Macro Comp Question 1 Answer Key, June 2013
(a) [10 points] Identify state and control variables for the household’s problem, and write
down the household’s Bellman Equation.
Answer: from the household point of view, the endogenous state variables as of the beginning of
period t are last period’s deposits Dt-1 and last period’s capital holdings KHt-1. The exogenous
state variables are productivity Zt , the rates of return on deposits Rt-1 and Rt, and the price of
capital Qt. Note that R and Q are exogenous from the household’s point of view even though
they are endogenous in the economy’s equilibrium.
The household’s control variables in period t are consumption Ct, deposits Dt, and capital
holdings KHt. By the budget constraint (2), any two of these control variables are sufficient to
fully determine the third; I will use deposits and capital holdings as the choice variables, in
which case the Bellman’s Equation is
Vt (Dt-1, KHt-1) = Max u(RtDt-1 + (Zt + Qt)KHt-1 – Dt – QtKHt - f(KHt)) + β EtVt+1(Dt,KHt)
(Dt, KHt)
Note that for convenience, I write V as an explicit function only of endogenous state variables;
the dependence on exogenous state variables is implicit in writing Vt and Vt+1.
[Note on grading: most students got 9 or 10 points on part (a). The most common reason
for having a point deducted was failure to recognize that there are exogenous as well as
endogenous state variables in the problem. You got 10 points if your answer was otherwise
good and you either verbally listed some exogenous state variables OR put a time subscript
in your value function V. You got a point taken off if you did neither].
(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.
Answer: First order condition with respect to Dt:
(A1)
-u’(Ct) + β EtV1t+1(Dt,KHt) = 0
First order condition with respect to KHt:
(A2)
-(Qt + f’(KHt)) * u’(Ct) + β EtV2t+1(Dt,KHt) = 0
Envelope condition with respect to Dt-1:
(A3)
V1t (Dt-1, KHt-1) = Rt u’(Ct)
Envelope condition with respect to KHt-1:
(A4)
V2t (Dt-1, KHt-1) = (Zt + Qt) * u’(Ct)
Updating (A3) by one period and plugging into (A1) yields the usual Euler Equation relating
intertemporal household consumption choices and the risk-free rate of return:
(A5)
u’(Ct) = β Rt+1 Et u’(Ct+1)
Updating (A4) by one period and plugging into (A2) yields the following Euler Equation:
(A6)
u’(Ct) * (Qt + f’(KHt)) = β Et [(Zt+1 + Qt+1) * u’(Ct+1)]
We can rewrite (A6) as follows:
(A7)
u’(Ct) = β Et RKHt+1 u’(Ct+1)
Where
(A8)
RKHt+1 = (Zt+1 + Qt+1)/(Qt + f’(KHt))
Note that RKHt+1 represents the household’s rate of return from investing in capital in period t.
On the margin, it costs the household Qt + f’(KHt) units of output today to obtain a unit of capital,
which in turn yields Zt+1 + Qt+1 units of output tomorrow.
[Note on grading: I gave 10 points for getting the FOCs correct, 5 points for getting the
standard consumption Euler equation (A5), and 15 points for getting the “nonstandard”
Euler equation (A6) and for correctly defining the rate of return (A8).]
(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.
Answer: plug (5) into the LHS of (6). Plug (4) into (7), update one period and plug the result
into the RHS of (6). Assuming that (6) holds with equality and using the notation hint above,
this yields
(A9)
θ [(1-σ) nt + Dt] = β Et [ηt+1 RKBt+1 ((1-σ) nt + Dt) - ηt+1 Rt+1 Dt ]
Collecting all terms involving Dt on the left and all terms involving nt on the right yields
(A10)
[θ - β Et ηt+1 RKBt+1 (RKBt+1 – Rt)] * Dt = [β Et ηt+1 RKBt+1 – θ] * (1-σ) n
which can easily be rearranged into the form (8). Implicitly we are assuming that the following
set of inequalities hold:
β Et ηt+1 RKBt+1 > θ > β Et ηt+1 RKBt+1 (RKBt+1 – Rt)
[Note on grading: the solution that my question guided you towards is similar to the
solution presented in a recent paper by Gertler and Kiyotaki, on which this question is
based. Some of you took a shortcut by exploiting equation (3) and arrived at a different
solution. That solution may or may not be equally valid, but I specifically told you to use
(4)-(7) to get the answer. If you instead used (3), but otherwise had correct math, you could
get 15 out of 30 points].
(d) [30 points] Verify the conjecture (7) using (8) and (3)-(5) and derive the formula for ηt.
Answer: plugging (8) into (5) we have:
(A11)
Qt KBt = [1 + φt ] * (1 – σ) nt
Updating (4) one period, plugging in (A11) and (8), we have:
(A12)
nt+1 = RKBt+1 [1 + φt] (1-σ) nt - Rt+1 φt (1-σ) nt
= [RKBt+1 + φt (RKBt+1 – Rt+1)] (1-σ) nt
Update (7) one period, plug in (A12), and plug the result into the last term of the firm’s Bellman
Equation (3). This yields:
(A13)
VBt = σ nt + β Et ηt+1 [RKBt+1 + φt (RKBt+1 – Rt+1)] (1-σ) nt
= [σ + (1-σ) β Et ηt+1 (RKBt+1 + φt (RKBt+1 – Rt+1))] nt
Which verifies that the conjecture (7) is correct and establishes a formula for ηt.
[Note on grading: some of you took a shortcut and used (6) to derive the answer. I
specifically told you to use (3)-(5) and (8); if you used (6), then you could get up to 15 out of
30 points if your math was otherwise correct. Note that if you got full credit for parts (a)
and (b) and half credit for parts (c) and (d), your overall score would be 70, which is a
passing grade, so it was possible to pass the question even if you took shortcuts in parts (c)
and (d). Another thing to note is that some of you tried to set up and solve a Bellman
equation to characterize optimal bank behavior in parts (c) and (d). This was not
necessary or helpful in this case; I already told you that (6) binds, meaning that the optimal
behavior of banks is to take in as many deposits and buy as much capital as possible; there
is no need to do anything else to solve for optimal bank behavior.]