Consider the following overlapping generations model with private debt. Consumers are heterogenous with respect to their preferences. The lenders' utility function is uL(c1,t , c2,t+1 ) = ln(c1,t ) + ln(c2,t+1 ). The borrowers' utility function is uB (c1,t ,c2,t+1 ) = ln(c1,t ) + 0.5ln(c2,t+1 ). All consumers are endowed with y1 = y2 = 1. There are equal numbers of lenders and borrowers in each generation. Assume the population of each generation is Nt = 1. Let lt and bt denote the individual quantities of lending and borrowing, and rt denote the gross real interest rate on loans. There are no assets other than private loans.
(a) Write the budget constraints when young and when old, and the lifetime budget constraint for both types of consumers. Write the market clearing conditions.
(b) Solve for the stationary competitive equilibrium (r, (cL 1 , cL2 ), (cB1 , cB2 ), b, l).
(c) Now suppose that we introduce ?at money into this economy. The money stock Mt grows at the rate z > 1. Find the value of ?at money vt .