Suppose a household has the following lifetime utility function:U=c1/2 + ẞc¹/212tt+1A) Find expressions for the partial derivatives of lifetime utility, U, with respect toperiod t and period t + 1 consumption. Is marginal utility of consumption in bothperiods always positive?B) Find expressions for the second derivatives of lifetime utility with respect to period tand t+1 consumption, i.e., 2U and a 20_Are these second derivatives always negative forac²²+1any positive values of period t and t+1 consumption?C) Derive an expression for the indifference curve associated with lifetime utility levelUo (i.e., derive an expression for C++₁ as a function of U₁ and c). What is the slope of theindifference curve? How does the magnitude of the slope vary with the value of c?

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Suppose a household has the following lifetime utility function: U = c1/2 + ẞc¹/2t+1 A) 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) Find expressions for the second derivatives of lifetime utility with respect to period t and t+1 consumption, i.e., a 2 and 320_Are these second derivatives always negative for aede any positive values of period t and t+1 consumption? C) Derive an expression for the indifference curve associated with lifetime utility level U. (i.e., derive an expression for C++1 as a function of U, and c). What is the slope of the indifference curve? How does the magnitude of the slope vary with the value of c

Suppose a household has the following lifetime utility function: U = c1/2 + ẞc¹/2t+1 A) 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) Find expressions for the second derivatives of lifetime utility with respect to period t and t+1 consumption, i.e., a 2 and 320_Are these second derivatives always negative for aede any positive values of period t and t+1 consumption? C) Derive an expression for the indifference curve associated with lifetime utility level U. (i.e., derive an expression for C++1 as a function of U, and c). What is the slope of the indifference curve? How does the magnitude of the slope vary with the value of c

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

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Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

Suppose a household has the following lifetime utility function:
U=c1/2 + ẞc¹/2
12tt+1
A) 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) Find expressions for the second derivatives of lifetime utility with respect to period t
and t+1 consumption, i.e., 2U and a 20_Are these second derivatives always negative for
ac²²+1
any positive values of period t and t+1 consumption?
C) 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 c?

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Step 1: Define utility function

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Step 2: Find partial derivatives of utility function

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Step 3: Find second derivatives of utility function

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Step 4: Derive indifference curve expression

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