Homework #1 1. In 1900 GDP per capita in Japan (measured in 2000 dollars) was $1,433. In 2000 it was $26,375. (a) Calculate the growth rate of income per capita in Japan over this century. (b) Now suppose that Japan grows at the same rate for 21st century. What will Japanese GDP per capita be in the year 2100? 2. In 2013 GDP per capita in the United States was $53,001 while GDP per capita in Bangladesh was $3,167. Income per capita in the United States has been growing at a constant rate of 1.9% per year. Calculate the year in which income per capita in the United States was equal to year 2013 income per capita in Bangladesh. 3. Relation between productivity (output per worker) and income per capita. Call L the number of workers (labour force), F the number of people of working age, and N total population. We usually assume no unemployment in the long run. Productivity, y; equals output per worker (y = Y=L) while income per capita, let us call it i; equals output per person (i = Y=N): The labour force participation ratio, l, is the proportion of people in the labour force to people of working age (l = L=F ) while the dependency ratio, d; is the proportion of total population to population of working age (d = N=F ). Therefore, Y N = Y L F L F N i = y l d (a) What is the relation between an increase in productivity and an increase in income per capita? (b) Suppose that population growth in Ameropa decreases from and, as a consequence, productivity increases by 3%. The population growth does not a ect the labour force participation ratio but it a ects the dependency ratio. Suppose that the dependency ratio decreases by 8% because there are fewer children. Calculate the total e ect on income per capita of the demographic change; i.e., calculate the combined e ect of the increase in productivity and of the change in the dependency ratio. 1
(c) Suppose again that population growth decreases and, as a consequence, productivity increases by 3%. However, suppose instead that the population becomes older and the dependency ratio increases by 6%. What is in this case the total e ect on income per capita of the demographic change: i.e., the combined e ect of the increase in productivity and of the change in the dependency ratio? (d) Suppose now that in case c. the government, worried about the aging of the population, encourages women to work outside the home with the e ect that the labour force participation ratio increases by 4%. What is the total e ect on the measured income per capita of the demographic change plus the government action? 4. The growth rate of aggregate productivity (sectoral composition). Suppose that the economy is composed of two sectors, agriculture and manufacture, whose production functions are, respectively, and Y A = B A L A Y M = B M L M where L A and L M (L A + L M = L) are the quantities of labour used in each sector and B A and B M the productivities of each sector. The total quantity of output is Y = Y A + Y M. The aggregate level of productivity in the economy is de ned as B = Y L = B AL A L + B M L M : L Show that the growth rate of aggregate productivity is a weighted average of the growth rates of the productivity in the two sectors where the weights are the sectors share of total output. (Hint: rst, calculate B _ keeping quantities of labour constant; second, divide by B to obtain B=B; _ nally, multiply and divide each term by the productivity in each sector.) 2
Homework #2 1. Consider the Solow model without technical progress. In class we derived the steady-state values for capital per worker (k ) and output per worker (y ) as a function of saving rate (s), population rate (n), and depreciation rate (d): (a) Suppose that s = 20%; n = 1% and d = 10%: Calculate k ; y and consumption per worker (c ). (b) Repeat the exercise for s = 20%; n = 4% and d = 10%. What do you observe as compared to your answer in part (a)? Comment. (c) Now suppose that saving rate raises permanently from 20% to 30% (n = 1% and d = 10%). How does this increase in the saving rate a ect k ; y and c? (Again, compare to your answer in part (a).) 2. Output per worker (labour productivity) is de ned as y = Y =L: Show that the growth rate in labour productivity depends on growth in total factor productivity and growth in the capital-labour ratio (k = K=L). In particular, show that _y y = B _ B + k _ k : 3. Suppose an economy described by the Solow model is in a steady state with population growth n of 1.1 percent per year and technological progress g of 1.5 percent per year. Total capital and total output grow at 2.6 percent per year. Suppose further that the capital share of output is 0.4. If you used the growth accounting equation to divide output growth into three sources capital, labour, and total factor productivity, how much would you attribute to each source? 4. In the economy of Solovia, the owners of capital receive 70% of national income, and the workers receive 30%. (a) The men of Solovia stay at home performing household chores, while the women work in factories. If some of the men decided to start working outside of the home so that the labour force increases by 4 percent, what would happen to the measured output of the economy? (Hint: use the growth accounting equation). (b) In year 1, the capital stock was 6, the labour input was 3, and output was 12. In year 2, the capital stock was 7, the labour input was 4, and output was 14. What happened to total factor productivity growth between the two years? 3
