Equilibrium in a Two-Period Endowment Economy with CRRA UtilityThe Constant Relative Risk Aversion (CRRA) utility function is a widely usedspecification of preferences in economics that captures risk aversion and intertemporalconsumption smoothing. The CRRA utility function has the desirable property that thedegree of risk aversion is constant and independent of the level of consumption. Thismeans that as a household's consumption grows, its willingness to take risks remainsthe same. The coefficient of relative risk aversion (σ) measures the extent to whichhouseholds are risk averse and prefer a smooth consumption path over time. A highervalue of σ indicates a greater degree of risk aversion and a stronger preference forconsumption smoothing. Consider a two-period endowment economy with a largenumber of identical households. Each household has the following lifetime utilityfunction:U(j) =Ct (j) ¹-0 -11-σ8 (C4+1(1) 1-0-1)+ Bwhere 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.a) Writing Down the Intertemporal Budget ConstraintWrite down the household's intertemporal budget constraint.b) Deriving the Euler EquationFormulate the household's problem and derive the household's Euler equationusing the CRRA utility function.c) Deriving the Optimal Consumption FunctionDerive the household's optimal consumption function in each period (Ct andCt+1) as a function of Yt Yt+1, and rt .d) Discussing the Equilibrium in the Endowment EconomyIn the endowment economy, aggregate consumption must equal aggregateendowment in each period. Using this condition and the optimal consumptionfunctions derived in (c), discuss how the equilibrium real interest rate rt isdetermined in this economy. Explain the role of the real interest rate in ensuringequilibrium in the lending market and why it is necessary for achieving thisequilibriume) Analyzing the Effect of Changes in Relative Risk AversionGiven the consumption functions, discuss how changes in the coefficient ofrelative risk aversion (σ) affect the household's consumption choices and thesensitivity of these choices to changes in the real interest rate. Provide anintuitive explanation for your findings.

Intertemporal choice, also known as dynamic choice or time preference, refers to the decision-making…

quilibrium in a Two-Period Endowment Economy with CRRA Utility he Constant Relative Risk Aversion (CRRA) utility function is a widely used pecification of preferences in economics that captures risk aversion and intertem onsumption smoothing. The CRRA utility function has the desirable property that

quilibrium in a Two-Period Endowment Economy with CRRA Utility he Constant Relative Risk Aversion (CRRA) utility function is a widely used pecification of preferences in economics that captures risk aversion and intertem onsumption smoothing. The CRRA utility function has the desirable property that

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.8P

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