Macro Comp Question 1, June 2013
Consider an economy consisting of large numbers of identical, competitive households,
and identical, competitive banks. Banks take deposits Dt from households at time t, which pay a
risk-free gross return of Rt+1* Dt in period t+1. Both households and banks have access to a
production technology that converts Kt units of capital obtained in period t into Zt+1 * Kt units of
output in period t+1, where Zt+1 is an i.i.d. stochastic productivity shock. The total stock of
capital is fixed and is normalized to 1; there is no investment in the aggregate and capital does
not depreciate. Letting KHt and KBt refer to capital obtained in period t by households and
bankers, we thus have that in equilibrium, KHt + KBt = 1.
While the supply of capital is fixed in the aggregate, an individual household or banker
can purchase or sell capital at a price Qt, where Qt varies stochastically in response to shocks.
Bankers are more efficient at using the production technology than households. Specifically,
bankers can acquire KBt units of capital at cost Qt * KBt, while households must pay
Qt * KHt + f(KHt)
in order to acquire KHt units of capital. f(K) is a positive, increasing and convex function
for all K > 0, and represents extra managerial costs that households must incur.
PART I: HOUSEHOLDS
Household preferences are given by
(1)

E0 ∑ t u(Ct)
t=0
where u(C) is positive, increasing and concave, 0 < β < 1 is a discount factor, and E0
represents time zero expectations. Households each period receive the gross returns on last
period’s bank deposits, the flow output from any capital obtained last period, and the resale value
of their capital holdings. They allocate this income between consumption and saving, where
saving can be accomplished by making bank deposits, and/or by acquiring capital and paying the
related managerial costs. The household’s flow budget constraint is thus
(2)
Ct + Dt + Qt * KHt + f(KHt) = Rt Dt-1 + (Zt + Qt) * KHt-1
Note that since households are competitive, they take market prices Rt and Qt as given,
although these prices are determined endogenously in the economy’s equilibrium.
(a) [10 points] Identify state and control variables for the household’s problem, and write
down the household’s Bellman Equation.
(b) [30 points] Take the first order and envelope conditions. Derive an Euler equation
relating intertemporal household consumption choices and the gross rate of return on riskfree deposits R. Derive a second Euler equation relating intertemporal consumption and
the household’s gross rate of return on holding capital, which you can denote RKH. Derive
an expression for RKH in terms of current and/or future values of Z, Q and KH.
PART II: BANKS
Bankers have linear preferences, and are required (by law, say) to consume exactly a
fraction σ of their net worth nt every period. If we let VB denote the value of a bank at the start
of the period, then the following recursive equation holds for banks:
(3)
VBt = σ nt + β Et VBt+1
The net worth of a bank entering period t consists of the flow output from capital
obtained last period, plus the capital resale value, net of promised repayments to depositors:
(4)
nt = (Zt + Qt) * KBt-1 – Rt * Dt-1
The bank consumes a fraction σ of its net worth, and uses the remaining net worth along
with newly issued bank deposits in order to buy capital for the next period. Thus,
(5)
Qt * KBt = (1-σ) nt + Dt
However, there is a financial friction that limits the size of banks. Specifically, at the end
of each period, a banker has the option of stealing a fraction θ of its capital, reneging on its
deposits, and going out of business. If a bank decides to “steal”, it’s payoff is θ * Qt * KBt. For
a bank to be deterred from stealing, the payoff to stealing must be less than the discounted
expected value of being in business at the start of next period. In other words, the following
incentive compatibility condition must hold in all periods to prevent banks from stealing:
(6)
θ * Qt * KBt ≤ β Et VBt+1
Households will only deposit funds with banks if (6) holds, because otherwise they will
expect the bank to renege. You may assume that (6) always holds with equality and that it puts
an upper limit on the deposits banks can accept. Intuitively, the value of stealing (the LHS of
(6)) increases in deposits, because higher deposits mean more capital that the bank can steal.
The value of remaining in business (the RHS of (6)) also increases in deposits, but not as quickly
as the value of stealing, because the extra revenue at t+1 coming from the higher capital is partly
offset by the extra cost of repaying depositors.
A reasonable conjecture (which you will verify below) is that the value of being a bank at
time t is a linear function of the bank’s net worth:
(7)
VBt = ηt * nt, where ηt is some time varying function.
(c) [30 points] Assume that (6) holds with equality. Use (4)-(7) to show that a bank’s
maximum deposits in period t are a linear function of its net worth:
(8)
Dt = φt * (1-σ) * nt,
Where φt is a time-varying “leverage multiplier” whose formula you will derive.
You may find it convenient to save notation by using RKBt+1 = (Zt+1 + Qt+1)/Qt to denote the
bank’s gross rate of return on capital held from period t to t+1.
(d) [30 points] Verify the conjecture (7) using (8) and (3)-(5) and derive the formula for ηt.