The consumer's problem: Logarithmic preferences This question appeared (verbatim) on the 2016 final exam. Question III...

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The consumer's problem: Logarithmic preferences This question appeared (verbatim) on the 2016 final exam. Question III (40 points): The consumer's problem in a dynamic model Consider a two-period consumption-saving model from Chapter 9. A consumer's income in the current period is exogenous. The incomes in the current and in the future periods are denoted by y and y'. There are no taxes. The consumer's utility function is giv en by = U(c,c') Inc Inc', 0 < <1, where denotes the discount factor. To answer questions a) to e), assume that the credit market is perfect. The real interest rate on borrowing and lending is denoted by r. a) (5 points) State in details the consumer's optimization problem. b) (5 points) Derive and interpret the two conditions that define the optimal choice of consumption in the current and in the future periods. c) (5 points) Find the analytical expressions for the optimal values of consumption in both periods, and optimal savings s. d) (5 points) Suppose y =480, y' =500 and =0.9. Consider the real interest rate r=0.1. What are the optimal values of consumption in both periods? Is the consumer a lender or a borrower? Explain. Represent the consumer's choice of c, c' and s on a graph. Make sure to label the axes and the points of the intercepts of the budget constraint with the horizontal and vertical axes. Indicate the value of the slope of the budget constraint, the endowment point, and the point of optimality. e) (10 points) Keep the values y=480, y'=500 and =0.9. Suppose the real interest rate increases to r=0.2. Find the new optimal values of consumption. Is the consumer a lender or a borrower? Explain. How is the consumer's welfare affected by the increase in the real interest rate? Compare with the case of r=0.1. Explain your response. Draw a new graph to illustrate the impact of a higher real interest rate on the consumer's choice. On this diagram, show the optimal consumption choice for the old and the new interest rates. Comment on the relative strength of the income and substitution effects of the real interest rate changes. The consumer's problem: Logarithmic preferences This question appeared (verbatim) on the 2016 final exam. Question III (40 points): The consumer's problem in a dynamic model Consider a two-period consumption-saving model from Chapter 9. A consumer's income in the current period is exogenous. The incomes in the current and in the future periods are denoted by y and y'. There are no taxes. The consumer's utility function is giv en by = U(c,c') Inc Inc', 0 < <1, where denotes the discount factor. To answer questions a) to e), assume that the credit market is perfect. The real interest rate on borrowing and lending is denoted by r. a) (5 points) State in details the consumer's optimization problem. b) (5 points) Derive and interpret the two conditions that define the optimal choice of consumption in the current and in the future periods. c) (5 points) Find the analytical expressions for the optimal values of consumption in both periods, and optimal savings s. d) (5 points) Suppose y =480, y' =500 and =0.9. Consider the real interest rate r=0.1. What are the optimal values of consumption in both periods? Is the consumer a lender or a borrower? Explain. Represent the consumer's choice of c, c' and s on a graph. Make sure to label the axes and the points of the intercepts of the budget constraint with the horizontal and vertical axes. Indicate the value of the slope of the budget constraint, the endowment point, and the point of optimality. e) (10 points) Keep the values y=480, y'=500 and =0.9. Suppose the real interest rate increases to r=0.2. Find the new optimal values of consumption. Is the consumer a lender or a borrower? Explain. How is the consumer's welfare affected by the increase in the real interest rate? Compare with the case of r=0.1. Explain your response. Draw a new graph to illustrate the impact of a higher real interest rate on the consumer's choice. On this diagram, show the optimal consumption choice for the old and the new interest rates. Comment on the relative strength of the income and substitution effects of the real interest rate changes.