2. A simple model of illiquidity Consider the threeperiod overlapping generations model of illiquidity studied in lecture. People are endowed with 3;; units of consumption when young and nothing in other 2 periods of life. Population grows at rate 1 + n: N: = [1 + n}Nt_1. There is a constant supply of at money M in the economy. There is also a physical asset, capital, which has a rate of return X 3.:- {1 + of two periods after it is created. We assume the quality of capital cannot he veried, so that it cannot bl pledged for consumption in the second period. [a] Argue that individuals never wish to hold money to consume in the third period. Using the above, the period budget constraints satisfy Cu + Vtmt + fit 3 y [5} I'32,:+1 E Vt+1mt [5} C3,t+2 E th [7} [b] Derive and interpret the lifetime budget constraint. The interpretation is that we represent future consumption in presentvalue terms by dividing by the rate of return. Assume the preferences of the household are Hearse-'33} =lecl +lse +'gluse {9} {c} Set up the Lagrangian for maximizing lifetime utility subject to the lifetime budget constraint. Compute 51;: 62,:+1, and c3,+2. How does population growth 1 + n. and the twoperiod return on capital X inuence consumption allocations? Suppose a bank borrows from the young and issue deposits. These deposits can be redeemed at a rate [1 + 1"} when middle-aged. The banks invest invest in capital and make a return X after two periods. Provided X 3; [1 + TV, the bank makes a prot, Moreover, deposits offer a greater return than holding cash provided that re-n. Assume}? ::-{1+r}2 :- {1+n}3. {d} With access to banking deposits, do individuals still choose to hold at money? Mat about capital for the third period? 1Write the budget constraints each period using these answers and formulate the lifetime budget constraint. (e) How does the availability of banking inuence investment and output