The Constant Relative Risk Aversion (CRRA) utility function is a widely used specifica- tion of preferences in economics that captures risk aversion and intertemporal consump- tion smoothing. The CRRA utility function has the desirable property that the degree of risk aversion is constant and independent of the level of consumption. This means that as a household's consumption grows, its willingness to take risks remains the same. The coefficient of relative risk aversion (σ) measures the extent to which households are risk- averse and prefer a smooth consumption path over time. A higher value of σ indicates a greater degree of risk aversion and a stronger preference for consumption smoothing. Consider a two-period endowment economy with a large number of identical house- holds. Each household has the following lifetime utility function: U (j) = C₁(j)¹-0-1 (C++1(j)¹–σ — 1` - +B -
The Constant Relative Risk Aversion (CRRA) utility function is a widely used specifica- tion of preferences in economics that captures risk aversion and intertemporal consump- tion smoothing. The CRRA utility function has the desirable property that the degree of risk aversion is constant and independent of the level of consumption. This means that as a household's consumption grows, its willingness to take risks remains the same. The coefficient of relative risk aversion (σ) measures the extent to which households are risk- averse and prefer a smooth consumption path over time. A higher value of σ indicates a greater degree of risk aversion and a stronger preference for consumption smoothing. Consider a two-period endowment economy with a large number of identical house- holds. Each household has the following lifetime utility function: U (j) = C₁(j)¹-0-1 (C++1(j)¹–σ — 1` - +B -
Chapter1: Making Economics Decisions
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Transcribed Image Text:The Constant Relative Risk Aversion (CRRA) utility function is a widely used specifica-
tion of preferences in economics that captures risk aversion and intertemporal consump-
tion smoothing. The CRRA utility function has the desirable property that the degree of
risk aversion is constant and independent of the level of consumption. This means that
as a household's consumption grows, its willingness to take risks remains the same. The
coefficient of relative risk aversion (σ) measures the extent to which households are risk-
averse and prefer a smooth consumption path over time. A higher value of σ indicates a
greater degree of risk aversion and a stronger preference for consumption smoothing.
Consider a two-period endowment economy with a large number of identical house-
holds. Each household has the following lifetime utility function:
U(j) =
C+(j) 1-0 - 1
1-σ
+ẞ
C++1(j) 1-0 -
1
1-σ
where C₁(j) and C++1(j) are consumption in periods t and t + 1 for household j, re-
spectively, ẞ is the discount factor, and σ > 0 is the coefficient of relative risk aversion.
All households are endowed with an exogenous amount of income, Y₁ in period t and
Y++1 in period t + 1. Households can borrow or lend at a common real interest rate, r.
Q1) Write down the household's intertemporal budget constraint.
Q2) Formulate the household's problem and derive the household's Euler equation
using the CRRA utility function.
Q3) Derive the household's optimal consumption function in each period (C+ and C++1) as
a function of Y, Y₁+1, and rt.
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