal to the ratio PD/(1 + PD) , where PD is a long-run average of the price-dividend ratio. The number used in our calibration was 0.9 and is the same one used Bollerslev et al. (2009). According to its formula, the value of κ 1 = 0.9 corresponds to a price-dividend ratio of around 10. This number, however, can vary depending on the data set considered. Indeed, the average price-dividend ratio computed for Brazil in the period 2001-2015 is around 40. Thus, we also assess the sensitivity of our results to the choice of κ 1 by comparing the estimates of γ t for another two different values of κ 1 : 0.98 and 0.95. These two values correspond, respectively, to price-dividend ratios of 40 and 20. Figure 7 shows the image of the function γ t -the inverse of the function in equation (4)-for plausible values of the variance premium and for nine pairs of ( ψ ,κ 1 ) , each one a possible combination of ψ = (0.5,1.5,3.0) with κ 1 = (0.90,0.95,0.98) . The estimated values of γ t differ depending on the chosen parametrization, particularly across κ 1 . At a variance premium of five, γ t can be 13, 16, or 23 if κ 1 is, respectively, 0.98, 0.95, or 0.90. On the other hand, the values of γ t do not vary much across ψ ; for any given κ 1 and value of variance premium, the range of possible values for γ t is less than one. Overall, at any pair of ( ψ ,κ 1 ) considered the estimates of γ t are higher than 10.

Predicting future returns
If the variance premium positively comoves with investors risk-aversion, it should predict future market returns: when risk aversion is high, prices are low; consequentially, future returns (after risk aversion reverts to its mean) should be high. Moreover, the risk aversion measure itself, computed in section 3.2, should also predict future returns. In this Section we test these predictions by regressing future market returns on both the variance premium and the risk-aversion measure. Table 4 shows the results of our main regression. The dependent variable is the return on the market portfolio 4 weeks ahead. To limit the overlapping of time-series, we reduce the frequency of our data set from daily to weekly by keeping only the last observation of the week. Additionally, to account for the remaining serial correlation in the error term, the standard errors are computed using Newey-West estimator. Columns (1) and (2) show that implied variance IVar t and expected variance σ 2 t alone are not very good predictors of future returns. On the other hand, Column (3) shows that the variance premium, resulting from a combination of both variables, IVar t − σ 2 t , strongly predicts future returns at the 4-week horizon. The estimated coefficient is positive, 0.089 , and significant at the 1% confidence level. Column (4) shows that the risk aversion measure also predicts future returns at the 4-week horizon. The estimated coefficient is positive, 0.180, and significant at the 1% confidence level.
The predictive power of the variance premium and the risk aversion measure remains after we include in the regression the divided yield log , another common predicting variable. Again, columns (5) and (6) show that implied variance and expected variance alone are poor predictors of returns. On the other hand, both the variance premium and the risk aversion measure do predict future returns. Column (7) shows a positive coefficient for the variance premium, 0.066, significant at the 5% confidence level. Column (8) shows a positive coefficient for the risk-aversion measure, 0.135, also significant at the 5% confidence level.
In columns (1) through (8) of tables 5 and 6, the regressions are the same as the one in Column (7) and (8), respectively, of Table 4, except for the horizon of future returns. As the significance and values of the estimates indicate, the variance premium predictability is stronger at the 4-week horizon-columns (7) and (8).
A concern is that the standard errors in the first eight Columns in tables 5 and 6 may be biased due to the presence of a persistent explanatory variable such as the log dividend yield (see for instance Stambaugh, 1999) combined with a persistent dependent variable (overlapping returns). To address this concern, columns (9) and (10) in both tables show the same regressions of columns (7) and (8) but based on non-overlapping 4-week returns. As we can see, the coefficients on the variance premium and risk-aversion remain positive and significant.
Another concern may be that the actual expected variance by market participants cannot be observed. Hence, our measure of expected variance depends on the model chosen by the econometrician. To address this concern, we also assess to which extent our results depend on the chosen variance model. Tables 7 and 8 show the estimates of several models. Table 7 brings the estimates of Corsi's HAR model (Corsi, 2009) in Column (1), with the addition of a 42-day realized variance lag in accordance with the frequency of the dependent variable. In columns (2), (3) and (4) we include the lagged implied variance, IVar t , that was shown to contain important predictive information. Columns (3) and (4) include leverage variables to account for the asymmetric response of variance to past negative returns. Table 4. PREDICTABILITY REGRESSIONS The table shows the estimates of predictability regressions. The dependent variable is the return on the market portfolio 4-weeks ahead. The explanatory variables are: i) σ 2 t , the expected variance on the next 8 weeks estimated by best model following the General-to-Specific selection method; ii) IV ar t , the expected implied variance on the next 8 weeks estimated from prices of options contracts at time t ; iii) IVar t − σ 2 t , the variance premium; iv) γ t is the risk aversion computed using the functional form in Bollerslev et al. (2009) and the variance premium; and v) log ( , the log dividend yield. Regressions are based on weekly observations. To account for error correlation, the standard errors are computed using Newey-West lags. The standard errors are reported in parenthesis. The corresponding p -values are denoted by * if p < 0.10, ** if p < 0.05 and *** if p < 0.01.

weeks
(1)  (9) and (10) are non-overlapping on the dependent variable and are based on monthly observations. To account for error correlation, standard errors in columns (3) is the log dividend yield. Regressions in columns (1) through (8) are based on weekly observations. Regressions in columns (9) and (10) are non-overlapping on the dependent variable and are based on monthly observations. To account for error correlation, standard errors in columns (3) Table 7. ROBUSTNESS: DIFFERENT VARIANCE MODELS The table shows the estimates of different models of expected variance. The dependent variable is the realized variance over the following 42 days, calculated from 5-minute returns on the Ibovespa portfolio. The explanatory variables are: i) IVar t is the expected implied variance on the next 8 weeks estimated from prices of options contracts at time t − 1; ii) RV (k ) t −1 is the realized volatility on the following k days at time t − 1, where k = 42, 21, 5, 1; computed iii) Lev (k ) t −1 is the cumulative negative 5-minute returns continuous component of the realized variance on the following k days at time t −1, where k = 42, 21, 5, 1. The standard errors reported in parenthesis. The corresponding p -values are denoted by * if p < 0.10, ** if p < 0.05 and *** if p < 0.01.

M1
M2 M3 M4 The table shows the estimates of different models of expected realized variance. The dependent variable is the realized variance over the following 8-weeks, calculated from 5-minute returns on the Ibovespa portfolio. The explanatory variables are: i) IVar t −1 is the expected implied variance on the next 8 weeks estimated from prices of options contracts at time t − 1; ii) C (k ) t −1 is the continuous component of the realized variance during the following k days at time t −1 , where k = 42, 21, 5, 1; iii) J (k ) t −1 is the jump component of the realized variance during the following k days at time t − 1, where k = 42, 21, 5, 1; and iv) Lev (k ) t −1 is the absolute of the sum 5-minute negative returns during the following k days at time t − 1, where k = 42, 21, 5, 1. The standard errors reported in parenthesis. The regressions are based on daily observations. The corresponding p -values are denoted by * if p < 0.10, ** if p < 0.05 and *** if p < 0.01.

M5
M6 M7 M8 In Table 8 we separate the realized variance into its continuous and jump components and use these variables instead. Column (1) shows the estimates of the GUM model, the starting model in the General-to-Specific selection method adopted in section 3.1. The GUM regression includes all the variables initially selected as candidate variables to forecast variance. Columns (2) through (4) are variants of this more general model.
As we can conclude by comparing the statistical properties of each regression in tables 3, 7 and 8, the GETS model has the lowest information criterion, BIC, as the selection method strongly penalizes the inclusion of variables and favors a more parsimonious model. Models M4, M5 and M6 have comparable R 2 to the GETS models, explaining more than 35% of the variation of the dependent variable, but with the inclusion of extra regressors.
We now assess how sensitive is our predictive regression to the selection of the variance model. For each one of the regression models shown in tables 7 and 8 we calculate a volatility premium as in equation (3). The results of the predictability regressions at the 4-week return horizon are shown in Table 9. In Column (1) we use a simple model to predict future variance and set σ 2 t = σ 2 t −1 following the definition of Bollerslev et al. (2009). Column (2) replicates our main regression that uses the GETS model to predict variance. Columns (3) through (10) show the predictability regressions for each of the 8 models presented in tables 7 and 8. As can be seen, the results are largely robust to the selection of the variance model.

CONCLUSION
This is the first article to propose an implied volatility index for the Brazilian stock market based on option and futures prices traded locally. The methodology we propose has to deal with the relatively low liquidity of contracts used. This is a first contribution of this paper.
We use our implied volatility index to calculate the so-called variance premium for Brazil. Assuming Bollerslev et al. (2009) economic structure, we also pin down a time-varying risk aversion measure of the representative investor in the Brazilian market. In line with international evidence, we show the variance premium strongly predicts future stock returns. Interestingly, we also find that our measure of risk aversion is a strong predictor of future returns with a slightly superior fit than the variance premium. To the best of our knowledge, this is the first analysis of this kind for an emerging market. This is the second contribution of this paper.
Further extensions of this work include applying our methodology to construct implied volatility indices for other markets with low liquidity. With respect to the risk aversion measure, different economic models and parameter calibration can be tested.
is the log dividend yield. Regressions are based on weekly observations. To account for error correlation, the standard errors are computed using Newey-West lags. The standard errors are reported in parenthesis. The corresponding p -values are denoted by * if p < 0.10, ** if p < 0.05 and *** if p < 0.01. (1)

The j adjustment
In this section we demonstrate how to obtain the adjustment term j . In the following derivations we refer to an out-of-the-money option as OTM, and to an in-the-money option as ITM.
Under the risk neutral measure, it can be shown that the variance is approximated by a portfolio of OTM calls and puts. However, in practice, the portfolio used is • K 0 is the strike closest to the futures price F ; • r t is risk-free rate from day t to day T , obtained from the daily settlement price of the futures interbank rate (DI); is the market price on day t of option with strike K i .
Since we don't necessarily have a call and a put at K 0 , an adjustment in the formula above is needed. The following 6 cases can arise: Case 1: If K 0 ≤ F and we have data on call and put prices at K 0 . This is the standard case set by Carr & Wu (2006). It follows from the Put-Call parity that: Therefore, substituting for the O (K 0 ) term in Equation (A-1), we obtain where, the last equality, follows from the assumption that ∆K 0 = F − K 0 . Substituting back in Equation (A-1) we obtain that the last term below is zero: , that is, all options are OTM. Equivalently, we can write the above equation as and C (K 0 ) is ITM. In Brazil, there are days when only a call or a put at K 0 is traded. Besides, we have to define K 0 as the option strike value which is closest to F ( t,T k ) and, because of that, we may have either . Given that, we have to create the following 5 additional cases.
Case 2: If F < K 0 and we have data on call and put prices at K 0 . In this case, P (K 0 ) is ITM and, by the Put-Call parity, we obtain analogously: , that is, all options are OTM. Equivalently, Case 3: If K 0 ≤ F , we have data on put prices and don't have data on call prices at K 0 . In this case, all options are OTM and no adjustment is needed. That is, we set j = 0 in the formula:

Case 4:
If K 0 > F , we have data on call prices and don't have data on put prices at K 0 . In this case, all options are OTM and no adjustment is needed. That is, we set j = 0 in the formula: Case 5: If K 0 ≤ F , we have data on call prices and don't have data on put prices at K 0 . In this case, C (K 0 ) is ITM and should be transformed into a OTM P (K 0 ) by the Put-Call parity.
Following the same steps of Case 1, we obtain Case 6: If K 0 > F , we have data on put prices and don't have data on call prices at K 0 . This can be solved similarly as Case 5 with j = 2 and P (K 0 ) ITM.
We build a simple two-period general equilibrium model with incomplete markets which incorporates reverse market mortgages without appealing to the complicated framework required by the infinite horizon models. Two types of agents are considered: elderly agents and investors. The former are owners of physical assets (for instance housing) who will want to sell them to investors. For that end the elderly agents, who are assumed to not have any bequest motive, issue claims against physical assets they own. One of the claims issued will be interpreted as reverse mortgage (loan for seniors) and the other one as a call option written on the value of housing equity. By assuming that both the elderly agents and the investors are price takers, and by applying the generalized game approach, we show that the equilibrium in this economy always exists, providing the usual conditions on utilities and initial endowments are satisfied. We end with a remark on efficiency of the equilibrium.

