Consider an overlapping-generations model with 400 lenders and 200 borrowers

Consider an overlapping generations economy where people live for two periods. Population and the stock of at money are constant. Assume the total stock of at money is M; = $1,000. Assume the population is N: = 300. The banking sector is competitive. Private banks are capable of borrowing from the consumers and investing in capital which provides a (gross, real) one-period rate of return :0 = 2.5 for each unit of goods saved in capital (capital takes one period to become output). There is no entrepreneurs and no direct investment in capital in this economy. If consumers deposit to a bank, there is a xed transaction cost of (:5 = 10 units of goods to withdraw. This cost is the same regardless of the amount of deposits withdrawn. There are three types of consumers in this economy and each type is endowed when young with a different amount of goods. Assume that there are 100 consumers of each type. All types are endowed with nothing when they are old. Also assume that these consumers want to consume only when they are old. They will choose to save in deposits or currency depending on which one offers the higher rate of return, after considering the transaction costs of deposits. There is no transaction cost for saving in currency. The types of consumers are classied in the following table _ endowments yl- goods Type 1 Type_ Type? 1. (5 pts.) What is the rate of return 'r on deposits that the banks will offer in equilibrium? Explain. 2. (10 pts.) Which type of consumers will choose to save in at money and which type will choose to save in deposits? Explain your answer and show your work. 3. (6 pts.) Write down the market-clearing condition for at money. What is the price level in this economy? Derive your answer. 4. (8 pts.) What is the total nominal money supply, (M 1);, in this economy? What is the money multiplier? Derive your answer. 5. (6 pts.) What is the GDP in this economy in period t? Derive your answer. 6. (12 pts.) From now on, suppose that there is an unanticipated and permanent decrease in the rate of return of capital in period t, i.e. the 10. return on capital :1: goes down to :3" = 1.5. What happens to the total amount of goods saved in deposits at time t? What happens to the money multiplier at time t? In answering this question, make sure to calculate the new value of each variable Ht, Qt, and money multiplier M Mt. (8 pts.) What happens to GDP at period 15+ 1? What happens to (M 1)t? What is the correlation between these two variables? In answering this question, make sure to calculate the new value of each variable Gna+1,(M1),. (5 pts. BONUS) Explain why there is a positiveegative correlation between GDP+1 and (ll/11);. (5 pts.) In an attempt to offset the decrease in productivity, the gov- ernment decides to (temporarily) print more at money in period t to increase (one time) the stock of fiat money to Mt = $3,000 (but keeping the growth rate of at money stock constant at z = 1, i.e. Mt\" = Mt+2 = = $3, 000). What is the effect of this increase on GDPHI? What is the effect on (M1)t and what is the new value of (M1)t? (5 pts.) Was the policy that the government implemented able to offset the effect of the decrease in productivity on GDP? Why changing Mt to induce a change in (M 1); is not enough to offset the lower productivity? What does this model imply about the effect of monetary policy on the real variables