Question

Consider an economy occupied by two households (i- A, B) who are facing the two-period consumption problem. Each household i - A, B is facing the following utility maximization problem: max subject to ci +biy(1+r)bo where Vi and US are household i's exogenous income in period t 1.2. cỈ and c are household i's consumption in period t 1,2. bo,bi is household i's bond holdings of which bo is exogenously given, r is the real interest rate, and 0 < β < 1 is the household discount factor. (a) Combine the period budget constraints to obtain the lifetime budget constraint, then write down the Lagrangian function and solve for the Euler equation (b) Interpret the Euler equation, tell me what it means intuitively (c) From this point forward, let the period utility function be u(c) - log(c). Solve for the closed form solution of c1, c2, and bi as functions of yi,y2, r, and 3, and bo (Since I know e did this in class I'm expecting to see your work) (d) From this point forward, let the exogenous income stream of each household i A, B 2(0,2) also each individual has bo - 0. For each household, what are the consumption and savings in each period as functions of r and ß. Which households are borrowing and saving? Why? (e) Write down the credit market clearing condition, and solve for the equilibrium interest rate.

Answer 1

The second period budget constraint is c_2 prime = y_2 prime + (1 + r)b_1 prime from first equation b_1 prime = y_1 prime + (1 + r) b_0^i - c_i^i Therefore c_2^i = y_2^i + (1 + r) [y_i^i + (1 + r) b_0^i -c^i_i] = y_2^i + (1 + r) y_i^i + (1 + r0^2 b^i_0 - (1 + r) c_i^i or c_1^i + c^i_2 /1 + r = y_i^i + y^i_2/1 + r + (1 + r) b^i_0 (1) The euler�s equation is given as: Mu [c_i]/Mu [c_2^i] = 1 + r or u Prime (c_i^i)/beta u prime (c_2^i) = 1 + r Therefore u prime (c_1^1) = beta (1 + r) u prime (c_2^i) The euler�s equation states that at equilibrium the consumer must be indifferent between consuming 1 more extra unit today and tomorrow. That is 1 more than dollar from optimum consumption must give the consumer same