Utility Optimization Problem Suppose a household has the following lifetime utility function: (a) Finding Partial Derivatives of Utility 1/2 U = c²/² + ẞc++ 1/2 Ct Find expressions for the partial derivatives of lifetime utility, U, with respect to period t and period t + 1 consumption. Is marginal utility of consumption in both periods always positive? (b) Finding Second Derivatives of Utility a²U Find expressions for the second derivatives of lifetime utility with respect to period t and t+ 1 consumption, i.e., 24 and 22. Are these second derivatives always negative for any positive values of period t and t+1 consumption? ac²+1 c) Deriving Indifference Curve Expression Derive an expression for the indifference curve associated with lifetime utility level Uo (i.e., derive an expression for C++1 as a function of U₁ and c). What is the slope of the indifference curve? How does the magnitude of the slope vary with the value of ą? d) Combining Budget Constraints Suppose that the household faces two within period budget constraints of the form: G+ S₁ = Y₁₁ C+1=Y+1+(1+rt) St Combine the two period budget constraints into one intertemporal budget constraint. e) Deriving Euler Equation Use the intertemporal budget constraint and this utility function to derive the Euler equation characterizing an optimal consumption plan. f) Deriving Consumption Function Use this Euler equation and the intertemporal budget constraint to derive a consumption function expressing & as a function of Yt, Y++1, and rt. (g) Finding Partial Derivatives of Consumption Function Use your consumption function to find the partial derivatives of c with respect to Yt, Yt+1, and rt.
Utility Optimization Problem Suppose a household has the following lifetime utility function: (a) Finding Partial Derivatives of Utility 1/2 U = c²/² + ẞc++ 1/2 Ct Find expressions for the partial derivatives of lifetime utility, U, with respect to period t and period t + 1 consumption. Is marginal utility of consumption in both periods always positive? (b) Finding Second Derivatives of Utility a²U Find expressions for the second derivatives of lifetime utility with respect to period t and t+ 1 consumption, i.e., 24 and 22. Are these second derivatives always negative for any positive values of period t and t+1 consumption? ac²+1 c) Deriving Indifference Curve Expression Derive an expression for the indifference curve associated with lifetime utility level Uo (i.e., derive an expression for C++1 as a function of U₁ and c). What is the slope of the indifference curve? How does the magnitude of the slope vary with the value of ą? d) Combining Budget Constraints Suppose that the household faces two within period budget constraints of the form: G+ S₁ = Y₁₁ C+1=Y+1+(1+rt) St Combine the two period budget constraints into one intertemporal budget constraint. e) Deriving Euler Equation Use the intertemporal budget constraint and this utility function to derive the Euler equation characterizing an optimal consumption plan. f) Deriving Consumption Function Use this Euler equation and the intertemporal budget constraint to derive a consumption function expressing & as a function of Yt, Y++1, and rt. (g) Finding Partial Derivatives of Consumption Function Use your consumption function to find the partial derivatives of c with respect to Yt, Yt+1, and rt.
Chapter4: Utility Maximization And Choice
Section: Chapter Questions
Problem 4.3P
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Transcribed Image Text:Utility Optimization Problem
Suppose a household has the following lifetime utility function:
(a) Finding Partial Derivatives of Utility
U = c1/2 + Bc1/2
Find expressions for the partial derivatives of lifetime utility, U, with respect to
period t and period t + 1 consumption. Is marginal utility of consumption in both
periods always positive?
(b) Finding Second Derivatives of Utility
a²U
Find expressions for the second derivatives of lifetime utility with respect to period t and
t + 1 consumption, i.e., 321 and 227. Are these second derivatives always negative for
any positive values of period t and t+1 consumption?
c) Deriving Indifference Curve Expression
Derive an expression for the indifference curve associated with lifetime utility level Uo
(i.e., derive an expression for C++1 as a function of U₁ and c). What is the slope of the
indifference curve? How does the magnitude of the slope vary with the value of Ę?
d) Combining Budget Constraints
Suppose that the household faces two within period budget constraints of the form:
Єt + St = Yt
C++1=Y+1 + (1 + rt) St
Combine the two period budget constraints into one intertemporal budget constraint.
e) Deriving Euler Equation
Use the intertemporal budget constraint and this utility function to derive the Euler
equation characterizing an optimal consumption plan.
f) Deriving Consumption Function
Use this Euler equation and the intertemporal budget constraint to derive a consumption
function expressing ct as a function of Yt, Yt+1, and rt.
(g) Finding Partial Derivatives of Consumption Function
Use your consumption function to find the partial derivatives of c with respect to Yt, Yt+1,
and rt.
Transcribed Image Text:(h) Analyzing MPC
Is the MPC positive and less than 1? Does it depend on the value of rt here?
(I)Analyzing Consumption and Interest Rate
Is consumption decreasing in the real interest rate?
(j) Analyzing Consumption and Future Income
Is consumption increasing in Y₁+1?
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