Problem Set 2: Answer Key

Transcription

1 ECO Economic Development: Macro Persp. Thomas Osang Problem Set 2: Answer Key Part I: 1. Steady-State solutions for k and y : k = ( sa n ) 1 1 a y = A How to derive the above results: 1 1 a First, with k = 0 in the SSE, we find ( s n ) a 1 a sak a = nk Next, dividing both sides of the above equation by k, s and A yields: k a 1 = n sa This can be rewritten as: k 1 a = sa n Getting rid off the exponent on the left hand side of the above equation yields: k = ( sa n ) 1 1 a Plugging the above solution for k into y = Ak a yields: y = A[( sa n ) 1 1 a ] a Simplifying the right hand side of the above expression produces the solution for y. 2. Relationship between k and A, s and n: k A = 1 a 1 a (sa n ) 1 a ( s ) > 0, all three product terms are positive n k s = 1 a 1 a (sa n ) 1 a ( A ) > 0, all three product terms are positive n k n = 1 a 1 a (sa n ) 1 a ( sa n 2 ) < 0, first 2 terms are positive, last term is negative

2 Part II: To illustrate the effect of technological progress, let us assume the following functional form of the economy s production function: Y = AK a L 1 a where A is a constant that represents the level of technology (or TFP). Written in intensive form, this function becomes y = Ak a The fundamental equation of the Solow model is then given by: dk = sak a nk The graph below shows how the initial steady state equilibrium value, k 0, is being determined by the intersection of the sa 0k a curve and the nk line, with the level of technology (or TFP) equal to A 0. nk, sy Figure 1: nk sa 1 k α sa 0 k α k0 k1 k When TFP improves from A 0 to A 1 (with A 1 > A 0, the entire savings per worker curve shifts upward to sa 1 k a. 2

3 Short term effects: During the transition process from the old to the new SSE the per capita variables (k and y) grow at a positive rate, and the aggregate variables (Y, K, I, S) grow at a rate higher than n, the rate of population growth. The ratio of capital to workers, k, rises continuously until it reaches its new equilibrium level, k 1. Long-term effects: Once the new SSE is reached, the per capita variables, k and y, cease to grow, and the aggregate variables (Y, K, I, S) will grow at rate n. In the new steady state equilibrium, output per capita, y, is higher than in the old equilibrium for two reasons: First, because A 1 > A 0 ; and second, because k 1 > k 0. Part III: Based on the Solow model, what are the implications of the following policy changes on aggregate as well as per capita variables both in the short and in the long run? 1. The government introduces an incentive program that permanently reduces the number of children per family. The smaller family size is equivalent to a reduction in the population growth rate, n (from n 0 to n 1 with n 1 < n 0 ). This means, as shown in Figure 2 below, that the nk line shifts to the right. At k0, the original steady-state level of the capital-worker ratio, there is now an excess of savings/investments over the amount of investment needed to endow the new worker with the level of capital used by each of the current workers (i.e., n 1 k0 < sf(k 0 )). As a result, dk > 0 which implies that next period s capital stock (per worker) will be higher than this period s one (i.e., k t+1 > k t ). This process continues until k reaches its new steady state level at k1. The economic implications of the decline in n are as follows. In the short run (i.e. during the transition period) the per capita variables (k and y) grow at a positive rate; the aggregate variables (K, Y, S, I, etc) grow at a positive rate greater than n. In the long run (i.e., the economy is in its new steady state equilibrium) the per capita variables remain constant at levels k 1 and y 1, respectively; the aggregate variables grow at rate n. 3

4 Figure 2: nk, sy n 1 k sf(k) k 0 n 0 k k 1 k 4

5 2. The government removes tax breaks on retirement savings plans. The removal of the tax incentives for retirement saving plans is going to reduce the overall savings rate, s, (from s 0 to s 1 with s 1 < s 0 ). This means, as shown in Figure 3 below, that the sf(k) curve shifts downward. At k0, the original steady-state level of the capital-worker ratio, there is now a shortfall of current savings/investments over the amount of investment needed to endow the new workers with the level of capital used by each of the current workers (i.e., nk0 > s 1f(k0 ). As a result, dk < 0 which implies that next period s capital stock (per worker) will be lower than this period s one (i.e., k t+1 < k t ). This process continues until k reaches its new steady state level at at k1. The economic implications of the decline in s are as follows. In the short run (i.e. during the transition period) the per capita variables (k and y) grow at a negative rate (i.e., they decline); the aggregate variables (K, Y, S, I, etc) grow at a positive rate less than n. In the long run (i.e., the economy is in its new steady state equilibrium) the per capita variables remain constant at levels k 1 and y 1, respectively; the aggregate variables grow at rate n. 5

