Consider the closed economy model with the consumption-savings
decision discussed in Chapter 9. Different from the baseline model, there are three periods.
Variables without a prime indicate first period values, those with a prime indicate second period values, and those with a double prime indicate third period values. There are
Translate Solutions to Calculu... PCRS C Chinese for Beginn... S MAT Page Question 2: (25 points) Consider the closed economy model with the consumption-savings decision discussed in Chapter 9. Different from the baseline model, there are three periods. Vari- ables without a prime indicate first period values, those with a prime indicate second period values, and those with a double prime indicate third period values. There are m identical consumers in the economy. Each consumer receives an endowment of y, ?', and y" in the first, second, and third periods, respectively. Each consumer pays a lump sum tax of 4, (', and (" in the first, second, and third periods, respectively. The government spending in the three periods are denoted as G, G', and G". The savings between periods I and 2 are denoted as s1, and the savings between periods 2 and 3 are denoted as $2. The government bonds between periods 1 and 2 are denoted as Bi, and the government bonds between periods 2 and 3 are denoted as B2. The risk-free real interest rate between periods 1 and 2 is given by ri, and that between periods 2 and 3 is given by 2. Consumers have identical preferences, given by: U(ce. c') - In(c) + 8in(c) + 37 In(c") where B E (0, 1) is the discount rate. 2.a. (3 points) Write down a consumer's (i) first period budget constraint, (ii) second period budget constraint, (iii) third period budget constraint, and (iv) lifetime budget, constraint. 2.b. (5 points) Write down the optimization problem of a consumer formally. Write down the associated Lagrangian. Derive the first order conditions with respect to c, c, and c". (Hint: You can call the lifetime wealth we not to carry the huge expression around.) 2.c. (6 points) Use the conditions you found in 2.b to solve for the optimal c, c', and c" as a function of the exogenous variables and the real interest rates alone. (Hint: Try to express c' and " as functions of c first; then use substitution.) 2.d. (3 points) Write down the government's (i) first period budget constraint, (ii) second period budget constraint, (iii) third period budget constraint, and (iv) present value budget constraint. 2.e. (8 points) Does the Ricardian Equivalence hold in this economy? If your answer is yes, prove it rigorously using your previous answers. If your answer is no, construct a counterexample using your previous answers to disprove. Show your work. m B A
