midterm_1B_with_answers

ECON-UA 12 — Intermediate Macroeconomics Professor: Jaroslav Boroviˇcka TA: Hyein Han Fall 2023 Sample midterm exam 1B — not for examination Suggested solution Do not turn this page until instructed! Read the instructions below first. Name: Signature: 1. This is a 70 minute midterm exam . It is worth 100 points . This booklet contains 13 pages. 2. At the beginning, you have 3 minutes to look through the exam. This time is not counted in the examination time above. No writing or note-taking is allowed during this period — only reading. 3. When the time is up, you have to stop writing and return the booklet to me within exactly 1 minute. This is your responsibility. 4. The exam is closed book, no notes. 5. No electronic devices are allowed . 6. Use empty pages at the end as scratch paper. Material written on scratch paper will not be considered for grading. 1

1 True or false? [24 points total] For each of the following statements, decide whether it is true or false, and explain your answer in at most two sentences using the material you learned in the course . No points will be given without an appropriate explanation! Question 1.1 [3 points] GDP per capita is not a useful measure of well-being because it has little to do with reasonable indicators of life quality like life expectancy, absolute poverty or health. Answer FALSE. On the contrary, we have seen plenty of empirical evidence that GDP correlates with many of other measures of well-being. Countries that are rich also have high life-expectancy, measures of health, education, etc. Question 1.2 [3 points] Under perfect competition, firms earn zero economic profits. The value of the stock market under perfect competition should therefore be zero. Answer FALSE. Economic and accounting profits are different concepts— accounting profits, for instance, include payments to all factors of production that the firm owns (both tangible and intangible capital). Therefore, even un- der perfect competition, the value of the stock market (which is the present discounted value of accounting profits) can and will be nonzero. Question 1.3 [3 points] The model of production could not explain the differences in output per capita across countries through the differences in capital per capita. However, the Solow growth model was much more successful in explaining the difference in output per capita through differences in saving rates. Answer FALSE. Differences in saving rates in the Solow growth model ex- plained even less of the differences in output per capita than differences in capital per capita. Question 1.4 [3 points] In the Solow growth model, there is no money, so households have no way of saving for the future. Answer FALSE. In the Solow growth model, households save by accumulat- ing real resources, represented by the capital stock. Question 1.5 [3 points] If the labor share was close to zero, then the model of production with Cobb–Douglas production technology would be able to explain the differences in output per capita through differences in capital per capita. Answer TRUE. The Cobb–Douglas production technology implies the rela- tionship ln y = ln A + α ln k . 2

2 Growth rates and returns to scale [13 points total] Question 2.1 [3 points] The Cobb–Douglas production function is given by Y = F ( K , L ) = AK α L β where A , α , β are strictly positive parameters. What restrictions do the parameters have to satisfy so that the production function exhibits constant returns to scale? When these restrictions are satisfied, is the marginal product of capital increasing, con- stant, or decreasing in K ? Provide calculations for both questions. Answer We have F ( 2 K , 2 L ) = A ( 2 K ) α ( 2 L ) β = 2 α + β F ( K , L ) and hence we require β = 1 − α . Under these restrictions MPK = ∂ ∂ K F ( K , L ) = α AK α − 1 L 1 − α which is a decreasing function of capital. Question 2.2 [4 points] Consider the production function given by Y = F ( K , L ) = aK α + bL β where a , b , α , β are strictly positive parameters. What restrictions do the parameters have to satisfy so that the production function exhibits constant returns to scale? When these restrictions are satisfied, is the marginal product of capital increasing, con- stant, or decreasing in K ? Provide calculations for both questions. Answer We have F ( 2 K , 2 L ) = a ( 2 K ) α + b ( 2 L ) β = 2 α aK α + 2 β bL β and hence we need α = 1 and β = 1. With these restrictions, MPK = ∂ ∂ K F ( K , L ) = a and hence the production function exhibits a constant marginal product of cap- ital. Question 2.3 [6 points] Consider growth in an economy with Cobb–Douglas production 4

