Department of Economics Shanghai University of Finance and Economics Intermediate Macroeconomics

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1 Department of Economics Shanghai University of Finance and Economics Intermediate Macroeconomics Instructor: Min Zhang Answer 2. List the stylized facts about economic growth. What is relevant for the world as a whole? And what is relevant for riches countries? Answer: Stylized facts are as follows, Fact Large variations in per capita income across countries and rise in Income Inequality. Fact 2 Large variations in growth rates of income across countries. Fact 3 A country s relative position in the world distribution of per capita incomes may experience large changes over time. Countries can move from "poor" to "rich", and vice versa. Fact 4 Kaldor facts (balanced growth) in most developed countries like the Unites States: the real rate of return to capital shows no trend upward or downward; the share of income devoted to capital and labor show no trend; the average growth rate of output per person has been positive and relatively constant over time. Fact 5 What s more, the growth rate of real GDP per person also declined, i.e. from 3.% per year for to.8% per year for This is sometimes called the productivity slowdown. The first three facts are relevant for the world as a whole and the Kaldor facts and the productivity slowdown are relevant for riches countries. 2. Verify if the following production functions have Constant Returns to Scale property? (K and N represent capital and labor, respectively. is a constant. ) Y = 5(KN) 3. Y = Y = K + 3(KN) 2 [0.5K + 0.5N ] Answer:Y = 5(KN) 3 is not constant return to scale. Both Y = K + [ ] 3(KN) 2 and Y = 0.5K + 0.5N are constant return to scale. Suppose α is a constant, we do fllowing calculations to show whether they are CRS.

2 Y = 5(KN) 3 is not constant return to scale. 5(αK αn) 3 = 5α 2 3 (KN) 3 5α(KN) 3 = αy. Y = K + 3(KN) 2 is constant return to scale. Y = [ 0.5K αk + 3(αKαN) 2 = αk + 3α(KN) 2 = αy N ] is constant return to scale. [0.5αK ] + 0.5αN = α [0.5K ] + 0.5N = αy. 3. Derive the growth accounting equation (3.4) by using the production function which is Cobb-Douglas in K and L with the capital share as Alpha. Answer: Since Y = AK α L α Take total derivative with respect to Y, we get following equation, Therefore, Y = K α L α A + αak α L α K + ( α)ak α L α L Y Y = Kα L α AK α L α A + αakα L α AK α L α Y Y = A A + α K K + ( α) L L Now we get the accounting equation 3.4, K + ( α)akα L α AK α L α L Y Y = A A + α K K + ( α) L L 4. What is MPK or MPL? What is the main property of MPK? Is the average product of capital (Y/K or y/k) diminishing with an increase in K or k? Why? Show it in a figure in (y,k) space. Ans: MPK is the marginal product of capital, which means the additional output produced from additional capital input with labour constant. Similarly, MPL is the marginal product of labour. Sure. The average product of capital (Y/K or y/k) does diminish with an increase in K or k. The reason lies in the diminishing MPK. Additional input brings less output as input increases, which will definitely lower the average output of the input. This is shown in the following figure. 5. Given the production function used in Chapter 3, derive the growth rate of capital per worker and growth rate of real GDP per worker. Show the growth rate of capital per worker in the Transition Dynamics. 2

3 Answer: Firstly we derive the growth rate of capital per worker and growth rate of real GDP per worker. We know that the change in capital stock = real saving, i.e. K = s(y δk),divide both sides by K, we get K K = s Y K δ.since k/k = K/K L/L,we have the growth rate of capital per worker k/k = s(y/k) sδ n = s(y/k) sδ n. As long as we get the growth rate of capital per worker, we could get the growth rate of real GDP per worker y/y = α k/k = α[s(y/k) sδ n]. Our analysis allows us to think of the process of economic growth in the Solow model as having two phases. In the first phase, there is a transition from an initial capital per worker to its steady-state value, k*. During this transition, the growth rate, K/K = k/k, is greater than zero but declines gradually toward zero. The growth rate of capital per worker is k/k = s(y/k) sδ n. Since K is rising, net investment is greater than zero, that is, gross investment, I, exceeds depreciation, δk. In the second phase, the economy is in (or close to) the steady state. In this phase, K/K = k/k = 0, net investment is zero, and gross investment, I*, just covers depreciation, δk*. 6. Explain in words the impact of an increase in Delta on the growth rate on transitional path and on the steady state value of k. Show the effect in the Transition Dynamics. Answer: An increase in Delta will decrease the growth rate on the transitional path and decrease the steady state value of k. This can be shown in the transition dynamics. The growth rate of capital per worker on the transitional path is k/k = s(y/k) sδ n = s(y/k) sδ n, 3

