D) Discussing the Equilibrium in the Endowment Economy: In the endowment economy, aggregate consumption must equal aggregate endowment ineach period. Using this condition and the optimal consumption functions derived,discuss how the equilibrium real interest rate rtis determined in this economy. Explainthe role of the real interest rate in ensuring equilibrium in the lending market and why itis necessary for achieving this equilibrium(e) Analyzing the Effect of Changes in Relative Risk Aversion:Given the consumption functions, discuss how changes in the coefficient of relative riskaversion (σ) affect the household’s consumption choices and the sensitivity of these choicesto changes in the real interest rate. Provide an intuitive explanation for your findings. Equilibrium in a Two-Period Endowment Economy with CRRA UtilityThe 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 ofrisk aversion is constant and independent of the level of consumption. This means thatas a household's consumption grows, its willingness to take risks remains the same. Thecoefficient 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 agreater 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:Ct(j)1-01-0 - 1U(j)1-σ+ B8 (Ce+1(1)1-0 -1)where Ct (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, Y+ in period t andY++1 in period t + 1. Households can borrow or lend at a common real interest rate, rt.intertemporal Budget ConstraintY₁ + 1+1 = C++C++11+rDeriving the Euler EquationSimplifying, we find the Euler equation:C = B(1+r)Which can also be written as:-0C++1Ct=· ß(1+r)Deriving the Optimal Consumption FunctionC++1 = (ẞ(1+r))³ C₁Final SolutionThe final expressions for optimal consumption in periods t and t + 1 are:Y++1Ye+++C+ =1+ (1+r)

The notion at discussion is the Constant Relative Risk Aversion (CRRA) utility function in a…

Answered: D) Discussing the Equilibrium in the…

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

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Publisher:NEWNAN

Chapter1: Making Economics Decisions

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D) Discussing the Equilibrium in the Endowment Economy: In the endowment economy, aggregate consumption must equal aggregate endowment in
each period. Using this condition and the optimal consumption functions derived,
discuss how the equilibrium real interest rate rt
is determined in this economy. Explain
the role of the real interest rate in ensuring equilibrium in the lending market and why it
is necessary for achieving this equilibrium

(e) Analyzing the Effect of Changes in Relative Risk Aversion:Given the consumption functions, discuss how changes in the coefficient of relative risk
aversion (σ) affect the household’s consumption choices and the sensitivity of these choices
to changes in the real interest rate. Provide an intuitive explanation for your findings.

Equilibrium in a Two-Period Endowment Economy with CRRA Utility
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:
Ct(j)1-0
1-0 - 1
U(j)
1-σ
+ B
8 (Ce+1(1)1-0 -1)
where Ct (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, Y+ in period t and
Y++1 in period t + 1. Households can borrow or lend at a common real interest rate, rt.
intertemporal Budget Constraint
Y₁ + 1+1 = C++
C++1
1+r
Deriving the Euler Equation
Simplifying, we find the Euler equation:
C = B(1+r)
Which can also be written as:
-0
C++1
Ct
=
· ß(1+r)
Deriving the Optimal Consumption Function
C++1 = (ẞ(1+r))³ C₁
Final Solution
The final expressions for optimal consumption in periods t and t + 1 are:
Y++1
Ye+++
C+ =
1+ (1+r)

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