Answer keys to HW3

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Problem Set 3
This problem set is due on December 3rd.
I. A Real Intertemporal Model with Investment (60 pts)
1) Having known that the firm makes an intertemporal decision on the investment given the typical
set-ups for the profits and production functions we assume over lectures, answer the following
questions.
a) (5 pts) State the firm’s maximization problem.
max V = π +
I
π′
,
1+r
where π and π ′ are the profits in the current and the future periods, respectively, and r is the
interest rate.
b) (10 pts) Solve the firm’s problem (You should find the marginal benefit and marginal cost of
investment and eventually derive the ’optimal investment rule (hereafter OIR)’).
Y ′ − w′ N ′ + (1 − δ)K ′
1+r
z ′ F ((1 − δ)K + T, N ′ ) − w′ N ′ + (1 − δ)((1 − δ)K + I)
= zF (K, N ) − wN − I +
1+r
V = Y − wN − I +
∂V
z ′ FK ′ (.) + (1 − δ)(1)
= |{z}
−1 +
=0
∂I
1
+
r
|
{z
}
MC
MB
⇔
z′F
K
′
′
′ (K , N )
+ (1 − δ)
1+r
=1
⇔ M PK ′ − δ = r (Optimal investment rule)
c) (10 pts) Further assume the firm’s problem with asymmetric information where there are two types
of firms (lenders and borrowers) and two types of interest rates (lending rate of r1 and borrowing
rate of r2 ). Borrowers are either good firms or bad firms, and lenders cannot distinguish bad firms
from good firms. Under this assumption, what is the effect of the financial crisis on the borrowers’
investment? (Show the firm’s optimal investment rules for borrowers and their optimal investment
schedules (hereafter OIS) before and after the crisis.)
OIR before the crisis: M Pk′ − δ − x1 = r1 ,
OIR after the crisis: M Pk′ − δ − x2 = r1 ,
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where x1 and x2 are the default premia before and after the crisis, respectively such that x2 > x1 .
The OIS’s are in Fig 1. Thus, eventually, given lender’s interest rate, the amount of investment
decreases for borrowers.
Fig. 1: The Effect of an Increased Default Premium on a Firm’s Optimal Investment Schedule
2) (10 pts) What is the effect of a decrease in the depreciation rate on the firm’s investment decision?
There are two opposing effects of a decrease in the depreciation rate. First, a direct effect - given
the marginal product of capital in period two, MPK ′ the net marginal product of capital, MPK ′ -δ
will increase when the depreciation rate decreases. For any given real interest rate, this effect shifts
the Optimal Investment Schedule to the right. This direct effect eventually results in an increase in
I given r. Second, an indirect effect - using the law of motion of capital, K ′ = (1 − δ)K + I , a
decrease in δ results in an increase in K ′ . This results in a decrease in I given r. Putting them
together, the overall effect of a decrease in δ on I is ambiguous.
3) (10 pts) There is a temporary increase in the relative price of energy. What is the effect of this on
the equilibrium?
In the labor market, labor demand curve shifts to the left due to the decrease marginal product of
labor. In the current good market, output supply curve shifts to the left, which leads to an increase
in r. With an increase in r, the labor supply shifts to the right. In sum, y decreases, r increases, N
is ambiguous, w decreases, C decreases, and I decreases.
4) (10 pts) Suppose that there is a permanent increase in total factor productivity. What is the effect of
this on the equilibrium? Also, show how the impact of this differs from the case where total factor
productivity is expected to increase only temporarily.
An increase in z ′ increases Y , r, and N ; and lowers w. Changes in C and I are ambiguous. A
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permanent increase in total factor productivity simply combines the effects of increases in z and z ′ .
Y increases. w and r may either rise or fall. N , C and I change ambiguously as well.
II. Money, Banking, Prices, and Monetary Policy (40 pts)
1) (5 pts) Fisher relation shows the relationship between the real and nominal interest rates: 1+r =
1+R
1+i ,
where r is the real interest rate, R is the nominal interest rate, and i is the inflation rate. Using the
equation mentioned, mathematically derive the short version of Fisher relation (r ≈ R − i).
Take log by both sides to yield
ln(1 + r) =ln
l+R
l+i
⇔ ln(1 + r) = ln(1 + R) − ln(1 + i)
⇔r ≈R−i
2) (5 pts) State the transaction and budget constraints in the monetary intertemporal model.
Transaction constraint:
P (C + I + T ) + B d = M−1 + B−1 (1 + R−1 ) + P X d
|
{z
} |
{z
}
expense for transactions
Monetary value available for transactions
Budget constrain:
P (C + I + T ) + B d + M d + P qX d = M−1 + B−1 (1 + R−1 ) + P Y
{z
} |
{z
}
|
expense
Wealth or Disposable income
3) (10 pts) Derive the demand for credit card balances (xd ) given the model set-ups we assume in
lectures.
The marginal benefit and marginal cost of using credit card balances in nominal terms are M B =
(1 + R)P and M C = P (1 + q), respectively. Using the two constraints in the precious step, we get
to that M d = P Y − P (1 + q)X d . The agents decide whether to use credit cards or not depending on
the size of MB and MC, that is, M B − M C = R − q , and we have 3 different cases: (1) f R > q ,
consumers and firms borrow as much as possible on their credit cards during the period (Use credit
card as much as possible). Then, M d = 0, M d = P Y − P (1 + q)X d = 0 ⇔ X d =
R < q,
Xd
Y
1+q ,
(2) if
= 0, (3) If R = q , indifferent between using credit card and using cashes. Thus, we
eventually have the following demand for credit card balances (xd , Fig. 2).
4) (5 pts) In the monetary intertemporal model, what is the effect of an increase in M s (supply for
money) on the equilibrium? Explain with the aid of diagrams.
Due to the neutrality of money, all the real variables stay the same in the labor and output markets
except for the nominal price (P ) and quantity of money (M ) in the money market with both being
increased. (Fig. 3)
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Fig. 2: Demand for credit card balances
(a) Labor market
(b) Output market
(c) Money market
Fig. 3: Effect of an increase in M s on equilibrium
5) (10 pts) In the monetary intertemporal model, what is the effect of an increase in z ′ (future total
factor productivity)? (Assume that the effect of an increase in Y dominates the effect of an increase
in r in the output market.) What would be the best monetary policy for the government (or the
central bank) if it pursues the stable price?
Refer to Fig. 4.
r, Y , N , M increase; w, P fall; c, I ambiguous. If the government pursues the stable price level,
it increases M s to the point where P reaches the original level.
6) (10 pts) In the Fridman-Lucas money surprise model, suppose that the central bank want to reduce
the price level. Suppose the central bank has two options: (i) announce in advance that the money
supply will decrease; (ii) surprise the public with a decrease in the money supply. Which option is
preferable? Explain with the aid of diagram.
If the money supply decrease is announced in advance, then the decrease in money is neutral. The
price level falls in proportion to the decrease in the money supply. This works as in Fig. 3, except
in reverse. If the money supply decrease is not announced, then this works as in Figure 12.14 in
your textbook except in reverse. Money demand decreases, which offsets the decrease in the price
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Fig. 4: Effects of an increase in z ′
level that occurs. Thus the price level will decrease, but less than in proportion to the money supply
decrease, and there is a decrease in output and employment, which is costly. As a result, if the goal
of the central bank is to reduce the price level, it should announce the money supply decrease in
advance.

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