function Y t = A t K α t L 1 − α t . Assume that there are two countries, C and D . They both have the same growth rate of TFP, denoted g A , and the same growth rate of capital, g K . Country C has a higher population growth rate than country D , g C L > g D L . Which of the two countries will have a higher growth rate of total output? Which of the two countries will have a higher growth rate of output per capita? Provide calculations (including the derivation of the output growth rate) for both questions . Answer The growth rate of total output is obtained by computing 1 + g K = Y t + 1 Y t = A t + 1 K α t + 1 L 1 − α t + 1 A t K α t L 1 − α t = A t + 1 A t K t + 1 K t α L t + 1 L t 1 − α = ( 1 + g A ) ( 1 + g K ) α ( 1 + g L ) 1 − α and the logarithmic approximation yields g Y = g A + α g K + ( 1 − α ) g L The growth rate of output per capita is then given by g y = g Y − g L = g A + α g K − α g L Hence, country C has a higher growth rate of total output g C Y − g D Y = ( 1 − α ) g C L − g D L > 0 but a lower growth rate of output per capita g C y − g D y = − α g C L − g D L < 0. 5

If it does constitute an arbitrage opportunity, what should investors do to take advan- tage of it? If it does not constitute an arbitrage opportunity, how should we properly interpret the interest rate R in the arbitrage equation? Answer This does not constitute an arbitrage opportunity because these are not comparable investments—investing into stocks is riskier. Hence the proper interest rate is one that includes compensation for stock market risk, called the risk premium. Hence R = risk-free rate + risk premium. Question 3.5 [3 points] Consider the opportunity to invest into two stocks, A and B . If stock A is considered to be riskier than stock B , what does the Gordon growth formula imply for the price-dividend ratio of stock A compared to stock B ? Answer The riskier stock A should earn a higher risk premium, hence a higher R . This implies that the price-dividend ratio p s , t d t = 1 R − g d should be lower for stock A . Question 3.6 [5 points] Let g s be the annual growth rate of the stock price, and g d the annual growth rate of the dividend. Show that if these growth rates are not the same (over long periods of time), then the price dividend ratio p s , t / d t will either grow to infinity or decline to zero. Empirically, how does the price-dividend ratio behave over long periods of time? Answer The one-period growth rate of the price-dividend ratio is p s , t + 1 / d t + 1 p s , t / d t = p s , t + 1 p s , t d t d t + 1 = 1 + g s 1 + g d . Over the course of T years, this accumulates to 1 + g s 1 + g d T This is a geometric sequence in T . When g s > g d , then the coefficient 1 + g s 1 + g d is larger than one and the sequence diverges to infinity. When g s < g d , then the coefficient 1 + g s 1 + g d is smaller than one and the sequence converges to zero. Empirically, the price-dividend ratio is roughly stationary (oscillates around a stable level), and hence over long periods of time, we have to have g s ≈ g d . 7

4 Economic policies in the Solow growth model [40 points total] Consider the Solow growth model, described by equations Y t = AK α t L 1 − α Δ K t + 1 = sY t − δ K t Question 4.1 [2 points] Using the equations above, state the steady state condition (one- line answer). Answer The condition is sY ∗ = δ K ∗ . Question 4.2 [6 points] Solve the steady state condition for the steady state level of cap- ital K ∗ as a function of parameters of the model. ( Hint : Substitute out output using the production function.) Further determine the steady state level of output Y ∗ and the steady state level of out- put per capita y ∗ as functions of model parameters. Answer We have sA ( K ∗ ) α L 1 − α = δ K ∗ and thus K ∗ = sA δ 1 1 − α L . To determine the steady state level of output, substitute K ∗ into the production function. Then Y ∗ = A ( K ∗ ) α L 1 − α = A sA δ α 1 − α L = A 1 1 − α s δ α 1 − α L . and y ∗ = Y ∗ L = A 1 1 − α s δ α 1 − α . (4.1) We are now interested in studying some proposed economic policies that you may have heard of on national news. (Bear in mind that what we consider here are simpli- fied abstractions that capture some but definitely not all of the impacts of the proposed policies.) Question 4.3 [5 points] Presidential candidate D proposes to deport a part of the residing population, thus decreasing the population size from L to L ′ (the proposal is to do it very quickly, which we approximate here as an instant change). Sketch the Solow diagram (with the stock of capital on the horizontal axis), including 8