Homework #3 1. We have seen in class Kaldor s stylized facts of growth in developed countries (a) The factor distribution of income shows no trend (b) GDP per capita exhibits steady and sustained growth (c) The ratio of capital to output shows no trend (d) The real rate of return to capital shows no trend (e) Wages exhibit sustained growth The Cobb-Douglas production function is used to replicate fact a. In this exercise, you are asked to show that the steady state in the Solow model with technological progress replicates facts b, c, d, and e. (Hints: start with the equations in p. 40; and remember the factor distribution of income.) 2. Problem 1 of chapter 3 (page 75) 3. Unemployment and growth. Consider how unemployment would affect the Solow growth model. Suppose that output is produced according to the production function Y = K [(1 u )L] 1, where u is the natural rate of unemployment. There is no technological progress. Assume again that the labour force equals population. (a) Express output per worker, y; as a function of capital per worker, k, and the natural rate of unemployment. Describe the steady state of this economy. (b) Suppose that some change in government policy reduces the natural rate of unemployment. Describe how this change a ects output both immediately and over time. Is the steady-state e ect on output larger or smaller than the immediate e ect? Explain (Your answer should include a graph). 4. Problem 6 in chapter 3 (page 77) 4
Homework #4 1. Consider the Solow model with population growth but no technical progress. Assume that the savings rate can take two di erent values s 1 and s 2 ; where s 1 > s 2. The savings rate depends on the level of output per capita (and therefore the level of capital per capita). Speci cally, the savings rate equals s 2 when output per capita is below a certain threshold, y < y; (when k < k) and increases to s 1 when output per capita is at or above this threshold, y y (when k k). Draw a diagram for this model. Assume that (n + d) k < s 1 y and that (n + d) k > s 2 y: Explain what the diagram says about the steady state of this model. 2. Consider the model of technology and growth presented in Chapter 5. Suppose that L = 1; = 0:2; = 1; = 1 and s R = 0:1: (a) Calculate the growth rate of output per worker. (b) Now suppose that s R is trebled. Calculate the new growth rate of output per worker. How many years will it take before output per worker returns to the level it would have reached if s R had remained constant? 3. Consider the original Lucas endogenous growth model where the production function is given by: Y = K (hl) 1 where h is average human capital (human capital per worker). As you can see h ful lls here the same role as A does in the original Solow model. Therefore, the growth rate of the economy (output per worker) in the balanced growth path equals the growth rate of the (average) human capital. Lucas assumes that human capital evolves according to _h = uh where u is the (average) fraction of our lifetime spent accumulating human capital. (a) What is the growth rate of (average) human capital? What is the growth rate of output per worker? (b) We have not studied this model as it is in class because empirical evidence rejects this model (we have studied a variation of it). Taking into consideration your answer in part a. and your knowledge about di erent growth rates across countries, explain why empirical evidence rejects this model as stated. 5
4. Population Growth and Technological Change since 1 million B.C. In his 1993 paper, Kremer introduces a couple of variations in the Solow model to replicate the time series of average annual population growth rate that he constructs and it is shown in page 92 of your textbook. The rst variation is to assume that output is a function of the xed stock of land and labour, rather than capital and labour. The second is to assume that population adjusts so output per person equals the subsistence level y: According to Echevarria (1998), the share of labour if the rst sector is 0.41. Normalize y to be equal to 1 unit and assume the quantity of land to equal 370 million units. (a) Solve analytically (mathematically) for the level of global population. (b) Suppose that there were no TFP growth in the rst sector (normalize TFP or labour e ciency to one). What would the steady state level of global population (the level around which population would have uctuated) have been given these parameters? (c) If TFP growth is positive, what is the relation between population growth and rate of technical progress in the rst sector, according to your answer to a. (d) According to Echevarria (1998), the rate of TFP growth in the rst sector is 0:0035 = 0:35%. What was the average growth rate of global population until these Malthusian/Darwinian population dynamics broke down? 6
