Suppose that the following economy consists of two types of people, A and B, with population 100 for each type. Time is infinite and labeled as t = 1, 2, . . . ,∞.Type A can only work in the even-period while type B can only work in the odd-period. When they work, they can produce 1 unit of apple. Eating an apple will yield 1 unit of utility. An apple is no storable and cannot be divided. Type A value apples more than Type B in the odd-period, i.e., they want to eat the apple more than Type B do. However, Type B value apples more than type A in the even-period.

(a) How does the socially optimal allocation look like? In the absence of medium of exchange, list the requirements to achieve the optimal allocation.

(b) Suppose that types A and B are anonymous and cannot commit. At the beginning of time t, each Type A agent is endowed with 1 unit of silver coin. Assume thatother than the Mint, no one can produce coins. The Mint will not supply morecoins at this moment. A coin can be either stored or consumed by people. Ifconsumed, it will generate 1 unit of utility as well. But after consumption, itsimply disappears and cannot be replaced. Explain the trading pattern and whycoins will not be consumed.

(c) What is the value of a coin? Is it still equal to 1 unit of utility?

(d) Suppose now the agent is able to produce 2 units of apples. But for some reason,1 silver coin can only exchange for 1 unit of apple. This exchange rate is fixed.Clearly, there is a liquidity-shortage problem here: people want to consume 2apples but they can only afford 1. Assume that there exists a bank which onlyhas a record-keeping technology. The bank can costlessly print (fiat) paper money,accept deposits/collateral and make loans. The bank promises to convert 1 unit ofpaper money to 1 unit of silver coin on demand. What is the role of a bank in thisenvironment? Write down the balance sheet of the bank with the unit measuredin terms of coins in period t = 1. What are the liquid assets and illiquid assetsin this economy? Which one is outside money and which one is inside money?Explain the process for the bank to create and destroy its paper money.

(e) Suppose in one of the even-periods, say t = 4, some of the Type B agents experience a preference shock: starting from t = 5, they will not consume anymore(think about they will pass away). What happens to the bank if the size of theshock is small which affects only 25 Type B agents? What happens to the bank ifthe size of the shock is large which affects 75 Type B agents? Determine the sizeof the shock affecting how many people would lead to the bank insolvent.

(f) Now the Mint can produce more coins. Can the Mint stabilize the banking sectorin the presence of shock? Explain.1