Write the aggregate production function in per-worker terms

Question

1. Binary Choice

Please circle legibly the letter that corresponds to your answer.

1. According to the Fisher equation, the nominal interest rate is
a. Equal to the real interest rate plus the expected inflation rate.
b. Equal to the real interest rate minus the expected inflation rate.

2. During periods of unexpected inflation, lenders are hurt while borrowers gain because the
a. Ex post real interest rate exceeds the ex ante real interest rate.
b. Ex post real interest rate is lower than the ex ante real interest rate.

3. The Solow growth model predicts that countries with higher population growth rates will have
a. Lower steady-state levels of output per worker.
b. Lower steady-state growth rates of output per worker.

4. If inflation falls from 6 percent to 4 percent and nothing else changes, then, according to the Fisher effect
a. The real interest rate falls by 2 percent and the nominal interest rate remains constant.
b. The nominal interest rate falls by 2 percent and the real interest rate remains constant.

5. If investment demand decreases in a small open economy
a. Net exports decrease.
b. Net exports increase.

6. If domestic investment is less than domestic saving, one would observe
a. Borrowing from abroad.
b. Lending to abroad.

7. Which of the following statements is not a valid explanation of why capital is flowing to the United States from many poorer countries?
a. Property rights are enforced more fully in the United States than in many poor countries.
b. According to the Cobb-Douglas production function, additional capital will be more productive in countries with more capital per worker.

8. If a computer costs $2,000 in the United States, how much will the same type of computer cost in euros if the nominal exchange rate is 0.8 euros per U.S. dollar?
a. 2500 euros
b. 1600 euros

9. Suppose that a country in a steady state implements policies to increase its saving rate. After the new steady state is reached
a. The level of output per worker will be higher than before.
b. Output per worker will grow more rapidly than before.

10. "Break-even" investment is the amount of investment
a. Required to keep the capital stock per worker constant.
b. At which the marginal product of capital is equal to its price.

Short Answer/Graphing Questions

Please provide your answer in the provided space. Remember to write legibly and take the time to organize your answer before you begin to write.

For short essay responses please remember to use complete sentences: sentences should have a subject, a verb, proper capitalization and proper punctuation. We reserve the right to lower your graph for responses that are not written in standard English with these components.

1. The labor market is in a steady state, if the number of people finding jobs is equal to the number of people losing jobs. Assume the labor force L is fixed, the number of people who are employed is E, the number of people who are unemployed is U, the rate of job separation is s and the rate of job finding is f.

Write down the condition for steady state in the labor market:

Derive the natural rate of unemployment:

If f = 0.7, s = 0.10, what is the natural rate of unemployment?

2. The quantity equation is MV=PY, where M is the quantity of money available in an economy, V is the transactions velocity of money, P is the GDP deflator, and Y is real GDP.

In 2005,
Real GDP = Y = $11,131 billion
The GDP deflator = P = 1.121
M = $6,539 billion

What is the value of V?

Recall that the %? in ( ) is approximated by the following: (Percentage change in ( )) ≈ %? in A + %? in B. Apply this approximation to both sides of the quantity equation and write down a formula which characterizes the relationship between the percentage change in M, the percentage change in V, the percentage change in P, and the percentage change in Y.

In 2008, the US government increased the money supply by 2%, the real GDP growth rate was 5% and there was no change in the transactions velocity of money. Provide an approximation of the inflation rate. Show your work.

3. Assume the economy is initially in long-run equilibrium. Explain briefly how short-run stagflation results from an adverse supply shock.

Suppose that stagflation occurs. To prevent a reduction in output in the short-run, should the government increase or decrease the money supply? Briefly explain your answer.

4. Define frictional unemployment and structural unemployment. Then, briefly explain one reason for each type of unemployment.

5. Suppose you are analyzing the U.S. economy using the model of aggregate supply and aggregate demand. Assume the economy is initially in long-run equilibrium. You observe that the U.S. government decreases the money supply. Holding everything else constant, will the aggregate demand curve shift? If so, which direction will it shift? Briefly explain your answer.

How does the decrease in money supply affect the price and output level in the short run and in the long run? Illustrate your answer with a graph depicting the effect of this change. Be sure your graph is clearly labeled (horizontal and vertical axis, any curve used in the graph, and any equilibrium points) and that you identify the initial equilibrium, the new short-run equilibrium and new long-run equilibrium.

Problems (points noted for each question: total points in this section equal to 30 points)

Hint: if you get discouraged by the algebra, make sure you answer the graphing parts of the problem.

(1) Assume aggregate production function Y_t = K_t^αL_t^{1 - α} for use in the Solow model with population growth. The subscript t stands for time, t = 0, 1, 2, ..., and so on. The labor force grows at rate n such that L_t = 1 + nL_{t - 1}. See the table below for a list of variables you should use for the problem; these should be familiar to you. Please leave all equations in symbolic form (don't substitute in numbers).

Definition	Variable	Per-worker
Output	Yt	yt
Capital	Kt	kt
Consumption	Ct	ct
Investment	It	it

Definition	Variable
Labor	Lt
Population growth rate	n
Depreciation rate	δ
Cobb-Douglas parameter	α
Savings rate	s

 

You can also assume that: n > 0, 0 < δ < 1, 0 < α < 1, and 0 < s < 1.

A) Write the aggregate production function in per-worker terms.

The "law of motion" refers to the change in a variable between time period t and time period t + 1.

B) Write the law of motion (?k_t) for the per-worker capital stock k_t.

C) Graph y_t, sf(k_t), and (n+δ)k_t versus k_t. Identify steady-state levels k_ss, c_ss, and i_ss on your graph. Do not solve for numeric values for these steady-state levels. Label your graph carefully and completely.

D) For this part onward, let Y_t = K_t^αE_tL_t^{1 - α}. Labor-augmenting technological progress Et grows at rate g such that E_t = 1 + gE_{t - 1}. Write the production function in effective units. For notation, you can use lower-case variables to stand for effective units now.

E) Write the law of motion (?k_t) for k_t in effective units.

F) Solve for the steady-state level of the capital stock in effective units, k_ss.

G) Solve for the golden-rule level of the capital stock in effective units, k_gr.

H) Solve for the golden-rule savings rate s_gr that maintains k_gr as a steady-state.

I) Graph y_t, sgrf(k_t), and (n+g+δ)k_t versus k_t. All variables here are in effective units. Identify golden-rule levels k_gr, c_gr, and i_gr on your graph. Draw the tangent line of y_t at k_gr and make sure that its slope is approximately consistent with the geometric interpretation of the golden-rule level of the capital stock.

2) You are provided with the following information about the foreign exchange market for US Dollars ($US) where e denotes the nominal exchange rate (Japanese Yen per US Dollar, ¥/$US). S$US refers to the quantity supplied of US dollars while D$US refers to the quantity demanded of US dollars.

S_$US = 5 + 2e
D_$US = 10 - e

A) Solve for the equilibrium nominal exchange rate, e*.

B) Suppose that the price ratio P_US/P_JPN = 320. Solve for the equilibrium real exchange rate, ε*.

C) Use the above information as well as the following information to answer this question. Assume the following model, treating the US as a small open economy:

Y = C + I + G + NX = 300

C = C‾ + 0.7Y - T

I = 50 - 200rworld

G = 50

T = 100

NX(ε*) = 75 - 40ε*

rworld = 0.10

Solve for C‾.

D) Solve for S_private, S_public, and S = S_total.

E) Solve for net capital outflows NCO using savings and investment. Compare this to your value for NX. Are they consistent?

F) Graph ε versus NCO and NX; identify NX*, NCO*, and ε* on your graph.