K ∗ K ∗′ t 0 time t K t y ∗ t 0 time t y t 0 t 0 time t N t Figure 2: Trajectories for capital K t , output per capita y t and net investment N t = sY t − δ K t for Question 4.4 . In the moment of the impact of the policy, capital per capita increases abruptly, because population falls and capital stays the same at that moment (it is a stock variable). Therefore, output per capita increases as well. However, over time, output per capita has to converge back to the original steady state because y ∗ does not depend on L (see equation ( 4.1 )). Net investment falls on impact of the policy, which starts off the process of capital decumulation. Over time, net investment converges back to zero, as it has to be zero in the new steady state. Question 4.5 [4 points] The proposed policy, if implemented fully, may decrease the pop- ulation by about 2%, i.e., L ′ L = 1 + g L where g L is the percentage change in population, with g L = − 0.02 = − 2%. Compute the percentage change in total capital and in output per capita that occurs • immediately at the moment of implementation of the policy (short run effect, before capital adjusts), • between the old and new steady states (long run effect, after capital adjusts). Hint : To compute the short run effect, for instance, for the output per capita, evaluate the ratio y ′ / y where y is given by the Cobb–Douglas production function (in per capita terms) before the policy change and y ′ after the policy change. Denote this ratio as 1 + g y and then take the logarithmic approximation to compute g y as a function of g L . For the long-run effect, proceed in the same way to evaluate y ∗′ / y ∗ using the formula for the steady state output per capita. Answer To compute the short run effects, first notice that there is no short run effect on total capital, because capital is a state variable. Further y ′ y = A ( k ′ ) α Ak α = k ′ k α = K / L ′ K / L α = L ′ L − α 10

and therefore 1 + g y = ( 1 + g L ) − α g y ≈ − α g L Therefore, a 2% decrease in L leads to a 2 α % = 2 3 % short-run increase in output per capita. The long-run effect on total capital is K ∗′ K ∗ = ( sA δ ) 1 1 − α L ′ ( sA δ ) 1 1 − α L and hence g K = g L . In the long run, total capital decreases by 2%. The long-run effect on output per capita is zero, since ( 4.1 ) shows that y ∗ does not depend on L . Question 4.6 [2 points] Presidential candidate H proposes to increase taxation of capital income. Here, we implement the capital tax as a fraction τ K of capital that firms have to pay to the government every period . For simplicity, we assume that the firm receives nothing in return and the resources paid to the government are destroyed. Modify the law of motion for capital to incorporate the policy into the model. Show that the introduction of the capital tax is equivalent to an increase in the depreciation rate to δ ′ . What is the relationship between δ and δ ′ ? Answer Now we have Δ K t + 1 = sY t − δ K t − τ K K t = sY t − ( δ + τ K ) | {z } δ ′ K t . Question 4.7 [5 points] Sketch a NEW Solow diagram, including curves for output, depre- ciation and investment. Appropriately label everything. In this diagram, clearly depict the impact of the proposed policy by shifting the relevant curves. Depict the new steady state level of capital K ∗′ Answer The answer is in Figure 3 . The increase in tax rate (which is equivalent to an increase in the depreciation rate to δ ′ = δ + τ K increases the slope of the ‘depreciation’ function (which now included the tax payments). The steady state level of capital decreases. Question 4.8 [6 points] Plot the trajectory (time path) of the following variables: • capital 11

who remain in the economy) better off in the short run? In the long run? Again, your answers should reflect the effects that the Solow model predicts. You should bear in mind that this is not a complete description of the overall impact of the policies. The purpose of this exercise is to clearly expose one particular channel (capital accumulation in the Solow model) through which policies impact the economy. Answer We can measure well-being in terms of consumption per capita. Since consumption per capita is a fraction 1 − s of output per capita, we can focus on output per capita. The ‘ D policy’ improves the well-being in the short run, since deported population leaves capital behind, which increases the capital per capita for the remaining population. This also temporarily increases output per capita. However, the economy starts decumulating capital, so that in the long run, the households are equally well off as before. The ‘ H policy’ leads to capital decumulation without a temporary increase in output per capita. The long-run steady state is lower than the initial steady state, which makes households worse off. Of course, we neglected many aspects of these policies that would be rele- vant for the evaluation of the overall effect. Deportations would be costly, the country may be using specific types of workers which are hard to replace by existing workforce, etc. The proceeds from capital taxation are typically not destroyed but the government can use them to finance government expendi- tures. They also may have appealing redistributional effects. These are just some examples of some other effects that one should consider. References Krugman, Paul. 1994. “The Myth of Asia’s Miracle.” Foreign Affairs 73 (6):62–78. 13