February 7, 2021, Christopher D. Carroll CRRA-RateRisk
Consider a consumer with CRRA utility whose only available financial asset has a risky return factor which is lognormally distributed, .
With market assets , the dynamic budget constraint is:
| (1) |
Start with the standard Euler equation for consumption under CRRA utility:
| (2) |
and postulate a solution of the form :
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which (finally) yields an exact formula for :
| (3) |
Since , fact implies that (using the definition ,
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Substituting in (3):
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which hold if is close to zero. Substituting into (4) and using and gives
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which, when , reduces to the usual perfect foresight formula .
This equation implies the plausible result that as unavoidable uncertainty in the financial return goes up ( rises) the level of consumption falls (because , so which multiplies is negative). The reduction in consumption as risk increases reflects the precautionary saving motive.1
The top figure plots the marginal propensity to consume as a function of the coefficient of relative risk aversion (for both the true MPC and the approximation derived above), under parameter values such that so that a change in does not affect the MPC through the intertemporal elasticity of substitution channel. As intuition would suggest, as consumers become more risk averse, they save more (the MPC is lower; that is, the plotted loci are downward-sloping).
The other way to see the precautionary effect is to examine the effect on the MPC of a change in risk. For a consumer with relative risk aversion of 3, the bottom figure shows that as the size of the risk increases, the MPC falls.