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Intermediate Macroeconomics Lecture 5 - Endogenous growth models Zsófia L. Bárány Sciences Po 2014 February

Recap: Why go beyond the Solow model? we looked at the Solow model with technological progress and found that it matches the Kaldor facts well we looked at why economists moved beyond the Solow model: capital does not move from rich to poor countries growth accounting technological improvements contribute significantly to growth development accounting there are large differences in the level of technology across countries this week we look at endogenous growth models 1. learning by doing 2. human capital 3. research and development and look at international technology transfer

Learning by doing

Learning by doing Based on Romer (1989). Main idea: skills or knowledge are accumulated during the production the skills or knowledge accumulation is free and is a by-product of production the marginal product of capital diminishes at the firm level BUT when a firm invests, other firms learn from its experience too, i.e. investment by a firm generates a positive externality for the economy no diminishing marginal product of capital at the aggregate level, i.e. AK for aggregate capital

Model A representative firm i s production function Y i = K α i (BN i ) 1 α α < 1 diminishing marginal product of capital B is the stock of economy-wide knowledge, the firm takes this as given The economy wide stock of knowledge is proportional to the economy-wide stock of capital B = λk λ > 0 represents the idea of a positive externality the higher aggregate capital, K, and thus aggregate output, Y, the higher is productivity, B

Since firm i is a representative firm, it represents aggregate output and capital K = K i and N = N i and Y = Y i Use B = λk in the production function: Y = K α (BN) 1 α = K α (λkn) 1 α = K(λN) 1 α this is an AK production function, with no diminishing marginal returns to capital at the aggregate level

The capital accumulation equation K K = sk(λn) 1 α dk Let s assume that the population is constant k k = s(λn) 1 α k dk the growth rate of capital per person k k k = s(λn) 1 α d = x = constant if x > 0, then there is long run endogenous growth this is satisfied if the saving rate, s is sufficiently high

Endogenous growth in the learning-by-doing model i s(λn) 1 α dk k Assuming s(λn) 1 α d > 0

Implications of the learning-by-doing model there is endogenous growth because the stock of knowledge is determined by the endogenous level of K through learning-by-doing the saving rate affects not only the level of income but also the growth rate, as x depends on s the growth rate is constant in both the short and the long run, so there is no convergence there are scale effects : the growth rate depends on the size of the population larger N implies stronger knowledge spillovers and therefore higher growth rate, x

Human capital

Human capital skills are required to put ideas or knowledge into practice for the OECD countries and in most parts of the world average years of schooling are increasing fraction of college graduates is increasing as opposed to the process of learning-by-doing, there are costs and returns to education

Human capital Based on Lucas (1988). introduce a production function for human capital the production of new human capital is proportional to existing human capital AK for the production of human capital no diminishing marginal product in the production of human capital how good someone is in accumulating human capital (A) might depend on his years of education prediction: positive growth in the long run

Model The consumer consumes all his wage income: C = wuh s w - real wage per efficiency unit of labor u - fraction of time devoted to working (exogenous) H s - stock of human capital supplied uh s total units of efficiency labor supplied Future human capital H s = b(1 u)h s depends on current human capital, H s on the time devoted to training and education, 1 u b - efficiency of human capital accumulation

Model The representative firm s production function: Y = zuh d z TFP uh d amount of efficiency units of labor in production the profits are: π = Y wuh d = zuh d wuh d = (z w)uh d

The competitive equilibrium: the labor market has to clear: H d = H s = H the goods market has to clear: C = Y = zuh human capital accumulation: H = b(1 u)h the equilibrium growth rate of human capital is: H H H = b(1 u) 1 if b(1 u) > 1 human capital increases forever there is endogenous growth

Endogenous growth in the human capital model H b(1 u)h 45 Assuming b(1 u) > 1 H

The role of u remember, u is the fraction of time devoted to working, while 1 u is the fraction of time devoted to studying, and C = zuh what is the effect of a decrease in u on Y and C? immediate effect fewer hours worked (at constant H) a decline in Y a decline in the level of C long run effect after the immediate change u is again constant, thus C C C = H H H = b(1 u) 1 lower u higher 1 u, more time devoted to studying higher growth rate of H and C

The role of u Is a lower u necessarily better?

Is there convergence? imagine country A and country B have the same characteristics u A = u B, z A = z B, b A = b B but country A has a higher initial level of human capital H A (0) > H B (0) will they converge?

Convergence in the Solow model in the Solow model what happens to two countries, A and B, which have the same characteristics s A = s B, n A = n B, d A = d B, same technology zf (K, N) but country A has higher initial capital per person than country B: k A (0) > k B (0)?

