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Problem Set 2
This problem set is due on October 31 (Friday) by 4:30pm., before class. No late homework will be accepted.
I. Economic Growth Model
1) (5pts) In the Maltusian model, suppose that there is a technological advance that reduces death rates.
Using diagrams, determine the effects of this in the long-run (steady state) and explain your results.
A reduction in the death rate increases the number of survivors from the current period who will still
be living in the future. Therefore, such a technological change in public health shifts the function
g(c) upward as shown in Fig 1b. Equilibrium per capita consumption decreases. In Fig 1a, we also
see that the decrease in per capita consumption requires a reduction in the equilibrium level of per
capita land. The size of the population has increased, but the amount of available land is unchanged.
(a)
(b)
2) In the Solow growth model,
a) (10 pts) In addition to the production function given by Y = zF (K, N ) = zK α N 1−α , assume that
economic environment is the same as the one we talked about in the lecture. Find an equilibrium.
(hint: find k ∗ , c∗ , y ∗ .)
(1−δ)k+szkα
. In the steady state, k =
1+n
α
sz 1−α
and c∗ =(1 − s)z( n+δ
)
are obtained
Using the law of motion of capital, first find k 0 which is
1
α
sz 1−α
sz 1−α
k 0 = k ∗ , which yields that k ∗ =( n+δ
) . y ∗ =z( n+δ
)
from k ∗ being plugged into equilibrium conditions.
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b) (5 pts) Find growth rates of k and K in the steady state.
%∆k = 0 and %∆K = n.
FYI, %∆K = %∆k + %∆N = 0 + n.
c) (5 pts) What is the effect of an increase in the total factor productivity (z ) in the Solow model?.
Per capita capital (k ), per capita output (y ), and per capita consumption (c) increase (see Figure 1b.).
(a) Effects of a decrease in n
(b) Effects of an increase in z
Fig. 1: Equilibrium effects of changes in n and z
d) (5pts) What is the effect of a decrease in the population growth rate (n) in the Solow model?.
Per capita capital (k ), per capita output (y ), and per capita consumption (c) increase (see Figure 1a.).
3) Suppose that the production technology is given by Y = zK (so called AK model, where K could
be considered as human capital.), where Y, z , and K are usual variables we defined in lectures. A
consumer consumes wuH s units, where u is a fraction of time devoted to working and uH s is the
number of efficiency units of labor. Human capital is assumed to evolve the following: (H s )0 =
b(1 − u)H s , where b represents the human capital technology. Given that economic environment,
answer the following questions.
a) (2 pts) Find an equilibrium consumption.
C = zuH
b) (3 pts) Find the growth rate of human capital and consumption.
Both have the same growth rate of b(1 − u) (or b(1 − u) − 1).
c) (5 pts) What is the difference in a convergence of economic growth between the model with
human capital accumulation and Solow model. Explain why.
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The model with human capital accumulation predicts no convergence in economic growth while
Solow model predicts convergence. The difference is coming from different characteristics of
marginal product of capital - M PHC in the model with HC does not decrease while M Pk decreases
in the Solow model.
II. Two-Period Model
1) Assume a consumer who has current-period income y =200, future-period income y 0 =150, current
and future taxes t=40 and t0 =50, respectively, and faced a market real interest rate of r=.05 or 5%
per period. The consumer would like to consume equal amounts in both periods, that is, he or she
would like to set c = c0 . However, this consumer cannot borrow at all, that is, s ≥0. Answer the
following.
a) (5 pts) Show the consumer’s lifetime budget constraint and indifference curve in a diagram.
Plugging the given information into the lifetime budget constraint gives c + 0.95c0 = 255.2.
In the figure below, the initial BC is given by BE1 D, with an initial endowment point E1 =(160,100).
b) (5 pts) Calculate his or her optimal current- and future-period consumption and optimal savings,
and show this in your diagram.
With perfect-complements preferences, the consumer picks point A in fig.2a below. Plugging
in c = c0 into the budget constraint and solving, we find that c = c0 = 130.9 and so s =
y − t − c=160-130.9=29.1. In this case, the fact that the consumer cannot borrow does not matter
for the consumer’s choice, as the consumer decides to be a lender.
c) (5 pts) Suppose that everything remains unchanged, except that now t=20 and t0 =71. Calculate the
effects on current and future consumption and optimal savings, and show this in your diagram.
When t = 20 and the consumer’s lifetime wealth remains unchanged at 255.2. However, the
budget constraint shifts to BE2 F in fig.2a, with the new endowment point at E2 = (180, 79).
This change does not matter for the consumer’s choice, again because he or she chooses to be a
lender. Consumption is still 130.9, but now savings is 49.1.
d) (10 pts) Now, suppose alternatively that y =100. Repeat parts (a) and (c), and explain any
differences.