INTRODUCTION
One of the great problems that many countries' social security systems faces is how to sustain the elderly people who are ageing later than in previous generations. In many countries, in order to reduce costs for social security, they had to increase either the retirement age or the time of contribution for social security. It is also well known that the older the population is, the more there is spent on medical care (see De Nardi, French, & Jones, 2010). Hence, it is the government that bears the costs of medical care, especially when the retirement pension or the saving rate is low like in Brazil or in many countries of Latin America.
On the other hand, in these countries, particularly in Brazil, there are many ways elderly people can borrow money. Among them are payroll deductions. Although these types of borrowing solve elderly people's problems in the short term, they may compromise their future consumption pattern leading to a loss of well-being in the long term. Before this problem, many developed countries like the USA and the UK had created markets for reverse mortgages, which are loans for seniors (see Cocco & Lopes, 2015 for a recent study). Due to the complexity of this new financial product, many countries were or are carrying out studies for its implementation (see for example Caetano & Mata, 2009 for the Brazilian case). In Brazil, reverse mortgages are offered as forms of investments which are sold as options to fund retirements (see for instance Macedo, 2015). Lastly, reverse mortgages differ from classical mortgages in several aspects. For the sake of completeness, the main characteristics are illustrated in the two following paragraphs The most important financial decision the typical household makes is buying a house. Such a decision depends on household wealth, housing prices, rental rates, the kind of financing, age and many other factors. In youth, the household decides whether to buy or rent a house and, if it buys, what sort of mortgage to choose. Whatever type of mortgage chosen, households would have a type of savings. To understand that we raise the following question: How does a mortgage move money around? First, a mortgage is a loan which transfers income from the future to the present. Second, after origination, a mortgage transfers money from the present to the future. When a borrower makes a mortgage payment, a portion of the payment goes to reduce the balance of the loan, thus increasing the net worth of the household. In this sense, a mortgage is a savings scheme, and for many households it is the main vehicle for life-cycle wealth accumulation.
However, the picture changes dramatically if we consider elderly homeowners. All elderly people face uncertainty regarding their lifespan, health, medical expenses and housing prices. Of course, they also have to make choices concerning consumption and financial saving. The housing decision is not as simple as it seems because it is both a consumption good and an investment good. Thus, the correct treatment of housing equity may not be very obvious in the retirement saving context. Whatever the situation, elderly homeowners need to make such decisions. Given the uncertainty faced by elderly persons, how should they finance such decisions about consumption and mainly, health care? Since the only asset that elderly homeowners in our model have is their homes, they should find a way to make them liquid. One option is to sell the house. The other is to fall into a reverse mortgage.
A reverse mortgage (or lifetime mortgage) is a loan available to seniors, and is used to release the home equity on the property in the form of one lump sum or multiple payments. The homeowner's obligation to repay the loan is deferred until the owner leaves (e.g., into aged care), the home is sold or the owner dies. When an elderly homeowner receives cash flows due to the reverse mortgage, it is as if they were spending some portion of their housing equity (wealth product from their past), thus increasing the net worth of the household. In this sense, a reverse mortgage is a dissavings scheme, and for many elderly persons it is the main vehicle for life-cycle consumption-health financing.
In spite of the great volume of both theoretical and empirical studies about reverse mortgages (see e.g. Cocco & Lopes, 2015), few are the papers which deals with it in a general equilibrium framework. These have mainly been carried out in a life-cycle setting under a partial equilibrium analysis. Our main contribution is to show that reverse mortgages are compatible with well-functioning markets. That is, there exists a competitive general equilibrium in an economy in which the elderly people engage in reverse mortgages in order to fund their consumption. This may include, in addition to food consumption, medical expenditures and leisure. To reach that objective we build a simple two-period general equilibrium model with incomplete markets which incorporate reverse market mortgages without appealing to the complicated framework required by the infinite horizon models.
To reach such a goal, several things are necessary and some simplifications must be imposed: first, we need to borrow the financial structure to accommodate our financial instruments from Allen & Gale (1991). Second, we must assume that elderly homeowners have no bequest motives 1 so that they can issue derivatives on the remainder (if there are any) after the reverse mortgage ends. 2 For an analysis of the consequences of the absence or presence of the bequest motive in the elderly dissaving context, see Ando, Guiso, & Terlizzese (1994). Third, although our model has been inspired by the model of Allen & Gale (1991)-in relation to the classification 3 of agents who interact in the economy-ours presents notable differences. Our model considers multiple goods and two kinds of claims, contrary to Allen and Gale who consider a variety of claims. Lastly, our concept of equilibrium maintains the characteristics of the Arrow-Debreu equilibrium in the context of general equilibrium with incomplete markets. That is, agents maximize their utility functions subject to their budget constraints, and all markets are clear. The concept of equilibrium used by Allen and Gale is a two-stage equilibrium which is more appropriate for the case of oligopolistic competition. We do not adopt this concept because we are assuming that both agents, seniors and lenders, are price takers. In our model physical assets are indexed by elderly agents. Even so, this does not break the anonymity of the markets because the investors are only interested in the durability of physical assets and not in the identity of the proprietors. Finally, we show the existence of equilibrium and briefly discuss its constrained efficiency.
The methodology used to reach our existence objective is the generalized game approach used by Arrow & Debreu (1954). More precisely, we define the generalized game played by a finite number of players. These players are elderly agents, lenders, and fictitious agents called auctioneers. In the first period, there is one auctioneer choosing first-period prices of goods and the prices of the two claims traded (reverse mortgage, call options) in order to maximize the total first-period excess demand. In the second period we have one auctioneer for each state who maximizes the excess demand of goods traded in the second period. Then, we demonstrate that equilibrium for the generalized game corresponds to equilibrium for our original economy.

Related Literature
One of the major financial innovations of recent decades, in markets of collateralized loans, has been to allow borrowers to use the collateral that backs their loans. For instance: mortgage markets and leasing. Since the pioneer work of Dubey (1995) up to their more recent version, Geanakoplos & Zame (2014), many theoretical works have been written; be they of finite horizon or of infinite horizon. With respect to the latter, and particularly to those GEI models with infinite horizons where agents do not live forever (e.g. overlapping generations), agents are not allowed to trade in the financial markets in the last period of their lives. An exception is the paper of Seghir & Torres-Martínez (2008), which uses collateral (like in Geanakoplos & Zame, 2014) to enforce the promises.
Another major financial innovation in recent decades has been the development of reverse mortgage markets where elderly borrowers receive a loan against housing equity and are allowed to stay in their homes enjoying all the benefits thereof. Due to the recent development of this market 4 a growing amount of literature has been interested in this topic. Studies on the potential demand for reverse mortgages goes back to Rasmussen, Megbolugbe, & Morgan (1995). See also Stucki (2005) who estimated the potential market at 13.2 million older households. For an ample and deep study about the recent expansion of the reverse mortgage market, see Shan (2011) and Nakajima (2012), respectively. Among the most recent papers about reverse mortgages, we highlight Cocco & Lopes (2015) who focus on the design of the reverse mortgage. The work of Nakajima & Telyukova (2011) also stood out It. In it, a rich structural model of housing and saving/borrowing decisions in retirement is used. This literature, about reverse mortgages, has been developed through three major trends. Namely, life-cycle and precautionary savings, housing and portfolio choices and discrete choices. Using mathematical programming with an equilibrium constraints approach, Michelangeli (2010) solves consumption, housing, and mobility decisions within a dynamic structural life-cycle model.
The paper is organized as follows. In the next section we describe the model. In section 3 we define the concept of equilibrium and state our result on the existence of equilibria. In section 4 we demonstrate our results and we provide a short section discussing the constrained efficiency properties of equilibria. Finally, we end by offering some concluding remarks.

THE MODEL
We consider a two-period economy, where agents know the present but face an uncertain future. That is, it is assumed that in period 0 (the present) there is just one state of nature while in period 2 (the future) there are S states of nature. There are L commodities in each period and in each state of nature. Thus, the consumption set is R L(S +1) + . Any element of this set will be denoted by a pair (a o ,ã) where a o ∈ R L + andã ∈ R S L + , with a s ∈ R L + . The price system of commodities is denoted by (p o ,p) and is assumed to belong to R L(S +1) + .

Agents
The economy is populated by a continuum of elderly agents and a finite number of investors (lenders). The elderly population is divided into a finite number V of age groups. To make things simpler, we consider a representative agent h ∈V in each age group so that we have a finite number of elderly agents. These agents own physical assets (e.g. their houses) as part of their initial endowments ω h o ∈ R L + and are interested only in the first-period consumption. Thus, each elderly homeowner h ∈ V is characterized by his/her utility function U h : R L + → R , his/her initial endowments, ω h o ∈ R + , and by the way he/she deteriorates his/her physical assets (e.g. his/her house). Let Y h s ∈ R L×L + be the linear transformation 5 which represents the deterioration of physical assets in state s . There are i ∈ I investors who value consumption in both periods. Thus, the utility function and initial endowments of each investor i ∈ I are u i : R L(S +1)

Financial Structure
Let C ∈ R L + be the unit of measure of the physical assets of the economy. For instance, if the physical assets were houses, then C would represent the unit of the constructed area (e.g. m 2 or square feet). Each unit bundle C ∈ R L + is deteriorated according to the linear transformation Y h s which depends on state s ∈ S and the elderly person 6 h ∈ V . Thus, each unit bundle C becomes another bundle Y h s C ∈ R L + . Each elderly person h ∈ V issues debts by using two kinds of financial instruments: 7 reverse mortgages and derivatives. For each unit of reverse mortgage (borrowing against each unit bundle, C ∈ R L + , owned) issued by elderly person h, he will have to return r s if the state s occurs. Hence, in the second period the reverse mortgage lender keeps the minimum between the value of deteriorated physical assets and the outstanding debt. That is, Since we are assuming no bequest motives, each elderly person also issues debts by selling derivatives on the remainder (if there are any) after the reverse mortgage ends. This means that for each unit of reverse mortgage issued, one unit of derivative is issued, whose payoff is Therefore, each elderly person should issue the same number of reverse mortgages and derivatives. If θ h and ϑ h are amounts of reverse mortgages and derivatives issued by the elderly person h, then the cost of issuing them is p o Cϕ h . It is useful to note that the derivative whose payoff is defined above is the same as a call option written on the unit of measure, C ∈ R L + , of bundles of physical assets. Let π h and q h be the prices of reverse mortgages and derivatives respectively. An economy with reverse mortgage is defined by where each h represents a representative elderly agent whose characteristics were given above and each i represents an investor whose characteristics were also described above. Finally, F represents the financial structure which consists of only two claims-reverse mortgages and derivatives whose payoffs were described above.

Elderly Agents
Given the prices of commodities, reverse mortgages and derivatives, (p o ,π h ,q h ) , each elderly agent h ∈ V chooses (x o ,ϕ,ϑ ) ∈ R L + × R + × R + in order to maximize his/her utility subject to the following budget constraints: In addition, must be satisfied. Equation (1) says that the consumption of elderly agents is financed by the value of wealth after having gone through the reverse mortgage process and after having issued the call option. Condition (2) says the financial structure (set of claims) must be binding. This last condition implies that for the financial structure (set claims issued) to be feasible (compatible with the physical asset), we must have ϕ = ϑ .

Investors
Given the prices of claims (π ,q) ∈ R V × R V , and commodity prices (p o ,p) ∈ R L(S +1) subject to the following budget constraints: Budget constraints (3) and (4) say that: in the first period each investor finances both consumption and investments via his/her initial endowments; and in the second period his/her consumption is financed by the value of initial endowments and the returns of his/her investments made in the first period.

Definition
The equilibrium for an economy, E rm , consists of commodity price system (p o ,p) , reverse mortgage prices π ∈ R V + , call option prices q ∈ R V + ; allocations such that the following conditions are satisfied: subject to the budget constraint (1) and the compatibility condition (2). (3) and (4).

Each
3. Commodity markets clear: (ii) Claim markets clear: Condition (i) says all commodity markets are cleared. Condition (ii) says that everything that is demanded by investors must be equal to everything that is sold in both reverse mortgage and derivatives markets.

Existence
In this section we will give sufficient conditions which guarantee the existence of equilibrium for an economy with reverse mortgage. More precisely, we have the following theorem.
Theorem 1. For an economy in which (i) for all agents h ∈ V and i ∈ I , ω h and ω i belong to R L ++ and R L(S +1) ++ respectively; (ii) the utility functions U h : R L ++ → R and U i : R L(S +1) ++ → R are continuous, strictly increasing and strictly quasi-concave; (iii) the unit of measure of the physical assets of the economy C ∈ R L + is different from zero and does not deteriorate completely. 9 there is always an equilibrium.
Remark: For the second part of item (iii) to be true it is sufficient, for instance, to assume that Y h s is non-singular.

RESULTS
Before proving our main result, we first establish the following lemma which allows us to bound the allocations satisfying the feasible conditions (market clear conditions) of the equilibrium definition. More precisely, we state and prove the following lemma.

Lemma 1. Under hypotheses (i) and (ii) in Theorem 1, allocations
in the E rm that satisfy the feasibility conditions of the equilibrium definition are bounded.
Proof. From (i) of item 3 in the definition of equilibrium, in period t = 0, one has: Therefore, for each l ∈ L the following holds: The right hand side of the previous inequality is lower than Since x i ol and x h ol are both positive, it follows that From (i) of item 3 in the definition of equilibrium, in period t = 1, one has: Therefore, for each l ∈ L the following holds: From the positivity of x i l s it follows that x i l s ≤ W ls . Lastly, from (ii) of item 3 in the definition of equilibrium, in period t = 0, one has: As ϕ h = ϑ h , ∀h ∈ V , one has that both θ i h and φ i h are bounded from above by where W o = max l ∈L W ol and W 1 = max s,l W sl . So, Lemma 1 follows. □ In order to reach our goal (proof of Theorem 1), first we will define a generalized game, as in Debreu (1952). Then we will show that such a game has a Nash equilibrium; and lastly we will demonstrate that the equilibrium for the generalized game corresponds to the equilibrium for our economy.

The Generalized Game
As said above we will prove Theorem 1 by establishing the existence of equilibrium in a generalized game with a finite set of utility maximizing agents (elderly agents and investors) and auctioneers in each period, maximizing the value of the excess demand in the markets. Thus, we define the generalized game G in the following way: (1) and (2) and in addition are bounded from above by constants which were obtained from Lemma 1.
Similarly, each investor i ∈ I maximizes U i in the constrained strategy set B i (p,π ,q) ∩ □ i which consists of all choices (x,θ ,φ) ∈ R L(S +1) (3) and (4) and in addition are bounded from above by constants which were obtained from Lemma 1.
2. The auctioneer of the first period chooses (p o ,π ,q) ∈ △ L+V +V −1 in order to maximize 3. The auctioneer of state s of the second period chooses p s ∈ △ L−1 in order to maximize The following lemma guarantees the existence of equilibrium of the generalized game G .

Lemma 2.
Under the hypothesis of Theorem 1 there exists a pure strategy equilibrium for the generalized game G .
Proof. Lemma 1 follows from the equilibrium existence theorem in the generalized game of Debreu (1952). In fact, the objective functions of the agents are continuous and quasi-concave in their strategies. Furthermore, the objective functions of the auctioneers are continuous and linear in their own strategies, and therefore quasi-concave. The correspondence of admissible strategies, for the agents and for the auctioneers, has compact domain and compact, convex, and nonempty values. Such correspondences are upper semi-continuous, because they have compact values and a closed graph. The lower semi-continuity of interior correspondences follows from hypothesis (i) in Theorem 1 (see Hildenbrand, 1974, p.26, fact 4). Because the closure of a lower semi-continuity correspondence is also lower semi-continuous, the continuity of these set functions is guaranteed. We can apply Kakutani's fixed point theorem to the correspondence of optimal strategies in order to find the equilibrium. □ Finally, the following lemma claims that the equilibrium of G corresponds to the equilibrium of our economy.

Lemma 3.
If there exists an equilibrium for the generalized game G , then there exists an equilibrium for the economy E rm .

Proof.
1. Feasibility: is an equilibrium for the generalized game, we have that Adding in h gives Summing in i we have Since Summing (10) and (12) and grouping terms one has Since (p o ,π ,q) solves the first-period auctioneer's problem, we have ∑ i ∈I Notice that (13) and (14) hold with equality since (6), (8) and (9) also hold with equality. This last follows from the monotonicity of the utility functions. Now we will show that (15), (16) and (17) hold with equality. In fact, suppose that there exists an l ∈ L such that ∑ i ∈I which implies that p ol = 0 and this in turn implies that the consumption x h o and x i o of both agents are the maximum available. This implies that both agents could increase their consumption, contradicting (15). Therefore (15) must hold with equality. Now suppose that there is h ∈ H such that (16) is a strict inequality. Then the price of this asset must be zero. That is, π h = 0. This motivates the investors to purchase the maximum amount available, which contradicts the bounds already obtained for θ i . Thus, (16) holds with equality. Similarly we can shown that (17) is an equality as well. This implies that ∑ i ∈I Now we only need to prove conditions which clear markets in the second period. Before that, however, we notice that (2) implies that θ h = ϑ h for all h ∈ V . From this it follows that the right hand side in (13) is zero. Therefore, one has From the fact that p s solves the second-period auctioneer's problem, we have ∑ Using the same argument to clear the goods markets in the first period, we prove that (22) holds with equality, which ends the feasibility.

