I distinguish between measures of risk aversion to betting in terms of dated consumption and measures of risk aversion to betting in terms of money. Bets in terms of money (wealth) are settled immediately before any irreversible consumer choice is made. In terms of Figure 1, the smooth consumption path a promises 3 units of consumption in each period.
The attitude towards betting in terms of money is determined by the property of the value function V(w). So under the time-separable utility function, the relative risk aversion for betting in terms of money is the same as the relative risk.
THE ATTITUDE TOWARDS RISK UNDER THE TIME-NON-SEPARABLE FUNCTION
An immediate implication is that the relative risk aversion to money bets does not depend on age: as the individual advances in age, the horizon, T+1, becomes shorter, but the consumption per period, w/(T+1), does not changes and therefore relative risk aversion does not change with age. Comparing (7) with (8) we see that when society is not separable in time, the measure of risk aversion to proportional bets in terms of money is different from the measure of risk aversion to proportional bets in terms of consumption. Note that the TNS utility function (1) achieves a separation between the attitude to money bets and the ordinal property of the utility function: The RAM coefficient does not depend on.
The limitations on the parameters imply limitations on the combination of IES and risk aversion measures we can take. This implies that a consumer may have a preference for betting in terms of money, but an aversion to betting in terms of consumption.
A TWO PERIODS SINGLE TREE ECONOMY
Since (7) implies RAM = 1 - α(1 + β), changing α will change it without affecting the asset's expected returns. It depends only on the expected growth rate of consumption (G) and the time preference parameter β. Thus, when the elasticity of substitution is different from unity, the price depends on the RAM parameter α.
This says that under the TNS (1) utility function, the price of the asset does not jump when we move from IES = 1 (ICD-IL function) to IES close to one (ICES function). Example: In the case of ICES, the price of an asset (16) will generally depend on the amount of total risk in the economy. In the second economy, G is a random variable that can have two possible realizations: 1 and 1.04 with equality.
We also note that when the elasticity of substitution is less than unity, the expected return on the asset in the risky economy is lower than the return on the asset in the risk-free economy.5 The predictions of the standard electric utility function are shown below. in the columns with RAM = 1. The predictions of the standard energy supply function (SP) are the same as in the columns with RAM = 1. In his presidential address, Lucas (2003) uses the standard energy supply function and argues for a relative risk aversion coefficient of unit.
Lucas' uses a well-known formula for an economy's average return on capital according to the preferences of the electricity function. Per capita consumption growth in the United States is about 0.02 and the after-tax return on capital is about 0.05, so the fact that. According to the TNS utility function (1) Claims 1 and 3 allow us to interpret Lucas's argument as an argument for IES = 1 and for the ICD-IL utility function.
A TWO PERIODS MANY ASSETS ECONOMY
As in the example in Table 1, the growth rate of aggregate yield (consumption) is 1 or 1.04 with equal probabilities and β = 1. The rate of return on the risk-free asset is lower and the difference (risk premium) increases with BET. We see that changes in the elasticity of substitution make a big difference both to the level of return and to the risk premiums.
Since requirement 3 applies to so many assets, the rates of return in Table 3 are the same as the rates of return under the standard power function (SP) that imposes RAM = 1/IES. The results for RAM = IES = 1 are the same in both tables because both share the IL utility function. Both the return and risk premiums are larger under the SP utility function.
As stated in the introduction, Selden (1978) proposed an unexpected utility function that separates between elasticities. Kreps and Porteus (1978) and Epstein and Zin (1989) extended Selden's analysis to the multi-period case in a time-consistent manner. I will now show that the Selden procedure, when IES = 1, can be observationally equivalent to the ICD-IL utility function.
In (32) the aggregator function is logarithmic and as in Epstein and Zin (1991), the certainty equivalence function is of the CES type. Thus, as in the log-expected utility case, the price of the asset depends only on current dividends (p = βy) and not on the certainty equivalent of future consumption. Under the ICD utility function, the relative risk aversion measure for bets in terms of second period consumption is: RAC = 1 - αβ.
