# intermediate macroeconomic theory

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**Problem Set 2**

**Problem 1: **

Consider the following extension of the Solow model with exogenous productivity growth. The expression for total output and the dynamics of capital per effective worker are given by the following expressions:

Y = KÎ± (AL) 1âˆ’Î±

Î”k = sy âˆ’ (Î´ + g)k

where g is the growth rate of A, y is output per effective worker, and Î´ is the depreciation rate of capital.

(a) Using the information given above, find the fundamental Solow equation. Find an expression for steady state capital per effective worker.

(b) What happens to steady state capital if the growth rate of productivity rises?

(c) Suppose that this economy consists of two types of households: laborers and capitalists. Capitalists own all the capital stock K and are paid a rental rate equal to the marginal product of capital. Find an expression for their share of national income.

(d) Assume that laborers save none of their income. If capitalists save **50% **of their income, the parameter Î± = 0.3, and the parameter Î´ = 0.1, what is the aggregate saving rate s? Will income inequality between capitalists and laborers worsen over time?

**Problem 2: **

Which of the following statements about the role of institutions in economic growth is TRUE and which of the following statements is FALSE? Explain.

(a) High expropriation risk due to low-quality governance and poor institutions might account for low levels of human and public capital in poor nations

(b) The quality of institutions in countries colonized by Europeans is correlated with whether climate/disease allowed European settlers to live in those areas

(c) Low-quality institutions make it difficult to protect property rights and enforce contracts

**Problem 3: **

Suppose the economy is on a balanced growth path in the Romer model, and then, in the year 2030, research productivity rises permanently to z1> z0.

(a) Solve for the new growth rate of knowledge and yt.

(b) Make a graph of yt over time (on a ratio scale).

(c) Why might the parameter z increase in an economy?

**Problem 4: **

In this problem, you are asked to use the labor supply – labor demand framework developed in class. Illustrate the effect of various shocks using a diagram of labor supply and labor demand.

(a) An unusually hot summer **decreases **workersâ€™ preference for leisure time. What effect will this shock have on employment and wages?

(b) Firms **bought new capital a lot **in the last period. What effect will this shock have on employment and wages?

(c) Congress passes a law funding national pre-kindergarten education by raising taxes on investment income. What effect will this shock have on employment and wages?

**Problem 5: **

Suppose labor income starts at $50,000 and then grows at a constant rate of 2 per cent per year after that. Let wt be labor income in year t, so that wt = w0(1 + g)t , where w0 = 50, 000 and g = 0.02.

(a) If the interest rate is R, what is the formula for the present discounted value (PDV from now on) today (in year 0) of labor income in a particular future year t?

(b) Now add up these terms from t = 0 to t = 45 to get a formula for the PDV of labor income. Your answer should look something like that in equation (7.12) of the textbook.

(c) Write your answer to part (b) so that it takes the form of the geometric series P DV = w0(1 + a + a2 + a3 + … + a45). What is the value of a that you find?

(d) Apply the geometric series formula to compute the PDV for the case of R = 0.04, R = 0.03, and R = 0.02. What happens when R = 0.02 and why?

**Problem 6: **

Suppose that college education raises a personâ€™s wage by **$20,000 **per year, from **$40,000 **to **$60,000**. Assume that the interest rate is 3 per cent and there is no growth in wages. Suppose you are a high school senior and deciding whether or not to go to college. Find the PDV of earnings in the following cases. Also, assume that the work time is still **31 years**, adding up to a **35-year **non-college work career.

(a) What is the PDV of your labor income if you do not go to college and start working immediately?

(b) As an alternative, you could pay $20,000 per year in college tuition, attend college for 4 years, and then earn **$60,000 **after you graduate. What is the PDV of your net earnings (adjusting for tuition) under this plan?

(c) Discuss the economic value of college education.

(d) What if you pursue the plan described in (b), but you also expect to be unemployed for 2 years after finishing college (knock on wood). During these years assume that you do not have access to any unemployment insurance. What is the PDV of your net earnings under this plan?

**Problem 7: **

Label the following statements true or false and justify your answer.

(a) Suppose that the money supply is growing at **5%**. If the velocity of money is falling at **5%**, then inflation is zero.

(b) Suppose that inflation equals the growth rate of the money supply. Then seigniorage revenues are growing over time.

**Problem 8: **

Consider a fast-growing emerging market economy where output is growing is at 10% per year. The central bank would like to choose a growth rate of the money supply to keep inflation at 0%.

(a) Assume that demand for money is constant and inelastic (does not depend on nominal interest rates). What should be the growth rate of the money supply to keep inflation at 0%?

(b) Due to the introduction of debit and credit cards, it turns out that the velocity of money is not constant. If more consumers are using these payment technologies, do you expect the velocity of money to be increasing or decreasing? Should the growth rate of the money supply be faster or slower relative to your answer in a)?

(c) Consider the following expression for the growth rate of the velocity of money. The growth rate of velocity now depends on the nominal interest rate. Do you expect the coefficient to be positive or negative? Why?

Î”V/V = 0.04 + Î»Î”i

(d) Suppose that the real interest rate remains constant. Use the expression in c) to derive an expression for inflation in terms of money supply growth, output growth, and expected inflation. If expected inflation is constant, what is the growth rate of the money supply that keeps inflation at zero?

(e) A number of commentators are worried that this country is â€œoverheatingâ€ and that inflation may start rising in the future. Should the central bank raise or lower the growth rate of the money supply today? Why?

**Problem 9: **

Suppose velocity is constant, the growth rate of real GDP is 3 per cent per year, and the growth rate of money is 5 per cent per year. Calculate the long-run rate of inflation in the following cases:

a) What is the rate of inflation in this baseline case?

b) Suppose the growth rate of money rises to 10 percent per year.

c) Suppose the growth rate of money rises to 100 percent per year.

d) Back to the baseline case, suppose real GDP growth rises to 5 percent per year.

e) Back to the baseline case, assusme that velocity of money rises at 1 percent per year. What happens to inflation in this case? Why might velocity change in this fashion?

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