Optimality:
We want to prove that ( (p,π ,p) . Suppose that what we have just said is untrue. Then, from the quasi-concavity of the utility functions of the agents and the interior of the solution in the generalized game on the part of agents, one finds a contradiction in the optimality of the players in G . This is because (x h o ,ϕ h ,ϑ h ) and (x i o ,x i ,θ i ,φ i ) satisfies the feasibility of Lemma 1. Therefore Lemma 3 follows. □

Remark on Efficiency
In this short section, we prove that the equilibrium allocations cannot be dominated in the Pareto sense by feasible allocations that satisfy the agents' budget restrictions, at equilibrium prices, at all states of the nature, except at a state where there may be some transference due to subsidies or taxes. More precisely, we prove the following result for the economy E rm dominates, in the Pareto sense, any feasible allocation that satisfies the budget restrictions of the agents at the original equilibrium prices.
Proof. Suppose by contradiction that there is a feasible 10 allocation that belongs in the agent's budget set and Then, it follows from standard arguments, such as the individual optimality of the equilibrium allocation and aggregation, that the allocation is not feasible. Thus, Theorem 2 follows. □ The efficiency of equilibrium allocations given by Theorem 2 is the weakly constrained efficiency, see Magill & Shafer (1991) for an ample discussion of this concept. In our model, besides the incompleteness of markets, there is another source of inefficiency which is introduced by the reverse mortgages. Trading of reverse mortgages entails a constrained transfer from lenders to borrowers (elderly people). However, this transfer could not be optimal since the housing property could increase above the value of the debt of the reverse mortgage. If this were the case, the lender would sell the housing to cover the debt and the surplus, which is the difference between the value of the house and debt, would be lost. The inefficiency introduced by the reverse mortgages may be corrected when a market for derivatives written on the value of the housing is created. Thus, the weakly constrained efficiency transfer from investors to homeowners is given by the equilibrium derivative prices. That is, the first-period income of the borrowers would increase due to the sale of the derivative by implying an increase in the investors' second-period income when the call option is in the money.

CONCLUDING REMARKS
In this paper we have constructed a simple two-period general equilibrium model which accommodates elderly agents who make use of the reverse mortgages to finance their consumption. We have also demonstrated the existence of equilibrium of this economy and have briefly provided some remarks about the constrained efficiency of the equilibrium allocation. Since we are not considering bequest motives, we postulate that the elderly agents issue derivatives whose payoff is the difference between the depreciated value of the housing and the value of the reverse mortgage. The methodology used to reach our goal has been that of the generalized game approach. In future research we will be analyzing the case in which senior citizens have descendants. We provide the microeconomic foundations of cheating in classroom through a static game with complete information. Our setting is composed by two students, who must choose whether or not to cheat, and a professor, who must choose how much effort to exert in trying to catch dishonest students. Our findings support the determinants of cheating found by the empirical literature, mainly those related to the penalty's level. It is also emphasized the importance of professors being well-motivated (with low disutility of effort) and worried about fairness in classroom. The several extensions of the baseline model reinforce the importance of the cost-benefit analysis to understand dishonest behavior in classroom. Finally, by relaxing the complete information assumption, we discuss the role of students' uncertainty about the professor's type and how low effort professors can send signals to create incentives for honest behavior.

INTRODUCTION
Academic dishonesty is a serious and widespread problem in the world. Although this practice may be found in institutions of all levels of education, it is better documented in colleges and universities. A recent survey conducted in the UK found that nearly 50,000 university students have been caught cheating from 2012 to 2015. The same data show non-EU scholars are the most likely to commit the offense, which suggests that student cheating is not restricted to a specific country (The Guardian, 2016). In fact, despite the absence of reliable data for regions such as Latin America, some studies have used alternative measures to estimate that violations of academic integrity have indeed risen in Latin American countries over the past two decades (García-Villegas, Franco-Pérez, & Cortés-Arbeláez, 2015). A substantial rise in student cheating practices has been found in colleges and universities in the United States as well. McCabe, Trevino, & Butterfield (2001), for instance, reports that the number of students that admit to engage in serious test cheating (e.g. copying from another student on the exam) increased from 39% in 1963 to 64% in 1993.
In this paper we propose a three-players static game with complete information in order to model the strategic relationship underlying the student's individual decision of cheating in classroom. Our setting is composed by two students and a professor, who must choose how much effort to exert in trying to catch dishonest students. In the baseline model, with two identical students, our findings highlight the role of the probability that the professor detects dishonesty in driving the student's decision. For instance, an equilibrium in which both students choose not to cheat requires that the probability of being caught committing the offense be large enough. This in turn requires a large level of professor's effort, which is mainly determined by his disutility of effort and the relative weight given to a fair classroom-without cheating-in his utility.
We also provide several extensions of the baseline model by relaxing some assumptions. First we analyze the basic static framework under incomplete information, when both students do not know the professor's type, whether lenient or severe. As a result of this modification, other equilibria may arise and the outcome depends mainly on how different the types are and the probability distribution adopted. Equilibria may also change when we relax the assumption that only the cheater student is punished and when we allow the "cheating technology" to be imperfect, such that the grade of student who copied is lower than that of the one who has the exam copied. In the former the possibility of an equilibrium in which both students cheat arises, while the latter increases the chance of a virtuous one. Yet, when we consider the case in which the utility of the student who has the exam copied is affected by the cheating behavior, and he has the option of avoiding that his classmate copies his exam, Nash equilibrium in pure strategies may no exist.
We also consider the case when there exists a further punishment for the dishonest student, that is, if the student is caught cheating, he is punished by losing a constant level of utility-due to failing grade in the course, suspension or expulsion, for example-in addition to zero grade in the exam. Under complete information, we find that a harder punishment decreases the minimum probability required to students choose not to cheat. This results resembles those of the classical analysis of Economics of Crime (Becker, 1968), in which the probability of being caught and the magnitude of the punishment drive the incentives of potential offenders. A more interesting extension is one that relaxes the assumption of complete information and allows the lenient professor to send a signal to students through a commitment that he will apply such a harder punishment in case of catching cheaters. In this scenario, when students believe that the probability of the professor being lenient is low, or there is a large difference between the two types of professor-reflected in their level of effort, for example-, or the cost of applying the harder punishment is not high enough, even the lenient professor can give incentives for both students to play fair.
A final extension allows students to be heterogeneous. They can be heterogeneous in terms of disutility of effort, for instance. This implies they choose different levels of effort and thus receive different grades. Different grades in turn implies different potential benefits for the cheater: the student with the lower grade has more to gain by cheating than the one with higher grade. This can be seen through the fact that the minimum probability that induces the student to play fair is decreasing in his own grade and increasing in the grade of the another student. Once again our model shows a feature of Economics of Crime, namely the higher the potential benefit of the offense, the more prone to commit it the individual is. In fact, the difference in terms of grades may be so large that playing not to cheat may be a dominant strategy for the student with higher grade regardless the probability of being caught cheating.
Although our framework presents similarities with the seminal model proposed by Gary Becker, there is an important distinction between them. 1 In our cheating game each student is at the same time a potential offender and a potential victim. For instance, the set of potential outcomes includes one in which both students choose to cheat, case in which they both copy the exam of the other student and have his own copied by the other. On the contrary, in the classical version of Becker's model (Becker, 1968) there is no active role for victims. Given that his framework is not a game, the only agent to play is the potential criminal. Observe that if we considered that each student has his own crib note, made previously at home, and he used it to cheat, his choice about cheating or playing fair would depend exclusively on the probability of being caught. Thus the other student's choice no longer would affect his decision and no student would be a "victim". Unlike our baseline model, this version would be very similar to the one provided in Becker's seminal paper. Furthermore, even police or law enforcer has no role in Becker's model. In our game, however, their role is played by the professor, which makes the probability of the cheater being caught endogenous. In fact, once we allow for the existence of two different types of professor, the value of the probability of catching cheaters depends on whether the "law enforcer in classroom" is lenient or severe. Even when the professor is severe, there is a possibility that his level of effort does not achieve the minimum required to create incentives for students to play fair. The actual professor type therefore affects directly the game's equilibrium. This feature is even more salient when we consider the extended model with incomplete information and possibility of signaling. In this case, the active role of the professor can also affect the level of uncertainty that students face-when the lenient chooses to signal in order to mimic the severe's behavior, for example. The existence of different types of "law enforcers", and the consequent uncertainty about which the true one is, is a novel contribution of our paper when compared to the standard literature on economics of crime, and to Becker's paper in particular.
Our contribution to the literature is to provide a theoretical framework that is able to capture all the strategic features of students' choice of cheating, and professor's choice of how much effort to exert in order to catch cheaters. To the best of our knowledge, the vast majority of literature on academic dishonesty adopt a psychological approach to investigate the determinants of student cheating (Macfarlane, Zhang, & Pun, 2014;McCabe et al., 2001). On the one hand, individual factors such as gen-der, grade point average (GPA), work ethic, competitive achievement striving and self-esteem have been found as having significant influence on the prevalence of cheating (Baird, 1980;Ward & Beck, 1990). On the other hand, contextual factors such as faculty response to cheating, sanction threats, social learning, and honor codes have also been shown to influence dishonest behavior (Michaels & Miethe, 1989). In fact, even those studies which perform economic cost benefit analysis often do it empirically, without a microeconomic model to support their results (Bisping, Patron, & Roskelley, 2008;Bunn et al., 1992).
An important exception is the study of Briggs, Workman, & York (2013), which uses game theory to analyze collaboration in academic cheating. The authors provide a relevant discussion about the use of mathematical utility modeling-and thus game theory-with respect to ethics. Based on the reasoning developed in works such as Gibson (2003), they argued that incorporation of games such as the Prisoner's Dilemma into the ethics issues-and thus in cheating as well-may be useful to better understand costs and benefits involved. However, their model focuses in collaboration in take-home tasks rather than in-class activities, and thus is substantially different from the framework we develop in this paper.
In particular, Briggs et al. (2013) studies collusion among students by presenting a game of team cheating, in which each student from a group of three people must choose whether or not to cheat, and must also choose whether or not to tattle on the cheater member to the professor. The probability of succeeding in cheating is endogenous, given that whenever at least one student tattles, the cheater is caught and punished. Observe that the model presented by those authors focuses on collaborative cheating rather than individual cheating, which makes it quite different from ours. Moreover, there is no active role for the professor, since the punishment is exogenously defined and the probability of cheaters being caught is determined by students's behavior.
The game-theoretic approach we employ allows us to provide both positive and normative conclusions. 2 First, our model fits several stylized facts found by empirical studies, such as the effect of higher penalties in decreasing prevalence of cheating, the inverse relationship between GPA and cheating behavior, and the importance of peer cheating behavior in explaining it (McCabe et al., 2001). Second, it also provides insights that can help reduce cheating on campuses. Some of them had already been found effective by the literature, such as harsh penalties imposed by both the institution and the professor. Others, however, have not received much attention, including hiring high effort professors, who value fairness in classroom. As we show below, a necessary condition for the existence of a virtuous equilibrium (without cheating) is that the professor do not be lenient.
The rest of the paper is organized as follows. In the following section we present our baseline model, composed by two identical students and a professor. We discuss the incentives each one faces and describe their processes of choice. This section also establishes necessary and sufficient conditions for the existence of a Nash equilibrium without cheating. Section 3 extends the model in several directions. Section 4 concludes and suggests some other extensions. The proofs of propositions omitted in the text are shown in the Appendix.

The baseline model
Our baseline model is composed by two identical students, A and B , and one professor (or teacher). There will be an exam in the course that the professor is in charge. Each student must choose his level of effort in studying and the professor must choose his effort to detect and punish in-class cheating. Whenever a student chooses to cheat, he does not study, such that his level of effort is equal to zero. All these actions are chosen before the exam happens, there is no communication among the players and the information is complete, such that we can model this strategic situation as a three-players static game.
We consider only one type of academic dishonesty, namely the action of one student of copying from the another student's exam without his consent or knowledge. Therefore, we rule out common cheating practices such as helping someone on the exam and using a crib note. We also do not consider academic cheating in take-home tasks, such as representing someone else's work as your own (e.g. sharing another's work, purchasing a term paper or test questions in advance, paying another to do the work for you). Given this assumption, whenever the professor detects a dishonesty action, he can punish only the student that copied the exam. We assume the punishment sets the dishonest student's grade equal to zero. If the student succeeds in cheating, his grade is equal to that of the other student.
The student's utility is a C 2 function and is given by where N i is his grade on the exam and e i ∈ [0,+∞) is his level of effort in studying, with i = A,B . We assume the marginal utility of the grade is positive, ∂U i /∂N i > 0, and the marginal utility of the effort is negative, ∂U i /∂e i < 0. Given the possibility of cheating, the grade of student i depends on his effort, the other student's grade, the probability of being caught cheating p ∈ [0,1], and mainly on his chosen strategy, whether cheating (C) or "playing fair" (PF). Moreover, given that players are identical, they have the same utility as well as the same grade function. There are four possible cases to consider: (i) Both students A and B choose to cheat: as in this case none of them exerts any effort in studying, both their grades are equal to zero, N A = N B = 0. (ii) Both students A and B choose to "play fair": in this case each player exerts his optimal level of effort e * i > 0, such that the grades are N A . Given the assumption of identical (1 − p) . (iv) student B plays fair while student A cheats: here we have the opposite of the case (iii), such that . Student's grade is increasing in his effort in studying. However, the return of the effort is decreasing. We also assume some other conditions on the behavior of this function, which may be seen equal to the Inada conditions. The assumption below summarizes and formalizes the features of the grade function.
Assumption 1. The student's grade is a C 2 function of his own effort e i , given by N i : [0,∞) → [0,10], and satisfies the following properties: Whenever student i chooses to play fair, he must maximize his utility by choosing the optimal level of effort e * i . The first order condition of his problem is then given by which can be understood as the equality of the marginal benefit of effort, through the increase in the student's grade, and its marginal disutility. We discuss the existence of such an optimal choice below.
Proposition 2. Suppose that the student's utility function has the following further characteristics: Then the first order condition of the student's problem (1) has an unique global maximizer at some interior point e * i .
The first assumption of the above result means that the total marginal effect of the effort is positive when e i = 0. This is equivalent to make the assumption that , that is, the marginal gain of utility due to the increase in the grade is higher than the disutility of effort when the level of effort is zero. Thus, our model rules out "very lazy" students. The proposition also assumes that both the disutility of effort and the marginal utility of the student's grade increase at increasing rates for all levels of effort and grades, ∂ 2 U i /∂e 2 i < 0 and ∂ 2 U i /∂N 2 i < 0, respectively. Although the assumptions made about the second derivatives are quite standard when one want to guarantee concavity, it is possible to think about at least two cases in which they could be relaxed. The first one is when there exists a minimum grade for passing the exam, sayN ∈ (0,10) . In this case, it is reasonable to assume that ∂U i /∂N i is not monotonically decreasing, instead there must exist a neighborhood ofN where the marginal utility of the grade is increasing. This means that ∂ 2 U i /∂N 2 i > 0 in such neighborhood, which may make the student's optimal choice be different from the one of our baseline model.
The second case happens when the student is of the type which always wants to achieve the maximum grade. This may fits students who want to graduate with honors, for example. Students with such behavior present increasing marginal utility of the grade, such that ∂ 2 U i /∂N 2 i > 0 for all N i ∈ (0,10) . The same comment about the previous case, concerning the potential changes in the optimal choice, applies to here. However, it is possible to guarantee the concavity of U if the magnitude of the second derivative is relatively small (see equation (A-2) in the proof of proposition 2). In fact, this alternative may also ensure a well-behaved solution in the case with a minimum grade.
The assumption that the mixed partial derivative is non-positive means that the marginal utility of the grade is higher when the level of effort is low than when it is high, ceteris paribus. The ideia behind this assumption is that the student gets a higher marginal pleasure when the grade can be achieved with less effort. The alternative assumption, namely positive mixed partial second derivative, suggests a behavior in which the student feels that his effort is rewarding, such that the marginal utility increases with the level of effort. Once again, it is possible to obtain the concavity of U with such alternative assumption as long as we impose bounds in the magnitude of the derivative.
Some comparative statics results may help us to understand the student's behavior.

