Money and banking economics homework question
ECON 4721 – Money and Banking
Homework 2: Money and the Price Level
Spring 2024
Exercise 1: Money Demand Consider a two-period, general equilibrium, endowment economy with no government and fixed labor supply (just like the one we studied in class). There is a representative household that consumes and holds real money balances. The household’s preferences are given by the following utility function:
U= c 1− σ c1t −σ Mt α +ψ + β t +1 1−σ pt 1−σ
The household’s budget constraints at periods t and t+1 are:
p t c t + p t s t + Mt ≤ p t w t
p t + 1 c t + 1 ≤ p t + 1 w t + 1 + ( 1 + i t ) p t s t + Mt
a) (5 points) Express the two budget constraints in real terms and use the Fisher equation to rewrite the t + 1 budget constraint as a function of both the nominal and real gross return on savings (you should follow your class notes closely in order to do this, since the procedure is exactly the same).
b) (5 points) Using the two budget constraints that you derived in part a), consolidate them into one single inter-temporal budget constraint
c) (10 points) Set up the household’s problem and take first order conditions. Use these optimality conditions to derive a clean expression for the demand of real money balances (let’s call this expression, simply money demand).
d) (10 points) Show that the money demand expression you derived in part c) is increasing in consumption and decreasing in the nominal interest rate
Exercise 2: Money Supply Consider a simple model like the one we studied in class, featuring three agents: a central bank, a representative private bank and a representative household.
The private bank starts with $1000 in loans, $300 in government bonds (we will simply call these, securities) and $150 in reserves. On the liability side, the bank faces $1000 in deposits.
The Central bank has $500 of securities while facing liabilities for $150 in reserves (the ones that the private bank has) and $250 in currency under circulation
The household holds currency for $250 and has deposits at the private bank for $1000. It faces no liabilities.
The reserves-to-deposit ratio that mandates the private bank’s lending behavior is given by ρ = ρ R + ρ E = Reserves 0.1 + 0.05 = 0.15 = Deposits . Where ρ R = 0.1 is the required reserve ratio (dictated by legal regulation) and ρ E = 0.05 is the excess reserve ratio, which means that the bank wants to hold slightly above the minimum requirement (so that it can successfully deal with both liquidity and credit risk). currency
On the household side, the key parameter is χ = currency+deposits = 0.2
Throughout this exercise, we will assume that both ρ and χ remain fixed.
a) (5 points) Write down the balance sheet (in T-account) for each of the three agents in this economy
b) (10 points) Consider now an open market operation in which the central bank purchases from the private bank, $100 worth in securities, and pays by increasing the private bank’s reserves in $100. Write down the three agents’ balance sheet immediately after this operation (the new balance sheets should only reflect the operation between the central and the private bank. The household’s balance sheet is actually unchanged).
c) (10 points) Assuming that the private bank wishes to keep its reserve-to-deposit ratio fixed and equal to ρ = 0.15 and also that the household wishes to keep its currency-to-money ratio fixed and equal to χ = 0.2, write down the final balance sheet for each of the three agents after all the rounds of borrowing and lending that would follow from the open market operation (make sure to use the formulae that we derived in class)
Exercise 3: Price Level and Inflation Consider an economy in a steady-state with growing money supply, growing output and a constant real interest rate. Money supply grows at a constant rate µ = 0.05, such that Mts+1 = (1 + µ) Mts . Output grows at a constant rate g = 0.02, such that yt+1 = (1 + g)yt . 2 Econ 4721-Money and Banking HW 2 Recall the equilibrium condition for money:
Mts = mdt (y, i ) pt where Ms is the money supply, md (y, i ) is real money demand and pt is the price level of the economy (think of this as the price index). Recall from our discussion in class, that we can think of the nominal interest rate as: it = rt + πt where π is the inflation rate. (For the following questions, you are advised to follow closely your class notes)
a) (10 points) Assume that real money demand takes the following form:
mdt (yt , it ) = yt ψ it 1 + it − σ1
Find the implied inflation rate as a function of the growth rate of money, the growth rate of output and the elasticity of money demand with respect to output (as we did in class, let’s call this term η).
b) (10 points) Repeat part a) but assume instead that the money demand is of the following form1 :
mdt (yt , it ) = yt (1 + it )−ε
c) (10 points) Repeat part a) but assume instead that the money demand is of the following form:
mdt (yt , it ) = yt F 2it α
where α = 0.5 (notice that this is just the Baumol-Tobin demand function that we derived in class).
Exercise 4: Federal Reserve Bank of Minneapolis Economic Policy Papers
(15 points) Read the following paper (written by Professors Chris Phelan and V.V Chari from our Econ Department, also serving as consultants at the Minneapolis Fed) and provide a brief summary of it (no more than 1 page long). The “Banks” We Do Need (https://www.minneapolisfed.org/article/2012/the-banks-we-do-need)