Now first-period income falls to 100. Wealth is now equal to W = 155.2. In the figure below, the
budget constraint for the consumer is AE1 D, so when the consumer chooses the point on his or her
budget constraint that is on the highest indifference curve, any point on the line segment BE1 will
do. Suppose that the consumer chooses the endowment point E1 , where c = 60 and c0 =100. This
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implies that s = 0, and the consumer is credit-constrained in that he or she would like to borrow,
but cannot. Now with the tax change, the budget constraint shifts to AE2 G, with the endowment
point E2 = (80, 79). Thus the consumer can choose c = c0 on the new budget constraint, and
solving for consumption in each period using the budget constraint. we get c = c0 = 79.5, and
s = 0.5. Here, notice that first-period consumption increased by almost the same amount as the
tax cut, although lifetime wealth remains unchanged at 155.2. Effectively, the budget constraint
for the consumer is relaxed. Therefore, for tax cuts that leave lifetime wealth unchanged, lenders
will not change their current consumption, but credit-constrained borrowers will increase current
consumption.
(a)
(b)
2) Consider the following effects independently.
a) (5 pts) Jane has just got promoted and expected her earnings to increase since the promotion.
How does it affect her optimal choice. (Determine changes in c, c0 , and s.)
Due to her promotion, her permanent income increases, which leads to increases in c and c0 , and
a decrease in s.
b) (5 pts) Assume that the real interest rate, r, decreases. Further, there are more lenders than
borrowers in the economy. What is the aggregate effects of a decrease in r? (Determine how C ,
C 0 , and S change.)
Given that the number of lenders is greater than the number of borrowers, we only consider the
case for a lender. With a decrease in r, C and S are ambiguous; and C 0 falls.
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III. Credit Market Imperfections
1) Suppose that there is a credit market imperfection due to asymmetric information. In the economy,
a fraction b of consumers consists of lenders, who each receive an endowment of y units of the
consumption good in the current period and 0 units in the future period. A fraction (1-b)a consumers
are good borrower who receive an endowment of 0 units in the current period and y units in the
future period. Finally, a fraction (1-b)(1-a) of consumers are bad borrowers who receive 0 units of
endowment in the current and future periods. The government sets G = G0 = 0, and each consumer
is asked to pay a lump-sum tax of t in the current period and t0 in the future period. Both the gov’t
and the bank cannot distinguish between good and bad borrowers.
a) (5 pts) Write down the government’s budget constraint, making sure to take account of who is
able to pay their taxes and who does not.
N (b + (1 − b)a)(t +
t0
)=0
1 + r1
where r1 is the deposit rate, N is the total population size in the economy.
b) (5 pts) Suppose that the government decreases t and increases t0 in such a way that the gov’t
budget constraint holds. Does this have any effect on each consumer’s decisions? Show, with the
aid of diagrams.
Yes, it affects consumer’s decisions since it changes lifetime BC. Specifically c increases due to
the shift-out of the lifetime BC toward more current consumption. Refer to Figure 10.2 in the
textbook (or you may draw only the borrower’s case since it does not affect the lender’s optimal
choice.)
c) (2 pts) Does Ricardian equivalence hold in this economy?
No!!!
2) Suppose that there is limited commitment in the credit market, but borrowers are uncertain about
the value of collateral. Each consumer has a quantity of collateral H , but from the point of view
of borrower, there are a 10% chance of losing H , a 70% chance of increasing the value of H by
10%, and a 20% chance of decreasing the value of H by 30% in the future.
a) (5 pts) Determine the collateral constraint for the consumer, and show the consumer’s lifetime
budget constraint in a diagram.
−s ≤
.91pH
⇐⇒
1+r
c≤y−t+
.91pH
1+r
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Notice that the expected value of collateral in the nominal value is 0.91pH , (=0.7×1.1+0.2×.7)pH .
Refer to Figure 10.5 in the textbook and/or lecture notes for a diagram.
b) (3 pts) How does the financial crisis affect the consumer’s current consumption? Explain your
result.
Financial crisis increases a chance of losing h, which certainly decreases the lifetime wealth and
leads the lifetime BC to shift-in. Thus c decreases.
3) (5 pts) Suppose a pay-as-you-go social security system is established where social security is funded
by a proportional tax on the consumption of the young. That is, the tax collected by the government
is sc, where s is the tax rate and c is consumption of the young. Retirement benefits are given out
as a fixed amount b to each old consumer. Can social security work to improve the welfare for
everyone under n>r ? Clearly mention the lifetime budget constraint.
Under this regime, disposable income for the young is y, but the price of current consumption is
(1+s). This implies that the lifetime budget constraint of the individual is now
c+
c0
y0
b
=y+
− sc +
1+r
1+r
1+r
In equilibrium, it must be that sc(1 + n) = b. The lifetime wealth in the absence of the social
security is less than the one with the social security under n > r. Thus individuals are better off
under n > r with the social security system.