Proposition 3. The student's optimal level of effort is a function that: (i) is decreasing in the marginal disutility of effort; (ii) is increasing in the marginal utility of his own grade; and (iii) is increasing in the return of the effort on higher grades.
The professor's utility depends on the grades of each student, the probability of catching students cheating in class, and his effort to catch in-class cheating θ ∈ [0,+∞) . We model it as a C 2 function given by W (N A ,N B ,p,θ ) . We assume the professor gets more satisfaction as students' grades increase, that is, ∂W /∂N A = ∂W /∂N B > 0. There is also an disutility of effort, such that ∂W /∂θ < 0. Finally, the professor wishes the fairest possible class, which means the marginal utility of the probability of catching any student cheating is positive, ∂W /∂p > 0. We must impose some further regularities in the professor behavior.
Assumption 4. The professor's utility function has the following further characteristics: (i) it is strictly concave for all levels of effort, that is, The characteristics of the probability function are quite standard and satisfy the Inada conditions, as we highlight below.
Assumption 5. The probability of catching any student cheating is a C 2 function of the professor's effort θ , given by p : [0,∞) → [0,1], and satisfies the following properties: We consider two types of professor, depending on the relative magnitude of his disutility of effort. The lenient professor is characterized by a very high disutility-or a very low marginal utility from the increase in the fairness of the class-when his level of effort is zero, such that dW L (a,b,0,0)/dθ ≤ 0 for all constant a,b ∈ [0,10]. This means that any effort to catch dishonest behaviors does not leave the lenient professor better off, regardless students' grades. In other words, for this type of professor the benefit from the increase in the probability of catching is not higher than the desutility of effort. 3 Observe that increases in professor's effort-and thus in probability of catching-do not positively impact the grades for any students' choices, so we can disregard such an effect in this case.
The other type is the severe professor, who is characterized by dW S (a,b,0,0)/dθ > 0 for all constant a,b ∈ [0,10]. Now an initial effort is worth, because the marginal benefit of increasing the probability p is higher than the disutility caused by such an effort. However, this case presents a further complexity, such that we may have to consider the effect that the higher probability has on the students' grades. For example, if only one of the students cheats, say student A, , and increases in p make a decrease. The impact on the severe professor's utility is then −∂W S /∂N A · N B p ′ < 0. We must later analyze whether such an effect is large enough to overcome the other two, which would make the severe professor mimic the lenient's choice.
As usual, the severe professor chooses his optimal level of effort θ * by maximizing his utility. The first order condition of his problem when student A cheats and student B plays fair is given by While marginal benefit of the professor's effort is only ∂W S /∂p · p ′ > 0, the marginal cost is composed by the direct disutility of effort, ∂W S /∂θ < 0, and the potential impact on the student A's grade, When student A plays fair and student B cheats, the professor's FOC is similar to (2), except by the exchange of subscripts. For the other two cases (both students cheat and both students play fair), the FOC is also similar to (2), but now without the effect on the grades.
Proposition 6. Suppose that the severe professor's utility function satisfies Assumption 4. Then the first order condition of his problem (2) has an unique global maximizer at some interior point θ * > 0.
The assumptions assumed in order to assure the existence of the global maximizer above are quite standard, as we have already discussed. We can now establish comparative statics results for the professor's optimal choice. Proposition 7. The severe professor's optimal level of effort is a function that: (i) is decreasing in the marginal disutility of effort; (ii) is decreasing in the marginal utility of students' grade; (iii) is increasing in marginal utility of the probability of catching students cheating; and (iv) is increasing in the marginal return of effort on the probability.
The above result states that when marginal benefits or marginal costs change, the optimal choice changes as well. While items (i) and (ii) are related to marginal costs, items (iii) and (iv) are associated to marginal benefits. This explains why in the former the relationship is inverse and in the latter it is direct. Item (ii) deserves some attention. Observe that the students' grades affect the professor's level of effort only if one of them chooses to cheat. In this case, there is a negative effect: higher effort implies higher probability, which in turn decreases the cheater's grade. Therefore, the optimal level of effort decreases when there are increases in ∂W /∂N i , with i = 12, because now the negative impact of increases in the probability of catching on the grades is higher.
We can sum up the game in the two payoff matrices below. For the sake of simplicity we henceforth set U (0,0) = 0 and W (0,0,0,0) = 0.

The student's decision
Let us consider the decision of student A. As we are assuming symmetry, the same results are valid for student B . First, suppose that the professor chooses θ * = 0 and student B chooses to cheat. In this case, student A chooses to play fair, because If the professor chooses θ * = 0 and student B chooses to play fair, student A now is better off by choosing to cheat, because where we once again use the symmetry assumption. Suppose now that the professor chooses θ * > 0. If student B chooses to cheat, student A chooses to play fair, because his payoffs are the same as those given by (3). However, if student B chooses to play fair, the choice of student A depend on the probability of being caught cheating. In fact, student A plays fair if and only if The next proposition helps us understand the student's behavior. 4 Proposition 8. There exists a probability of being caught cheating p min ∈ (0,1) such that for i,j = A,B and i j .
As the proof of the proposition shows-see Appendix, section A.1-, function is increasing in p : the higher the probability of being caught, the stronger the incentive to play fair. Thus, the above result implies playing fair is a strictly dominant strategy for student A if and only if p * > p min . Recall that we are assuming identical students, such that the same reasoning can be used to show that student B chooses to play fair regardless the another student's choice if and only if p * > p min . Yet, when p * = p min both students are indifferent between playing fair and cheating. We can sum up the students' best choices with the support of the above two payoff matrices. In order to do so, let us disregard the professor's decision for a while. In the first matrix, given θ = 0, there are two equilibria: student A plays fair and student B cheats and; student A cheats and student B plays fair. In the second one, given θ * > 0, there are three cases to consider. First, if p * < p min , then we have the same two equilibria found in matrix θ = 0. Second, if p * > p min , both students choose to play fair. Third, if p * = p min , then there are three equilibria: both choose to play fair; student A plays fair and student B cheats and; student A cheats and student B plays fair. We must now study the professor's best choice.

The professor's decision
Let us start with the lenient one. One can readily see that this type has a dominant strategy, namely θ * = 0. The assumption that his disutility of effort is relatively high implies . This covers all possibilities.
Despite the larger complexity of the behavior of the severe professor, there are two straightforward cases. First, suppose that both students choose to cheat. In this case we have W S (0,0,0,0) = 0 < W S (0,0,p * ,θ * ) , such that his best choice is θ * > 0. When both students choose to play fair, his best choice is once more , which is implied by the assumption that defines the severe type.
Suppose now that student A cheats and student B plays fair. The severe professor's best choice is θ * > 0 if and only if Recall that the definition of severe professor states that p ′ · ∂W S /∂p + ∂W S /∂θ > 0 when θ = 0, which means that he makes a positive effort whenever it has no effect on the students' grades. Thus, we must check whether the marginal effect of the effort on the grades, namely their decrease as a result of the increase in the probability of catching dishonest students, is enough to overcome the benefit measured by the above derivative. Formally, the severe professor's best choice is θ * if and only if ( Notice that when the above inequality is strict θ * > 0 strictly dominates θ * = 0, and when the equality holds the professor is indifferent between the two strategies. Furthermore, one can readily see that when student A plays fair and student B cheats, the severe professor's best choice is θ * > 0 if and only if ( . Before analyzing game's equilibria, an important remark must be made. A quite strong assumption that we made is that the effect of the students' grades on the professor's utility is independent from the way the grade is obtained, whether fairly or through cheating. Although this drawback is partially addressed by the presence of the probability p in the professor's utility, which means that there is a direct effect of fairness on his welfare, we must discuss how our results would change when such assumption is relaxed. Observe that our original assumption allows the existence of a kind of "pact of mediocrity", in which professor may choose not to exert any effort to try to catch dishonest students, and thus to increase their grades. In fact, this is the case of the lenient professor and, as we commented in footnote 3, this situation is not rare. Let now the professor utility function be given by W (N A ,N B ,θ ,p) − ψ , where ψ > 0 whenever at least one student cheats and ψ = 0 whenever both play fair. Here we are assuming that, once the professor finds out that there is cheating in classroom, he is not able to identify who the offender is. Thus, his disutility depends only on the presence of cheating, and not on the number of cheaters. As it is possible to note below, if we relaxed such assumption, there would be no change in our analysis.
For the severe professor, when student A plays fair and student B cheats, the best choice is θ * > 0 if and only if The above expression shows that there is no change in the professor's choice. In fact, once both matrices (those associated to θ * = 0 and to θ * > 0) are affected equally, there is no change in the comparisons made by the professor and so the equilibria remain the same. This conclusion also holds when the professor's utility is affected in different ways (by a decrease in the impact of cheater's grade, for instance), since any disutility would have the same effect in both matrices. Thus, we can invoke the ordinal property of the utility function and keep our original simplifying assumption.

Discussion on Nash Equilibria
We have already found all the players' best choices. Now we are able to compute all the several possible Nash equilibria of the baseline game. However, most of those outcomes involve at least one student cheating. In this section we are particularly interested in finding conditions to guarantee existence and uniqueness of what we call the virtuous equilibrium, a Nash equilibrium in which both students play fair. Our interest in this equilibrium can be explained by the fact that it is Pareto-dominant when θ * > 0: once this outcome is reached, the professor would have a loss in his utility if we tried to increase welfare of some student. Although this is not the case when θ * = 0, we can justify its importance by arguing that θ * > 0 is the most common case in practice. In order to do so, we first establish an important condition: ( which can be seen equal to (8) when the students are symmetrical-in this case Our main result in this section is given by the next proposition. 5 Proposition 9. The baseline game played by student A, student B , and the professor has the virtuous equilibrium if and only if p * ≥ p min , the professor is severe and condition (10) holds. Furthermore, if the above inequality is strict and condition (10) holds with strict inequality, then the equilibrium is unique.
The idea underlying the above result is that the virtuous equilibrium can only happen if professor gives students incentives to behave honestly. This is done by increasing the probability of detecting cheating above the threshold p min , which in turn requires that the professor make a positive effort, which is impossible when he is lenient. Thus, the equilibrium with no cheating is only possible when the professor is severe and his marginal disutility from decreasing grades is relatively low, as established by condition (10). The virtuous outcome is the only Nash equilibrium of the game whenever none of the players is indifferent between their strategies, which is guaranteed when p * > p min and condition (10) is satisfied with strict inequality.
Although the virtuous equilibrium is the most relevant, it is important to study the other equilibria involving cheating. Let us start with the case in which the professor chooses θ * = 0, that is, he is lenient or severe and condition (10) does not hold. As can be seen in the matrix presented above, the equilibria in this scenario are (play fair,cheat,0) and (cheat,play fair,0) . Once the professor provides no incentive to study through punishment, the possiblity of virtuous equilibrium is ruled out. However, given that assumption that excludes "very lazy students", studying is always preferred over having grade equal to zero. As students are homogenous, there is a coordination problem about defining who will be the cheater and who will be the studying one.
If the game were repeated, some alternatives to solve the coordination problem would be to allow mixed strategies and to consider the learning effects on the students' behaviors. Yet, in a static game like ours we must consider other options, such as pre-exam communication and focal point. Suppose that one day before the exam begins, students can talk and reach a self-enforcing agreement about who will cheat and who will study. This would solve the problem, but observe that the characteristic that would determines the students' roles must be one that is not taken into account in students' utility, and it is not a straightforward task, as an example below shows.
Something similar happens when one think about possibilities of focal points: some characteristic that is not reflected in their welfare must be the criterion to choose between the two students. In a one shot game in which players are homogeneous, it is not easy to find such a characteristic. One can think that one student, say A, gives more value to ethics than his classmate and both know that, such that the focal point would indicate the equilibrium (play fair,cheat,0) . However, it is reasonable to think that this difference between students must be reflected either in their disutility of effort or utility of grades. Other alternative is to consider that students have different seating positions in classroom, and one of them is in a position that makes cheating easier. Once again, this may solve the coordination problem, but different positions in classroom may also indicate that the probabilities of being caught cheating are different as well, which would imply that students are not homogeneous.
The same coordination difficulties arise when we study the other possible equilibria. For example, when professor chooses θ * > 0 but p * < p min , we have (play fair,cheat,θ * ) and (cheat,play fair,θ * ) as equilibria, and the same reasoning made above applies. When his choice is θ * > 0 and p * =p min , we have the two previous equilibria plus the virtuous one. Once again there is a coordination problem and now it is even more severe, once the choice is among three different results. Notice that in this latter case, the multiplicity arises because the punishment level makes students be indifferent between playing fair and cheating. One possible focal point here is the existence of some cultural aspect which states that studying is more valued by society than cheating. Nevertheless, the difficulties aforementioned remain.
Finally, there are scenarios in which the severe professor is indifferent between θ * = 0 and θ * > 0. The interpretation of these results is quite the same as before. Furthermore, when this is the case, there is a possibility of a result with five Nash equilibria, namely when p * = p min . The coordination problem is even more severe now, but we can think of a focal point involving some cultural aspect about the professor's behavior. Society may be consider high effort worthy, which would lead professor and students to choose the virtuous equilibrium.