6 Figure 3: nk, sy nk s 0 f(k) s 1 f(k) k1 k0 k 6

7 Part IV: Sketch the graph of the Solow model with variable factor productivity. 1. Show that the Solow model with variable factor productivity can give rise to three equilibria two of which are stable. Identify the stable equilibria as well as the low-level equilibrium trap. As shown in Figure 4 below, there can be three steady state equilibria in the Solow model with variable factor productivity, Both k I 0 and k III 0 are stable steady-state equilibria, while k II 0 is an unstable equilibrium. Figure 4: nk, sy nk s f(k) k k I 0 kii 0 kiii 0 7

8 2. Assume that the economy begins initially in the high-level equilibrium. Now illustrate the effects of medical assistance from the developed countries that lowers the crude death rate. In particular, show that it is possible that because of the benign actions of the developed countries the economy could end up in a low-level equilibrium trap. Discuss the transition from the old to the new steady state equilibrium. Since the economy is originally in the high-level equilibrium, its initial capital stock per worker is kiii 0. The decline in the CDR leads to an increase in the population growth rate n (from n 0 to n 1 with n 1 > n 0 ). As shown in Figure 5 below, a large enough increase in the country s population growth rate may lead to the disappearance of two of the equilibria shown in Figure 4, with the new, unique equilibrium being determined by the single crossing of the n 1 k line with the sf(k) curve. Since the new steady-state equilibrium at ki 1 is to the left of ki 0, it can be characterized as a low-level equilibrium trap. The transition from the old to the new equilibrium is straightforward. At k III 0, the original steady-state level of the capital-worker ratio, there is now a shortfall of savings/investments vis-a-vis the amount of investment needed to endow the new workers with the level of capital used by each of the current workers (i.e., n 1 k III 0 > sf(k III 0 )). As a result, dk < 0 which implies that next period s capital stock (per worker) will be less than this period s one (i.e., k t+1 < k t ). This process continues until k reaches its new steady state level at at k I 1. 8

9 Figure 5: n 1 k nk,sy k I 1 k I 0 k II 0 k III 0 n 0 k s f(k) k 9

10 Part V: Identify and discuss the key insights as well as the main limitations of each of the following models: 1. Solow s model with variable factor productivity Key insights: Like the basic Solow model, the Solow model with variable factor productivity explains an economy s growth process through capital accumulation. Unlike the basic model which features a unique steady-state equilibrium, the model with variable factor productivity can have three equilibria (two stable, one unstable). One of the stable equilibria has the characteristics of a low-level equilibrium trap. In contrast to the basic model, initial conditions (the initial amount of capital per worker available in the economy) matter for the long-term levels of the per capita variables (k and y). In particular, if k 0 is to the left of kii 0, the economy will end up in the low-level equilibrium. If k 0 > kii 0, the economy will eventually reach the high-level equilibrium. Main limitations: Like the basic Solow model, the Solow model with variable factor productivity ignores important elements of economic growth and development such as human capital formation and R&D expenditures by firms. As a result of these shortcomings, it cannot explain long-run growth in per capita variables and limits the long-run growth rate of all aggregate variables to the population growth rate. In addition, it abstracts from market imperfections such as unemployment. 2. Lewis two-sector model Key insights: The model explains the transformation from an essentially agrarian society with low growth and savings rates to an economy with two sectors (one traditional, the other modern) with high growth and savings rates. One crucial insight of the model is that surplus labor of the T-sector can be used to feed a self-sustaining growth process in the M-sector. The other essential insight of the model is that the government may have to play a crucial role during the early stage of economic development by providing financial assistance to firms in the M-sector. Lewis suggests that the government should raise its revenues by imposing an indirect tax on farmers in the T-sector. Main limitations: Like the Solow model, the Lewis model ignores the importance of human capital formation and R&D expenditures for economic growth. The model also does not recognize the importance of technological change and growth in the T-sector as an important element of economic development. In addition, Lewis idea of an indirect tax on farmers as the main source of government revenues is likely to lower T-sector output and raises the possibility that the country becomes dependent on food imports at some point. 10