INCOMPLETE MARKETS
Therefore, a unit change in RAC is equivalent to about 2 unit changes in RAM, and this will make the RAC measure of risk aversion appear more important than our RAM measure. We can think in terms of three independent lotteries that occur at the beginning of period 1. The second is a zero-sum lottery that determines e, and the third is a zero-sum lottery that determines u.
A state of nature s is a description of the outcome of all three lotteries.7 I assume ah = 0 and bh = 0.7 for all h. Using symmetry all consumers will make the same first period consumption choice and therefore clearing the first period consumption market requires: C0 = y. The first-order condition (24) must hold for all h and therefore I suppress the superscript h and write C1 s = Ds + us for the representative consumer.
For each realization of the total consumption, we add a bet in which the typical household can win or lose 0.08 units. The rate of return on the market portfolio now decreases in the RAM coefficient. Thus, it appears that the provision of incomplete markets will affect our estimate of β but have little or no effect on our estimate of the risk premium.
8 Deaton and Paxson (1994) find that the variance of log consumption within each age group increases by 0.07 each decade in the US (page 446). Their random walk assumption in equations (1) - (3) implies a variance in consumption of 0.007 per year which is roughly equal to a standard deviation of 0.08.
WELFARE CALCULATIONS
The second column of Table 5 reports the required percent offset (100λ) for different levels of RAM. In this case, the compensation required is significantly higher compared to the single shot case, but it is still a fraction of a percentage. If by "good policy" we eliminate aggregate risk, we can also greatly reduce the number of markets required for completeness.
RATES OF RETURN FOR HYPOTHETICAL CLAIMS UNDER THE ICD-IL FUNCTION
The time-invariant coefficients ai and bi can therefore be estimated from running the regression (46). Note that multiplying the coefficients ai and bi by the same constant does not change the expected return (28). Therefore, after estimating the regression coefficients in (46), we can enter the coefficients directly into (28) (without multiplying them by dt i) to calculate the predicted gross return on asset i.
Equation (46) requires data on gross rates of change in flows (fruits), and this data is easier to obtain than data on prices. However, we can predict the gross rate of return on human capital even without observing its price. But still, we can predict the rate of return on it by observing the stream of profits it brings.
Details of the calculations of these variables and the description of the data are in appendix c. Table 7*: Regressions of the rate of change of asset i on the rate of change of consumption. The expected return on the market portfolio (R1) is a good proxy for the returns on wage bill claims, wage income, and GDP.
The expected return on the riskiest portfolio (R2) is an estimate of the return on a claim on corporate earnings. The expected rates of return in Table 8 are consistent with the estimates in McGrattan and Prescott (2003), who took a clear account of taxes and frictions and found average returns in the 4–5 percent range.
CONCLUDING REMARKS
Lucas' observations on average interest rates across countries are consistent with the prediction of the ICD-IL utility function with IES = 1. In the ICD-IL utility function, the expected rate of return on the market portfolio is G/β regardless of the RAM coefficient and the amount of total risk. Risk premiums under the ICD-IL utility function are less sensitive to changes in the RAM coefficient than risk premiums under SP.
This can be explained by the fact that when we change the parameter in the SP function, it is. This may be a reason why risk aversion seems more important when using the default energy supply function. Allowing incomplete markets does not change the risk premiums in the ICD example we worked out.
Lucas objects to a power parameter of the SP function that deviates substantially from unity on the grounds that it conflicts with the above cross-country observation. The ICD utility function allows for variations in the RAM coefficient that do not violate the observations between countries. The predictions of the ICD-IL utility function are consistent with the findings of McGrattan and Prescott (2003), but cannot explain Mehra and Prescott's (1985) original puzzle.
In Table 1 we have seen that the prediction of the log utility function about the average return in the economy is the same as the prediction of the Cobb-Douglas functions. We can now consider the family of utility functions that are monotonic transformations of the log utility function. Yaari, Menachem., "On the Role of 'Dutch Books' in the Theory of Choice Under Risk" Nancy Shwartz Memorial Lecture, Northwestern, 1985.