The baseline model with incomplete information
A natural extension of our baseline model is to consider that students do not observe the type of the professor, whether lenient or severe. In order to see how this feature affects the results obtained in the baseline case, let us assume that both students know-in fact, it is common knowledge-that the professor is lenient with probability q ∈ (0,1) and severe with probability 1 −q . In addition, we restrict to the case in which condition (10) holds, since when it does not, the professor's expected level of effort is null, and thus the result is trivial. The professor's expected level of effort is therefore (10) is assumed to be satisfied. Associated to this effort, there is the expected probability of being caught cheating, namely . Given that the probability is an increasing function of θ , it is straightforward to notice that p E < p * . The first conclusion we can draw is that uncertainty increases the chances of cheating, when compared to the case with θ * > 0, because now students expect a lower probability of being caught. For example, if the severe professor is such that p * > p min , then for high enough q we can have p E < p min . Thus, the severe professor is worse off, since now a higher level of effort is necessary to make students' best choice be playing fair. This implies that the virtuous equilibrium will be reached only when the severe professor has lower disutility of effort or higher utility from the fairness in classroom.
From the lenient professor's point of view, uncertainty may bring fairness to the classroom: when students are sure about his type, they know that p = 0 and then will cheat (in equilibrium, only one of them will, as we have seen); but with incomplete information, it is possible that the severe professor is such that p * > p min and q is low enough, such that p E > p min , which make students play fair. Therefore, the presence of uncertainty may change students' behaviors and even the game's equilibria. In particular, while it may be harder for the severe professor to reach a virtuous outcome, it may facilitate the occurrence of such equilibrium when the professor is lenient.

Professor punishes both students
One can argue that the professor faces a further difficulty in trying to catch dishonest students: once he finds out that there is cheating in classroom, the task of identifying who the offender is may be hard. Let us investigate how our results change when at least one of the students is caught cheating and both are punished. Initially, observe that the expected grades when both cheat and both plays fair do not change. Instead, when student A plays fair and student B cheats, their grades now are (1−p) . Further, when students switch their roles, we have (1 − p) . It is straightforward to see that, assuming the above modification, there is no change in the results of the baseline model when the professor is lenient or is severe and condition (10) does not hold. Thus, we must analyze the case in which the professor's choice is θ * > 0. In order to do so, first suppose that B plays fair, and notice that the necessary and sufficient condition for A to play fair is the same of the baseline case. When B cheats, playing fair continues to be the best response of student A, because . The conclusion is that there is no change in the equilibria, which implies that our original assumption may be hold without loss of generality.
However, the results of the baseline model may change when we add a disutility from being unfairly punished. Assume that student i has an extra desutility c > 0 when he is punished by the professor and he did not cheat. In this case, the payoff of student A when he plays fair and his classmate /p * , student A chooses to cheat. Thus, assuming that students are homogenous, we would have an equilibrium in which the students' best choices are (cheat,cheat) .

Imperfect cheating technology
Suppose now that when one student copies his classmate's exam he is not able to achieve the same grade that the classmate does. We can justify such assumption by arguing that, due to the illegal character of cheating activities, in most of the cases students have difficulty to copy the whole exam, and thus his grade must only partially reflect the grade of the other student. We can model this by including a "discount factor" δ ∈ (0,1) in the cheater's grade. For example, when A cheats and B plays fair, the grade of student (1 − p)δ . Observe that δ can be seen as a measure of the "cheating technology", such that, when the technology is perfect, δ = 1, we have the baseline case. Throughout this section we assume that δ is the same for both students. The case in which δ A δ B is quite similar and does not change the main result below.
The first aspect to be noted in this scenario is that student's the best response when his classmate cheats continues to be paying fair. However, now when student B cheats and θ * > 0, A plays fair if and .
This implies that p min δ < p min , where p min δ is the minimum probability that makes student A plays fair when the cheating technology is imperfect. The conclusion is that now there is a stronger incentive to play fair. Once cheating is not effective as it is in the baseline model, this result is somehow expected.
The interesting novel result appear when one considers the case in which θ * = 0. Recall that in the baseline model, student A chooses to cheat if student B chooses to play fair, because U A . With imperfect cheating technology, this best response may change. Observe that the student A chooses to cheat if the student B chooses to play fair if and only if where we continue to assume that students are homogeneous. Thus, for low enough δ , (12) holds with strictly inequality, which means that equilibria when θ * = 0 may change. This result is summarized in the next proposition.
Proposition 10. Suppose that the cheating technology is imperfect, that is, δ ∈ (0,1) , such that the cheater's grade is strictly lower than the one of his classmate, who had the exam copied. Then, there existsδ ∈ (0,1) that makes playing fair be a dominant strategy for both students. Moreover, in this case, the virtuous equilibrium is the unique Nash equilibrium for each type of professor.

Direct impact of cheating on "victim's" welfare
Suppose that the student whose exam is copied has a further disutility from the fact of being victim of his classmate. We can justify this by assuming that he considers unfair that someone else benefits from his own effort. The simplest way to model such situation is to subtract from the students' utility function a disutility d > 0, such that, for example, when student A plays fair and student B cheats, the welfare of the former becomes U A −d while there is no change for the latter. One can note that this modification is quite similar to one in which there is a disutility from being unfairly punished. In fact, the results of both extensions are identical: if d is low enough, there is no change in equilibria; however, if p * < p min and d is large enough, then equilibrium involving (cheat,cheat) emerges.
Let us now consider the case in which the student who decides to play fair can avoid that his classmate succeeds in the attempt of copying the exam. A possible way to avoid the successful cheating is to cover the exam with his body, such that the classmate would be not able to see it. Other alternative is to write the answers with a small handwriting, which would make it difficult the task of copying. In any case, the student who is trying to avoid cheating exerts an additional effort, and this can divert his attention from the exam itself to the protection task. We model the effort's effect on his utility by assuming that his grade decreases whenever he engages in avoiding cheating, such that now γ , where γ ∈ (0,1) is a discount factor. Observe that γ can be understood as a measure of how efficient the protection effort is. We also assume that once an effort is exerted, the cheating behavior does not succeed, that is N B = 0. Therefore the studying student faces the following trade-off: he may exert an effort in order to guarantee that his exam will not be copied, and thus he will not have the disutility d ; or he may do nothing, which implies he has no cost in terms of grade, but his utility will be affected by d .
As a simplifying assumption, let γ be the same for both students. We also assume that d is not large enough, such that cheating is never a best response when the other student also cheats. Now, each student has a set of strategies with three elements, namely playing fair; playing fair, avoiding the copy; and cheating. Let us start by studying the best choices of student A when the professor is lenient or severe and condition (10) does not hold, that is, when θ * = 0. When B plays fair and avoids the copy, A prefers to play fair without avoiding the copy, because U A > U A (0,0) . When B plays fair and does not avoid the copy, the best choice of student A is to cheat, since that , where once again we consider that students are identical.
The most interesting case appears when student B cheats. Now A chooses to play fair and avoid to have the exam copied if and only if which can be rewritten as .
The conclusion is that student A avoids the copy if and only if the disutility of having the exam copied is relatively higher than the utility loss caused by the discount factor γ . Furthermore, the higher γ , the lower the right-hand side of (13), and thus the lower is the mininum value of d required to choose to avoid the copy. Other important result is that when (13) holds with strict inequality, there is no Nash equilibrium in pure strategies. In fact, there is no equilibrium which involves the avoiding behavior. Therefore, when (13) does not hold -or holds with equality-equilibria of the case θ * = 0 are (cheat,play fair and does not avoid,0) and (play fair and does not avoid,cheat,0) , which are the same of the baseline model. When θ * > 0, the best response of student A for when student B chooses to play fair and avoids the copy is the same of the previous case, since that the payoffs does not change. When B cheats, once again the payoffs of A does not change, such that his best response depends on whether the condition (13) holds, like in the case in which θ * = 0. Finally, when student B plays fair and does not avoid the copy, the choice of student A is to play fair if and only , like in the baseline case. This allows us to sum up the conclusions of this section: whenever p * < p min and (13) holds with strict inequality, there is no Nash equilibrium in pure strategies; if p * < p min and (13) does not hold-or holds with equality-, we have the same equilibria of the baseline case; if p * > p min we also have the same equilibrium of the baseline case. Therefore, the possibility of avoiding the copy depends on the magnitudes of d and γ , and when it appears to be optimal to avoid, there is no Nash equilibrium.

Harder punishment
Consider the case when there exists a further punishment for the dishonest student. For example, assume that the institution (school, college or university) has a code of ethical conduct (academic honor code) that establishes punishments such as failing grade in the course, suspension or expulsion. Let F > 0 be the constant disutility of this punishment, such that now the student's expected utility when he chooses to cheat is U i Assume also that this further punishment does not affect the professor's utility-there is no cost to implement it, for example.
The best choice analysis here is quite similar to that of section 2.1.1. In fact, when θ * = 0, there is no difference in the equilibria of the above matrices: given the professor's choice, student A plays fair and student B cheats and; student A cheats and student B plays fair. However, notice that when θ * > 0 students face a further incentive to play fair, namely the punishment F . This new feature does not change the way they make their best choices. For example, student A chooses to play fair if and Therefore, given p , the equilibria of the second payoff matrix are still the same, but now the value of p min is different, as the next result states.
Proposition 11. Suppose that if the student is caught cheating, his grade is set to zero and he is punished by losing F > 0 of utility. Then there exists a probability of being caught cheatingp min ∈ (0,1) such that for i,j = A,B and i j . Moreover, dp min /dF < 0, and in particular,p min < p min , where p min is defined in the proposition 8.
In fact, this further punishment may be large enough to make both students choose to play fair regardless the probability of being caught cheating, as the next corollary shows.
Corollary 1. For any given probability of catching students cheating p ∈ (0,1], there exists a punishment level F min > 0 such that if F > F min , then playing fair is a strictly dominant strategy for both students. The above results are similar to those of seminal study of Becker (1968), in particular, if the cost of committing an offense increases, ceteris paribus, potential offenders will be less prone to do it. Further, the punishment may be large enough to make all of them choose not to commit the offense. This may make us conclude that having a fair class, without cheating, is just a matter of choosing the correct level of punishment F . However, there is an underlying assumption in our framework that may be contested, namely there is no cost for the professor to implement this further punishment. As the empirical literature reports (e.g. McCabe et al., 2001), a considerable number of professors claim to treat in-class cheating lightly because of the bureaucratic costs associated to all the steps of a process of punishment.
Thus, in order to make the above analysis more realistic and interesting, let us consider that, before the exam starts-in the beginning of the course, for example-, the two types of professor (lenient and severe) can choose to send a signal about their behavior to the students. The professor who decides to send the signal says that the student caught cheating will have the further punishment F > 0, like those cited above. However, now there is a cost to implement the punishment ξ > 0, which creates a trade-off for the professor. In addition let us assume that the professor has a disutility ψ > 0 whenever at least one student cheats.
It is fundamental in our model that when the professors announce the punishment in the beginning of the course, they are able to commit to carrying out their promises. It may be hard to envision how such commitments would be possible in a one-shot course, so students might be suspicious of those threats. However, professors might have an incentive to carry through on their promises if they teach this course every year and if they care about their reputation for telling the truth. One possible manner of making such promise credible is to state it formally in writing in the course's syllabus. The one-shot course with commitment can be viewed as a kind of analytical shorthand for a repeated game in which professors value their reputations.
Other important assumption we make is that it is common knowledge that professor is lenient with probability q and severe with probability 1 − q. We also assume that the severe professor always sends the signal F > 0, that is, he always announces that there will be a harder punishment for cheaters. We can justify this by arguing that due to his characteristics, in particular his liking by fairness in classroom, his cost to implement the punishment is very low, or even zero. We also saw in section 3.1 that the severe professor is worse off when there is uncertainty, such that he always has incentives to distinguish himself from the lenient one. Therefore, our focus is to study under which circumstances the lenient professor chooses pay the cost to mimic the behavior of the severe.
Besides the prior q , students share a belief that the lenient professor chooses to send the signal F > 0 with probability µ , that is, Prob(F > 0|lenient) = µ . The timing of the game is the following: nature assigns probabilities for both types of professors; the lenient one chooses whether to send the signal F > 0; students observe the signal and so update their beliefs; then a static game identical to one of our baseline model is played. Notice that if the lenient professor chooses not to send the signal, then students knows for sure his type and the Nash equilibrium is (play fair,cheat,0) and (cheat,play fair,0) .
When the lenient professor chooses F > 0, after observing the signal, students believe that the professor is indeed the lenient with probability Prob(lenient|F > 0) = µq/[µq + (1 − q)]. By doing the same reasoning, they also believe that Prob(severe|F > 0) = (1 − q)/[µq + (1 − q)], where we use the Bayes rule in both cases. This set of beliefs allows to build the expected probability of being cheating: and the inequality holds because p(·) is a increasing function. Observe that we can compare this result to one of the case without signaling (section 3.1): This is other piece of evidence that the severe professor has incentive to send the signal whenever his cost is low enough.
We must now analyze students' best responses. First, because lazy students are ruled out, when student B cheats, student A plays fair, since that . Second, when student B plays fair and is defined as in corollary 1, then playing fair is the best response for student A as well.
and student B plays fair, student A plays fair if and only if condition (14), withp E replacing p * , holds, which can be seen asp E ≥p min . We have therefore two cases to analyze.
The first one is when the severe professor is such that p * ≤p min . This implies thatp E <p min , which means that the lenient professor does not have incentives to send the signal of harder punishment-and thus pay the cost ξ -, because the Nash equilibrium will be (play fair,cheat,0) and (cheat,play fair,0) by all means. Observe that this result is independent on the values of the parameters q and µ .
The second case is when p * >p min . Now, it is possible thatp E ≥p min as long as (1−q)/[(1−p) +qµ] is close enough to 1-for instance, when q or µ are close to zero. If the values of q and µ are such that p E <p min , then we have the same result discussed above, namely the lenient professor chooses not to send the signal.
Finally, let us study the professor's choice. For, suppose thatp E ≥p min . With this expected probability of being caught cheating, each student chooses to play fair. Thus the lenient professor chooses to send the signal that a harder punishment will be implemented if and only if Observe that in the case of homogeneous students, with , the condition above can be rewritten is a simpler way, namely ξ ≤ ψ . We can sum up the conclusion about the lenient professor's behavior in the following way: he will send the signal and adopt a further punishment (pretending to be of the severe type) if and only if two conditions are satisfied, namely (i)p E ≥ p min , which happens whenever µ and q are low enough, or p * is high enough; and (ii) the cost of signaling is lower than the disutility from having someone cheating in classroom.
In Bayesian games terms, we can say that the existence of a pooling equilibria, in which the lenient professor mimics the behavior of the severe one by sending a signal that he will also adopt a harder punishment, depends on several factors: (i) before observing the signal, students must believe that the chance of the professor being lenient is low; (ii) students must believe that the probability that a lenient chooses to send a signal is low as well; (iii) there is a large difference between the severe and the lenient professor, reflected in the large probability of catching cheaters for the severe; and (iv) the magnitudes of the costs ξ and ψ . Therefore, even when ξ < ψ there is no guarantee that the lenient professor will send a signal. Finally, one can note, once again this kind of professor is better off when the uncertainty is high, reflected in low values of µ and q , for example.