4 a larger δ will bring a smaller k/k.the steady state of capital per worker is s(y /k ) = sδ + n,a larger Delta needs a lager y /k. Because of diminishing average capital product, this means a lower steady state of capital per worker. 7. What are the two determinants of economic growth? Answer: The two determinants emphasized are the initial value of the capital per worker and the steady state value of capital per worker. Since the growth rate of capital per worker is k/k = s(y/k) sδ n, k/k is the distance between s(y/k) and sδ + n. s(y/k) is determined by the initial value of the capital per worker, and the steady state value of capital per worker is determined by s(y /k ) = sδ + n. Thus the larger the distance between initial and steady state, the bigger the growth rate will be. 8. Is the statement right that poor countries with low capital stock would catch up with rich ones? Explain your answer. Answer: The statement is wrong. Although the Solow model says that a poor economy with low capital and real GDP per worker grows faster than a rich one because of the diminishing average product of capital, y/k. This conclusion depends on the assumption that the economies have the same steady state or the determinants of the steadystate capital per worker, k*, were the same for all economies, which is reasonable for similar economies but is less reasonable when applied to a broad sample of countries having sharply different economic, political, and social characteristics. In particular, the assumption is unreasonable for the worldwide group of countries considered. Therefore, poor countries with low capital stock may not catch up with rich ones because of the gap between their steady state. 9. Is the prediction from the Solow model regarding convergence consistent with what is observed in reality? Explain with examples. Answer: The prediction only partly consistent with data. From the data, we learned that similar economies tend to converge, whereas dissimilar economies display no relationship between the level of real GDP per person and the growth rate. Thus, the convergence pattern is strongest for regions of advanced countries (the convergence between the states of the United States), next strongest among the group of rich countries (the convergence between OECD countries), and weakest in fact, absent for the full worldwide sample of countries which including both rich countries and poor countries. 0. Suppose poor countries and rich ones differ in their saving rate, explain in words why poor countries with low capital stock cannot catch up with rich countries. Show your reasoning in the Transition Dynamics. Answer: Suppose an economy with lower initial level of capital per worker also has a lower saving rate than a rich economy called economy 2. This means economy 2 has a higher level of steady state. In this case, it is uncertain which economy grows faster initially because it is uncertain which of the distance between s(y/k) and sδ +n is larger. The two countries converge to their respective 4

5 steady state, and the poor country with low capital stock cannot catch up with rich countries because of its lower steady state.. Derive the value of growth rate of capital per worker in the Solow model with exogenous technological progress. Answer: Suppose technology grows following A/A = g, Y/Y = A/A + α( K/K) + ( α)( L/L) Y/Y = A/A + α( K/K) + ( α)( L/L) = g + α( K/K) + ( α)n y/y = g + α( k/k) at the steady state, the growth rate of capital per worker k/k = s(y/k) sδ n is a constant, therefore y/k is constant y/y = k/k = g/( α) Exogenous technological progress at the rate A/A = g leads to long-term growth in real GDP and capital per worker, k and y, at the rate g/( α). 5

6 2. Derive the value of growth rate of real GDP per worker in the Solow model with endogenous technological progress. Answer: Suppose that labor force is the single production factor and the production function has the form of Generalized Cobb-Douglas function without capital Y (t) = A(t)( α L )L(t), where α L is exogenous and dl(t)/dt = n. Also suppose the evolution function of knowledge is thus da(t)/dt = B[α L L(t)] r A(t) θ, B > 0, r 0, g A = B[α L L(t)] r A(t) θ, since at the steady state g A becomes a constant, we have which turn out to be dg A /dt = rn + (θ )g A = 0, g A = rn θ. the growth rate of real GDP per worker in the Solow model with endogenous technological progress is If r =, θ = 0, we have g y = g A = rn θ. da(t)/dt = Bα L L(t). Since n = 0, L(t) = L, A = da(t)/dt = Bα L L(t) is a constant. Over time, the stock of knowledge increases continuously. However, the increase in A, i.e. A does not change and becomes a smaller and smaller fraction of the total stock A. Therefore, g A = Ȧ A decrease over time and converges to zero although it is always greater than zero. 6