1. Consider the model of liquidity and financial intermediation presented in class. The economy has an infinite horizon and is populated by a large number of individuals who each live for three periods. In period t there are N t individuals born where N t = n N t 1 and n 1 . Individuals receive utility from consuming in middle age and when old. That is, individuals have utility function U ( c 2 , t + 1 , c 3 , t + 2 ) = ln c 2 , t + 1 + ln c 3 , t + 2 where ( 0 , 1 ) . For simplicity you can assume that n = 1 and = 1 . Individuals have one unit of time that is supplied inelastically as labor, but possess different productivities denoted by [ 0 , 1 ] . An individual with productivity produces units of the consumption good when young. Assume that the productivity of the young generation is uniformly distributed over the interval [ 0 , 1 ] in every period. There is a risk-neutral bank takes deposits and commits to a one-period return denoted by r d , so that one unit of the consumption good deposited with the bank yields the gross return r d units of the consumption good in t + 1 . When an individual withdraws from the bank, they pay a fixed cost > 0 , measured in units of the consumption good. This fixed cost has many interpretations including the annual fee on a deposit account or the time required to go to a bank (or bank machine) to withdraw (in fact during COVID-19 you could also consider the risk associated with making an in-person withdrawal as a component of this fixed cost). Individuals have three stores of value available; fiat money ( m ) , deposits ( d ) , and capital ( k ) . There is a supply of fiat money available in period t denoted by M t , where M t = z M t 1 for all t where z 1 . Capital has a two period return X > n 2 and a one-period return x = X 1/2 , however, unmatured capital cannot be sold, traded or borrowed against. For parts (a) - (d) you can assume that the real rate of return on fiat money in a stationary equilibrium is n = 1 , and the return on deposits satisfies r d ( n , x ) , where x = X 1/2 . Further for parts (a) - (e) you can assume that the money supply is constant so that z = 1 and M t = M for all t . Please answer the following questions and assume a stationary equilibrium (i.e. c 2 , t ( ) = c 2 ( ) for all t and c 3 , t ( ) = c 3 ( ) for all t ) . (a) Write down the utility maximization problem of a type worker including all budget constraints. Explain how individuals in the economy will use money, deposits and capital. (b) Write down the Lagrangian and solve for the optimal choices of capital, deposits, real money and consumption for any type individual. If it is easier, you an solve the problem by considering a type w worker and doing the following: (i) Assume the bank does not exist so that individuals do not make deposits and can only use fiat money and capital. Solve for the equilibrium levels of consumption ( c 2 , t + 1 and c 3 , t + 2 ) , capital ( k t ) , and individual real money demand ( t m t.

1. 1. Consider the model of liquidity and financial intermediation presented in class. The economy
has an infinite horizon and is populated by a large number of individuals who each live for three
periods. In period t there are N t individuals born where N t = n N t 1 and n 1 . Individuals
receive utility from consuming in middle age and when old. That is, individuals have utility
function U ( c 2 , t + 1 , c 3 , t + 2 ) = ln c 2 , t + 1 + ln c 3 , t + 2 where ( 0 , 1 ) . For simplicity
you can assume that n = 1 and = 1 . Individuals have one unit of time that is supplied
inelastically as labor, but possess different productivities denoted by [ 0 , 1 ] . An individual with
productivity produces units of the consumption good when young. Assume that the productivity
of the young generation is uniformly distributed over the interval [ 0 , 1 ] in every period. There
is a risk-neutral bank takes deposits and commits to a one-period return denoted by r d , so that
one unit of the consumption good deposited with the bank yields the gross return r d units of the
consumption good in t + 1 . When an individual withdraws from the bank, they pay a fixed cost
> 0 , measured in units of the consumption good. This fixed cost has many interpretations
including the annual fee on a deposit account or the time required to go to a bank (or bank
machine) to withdraw (in fact during COVID-19 you could also consider the risk associated with
making an in-person withdrawal as a component of this fixed cost). Individuals have three stores
of value available; fiat money ( m ) , deposits ( d ) , and capital ( k ) . There is a supply of fiat
money available in period t denoted by M t , where M t = z M t 1 for all t where z 1 . Capital has
a two period return X > n 2 and a one-period return x = X 1/2 , however, unmatured capital
cannot be sold, traded or borrowed against. For parts (a) - (d) you can assume that the real rate of
return on fiat money in a stationary equilibrium is n = 1 , and the return on deposits satisfies r d (
n , x ) , where x = X 1/2 . Further for parts (a) - (e) you can assume that the money supply is
constant so that z = 1 and M t = M for all t . Please answer the following questions and assume a
stationary equilibrium (i.e. c 2 , t ( ) = c 2 ( ) for all t and c 3 , t ( ) = c 3 ( ) for all t ) . (a) Write
down the utility maximization problem of a type worker including all budget constraints. Explain
how individuals in the economy will use money, deposits and capital. (b) Write down the
Lagrangian and solve for the optimal choices of capital, deposits, real money and consumption
for any type individual. If it is easier, you an solve the problem by considering a type w worker
and doing the following: (i) Assume the bank does not exist so that individuals do not make
deposits and can only use fiat money and capital. Solve for the equilibrium levels of
consumption ( c 2 , t + 1 and c 3 , t + 2 ) , capital ( k t ) , and individual real money demand ( t m
t ) . Note: these should depend on . (ii) Assume now that there is no money so individuals are
only able to make deposits or hold capital. Solve for the equilibrium levels of consumption ( c 2 ,
t + 1 and c 3 , t + 2 ) , capital ( k t ) , and deposits ( d t ) in terms of r d . Note: these should
depend on w . (iii) Now consider the case with both money and bank deposits. Do individuals
hold money, make deposits, or both? If both, find an expression that determines what individuals
acquire money and what individuals deposit their goods with the bank. Explain the result.