Heterogeneous students
Let us now relax the assumption of symmetrical students. If students are no longer identical, they choose different levels of optimal effort. Without loss of generality, we assume that e * A > e * B . This can happen when they have similar utility functions, but different marginal disutility of effort, with the student A's disutility lower than the one of student B . We might also have assumed that either of other two items cited in proposition 3 differ among them, such that we would have the same result. Given that the grade function is increasing in the student's effort, . This inequality indicates that now the incentive for student A to play fair is different than the one for student B . In fact, although student B 's best choices are the same as they are in the symmetrical case, student A's decision depends on the magnitude of his disutility of effort as compared to the benefit from the increased grade. We must detail this difference below.
First observe that student B behaves exactly in the same way he does in the baseline model: when professor chooses θ * = 0, he plays fair if student A cheats, and cheats if student A plays fair, and; when professor chooses θ * > 0, he plays fair if student A cheats, and plays fair if student A plays fair if and only if p * ≥ p min . Because of the difference between students, we now call this minimum probability p min B . The next result shows that the existence of such an minimum probability is not guaranteed for student A.
. Then there exists a probability of being caught cheatinĝ p min A ∈ (0,1) for student A such that

if and only if U
A > 0 and ∂U A /∂e A < 0, the condition above is equivalent to saying that student A's marginal disutility of effort must be high enough to overcome the marginal benefit from the increase in his grade, when his level of effort is zero and his grade is . Whenever this condition fails, the utility from playing fair is higher than the one from cheating, regardless the professor's effort and the probability associated to this effort. In this case, when professor chooses θ * > 0, playing fair is a dominant strategy for student A, since we know that he makes the same choice if the another student cheats (recall that U A > U A (0,0) , such that student A's best choice is to play fair, and this is completely independent of p * .
, which indicates a possibility of cheating, depending on the probability of being caught. The intuition underlying proposition 12 is that student A's grade may be much higher than the one of his classmate that the risk of cheating is not worth taking, even when there are very low chances of being punished.
Proposition 12 also states that whenp min A exists, it is lower thanp min B . Once again, since N A , student A's gain by making a positive effort is higher than the one of the another student, as a result he is "more prone" to play fair than B . This is reflected in the minimum probability that induces the student to behave honestly. Thus, professor's best choice θ * > 0 can now result in several possibilities: As the FOC of professor's problem (2) shows, the chosen case will depend on his marginal utilities, mainly his disutility of effort. For example, if condition (2) holds when −∂W S /∂θ is very large, then the concavity of W implies that θ * will be very low, such that we may have the fifth case.
The potential nonexistence of p min A creates a multiplicity of Nash equilibria in the cheating game with heterogeneous students, a number even larger than the one of the baseline model. Due to this, we constrain our main analysis only for the case in which p min A exists.
,0 ) , student A has a weakly dominant strategy, namely playing fair, such that it suffices to study the best choices of the other two players, and the results are quite similar to those of proposition 9.

Proposition 13. Suppose that e *
A > e * B . Then the game played by heterogeneous students A and B and the professor has a virtuous equilibrium if and only if p * ≥ p min B , the professor is severe and condition (10) holds. Furthermore, if the above inequality is strict and condition (10) holds with strict inequality, then the equilibrium is unique.
The intuition of the above result is quite similar to one of the proposition 9, except by the difference between the minimum probabilities of being caught cheating necessary to make students A and B to play fair. Once again, a virtuous equilibrium requires that the professor exert a level of effort high enough to the probability associated be higher than that minimum level. As the higher the grade the lower the p min , a class composed by high effort students-because of their low disutility of effort, for example-makes the professor's task of maintaining fairness easier.
A possibility that is not exploited in this paper is the presence of more than two studentsidentical or not-in the classroom. This modification would make each student has more potential "victims", but at the same time there would be more potential "offenders" as well. Issues such as how to set students' seating position in order to minimize the chance of cheating would be possible to be studied when there are more players in the game. As we have seen in this section, if these students were heterogeneous, a plenty of possible outcomes would emerge and the differential between students' grades would have an important role.

CONCLUDING REMARKS
The novelty and main contribution of this paper is to highlight that cheating may be seen as a strategic choice, which involves cost-benefit analysis. In fact, our framework provides the microeconomic foundations of both student's choice of cheating or not and professor's choice of trying to catch dishonest students. By applying game theory's tools, we are able to better understand the determinants of academic dishonesty found by the literature, in a similar way the theoretical model of Becker (1968) has done with Economics of Crime. Our findings also provide further policy implications for cheating control in classroom. In particular, we emphasize the importance of professors being well-motivated (with low disutility of effort) and worried about fairness in classroom. Finally, we discuss the role of the students' uncertainty about the professor's type and how low effort professor can send signals to create incentives for honest behavior.
Our model is the first step towards a rigorous treatment of the strategic relationships underlying the students' choice of cheating or not. Therefore, there are several directions in which it can be extended. One that we believe to be promising is to explore even more the model with incomplete information. A student may be unsure about the other student's true grade-or his true effort level, his type ultimately-, which can be modeled by assuming that he has only a belief about it. Such an extension would allow the study of issues such as signaling, which in turn allows us to understand how professor can use the daily contact with students to prevent cheating. In this regard, repeated games also seem to be a good alternative to model the dynamics of the behaviors of both students and professor, and thus their daily contact. Other interesting extension involves to allow collaboration among students, both in take-home and in classroom tasks-with one deliberately trying to help other in the exam, for example-, which would contribute to the incipient literature started by Briggs et al. (2013). We believe that this approach may also provide alternative solutions to the coordination problems found in our model.

A.1. Proposition 2
Observe that Thus, by conditions (iii) and assumption 1, lim e i →+∞ dU i /de i = lim e i →+∞ ∂U i /∂e i = −∞. Recall that U i is C 2 , such that dU i /de i is continuous. Moreover, by condition (i), dU i /de i (0,0) > 0. Thus, we can invoke the intermediate value theorem and conclude that there exists an interior point e * i that satisfies (1). We must now show that that e * i is an unique global maximizer of the student's optimization problem. For, note that for all e i , because of condition (ii), assumption 1 and ∂U i /∂N i > 0. This implies that U i is strictly concave in e i . Therefore, the first order condition is sufficient to guarantee that e * i is an unique global maximizer. □

A.2. Proposition 3
To prove the first claim of the proposition, suppose that student A's best choice e * A satisfies (1). Suppose also that ∂U Thus, given that d 2 U i /de 2 i < 0, we must have e * B < e * A . Second and third claims can be proved by using the same reasoning. For, suppose that Now observe that at e B = e * A we once again have (A-3), such that we can conclude that e * B < e * A . It is straightforward to see that the same inequality is found when we suppose that students have the same marginal utilities, but different returns of the effort on grades. □

A.3. Proposition 6
Item (i) of assumption 4 implies dW S /dθ is strictly decreasing in θ . Moreover, by the definition of severe professor, dW S /dθ > 0 when θ * = 0. From item (ii) of assumption 4 we also have lim θ →+∞ dW S /dθ = −∞ < 0. Finally, given that W is a C 2 function, its derivative is continuous. Thus, the intermediate value theorem applies and there exists an interior point θ * that satisfies (2). This point is an unique global maximizer because W is strictly concave in θ . □

A.4. Proposition 7
We employ the same reasoning of the proposition's 3 proof. We prove the proposition for the case when student A cheats and student B plays fair. Proofs for the remaining cases are straightforward and very similar to this one. Suppose that θ * satisfies the severe professor's FOC (2). For the item (i), assume that all the derivatives of his utility function are fixed except his marginal disutility of the effort, which now is ∂Ŵ S /∂θ < ∂W S /∂θ . This implies that at θ = θ * Thus, given that d 2 W S /d 2 θ < 0, we must haveθ < θ * , whereθ solves For item (ii), assume that the only derivative that is not fixed is One can see that at θ = θ * we once again have dW S /dθ < 0, which impliesθ < θ * . We can repeat the procedure for the other two items and find that dW S /dθ > 0 when θ = θ * . Therefore, in those caseŝ θ > θ * . □

A.5. Proposition 8
First, define a function f : . Then, observe that f is continuous, because so is U i , and that is, f (p) is strictly increasing for all p ∈ [0,1]. Now we can compute where we use the symmetry of the students and the fact that U i (0,0) = 0. Therefore, given that p ∈ [0,1], the continuity of f and f ′ (p) > 0, then there exists p min ∈ (0,1) such that f (p min ) = 0, which is what had to be proven. □

A.6. Proposition 10
Recall that, because our assumption that rules out "very lazy students", namely ∂U i /∂e i (0,0) > 0, when student j chooses to cheat, student i is better off by exerting some effort and thus chooses to play fair. If we show that there existsδ ∈ (0,1) such that ∂U i /∂e i ( N jδ ,0 ) > 0 for i = A,B and j i , then student i will exert a positive level of effort in every possible case, which implies that the strategy playing fair is dominant for both students and the profile (play fair,play fair) will be part of the equilibrium both when θ * = 0 and θ * > 0.
Let N j be a fixed grade and define where δ k = δ /k with δ ∈ (0,1) fixed and k ∈ N, and observe that ( δ k N j ,0 ) → (0,0) as k → ∞. By using the continuity property, we have because ϵ > 0 was arbitrary. Finally, we have to prove that the virtuous equilibrium is unique for each type of professor. For the lenient one is trivial, once both students have playing fair as dominant strategy. Thus, the equilibrium is (play fair,play fair,0) . The severe professor compares the following payoffs: ) ,θ * ,p * ) .
Given our assumptions and definition of severe professor, he chooses θ * > 0 and the unique equilibrium is (play fair,play fair,θ * ) . □

A.7. Proposition 11
The proof is similar to that of proposition 8. First, define a function д: [0,1] → R, given by and recalling that д(p) is continuous, because so is f (p) , we can once more invoke the intermediate value theorem to conclude that there exists ap min ∈ (0,1) such that д(p min ) = 0.
We must now show thatp min is decreasing in F . For the particular case p = p min , we have д(p min ) = f (p min ) + p min F = p min F > 0, where p min is also defined in proposition 8. Given that д ′ (p) > 0 for all p , it must be the case thatp min < p min . For the general case, one can see that dp where we used the implicit function theorem. □

A.8. Corollary 1
Letp ∈ (0,1] be a given constant. Now we consider the function h: R * → R, given by Observe that if F > F min , then h(F ) > 0, that is, playing fair is the best choice when the another student plays fair as well. Given that student i also chooses to play when student j cheats, the strategy is dominant when the above condition holds. Now we have to consider three cases. First, ifp > p min , then the numerator of the above expression is negative, and so is F min . Therefore, in this case any punishment F ≥ 0 guarantees that playing fair is a dominant strategic for both students. Second, ifp = p min , then F min = 0, such that any positive punishment is sufficient for the result. Finally, ifp < p min , then F min > 0 and we need that F > F min . □

A.9. Proposition 12
and consider again function

Now, suppose that there exists ap min
. The final step of the proof is to demonstrate thatp min A <p min B . This can be shown by calculating the following derivative: that is, the minimum probability of student i decreases when his own grade increases, ceteris paribus. Therefore, if when the students are identical we havep min A =p min B , now the one with higher grade must have a minimum probability lower than his classmate. □ Identificamos os determinantes da inadimplência no Sistema Nacional de Crédito Rural do Brasil, utilizando o procedimento ARDL defasagens distribuídas testes de limites para cointegração de Pesaran, Shin & Smith (2001) e testes de causalidade de Granger de Toda & Yamamoto (1995). Os resultados mostraram que taxa de juros de referência, setor externo e ciclo de negócios não afetam inadimplência; maior razão preços pagos por preços recebidos pela agricultura aumenta inadimplência; os processos políticos de renegociação da dívida rural induzem endividamento, risco moral e seleção adversa; e a relação inadimplência e determinantes retorna ao equilíbrio de longo prazo em 19 dias.
We identify the determinants of default rate in the National Rural Credit System of Brazil using the ARDL bounds testing approach for cointegration by Pesaran et al. (2001), and Granger causality tests by Toda & Yamamoto (1995 (BNDES), bancos privados e estaduais, caixas econômicas, cooperativas de crédito rural e sociedades de crédito, financiamento e investimentos; e como instituições articuladas os órgãos oficiais de valorização regional e de prestação de assistência técnica (Martins, 2010).
Para uma investigação ainda mais detalhada dos determinantes da inadimplência do crédito rural no Brasil seria recomendável que em trabalhos futuros fossem levados em conta fatores geradores de risco idiossincrático devido às características dos contratos e características individuais dos mutuários. No entanto, para uma abordagem nesses moldes seria necessário dispor de microdados em painel que trouxessem características especificas dos empréstimos, dos mutuários e dos dados agregados da economia, nos moldes dos trabalhos de Bonfim (2009) e Louzis et al. (2012. Assim, como sugestão para pesquisas futuras que busquem mensurar os determinantes da inadimplência e risco de crédito, recomenda-se esse tipo de abordagem, ainda que a obtenção dos dados necessários a sua viabilização seja um desafio. We apply the tools of development accounting to a broad panel over the period 1970-2014. However, we depart from the traditional Cobb-Douglas hypothesis with Hicks-neutral technological change, and assume a CES technology, which allows for a constant but non-unitary elasticity of substitution, and for non-neutral technological change. For different values of the elasticity of substitution, and different representations of technological change, we find that the cross-country variation in GDP per worker accounted for by factor inputs is decreasing over time until the mid-2000s, when it reverses its trend. In addition, we find that in the recent period technology accounts for up to 80% of the cross-country variation in GDP per worker.

