Understanding the world economy Sample quantitative exercise

Transcription

1 SciencesPo Master E&B Coeurdacier Nicolas Understanding the world economy Sample quantitative exercise Steady-state income per capita and the role of savings More difficult for those willing to practice more quantitative models. Parts 1 and are still fairly close to the course. Part 3 is more difficult. Output (Y) is produced in an economy using physical capital (K) and labour input (L) according to the following Cobb-Douglas production function 1 : Y =AK 0.4 L 0.6 where A is total factor productivity (TFP). For simplicity, we consider variables per capita; then, Y denotes output per capita, K denotes physical capital per capita and L=1. We also assume to simplify calculations thata=1. The output per capita is then: Y =K 0.4 Assume that saving which equals investment is equal tos% of output (Y). The depreciation rate of capital is5% per period. 1) Explain why this economy converges towards a steady-state without growth. Show that in the steady-state, we must have: sy =0.05K where (*) denotes the variable at the steady-state value. ) Deduce from 1) that the steady-state capital stock and output per capita must verify: K =( s 0.05 ) ; Y =( s 0.05 ) 3 3) Use excel to calculate Y for different values of s [choose at least 5 values between 0 and 1 (excluding 0)]. Plot on a graph the logarithm of Y (log(y )) as a function of the logarithm of s (log(s)) [x-axis: log(s); y-axis: log(y )]. Comment. What is the slope of the straight line you obtain? Part : A simple test of the Solow model We propose a simple test of the theory by looking at the relationship between the log of income per capita and the log of savings across countries. We plot on a graph the log of income per capita as a function of the log of savings [x-axis:log(s); y-axis: log(y)] (see figure below). 1 This production function is consistent with a share of capital income in value added of 40%. 1

2 1 ln Y y = 1.568x R = ln s s in % 4)What is the slope obtained in the data? Compare thevalueof the slope obtained with realdata and the one predicted by the model (question 3). How would you explain the difference? Part 3: A Solow model augmented with Human capital We want to improve the theory by considering human capital (H) per capita as an input of production (H denotes the education and the skills of a worker...). Output (Y) is produced using physical capital (K) and human capital (H) according to the following production function: Y =AK 0.4 H 0.6 Like in the first part, we consider variables per capita; Y denotes output per capita, K physical capital per capita and H human capital per capita. We assume that when physical in the economyis higher, so is the levelof human capitalsuch that: H=λK b whereλandbare positive numbers. For instance, this happens because everytime money is spent on acquiring new machines it enhances the skills of the workers that use them.

3 5) Explain why with b = 1, allowing for the interaction between physical and human capital yields a constant marginal product of capital schedule. Does the economy converge towards a steady-state whenb=1? We now assumeb=1/3. To simplify algebra, we also assumea=λ=1. 6)Showthatthisimpliesthefollowingrelationship: Y =K 0.6. Doestheeconomyconvergetowards a steady-state? 7) Following the same steps as in questions 1), ) and 3), compute the steady-state capital stock (K ) and the steady-state output per capita (Y ) as a function of the savings rate and use excel to calculatey for different values ofs. Plot on a graph the logarithm ofy (log(y )) as a function of the logarithm ofs(log(s)) [x-axis: log(s); y-axis: log(y )]. What is the slope of the straight line you obtain? Compare with the data. Comment briefly. 3

4 Correction Part 1: The standard Solow model Output (Y) is produced in an economy using physical capital (K) and labour input (L) according to the following Cobb-Douglas production function : Y =AK 0.4 L 0.6 where A is total factor productivity (TFP). For simplicity, we consider variables per capita; then, Y denotes output per capita, K denotes physical capital per capita and L=1. We also assume to simplify calculations thata=1. The output per capita is then: Y =K 0.4 Assume that saving which equals investment is equal tos% of output (Y). The depreciation rate of capital is5% per period. 1) Explain why this economy converges towards a steady-state without growth. Show that in the steady-state, we must have: sy =0.05K where (*) denotes the variable at the steady-state value. The model converges towards a steady-state because of diminishing marginal productivity of capital (MPK): dy dk =0.4K 0.6 is decreasing with the stock of K. The capital accumulation equation betweentandt+1 tells us: In the steady-statek =K t+1 =K t, then: K t+1 =K t +I t 0.05K t I =0.05K where (*) denotes the variable at the steady-state value. Aggregate investment must equal aggregate savings, soi =sy sy =0.05K ) Deduce from 1) that the steady-state capital stock and output per capita must verify: sy =0.05K gives: K =( s 0.05 ) ; Y =( s 0.05 ) 3 s(k ) 0.4 = 0.05K (K ) 0.6 s = 0.05 K = ( s 0.05 ) This production function is consistent with a share of capital income in value added of 40%. 4

