Problem Set 2

ECON2220 C, E

Intermediate Macroeconomics

Spring 2025

Deadline: 23:59, March 24 (Monday)

Please submit a softcopy of your answers on Moodle.

Question 1. (6 points each)

(1) Kids of rich parents tend to have higher test scores. Can you conclude that higher income leads to better test scores?  What can be wrong with this argument? Provide one example that makes this argument invalid (Max 10 sentences. Hint: Omitted variable bias. You can make any story/example.).

(2) The figure below shows the relationship between the level of education and GDP per capita growth between 1960 and 2000. Specifically, the x-axis measures average years of schooling and the y-axis measures average growth rate of GDP per capita. Each point shows the data for a country. What can you say from this figure about the importance of education for economic growth? From this figure, can you say that higher education level leads to economic growth, so the government should raise the education level for economic growth? (Max 10 sentences)

Question 2. (4 points each)

Consider unsophisticated Robinson’s consumption decision. His consumption today (period 1) is determined by disposable income today as follows.

where C1 is consumption today, Y1 is income today, T1 is a lump-sum tax today, C is the minimum level of consumption Robinson needs and MPC is the marginal propensity to consume (0 < MPC ≤ 1).

(1) What is the effect of decrease in the lump-sum tax today on Robinson’s consumption today? How much does consumption today change when the lump-sum tax T1 decreases by one unit?

(2) What is the effect of decrease in a lump-sum tax tomorrow (say, in period 2) on Robinson’s consumption today? You can assume that the lump-sum tax today does not change. Explain the difference between your answer to this question (2) and question (1). Based on your answer, do you see any problem with this consumption function? Explain the problem if there is any.

Question 3. (4 points each)

Consider sophisticated Robinson’s two-period consumption decision problem. The notations of variables are the same as those in the lecture slides. His lifetime utility is given by

U(C1, C2) = lnC1 + lnC2

Here we assume a discount rate β=1 implicitly. Budget constraints are given by

C1 + B = Y1 一 T1

C2  = Y2  一 T2  + B

in period 1 and 2 respectively. So, we assume the interest rate is zero (r = 0) implicitly. T1 , and T2 are lump-sum taxes in period 1 and period 2 respectively.

(1) Derive the intertemporal budget constraint for Robinson (Hint: Combine two budget constraints to eliminate B.). Provide the interpretation of the intertemporal budget constraint (i.e. what does the intertemporal budget constraint imply?).

(2) Solve the utility maximization problem and derive the optimal level of consumption in each period, C1 and C2 (i.e. Express C1 and C2 in terms of Y1, Y2, T1, and T2.).

(3) Now suppose Robinson cannot borrow. He can only save. Namely, B ≥ 0. What is the optimal level of consumption in each period? Answer this question in each of the following two cases.

(i) Y1 — T1  ≥ Y2 —T2 (i.e. his disposable income is larger in period 1)

(ii) Y1 — T1 < Y2 — T2 (i.e. his disposable income is smaller in period 1)

Provide intuition for your answer. (Hint: With the optimal level of consumption you find in question (2), what is the implied borrowing B? Does it satisfy the borrowing constraint B ≥ 0? If not, what is the best for Robinson to do? Is it optimal to set B = 0, namely consuming everything you have in period 1?)

(4) Robinson still cannot borrow. Suppose the government plans to cut the lump-sum tax in period 1 by increasing the lump-sum tax in period 2, keeping the present value of lump-sum tax constant (i.e. T1 goes down while T2 goes up by the same amount so there is no change in T1 +  T2.). What is the effect of this policy on the optimal level of consumption? Again, answer this question in each of the following two cases.

(i) Y1 — T1 ≥ Y2 —T2

(ii) Y1 — T1  < Y2 — T2

Does this policy increase or decrease Robinson’s utility in each case? Explain your answer. For simplicity, in case (ii), assume the condition does not change from (ii) to (i) when T1 decreases.  (Hint: Robinson wants to smooth his consumption over time to maximize his utility. Does the tax cut help him do it?)

(5) Does Ricardian equivalence proposition hold in each case? Provide intuition for your answer.  (Hint: The Ricardian equivalence proposition says the timing of lump-sum taxes does not affect a consumption plan. Does it hold in each case in question (4)?)

Question 4. (6 points each)

How would each of the following affects equilibrium saving, investment, and the real interest rate? Explain using the saving-investment diagram. State your assumptions if necessary.

(1) Income is expected to rise in the future.

(2) Productivity is expected to drop next year.

Question 5. (4 points each)

Consider the Solow model. Production function is given by

The notations of variables are the same as those in the lecture slides. The depreciation rate d is  0.1, the population growth rate n is 0.1, and the saving rate s is 0.2. The level of productivity is constant, so At  = 4 all the time.

(1) Compute the steady-state level of capital per person k* .

(2) Compute the steady-state level of output per person y* .

(3) Compute the steady-state level of consumption per person C* .

(4) What is the growth rate of total output yt at the steady state? (Hint: yt  = ytNt. See the lecture slides how to compute the growth rate of a product of two variables.)

(5) What is the Growth Accounting equation for this economy? Apply the equation to this economy when the economy is at the steady-state, i.e. what is the contribution of capital, labor, and productivity to the growth rate of total output yt at the steady-state respectively? (Hint: You simply need to express the production function in growth rates and use the steady-state growth rates of y, A, K, and N.)

(6) Suppose the government wants to maximize consumption per person at the steady state by changing the saving rate s. Compute this saving rate that maximizes consumption per person (i.e. Golden rule saving rate) and the associated consumption per person. (Hint: Express C* in terms of parameters including S. Then, take the derivative of C* with respect to S to maximize C* .)

(7) Is the policy to change the saving rate from 0.2 to the one you find in question (6) always good for households? Explain a potential problem if any. (Hint: Consider the transition from the steady-state in question (3) to the steady-state in question (6). What happens to C during the transition, especially immediately after S changes? Notice Ct  = (1 — S)yt.)

Question 6. (5 points each)

Consider a Solow economy that begins with a capital stock per capita of $7 million, and suppose its steady-state level of capital per capita is $10 million. So, the economy is during the transition.

(1) Plot the paths of capital per capita k, output per capita y, and consumption per capita C over time in the graphs below (you can draw your own graphs.). Use the Solow diagram (i.e. the graph with the key equation in the Solow model) and the mathematics of the Solow model to explain what happens to the economy, both during the transition and in the long run. (Hint: Check the numerical example in the lecture slides. You can assume the per capita production function takes the form. y = Aka to plot y.)

(2) To its pleasant surprise, the economy receives a generous gift of foreign aid in the form of $1 million worth of capital per capita (electric power plants, machine tools, etc.) at the beginning when the amount of capital per capita is $7 million. So, the amount of capital per capita becomes $8 million. Answer the same questions as in question (1).

(3) Suppose instead of starting below its steady state, the economy begins in the steady state, with a capital stock equal to $10 million. Again, to its pleasant surprise, the economy receives $1 million worth of capital per capita at the beginning. So, the amount of capital becomes $11 million. Answer the same questions as in (1).

(4) Summarize what this question teaches you about the possible consequences of foreign aid. Does foreign aid help the country in the short run and in the long run? (Max 10 sentences)