INTRODUCTION
The current consensus in the Development Accounting literature establishes that the breakdown technology vs. inputs is "50-50", (see Caselli, 2005, for instance). That is, 50% of the cross-country variance in GDP per worker can be accounted for by cross-country differences in technology, and the remainder 50% can be accounted for by cross-country differences in factor inputs. However, this consensus rests largely on cross-section exercises with the Cobb-Douglas assumption.
In fact, the Cobb-Douglas (CD) production function is the number one choice to represent the aggregate technology in development accounting exercises. In general, one justifies the CD assumption on grounds that its property of constant factor shares matches the data. However, the evidence in Bernanke & Gurkaynak (2001) suggests that labor shares vary substantially across countries. If indeed factor shares vary across countries, then the CD assumption may not be the best representation for the aggregate technology.
In addition to its property of constant factor shares, the CD production function restricts the elasticity of substitution between capital and labor, henceforth denoted by σ , to be constant and equal to one. Whether or not σ is unitary is an empirical question. And, the empirical evidence does not support an unitary σ . For instance, for a panel of 82 countries over the period 1960-1987, Duffy & Papageorgiou (2000 find evidence that σ is well above unity, whereas Mello (2015), for a panel of 100 countries over the period 1970-2008, finds estimates of σ that are below unity. The value of the elasticity of substitution matters for development accounting exercises and, therefore, getting the appropriate value for σ is important.
Another restriction of the CD production function is that differences in technology arise in a neutral, or bias-free, form. This restriction derives from the property of the CD, which is the only production function in which the three forms of technological change-Hicks neutral, Solow neutral, and Harrod neutral-are equivalent. This can be shown as follows. Take a CD with Harrod-neutral (laboraugmenting) technological change: It is easy to see that this CD is equivalent to a CD with Hicks-neutral technological change, such as this Y = A 1−α K α ( hL ) 1−α , which is also equivalent to a CD with Solow neutral (capital-augmenting) technological change, That is, in the three cases above, technology enters equivalently in a multiplicative form.
One of the implications of this restriction is that if one country is technologically more advanced than another is, then it must use all its factor inputs more efficiently than the other country does. Therefore, a situation in which one country uses capital more efficiently than the other does, while it uses human capital less efficiently, cannot be identified when one assumes a CD production function.
The problem with this is that the evidence suggests that the efficiency with which factor inputs are used varies across countries. According to the evidence in Caselli & Coleman II (2006), rich countries use skilled labor more efficiently than poor countries do, whereas poor countries use unskilled labor more efficiently. Similarly, Caselli (2005) presents evidence that rich countries use human capital more efficiently than poor countries do. In order to identify these differences we need to depart from the CD world.
If indeed the elasticity of substitution differs from unity and factor-efficiency is non-neutral, as the empirical evidence suggests, then performing development accounting exercises relaxing these two constraints may change the consensus view, and, consequently, may change any policy implications derived from the exercise. These two restrictions-unitary elasticity of substitution and factor neutrality-can be relaxed by assuming a Constant Elasticity of Substitution (CES) production function as representative of the aggregate technology. The CES is the simplest production function that allows for a constant but non-unitary elasticity of substitution and non-neutral technological progress.
In this article, we perform a series of development accounting exercises for a broad panel of countries assuming a CES aggregate technology that allows for different values of the elasticity of substitution and factor non-neutrality in technological progress. Additionally, we explore the time variation in the data by applying the tools of development accounting on the time series for GDP per worker from 1970 to 2014, instead of focusing on a specific year as in traditional cross-section studies in the literature.
We construct a panel with data on 84 countries over the period 1970-2014 from the latest version of the PWT, version 9.0. Our estimates suggest that the proportion of the cross-country variability in GDP per worker that can be accounted for by the cross-country variability in factor inputs exhibits a persistent decreasing trend. However, from 2005 towards the end of the sample period, it exhibits a soft increasing trend. In the more recent period, the technology-input breakdown is about "80-20" in favor of technology as the key factor behind the huge observed international variation in GDP per worker. This is a big departure from the "50-50" consensus. Moreover, this finding is robust to different values of the elasticity of substitution, and different representations of technological progress.
Additionally, as a robustness check, we construct two panels with data from PWT 8.1 and PWT 7.0 and apply the same development accounting tools to these panels. Our initial findings are corroborated when we use data from PWT 8.1, and corroborated to a lesser extent when we use data from PWT 7.0. Interestingly, the explanatory power of factors of production as a key determinant of the cross-country variance in GDP per worker is greater when we use data from PWT 9.0 and PWT 8.1.
We contribute to the debate by shedding light on the proximate causes of economic growth. In particular, our study relates to Caselli (2005), Aiyar & Dalgaard (2009), Mello (2009), Ferreira, Pessoa, & Veloso (2008, and Arezki & Cherif (2010). We use the traditional tools of development accounting exercises applied to cross-sectional data, as in Caselli (2005), and apply them on a panel data setting, exploring the time variation in the data as in Mello (2009), Ferreira et al. (2008, and Arezki & Cherif (2010). Moreover, we study the sensitivity of development accounting exercises with respect to the value of the elasticity of substitution in the representative aggregate technology as in Aiyar & Dalgaard (2009), and the effects of non-neutral technological change as in Caselli & Coleman II (2006), and Arezki & Cherif (2010). Additionally, we explore the latest version of Penn World Tables dataset (version 9.0), as well as earlier versions of this dataset (versions 8.1 and 7.0).
We structure this article as follows. In section 2, we briefly review the literature. In section 3, we present our methodology, describing how we can decompose a CES production function into a factoronly component, and a technology component. In section 4, we present our data. In section 5, we present our estimates of the successes measures of the factor-only model for the PWT version 9.0. On section 6, we present estimates of the measures of success of the factor-only model for data from PWT versions 8.1 and 7.0, as a robustness check. Finally, section 7 concludes.

RELATED LITERATURE
The debate about the determinants of the huge observed cross-country income differences, whether it is the technology or factor inputs, goes back to the late 1960s (Caselli, 2008). However, it was not until the publication of Klenow & Rodríguez-Clare (1997) that the tools and tricks of development accounting were popularized.
Development accounting is to cross-section data, what growth accounting is to time series data. In a growth accounting exercise one computes, over a period of time, the growth rate of output and factor inputs, and estimate the growth in total factor productivity (TFP) as a residual. The exercise is helpful to identify the sources of growth, whether it comes from inputs or TFP. If growth in output comes from inputs, then it is likely to be temporary, whereas if it comes from TFP then it can be long lasting.
In a development accounting exercise, one has a cross-section of countries, and performs the decomposition of the level of GDP per worker into factor inputs and technology (TFP). Then one examines to what extent the cross-country variability in factor inputs vis-à-vis variability in TFP can explain the cross-country variability in GDP per worker.
The decomposition exercise can give insight into the proximate causes of growth. By identifying the sources of cross-country variability in GDP per worker one can think about policies aimed at reducing inequality among nations. For instance, if one finds that the quantity of factor inputs can account for a large portion of the cross-country variability in output per worker, then, instead of focusing on technology, policy makers should look into the causes of low accumulation of factor inputs across countries. Caselli (2005), the most cited survey in the literature, performs a series of development accounting exercises for a cross-section of 94 countries in the year 1996 with data from Penn World Tables version 6.1. His estimates suggest the breakdown factor inputs versus technology is about "50-50". According to Hsieh & Klenow (2010), another recent survey, the current consensus establishes that technology accounts for 50-70% of the cross-country differences in GDP per worker.
In our decomposition exercise, we break down GDP per worker in terms of the capital-output ratio, as in Klenow & Rodríguez-Clare (1997), among others. Moreover, to assess the role of factor inputs vis-à-vis technology in accounting for cross-country output differences, we use the methodology in Caselli (2005). Furthermore, we follow Mello (2009), Ferreira et al. (2008, and Arezki & Cherif (2010) in constructing a panel and exploring the time variation in the data, instead of looking at a single point in time as in much of the literature.

METHODOLOGY
We represent the aggregate technology by a CES production function as follows: where Y is output, K is physical capital, A is Harrod-neutral (labor-augmenting) technological progress, h is human capital per worker, and L is the number of workers. The elasticity of substitution given by the parameter σ . If σ = 1 we are back to the Cobb-Douglas world, in which output is given by Y = K α (AhL) 1−α . Aiyar & Dalgaard (2009) also adopt the above functional form. Based on the production function in (1), we can break down output per worker into two components, a factor-only component, and a technology component: In this case, the factor-only model is given by In the CD case, i.e., if σ = 1, the factor-only model is given by Equation (4) is the well-known Klenow & Rodríguez-Clare (1997) break-down. The specification in (1) assumes that technological change is Harrod-neutral or labor augmenting. However, we can extend this specification to include non-neutral technological change. Following Aiyar & Dalgaard (2009), we assume a CES that allows for Harrod (labor) and Solow (capital) neutral technological change: where B denotes the Solow neutral (capital augmenting) technological change, and A denotes the Harrod-neutral (labor augmenting) technological change. We can rearrange equation (5) and break it down into two components, just like we did with equation (1). We obtain the following expression: Ah.
The problem with the above decomposition is that the capital augmenting parameter B is included in the factor-only component. That is, in practice, we have not separated the technology component from the factor-only component. In order to obtain a feasible decomposition based on equation (6), we need to find a way to estimate or "fix" the capital augmenting parameter B . We follow the strategy in Caselli & Coleman II (2006), which is also adopted by Aiyar & Dalgaard (2009). The idea is, first, to fix the parameter B at the "technological frontier", which is taken to be the U.S. level. Second, given competitive markets and the production function in equation (5), we can write the capital share as follows: where S k denotes the capital share. The trick here is to assume that all countries have access to the technological frontier. That is, all countries have access to the same (U.S.) parameter B . From equation (7), we can estimate the parameter B as follows: where the variables with the superscript denote their U.S. levels. Implementing this strategy, when technological change is both Harrod and Solow neutral, the factor-only model, denoted by y AB KH , is given by: With the exception of the non-observable parameter B , which we estimate with equation (8), all other variables in equation (9) can be directly obtained from PWT dataset, or can be constructed from the variables therein. The first measure of success of the factor-only model we look at is the ratio of the variance of the log of the factor-only model to the variance of the log of GDP per worker. We denote this measure of success by S1: Var ( Log(y) ) .
As correctly pointed out in Caselli (2005), the S1 measure is sensitive to extreme values, which may contaminate the analysis. In this sense, Caselli (2005) also considers a second measure of success, denoted by S2, which takes the ratio of the 90 th to 10 th percentile ratio of the factor-only model to the 90 th to 10 th percentile of the observed GDP per worker. The S2 measure is given by where y 90th KH and y 10th KH denote, respectively, the level of GDP per worker of the factor-only model at the 90 th and the 10 th percentile, and Y 90th and Y 10th denote the observed level of GDP per worker at the 90 th and 10 th percentile, respectively.

DATA
We construct our main panel with data from the latest version of Penn World Tables (PWT) dataset version 9.0. Our panel includes 84 countries for which population is equal to or greater than 1 million in 1985, and the time series for the variables we use are complete over the period 1970-2014.
Our measure of output is the variable RGDPO (output-side real GDP at chained PPP in millions of 2005 USD), the measure for the aggregate stock of capital is the variable CK (capital stock at PPP in millions of 2005 USD), and the measure of workers is the variable EMP (number of individuals engaged in production). GDP per worker is calculated as the ratio of RGDPO/EMP , and the capital-output ratio is computed as CK/RGDPO .
Our measure of human capital is the variable hc in PWT 9.0, which is an index of human capital per person, based on years of schooling, from Barro & Lee (2010) dataset, and returns to education, from Psacharopoulos (1994). This measure of human capital is also used in the PWT 8.1.
In addition to the PWT version 9.0, as a robustness check, we work with two other versions of PWT, versions 8.1 and 7.0. For the PWT 8.1, our panel includes 77 countries for which population is greater than or equal to 1 million in 1985, and the time series for the variables is complete over the period 1970-2011.
For the PWT 7.0, we construct a panel with 85 countries for the period 1970-2008. For this panel, we compute the number of workers as RGDPCH *POP/RGDPWOK , where we denote the variables by their PWT 7.0 codes. Real GDP (Y ) is constructed by multiplying the series RGDPWOK2 by the number of workers. The series RGDPWOK2 is given by RGDPL2WOK = RGDPL2*POP/Workers, where RGDPL2 is an updated version of RGDPL which is real GDP (Laspeyre index).
In order to construct the time series for the physical capital stock, we follow Mello (2009) and use the perpetual inventory method. The initial value of aggregate capital is set at I 0 /(д + δ ) , where I 0 is initial investment (measured as the investment in the first year for which data is available), д is the average growth rate in investment for the first year for which data is available until 1970, and δ is the depreciation rate which we set at 6%. Given K 0 , K t evolves according to the capital accumulation equation, namely, K t = (1 − δ )K t −1 + I t . To ensure the quality of capital stock estimates, we initiate the series on the first year for which data is available and discard all observations until 1969. By discarding the initial years, we guard ourselves against a bad initial guess. See Mello (2009) for more details.
Our measure of human capital for the PWT 7.0, uses the average years of schooling for the population 25 years old or older obtained from Barro & Lee (2010) dataset. Specifically, we assume that human capital H is given by H = e 0.1 * u L, where u is the average years of schooling and L is the number of workers. That is, we assume that the Mincerian coefficient of returns to education is 0.1 for all countries.
In the Appendix, we provide the complete list of countries in the three panels that we use, PWT 9.0, PWT 8.1, and PWT 7.0, as well as the list of countries considered rich/poor, as defined in the next section. Our dataset is available upon request.