Research and development

Research and development new ideas and knowledge are developed in the market through devoting resources to research and development for the OECD countries R&D spending as a fraction of GDP is increasing over time number of researchers as a fraction of the total employment is increasing over time Research and Development

What is the key difference between ideas and physical capital? Like capital, ideas are economic goods there is a cost to producing them they can be used in production there is a price for them (the value for a patent) There are some differences along the following attributes rivalrous vs non-rivalrous degree of excludability Can you come up with examples for each?

Examples rivalrous non-rivalrous excludable non-excludable

One country R&D model Based on Romer (1990). introduce a production function for ideas production of new ideas is proportional to the existing stock of ideas, i.e. AK for the production of ideas the coefficient depends on, for example, the number of researchers there is no diminishing marginal product in the production of ideas

Model The production function is Y = AL Y where L Y is the number of workers engaged in producing output and A is the level of knowledge (or technology) The output per worker is then y = Y L = AL Y L = A L ( ) Y L A = A 1 = A(1 γ A ) L Y + L A L A + L Y where γ A is the fraction of workers engaged in R&D

Model We assume that the production of new knowledge leads to the following growth rate of knowledge Â A A A = γ a µ L proportional to the number of workers engaged in R&D, γ A L µ captures the cost of new inventions If γ A is constant y is proportional to A since y = A(1 γ A ) the growth rate of y is the same as the growth rate of A ŷ = Â = A A A = γ a µ L

The role of γ A γ A is the fraction of labor which works in R&D what is the effect of an increase in γ A? Immediate effect fewer people working in production a decline in Y a decline in C long run effect the growth rate of A increases the growth rate of Y increases

The role of γ A One-Country Model to to R&D Ln(A) R&D One-Country Mode Shifting Labor into R&D Ln(y) of drawing figure in log from handout 1 Slide #33

Summary of the one-country model there are scale effects the growth rate depends on the size of the population, L a larger L implies more workers engaged in R&D (for given γ A ) this implies that countries with larger populations have higher growth rates, higher levels of technology and are richer are these good predictions? one interpretation of the model is that a country s level of technology depends on R&D done around the world this is a reasonable assumption if there is international technology transfer two-country model

Two country R&D model There are two countries, labelled 1 and 2. The production function for each country j = 1, 2 is Y j = A j (1 γ Aj )L j Countries acquire new technologies either by invention or by imitation. The option of imitation is open only to the less developed country, the technology follower. Assume L 1 = L 2 = L and γ A1 > γ A2. country 1 will be the technology leader and country 2 the follower in the steady state

The cost of imitation Assume that the cost of imitation is µ c = c ( A1 A 2 ) c( ) is downward sloping c( ) tends to zero as A 1 /A 2 tends to infinity c( ) tends to the cost of invention, µ i as A 1 /A 2 tends to one the cost of imitating a given technology is less than the cost of reinventing the technology µ c < µ i

The cost of of imitation Cost of Imitation Slide #38

The steady state What can the growth rates of technology (and hence output) be in the two economies? R&D Two-Country Model the growth rate has the same form as before: Âj = γ A j country 1 is the leader µ 1 = µ i Steady country State 2 is the follower µ 2 = µ c if A 2 < A 1 the steady ate, the growth tes of A 1 and 2 are equal. here is a steady ate level of 1 /A 2. 1 A, L i 2 A, L c Growth rate of technology µ j L

The steady state In the steady state, the growth rates are the same γ A1 µ i L = γ A 2 µ c L µ c = γ A 2 γ A1 µ i The steady state level of relative technologies, A 1 /A 2 can be found using c( ) ( ) A1 c = µ c = γ A 2 µ i γ A1 A 2

The role of γ A2 what happens if the follower increases R&D effort, i.e. γ A2 increases? steady state effect more resources in R&D growth curve shifts up lower level of steady state A 1 /A 2

Slide #41 The role of γ A2 what happens if the follower increases R&D effort, i.e. γ A2 increases? R&D Two-Country Model steady state effect more resources in R&D growth curve shifts up lower level of An steady increase in in state in A R&D 1 /A in the follower 2 An increase in γ A2 shifts up the growth rate of A 2. The new steady state level of A 1 /A 2 is lower.

The role of γ A2 immediate effect fewer people in production drop in output, Y then temporary increase in the growth rate of the follower

the for r The role of γ A2 Slide #41 immediate effect fewer people in production drop in output, Y then temporary increase in the growthr&d rate of the Two follower &D Two-Country Model ase in in R&D in in the follower e in a Two-Country Model An increase in in R&D in in the follower Ln(y) does this compare to the one-country model? Slide #42 S

Recap of economic growth models 1. Malthusian Model population growth increasing in per capita consumption stagnation in the long-run 2. Solow Model capital accumulation no (endogenous) growth in the long-run conditional convergence model with labor-augmenting technological progress matches the Kaldor facts well 3. Endogenous Growth Models role of technology and its origin no convergence