Lately, your country has been discussing the impact of less immigrants due to the spread of COVID-19.
Suppose that you are hired as an economic advisor. Your project is to do this analysis on the basis of
the optimal growth model. In your model, there is no productivity growth, that the population grows at a
constant rate n, and depreciation is the constant at δ. Utility function is U (C) = C
1−σ
1−σ
and the production
function is Y = K
αN
1−α
.
Answer following questions.
(a) (10 Points) Derive conditions for finding the dynamics of {Ct
,Kt}
∞
t=0
and the steady state (Warning:
Not get a full credit if you write down the conditions only).
(b) (25 Points) Suppose that the economy of your country is initially in the steady state. Here is your
idea: the spread of COVID-19 will be temporary, so that immigrants will enter into your country in the nottoo-distant future. Put differently, there is a one-time decrease in the labor force from immigration (N
0 < N),
but the population growth rate n remains constant. Analyze the short-run and long-run effects of this change
in the levels of per-capita capital, consumption and aggregate output. How about the short-run and long-run
effects of growth rates of per-capita output and aggregate output?
Now, you finish presenting your analysis in the conference.
(c) (15 Points) After your presentation, some economists strongly disagree with your conclusion. They
consistently point out that it will take time to eliminate this pandemic, and thereby there should be an
decrease in immigration as a continuing process. That is, n decreases to a lower value n
0
for the time being.
Analyze the short-run and long-run effects of this change in the levels of per-capita capital, consumption,
aggregate output, growth rates of per-capita output and aggregate output if you accept their concerns.
(d) (20 Points) After your presentation, the prime minister of your country suggests that some immigrants should be treated as human capital, which affects productivity. So, less immigrants decrease productivity. For this analysis, you need to relax the assumption in your model that there is no productivity growth.
Given that he aggresses that the spread of COVID-19 will be temporary, does his suggestion potentially
change your answer to (b)? Or, does the result stay the same? Explain your statement as fully as you can.
2
Question 2 (30 Points)
Consider a two-period consumption-savings problem under uncertainty where a consumer has preferences:
logC0 +E0 [βlogC1],
where E0 is the expectation formed as of date 0. Income at date 0 is known, but income, consumption,
and asset payoffs in date 1 are random variables. Suppose the consumer can buy or sell an asset at price P
in date 0 which yields a stochastic payoff X in the next period. She gets the exogenous, possibly stochastic
income yt each period. If θ are her purchases of the asset at date 0, she thus faces the budget constraints:
C0 +Pθ = y0,
C1 = y1 +θX.
Answer following questions.
(a) (10 Points) Find the optimality condition (Euler equation) for the choice of asset holdings θ (or
equivalently the choice of consumption) (Warning: Your score will be lower if you derive Euler equation
incorrectly).
(b) (5 Points) Now suppose that there are many such identical consumers, and that the asset is in zero net
supply, so in equilibrium Ct = yt
. Moreover suppose that y0 is known and there are two states of the world
in period 1 which determine (y1,X). With probability 0.5, y1 = (1−∆) y0 and X = X1, and with probability
0.5 and y1 = (1+∆)y0 and X = X2, where ∆ > 0 is a constant and X1,X2 are specified payoffs in each state.
Now find an expression for the equilibrium asset price P.
(c) (5 Points) Consider three different assets which differ in their payoffs: asset A has X1 = X2 = 1, asset
B has X1 = 1−∆ and X2 = 1+∆, and asset C has X1 = 1+∆ and X2 = 1−∆. Find and rank the prices of
these three assets.
(d) (10 Points) Interpret your answer to (c) by using the following formula:
E[Y Z] = E[Y]E[Z]−COV (Y,Z),