Homework #5 1. Consider an extension of the Solow model without technological progress that encompasses a second type of capital, government capital, which consists of publicly funded infrastructure such as roads and ports. Let x denote the quantity of government capital per worker. The economy s production function (in per-worker terms) equals y = k x : The government collects a fraction of national income in taxes and spends all of this revenue producing government capital. The laws of motion" for government and physical capital are thus _x = y x _k = s(1 )y k: (a) Solve for the steady state level of output per worker (b) What value of will maximize output per worker in the steady state? 2. This problem, similar to Problem 5 in chapter 6, considers the e ect on an economy s technological sophistication of an increase in the investment in human capital. It looks at the short-run and long-run e ects on h of an increase in u (fraction of an average individuals s working lifetime spent learning skills). As in the previous problem, assume = 1: (a) Starting from the steady state, analyze the short-run and the longrun e ects of an increase in u on the growth rate of h. (b) Plot the behaviour of h=a over time. (c) Plot the behaviour of h(t) over time. (d) How does the model explain the di erences in growth rates that we observe across countries? 3. The drag of a xed resource. Let us assume that a xed amount of land, N, is available for production each period and output is produced according to Y = BK N L 1 : Nordhaus (1992) calculates land s share of output to be 10%. According to national accounts, gross capital income is about 30% of GDP in industrialized countries. Both productivity (output per worker) and labour have grown roughly at 2% during the last century in these countries. It is one of the Kaldor s facts that capital and output grow at the same rate in industrialized countries. 7
(a) Compute TFP (total factor productivity) growth in industrialized countries for the last century. (b) Now use the usual Cobb-Douglas production function with just two factors, capital and labour, to perform the same exercise. Is your measure of TFP growth in this case higher or lower than in case a? Are we underestimating or overestimating TFP growth by not taking into account the e ects on growth of a xed resource? By how much according to your calculations? 4. A technology transfer model. According to the model in chapter 6, income per capita equals y = k h 1 ; where h refers to human capital per person. New technologies can only be adopted if the average worker has enough human capital to use them; thus, as a working assumption, we equate technological level to human capital and, therefore, h refers to the technological level as well. New technologies are developed in the frontier: A refers to both the labour e ciency and the technological levels at the frontier. Improvements in the country s technology, h, on the other hand, are made by learning from the frontier s technology: _h h = e u A : h (a) According to the same model, the equation for the technological level, h, is as follows: 1= h = g e u A; where refers to the degree of openness of the economy and is the parameter that determines transfer of knowledge, depending on the distance to the frontier. Use the above equations for income per capita and the technological level to generate an expression of growth in income per capita as a function of growth of technical progress at the frontier, growth of capital per capita in this country, and change in average years of schooling. (Assume and g to be constant.) (b) Two Spanish researchers, de la Fuente and Domenech (2002), regress growth in per capita income versus growth in capital per capita and change in average years of schooling. They nd the coe cient for change in average years of schooling to be 0.5. Use this value and your answer to part a of this problem to nd an estimate of : (c) Suppose that the distance to the frontier decreases by 2%. What happens to the rate of technological progress in this country? (Hint: remember that the exponent in an exponential function is close to the elasticity). 8
5. A renewable resource. The natural growth of sh in a lake, G = S (100 S) ; 100 where S is the stock of sh; i.e., natural growth is a hump-shaped function of the stock, as with many renewable resources. The change in stock _S = G H where H refers to the quantity of sh harvested. Thus, to keep the stock stable, H should equal G and this harvest then constitutes a sustainable yield. (a) Calculate the maximum sustainable yield. (b) The stock that yields this maximum is called the optimal stock. What is the optimal stock of sh in this lake? 9