ESTIMATES
We initially analyze estimates of the S1 measure for the case in which technological change is Harrod neutral only. We assume different values for the elasticity of substitution according to the empirical evidence. The exercises are performed for σ = 1.5 , according to the evidence in Duffy & Papageorgiou (2000), for σ = 0.8 according to the evidence in Mello (2015), and Aiyar & Dalgaard (2009), for σ = 0.5 according to Antràs (2004), and as a benchmark for σ = 1, which is the Cobb-Douglas case. 1 Figura 1 displays the S1 measure for a CES with Harrod neutral technological change for data from PWT 9.0. It contains at least four salient features. First, we observe that the higher the elasticity of substitution the higher the explanatory power of the factor-only model. In particular, the explanatory power of the factor-only model with σ = 1.5 is about twenty percentage points higher than with σ = 0.5 . Second, for values of the elasticity of substitution equal to or less than one, the factor-only model explains a much lower percentage than the 50% consensus. For instance, for σ = 0.80 , the factor-only model explains about 30% of the cross-country variation in GDP per worker in the mid-1970s, and it decreases to about 10% in the mid-1990s. Only if we assume that σ = 1.5 , that the S1 measure comes close to the 50% consensus, but it still trails below the 50% for most of the sample period.
Third, the explanatory power of the factor-only model decreases over time. For instance, in 1970, for σ ≤1, the factor-only model explains about 35% of the cross-country variation in GDP per worker. Moreover, in 2000, for σ ≤1, its explanatory power drops to less than 20%. For σ = 1.5 , the explanatory power of the factor-only model drops by about 20 percentage points over the period 1970-2005. This finding is consistent with the estimates in Ferreira et al. (2008), and Arezki & Cherif (2010), who also find that the explanatory power of the factor-only is decreasing over time. Fourth, in the last eight years of the sample period, 2006-2014, the explanatory power of the factor-only model increases somewhat. For example, in the case of σ = 1.5 , it increases by more than 10 percentage points, while for σ = 1 it increases by a few percentage points.
The observations above are confirmed by examining the S2 measure of success. As can be seen in Figura 2, the pattern of S2 mimics that of S1, so that the four observations we made about S1 in Figura 1 also apply to S2 in Figura 2. One noticeable difference is that, for the entire sample period, according to the S2 measure, the explanatory power of the factor-only model is, on average, five percentage points greater than compared to the S1 measure.
In order to learn more about the cross-country variability in GDP per worker, we segment the sample in three parts: rich, middle-income, and poor countries. We consider rich the 21 countries (top 25%) in our panel with the highest level of GDP per worker in the year 2000. The list of rich countries can be found in Tabela A-1 in the Appendix. Figura 3 displays the S1 for measure for the sub-sample of rich countries.
We only report the S1 measure for rich countries for σ ≤1, because S1 estimates for σ = 1.5 generate too much variability, well above the observed variability in the data. For instance, in the year 1991, the variability generated by the S1 measure for σ = 1.5 is a factor of 12 of the observed variability in observed GDP per worker. In order to avoid any distortion in the figure with such large realizations we omit the S1 estimates for σ = 1.5 . These estimates are available upon request.
The S1 measure for σ = 0.8 and σ = 1, as shown in Figura 3, generates more variability than what is observed in the data for most of the sample period. In particular, for σ = 0.8 the factor-only model fully accounts for the variability in the data until 2002, and for σ = 1, until 2003. For σ = 0.5 , the S1 measure practically explains all of the variability in the data from 1978 to 2000. Interestingly, starting in the early 2000s, the S1 measure for rich countries for any value of σ loses explanatory power fast, reaching 2014 in the range 10%-24%.
The finding that the factor-only model has a higher explanatory power for rich countries is intuitive. After all, for rich countries the observed cross-sectional variance of GDP per worker must be smaller than for the panel as a whole. Additionally, it is plausible to assume that rich countries have access to same technology, and, if so, then cross-sectional differences in GDP per worker must come from cross-sectional differences in factor inputs.
If indeed the source of the cross-sectional differences in GDP per worker are cross-sectional differences in factor inputs, then it is easier for the policy maker to design policies to reduce income inequality. The reason being is that differences in technology can come from many sources, such as credit market imperfections or judicial uncertainty, while differences in factor inputs can be reduced via accumulation of capital, with a high investment rate.
Figura 4 displays the S2 measure for rich countries, including S2 estimates for σ = 1.5 . The explanatory power of the factor-only model is greater for σ = 1.5 . The range of S2 estimates for σ = 1.5 goes from 1.04 to 2.75, while the range of S1 for rich countries, for σ = 1.5 , goes from 1.47 to 13.2, which suggests that the S1 measure is in fact contaminated with extreme values and a higher elasticity of substitution magnifies the effects of outliers. Other than that, the general pattern exhibited by the S2 measure in Figura 4 for σ ≤1 mimics the pattern we observe for the S1 measure in Figura 3.
As in the case for the panel as a whole, the explanatory power of S2 decreases over time. However, until 2001 it generates enough variability to match the data. Interestingly, the loss in explanatory power is small and it only occurs for σ ≤1. In general, we can say that the factor-only model accounts well for the cross-sectional variability in GDP per worker for rich countries. Figures 5 and 6 display the S1 and S2 measures for poor countries, respectively. We classify as poor countries the bottom 21 countries (25% of the panel) ranked according to their GDP per worker in year 2000. The list of countries classified as poor is in the Appendix.
Figura 5 displays the S1 measure for poor countries. Again, we omit from the S1 estimates for σ = 1.5 due to extreme values. The estimates in Figura 5 show a hump-shaped form, with the hump formed between 1985 and 2000. For the years 1970-1984, the S1 estimates fall in the range 35%-60%, and increase over time. In the second period, 1985-2000, for σ = 1, S1 increases and even goes above 1; for σ = 0.8 , S1 estimates hover around 60%; and for σ = 0.5 , S1 estimates decreases below 40%. For the end of the sample period, 2001-2014, S1 estimates decrease fast, reaching low 10s% and 20s%, just to increase slightly in the final three years of the sample period.  1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998   1,6 ‡ A ìUñ ‡ A ìUô ‡ A íUì Figura 6 displays the estimates of S2 for poor countries, excluding estimates for σ = 1.5 . Again, S2 estimates for σ = 1.5 exhibit a lot of variability, being above one for most of the sample period, 1981-2005. For other values of the elasticity of substitution, the pattern we observe in Figura 6 is consistent with the one in Figura 5. That is, for σ ≤1, we observe a hump-shaped form, with the hump between the years 1985-2000. Additionally, the range of estimates is similar to the range of estimates in Figura 5, with the explanatory power of the factor-only model oscillating around 50% in the beginning of the sample period, reaching 100% for σ = 1 around 1990, and settling at 35%-45% at the end of the sample period.
Based on figures 5 and 6, we conclude that the factor-only model, for σ ≤1, accounts for 35%-45% for the cross-country variation in GDP per worker, and for σ = 1.5 , it accounts for 50%-70%. That is, the elasticity of substitution does affect the explanatory power of the factor-only model. However, it does not lead it too far off from the "50-50" consensus.
Figura 7 displays S1 estimates for the case in which technological progress is non-neutral, that is, it is Harrod and Solow neutral. Estimates of S1 for σ = 0.5 and σ = 0.8 , in Figura 7, exhibit a decreasing trend, starting in the low 40s% and falling in the range of 10s% and 20s% towards the end of the sample period. Estimates of S1, for σ = 1.5 , also exhibit a decreasing trend. However, it starts in the low 20s% and reach the 10% around year 2000, when it starts to increase towards 15%. Additionally, for most of the sample period the explanatory power of the factor-only model is greater when σ = 0.8 . This is  1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 Sucess 1 -Poor, PWT 9.0 ‡ A ìUñ ‡ A ìUô ‡ A íUì in sharp contrast with the Harrod neutral only case, in which the explanatory power of the factor-only model is greater with σ = 1.5 . Moreover, the S1 estimates intersect at several points, suggesting that S1 is not monotonic with respect to the elasticity of substitution. The S2 measure for the Harrod and Solow neutral case, shown in Figura 8, mimics the pattern seen in Figura 7. As before, the explanatory power of the factor-only model is decreasing over time, and it is greatest when σ = 0.8 . More specifically, the explanatory power of the factor-only model starts in the range 30%-50%, and decreases over time to the range 10%-20%. The fall in the explanatory power of the factor-only model starts in the mid-1980s and it continues until the year 2000, just to increase slightly until 2014 and finish it around 15%.
Recall that to compute the S1 measure for the case of Harrod and Solow neutral technological change, we assume that all countries have access to the U.S. capital augmenting technology. Note, however, that countries still differ in their labor augmenting technological change. Therefore, all the cross-country variability in technology comes from the cross-sectional variability in labor-augmenting technology. Thus, the decreasing explanatory power of the factor-only model is exactly matched by a larger role of cross-country differences in labor augmenting technology in accounting for cross-country income differences.
The picture that emerges from the above estimates is that the "50-50" consensus seems to be valid until the late 1980s or early 1990s. However, currently, the estimates suggest that the bulk of  1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998  cross-country differences in GDP per worker are due to cross-country differences in the efficiency with which inputs are used. In particular, our estimates suggest that the current breakdown is "80-20" in favor of technology. These findings seem to be robust with respect to the different values of the elasticity of substitution and the form of technological change. Any policy prescription aimed at reducing cross-country income differences should focus on the ability to convert inputs into output, that is, on efficiency, rather than on fostering the accumulation of inputs.

ROBUSTNESS CHECK
In this section, we check for robustness of our estimates by computing the S1 and S2 measures for previous versions of the PWT dataset, namely, versions 8.1 and 7.0. We omit some of the figures here to economize on space (they are available upon request). We construct the panels using the same criteria as in section 4, that is, by selecting countries for which the population in 1985 is above 1 million, and data was available for the entire sample period, 1970for PWT 8.1, and 1970 Figura 9 displays the S1 measure for the PWT 8.1. First, the higher the elasticity of substitution the higher the explanatory power of the factor-only model. In particular, the explanatory power of the factor-only model with σ = 1.5 is about 20 percentage points higher than with σ = 0.5 . Second, for values of the elasticity of substitution equal or less than one, the factor-only model explains a much lower percentage than the 50% consensus. For instance, for σ = 0.80 , the factor-only model explains  1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 Sucess 1 -PWT 8.1 Sucess 2 -PWT 8.1 about 30% of the cross-country variation in income per worker in the mid-1970s and it decreases to about 10% in the mid-1990s. Only if we assume that σ = 1.5 , that the S1 measure comes closer to the 50% consensus, but it still trails below the 50% for most of the sample period. Lastly, estimates in Figura 9 are consistent with the ones we obtain with data from PWT 9.0. We confirm the above observations by examining the S2 measure of success for PWT 8.1, as shown in Figura 10. The pattern of S2 over time mimics that of S1. Therefore, the same observations we made for Figura 9 are also valid for Figura 10. One noticeable difference is that according to the S2 measure, the explanatory power of the factor-only model averages about five percentage points higher than compared with the S1 measure. Figures 11 and 12 display, respectively, S1 and S2 estimates assuming a CES with non-neutral technological change, and using data from PWT 8.1. The overall pattern of S1 in Figura 11 is consistent with the one in Figura 7. However, the explanatory power of the factor-only model is substantially reduced when compared to Figura 7, which was constructed using data from PWT 9.0.
Probably the most interesting aspects in figures 11 and 12 are the decreasing trend in S1 and the end of period kick back. These two aspects that are also present in estimates from PWT 9.0.
As mentioned above, we do not present all the corresponding figures for S1 and S2 constructed with data from PWT 7.0. Below, we present some of the estimates from PWT 7.0, comparing them with estimates from the more recent versions of the PWT.
Assuming Harrod neutral technological change only, Figura 13 displays S2 estimates for σ = 0.8 for three versions of the PWT we work with. As can be seen in Figura 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 Sucess 1  crease. Interestingly, the explanatory power of the factor-only model is quite low when we use data from PWT 7.0. The explanatory power of the factor-only model is greater with data from PWT 9.0, although from the mid-1990s towards the end of the sample period the S2 measure has more or less the same value whether computed with data from PWT 9.0 or PWT 8.1. These observations are also true for S2 estimates for σ = 1.5 assuming Harrod neutral technological change as shown in Figura 14.
We also compute the S2 measure, constructed assuming Harrod and Solow neutral technological change, using data from the three versions of the PWT we work with. Figura 15 displays S2 estimates for σ = 0.8 . The decreasing trend in S2 is present in all cases, less pronounced when we use data from PWT 7.0, but very strong when we use data from PWT 9.0 and 8.1. Again, the explanatory power of the factor-only model is greater when we use data from PWT 9.0. Interestingly, around 1995 we observe a convergence in S2 for data from PWT 9.0 and 8.1, with the explanatory power of the factor-only oscillating between 10% and 15%. Furthermore, we observe the increase in the explanatory power of the factor-only from 2005. These observations are also valid for Figura 16, which displays estimates of S2 assuming Harrod and Solow neutral technological change and σ = 1.5 . We should only add that for σ = 1.5 , the explanatory power of the factor-only model is some 20 percentage points lower than when σ = 0.8 .  1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 Comparing S2

CONCLUSION
We construct a broad panel with 84 countries over the period 1970-2014 with data from the latest version of PWT 9.0, and apply the tools of development accounting. We depart from two traditional assumptions commonly employed in the literature, namely, the Cobb-Douglas assumption and neutral technological change.
We adopt a CES production function that allows for a constant but non-unitary elasticity of substitution and non-neutral technological change. Our estimates suggest that the explanatory power of the factor-only model exhibits a decreasing trend, with a soft kick back from 2005 to 2014. Additionally, when technological change is Harrod neutral the explanatory power of the factor-only model is greater for σ = 1.5 , whereas when technological change is non-neutral the factor-only model explains more for σ = 0.8 for PWT 9.0 data, and for σ = 0.5 for PWT 8.1 data.
Finally, and perhaps most importantly, we find that in the more recent period, the 2000s, crosscountry differences in technology can account for up to 80% of the cross-country variation in GDP per worker. This suggests that countries should be primarily concerned with the efficiency in which their factor inputs are used, rather than the accumulation of factor inputs.  A intuição básica da hipótese de causalidade reversa aqui formulada é a de que os juízes se sentem cada vez menos confortáveis em dar o enforcement conforme a taxa de juros do contrato se eleva. Por exemplo, o juiz está mais propenso a mandar pagar rigorosamente o que está previsto em contrato quando a taxa de juros estipulada é de 12% ao ano do que quando é de 12% ao mês. Em outras palavras, a hipótese de causalidade reversa é a de que, quando se discute em juízo a validade de um contrato de financiamento, os integrantes do Poder Judiciário têm maior propensão a julgar favoravelmente aos devedores conforme aumenta a taxa de juros do contrato.
Em particular, a rigidez da política monetária brasileira seria compreensível sob o regime de câmbio fixo que prevalecera até 1999, mas sua permanência após a flutuação do câmbio continuaria ainda sem explicação. A conjectura dos autores foi, assim, a de que distorção seria e enorme dificuldade para o enforcement de contratos. Confira-se: It is an uncertainty of a diffuse character that permeates the decisions of the executive, legislative, and judiciary and manifests itself predominantly as an anti-saver and anti-creditor bias. The bias is not against the act of saving but against the financial deployment of savings, the attempt to an intertemporal transfer of resources through financial instruments that are, in the last analysis, credit instruments. (Arida et al., 2005, p.270, grifo meu) Jurisdictional uncertainty worsened after the 1988 Constitution introduced the possibility of changes in the interpretative emphasis between conflicting constitutional principles, particularly the subordination of private property to its social function. The Constitution of 1988 is a striking example of how the paternalistic attempt to substitute the government for the market in the allocation of long-term resources aggravates jurisdictional uncertainty. (Arida et al., 2005, p.272) O grande indício da existência de um problema de incerteza jurisdicional particularmente grande no Brasil estaria, então, na existência de mercado de títulos de longo prazo para devedores brasileiros apenas fora, mas não dentro, do Brasil. De resto, a fundamentação de Arida et al. pouco diferiu do que já se vinha mencionando em trabalhos anteriores. Assim, o texto formulador da hipótese de incerteza jurisdicional referiu novamente o questionário de Lamounier & De Sousa (2002) e mencionou um estudo de Amadeo & Camargo (1996) que retratara a parcialidade da Justiça do Trabalho -cuja versão inicial, aliás, já houvera sido mencionada no trabalho seminal de Pinheiro & Cabral (1998). Também não faltaram evidências anedóticas, como segue: The bias is transparent in the negative social connotation of figures associated to the moneylender -"financial capital" by opposition to "productive capital", "banker" as opposed to "entrepreneur". The debtor is viewed on a socially positive form, as an entity that generates jobs and wealth or appeals to the bank to cope with adverse life conditions. This bias may be observed more or less everywhere, but it is particularly acute in Brazil, probably because of the deep social differences and the high levels of income concentration in the country. Cultural and historical factors could also have facilitated the dissemination of this anti-creditor bias. (Arida et al., 2005, p.271) Yeung & Azevedo (2015) procuraram testar de forma rigorosa a hipótese de favorecimento sistemático do Judiciário aos devedores nas relações contratuais. O trabalho partiu de uma base de 1.687 decisões do Superior Tribunal de Justiça (STJ) entre os anos de 1998 e 2008. Foram extraídas variáveis sobre o tipo de recorrido ou recorrente, ou seja, se a parte era pessoa jurídica, pessoa física ou instituição financeira e tipo de dívida. Os resultados mostraram que não há viés explícito pró-devedor (que poderia ser identificado apenas nas estatísticas descritivas).

Metodologia
O text mining é um processo computacional de obtenção de informação de alta qualidade a partir de textos. Um software foi programado para ler milhares de decisões disponíveis online usando um algoritmo de classificação das sentenças a partir de uma variação da técnica Term Based Method. A linguagem de programação foi Python, sendo que a principal biblioteca utilizada para processamento dos dados foi a Natural Language Toolkit (NLTK).