5 Using the production function givesy. Y =(K ) 0.4 =( s 0.05 ) 3 3) Use excel to calculate Y for different values of s [choose at least 5 values between 0 and 1 (excluding 0)]. Plot on a graph the logarithm of Y (log(y )) as a function of the logarithm of s (log(s)) [x-axis: log(s); y-axis: log(y )]. Comment. What is the slope of the straight line you obtain? ln Y.5 y = x ln s s in % Obviously taking log of the previous expression gives: ln(y )= 3 ln(s) 3 ln(0.05) So the slope on the graph must be /3. In the steady-state, the only difference in income per capita is driven by differences in savings: higher savings increases the steady-state capital stock, which increases income per capita. Part : A simple test of the Solow model We propose a simple test of the theory by looking at the relationship between the log of income per capita and the log of savings across countries. We plot on a graph the log of income per capita as a function of the log of savings [x-axis:log(s); y-axis: log(y)] (see figure below). 5

6 1 ln Y y = 1.568x R = ln s s in % 4)What is the slope obtained in the data? Compare thevalueof the slope obtained with realdata and the one predicted by the model (question 3). How would you explain the difference? Slope = The slope is higher than the one predicted by the model. GDP per capita and savings are positively related as predicted by the model. Obviously the correlation is not perfect like in the model because many other factors than the savings rate affect income per capita (although differences in savings still explain a large part of the variability in income per capita, R close to 50%). The slopeis toohigh compared towhatthe simplesolowmodelpredicts (1.5versus /3). In other words in the data the impact of savings on income per capita is larger than what is predicted by the model. This simple model does not account for many variables that are both correlated with the savings rate and income per capita. For instance, in countries where TFP is high, the savings rate might be higher (because of higher return on capital) and so is the income per capita. Another potential missing variable is human capital: when savings are high, so is capital. Higher capital stock raises the return on education and increases human capital. As a consequence income per capita is even higher and the impact of savings is reinforced. Development of financial markets can be an another obvious candidate (increases both savings and income per capita). 6

7 Part 3: A Solow model augmented with Human capital We want to improve the theory by considering human capital (H) per capita as an input of production (H denotes the education and the skills of a worker...). Output (Y) is produced using physical capital (K) and human capital (H) according to the following production function: Y =AK 0.4 H 0.6 Like in the first part, we consider variables per capita; Y denotes output per capita, K physical capital per capita and H human capital per capita. We assume that when physical in the economyis higher, so is the levelof human capitalsuch that: H=λK b whereλandbare positive numbers. For instance, this happens because everytime money is spent on acquiring new machines it enhances the skills of the workers that use them. 5) Explain why with b = 1, allowing for the interaction between physical and human capital yields a constant marginal product of capital schedule. Does the economy converge towards a steady-state whenb=1? Withb=1, we have: Y =AK 0.4 H 0.6 =AK 0.4 (λk) 0.6 =Aλ 0.6 K Then : dy dk =Aλ0.6 =cst. We have a constant marginal productivity of capital. The interaction beween physical and human capital generates constant return to capital (even if each factor exhibits decreasing MPK). This prevents convergence towards a steady-state. To see this, rewrite the capital accumulation equation: K t+1 K t =I t 0.05K t =sy t 0.05K t =saλ 0.6 K t 0.05K t or rewriting in terms of the growth rate ofk (=g K ): g K = K t+1 K t K t =saλ AssumingsAλ >0,thecapitalstockgrowswithoutboundataconstant(positive)growth rate: a case of endogenous growth. We now assumeb=1/3. To simplify algebra, we also assumea=λ=1. 6)Showthatthisimpliesthefollowingrelationship: Y =K 0.6. Doestheeconomyconvergetowards a steady-state? b=1/3 anda=λ=1.implies: Y =AK 0.4 H 0.6 =K 0.4 K 1/3 0.6 =K 0.6 Yes convergence towards a steady-state because decreasing MPK! 7

8 Saving which equals investment is still equal to s% of output. The depreciation rate of capital is 5% per period. 7) Following the same steps as in questions 1), ) and 3), compute the steady-state capital stock (K ) and the steady-state output per capita (Y ) as a function of the savings rate and use excel to calculatey for different values ofs. Plot on a graph the logarithm ofy (log(y )) as a function of the logarithm ofs(log(s)) [x-axis: log(s); y-axis: log(y )]. What is the slope of the straight line you obtain? Compare with the data. Comment briefly. In the steady-state: And: Then taking log-, we have: sy = 0.05K ands(k ) 0.6 =0.05K (K ) 0.4 s = 0.05 or: K =( s 0.05 ) Y =( s 0.05 ) =( s 0.05 )1.5 ln(y )=1.5[ln(s) ln(0.05)] The slope now must be equal to 1.5! Allowing for a broader definition of capital (human and physical capital) makes the model more consistent with the data (in the data we found 1.56). The reason is the following: the interaction between human and physical capital reinforces the impact of savings on income per capita; higher savings, implies higher steady-state physical capital stock which increases income per capita (standard effect of the Solow model). But on the top of that, higher capital stock raises human capital which in turns raises even more income per capita. 8

9 ln Y 9 8 y = 1.5x ln s s in % 9