# Endogenous Growth Models

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1 Endogenous Growth Models Lorenza Rossi Goethe University

2 Endogenous Growth Theory Neoclassical Exogenous Growth Models technological progress is the engine of growth technological improvements are automatic and unmodeled (exogenous) Endogenous Growth Models Try to explain the engine of growth It is important to understand the economic forces underlying technological progress

3 Endogenous Growth and Learning IDEA: Capital accumulation embeds technological improvements (Arrow 1962 =)Romer 1982) Firms production function Y (i) = AK (i) α L (i) 1 α where A is the Total Factor Productivity (TFP). Technology A depends on Capital Stock. The higher the capital stock the more the economy is able to use new technologies A = BK 1 α where K is the aggregate level of capital stock and B is the learning factor (positive externality). Imposing symmetry across rms and substituting in the production function, we get the aggregate production function Y = BKL 1 α

4 Endogenous Growth and Learning Assuming that population L is constant and equal to 1. Then, the aggregate production function becomes, Y = BK This production function is characterized by constant return to scale. The marginal productivity of capital is constant and equal to the average productivity of capital and is B. The low of motion of capital is K = sy hence the growth rate of capital is K K = s Y K given that Y K = B =constant, K is positive. K = Ẏ Y dk d = sb d. If sb > d =) the growth rate

5 Endogenous Growth and Learning NOTICE!!!! IMPORTANT!! The rate of growth of A is Ȧ A = (1 K α) K = (1 α) (sb d) Contrary to the Solow model, the rate of growth of technology depends on the rate of growth of capital. At the same time technology a ects capital. Growth is an endogenous process. No transitional dynamics An increase in savings means that the growth rate increases permanently.

6 Endogenous Growth and Learning How to introduce a transitional dynamics Suppose that A = B 0 + B 1 K 1 α then Y = B 0 K α + B 1 K and the rate of growth of capital K K = sb 0K α 1 + sb 1 d the rate of growth of K is decreasing in K and converges to sb 1 d.

7 Endogenous Growth and Learning Endogenous growth plus transitional dynamics

8 Endogenous Growth and Learning Human capital and Endogenous Growth (Lucas 1988). The production function Y = K α (AL) 1 α where A = H human capital increases labor productivity, with L = 1 Y = K α H 1 α

9 Endogenous Growth and Learning De ne s K as the amount of GDP spend for capital accumulation. For simplicity and without loss of generality, we now assume that the capital depreciation rate is d = 0. Hence, K = s K Y = s K K α H 1 α De ne s H as the amount of GDP spent for human capital accumulation. Ḣ = s H Y = s H K α H 1 α

10 Endogenous Growth and Learning De ne γ = H K. substituting in the low of motion of capital and dividing by K K K = s K γ 1 α Similarly Consider that Ḣ H = s H γ α γ γ = Ḣ H Ḣ If H > K K =) γ K γ > 0 and γ increases. If γ increases K increases, while Ḣ H reduces, so that γ γ decreases. On the contrary if Ḣ H < K K =) γ γ while Ḣ K K < 0 and γ decreases. If γ decreases K K H increases, so that γ γ Ḣ H = K K and γ γ = 0. decreases, increases. The process stops only when

11 Endogenous Growth and Learning If γ γ = 0, then γ is equal to its steady state value, which is obtained taking Ḣ/H K /K = s H γ α s K γ 1 α = 1 solving for γ γ = s H s K Substituting this value in the low of motion of physical and human capital K = s α K K s1 H α = Ḣ H

12 Endogenous Growth and Learning The GDP growth rate is hence in the steady state of γ Ẏ Y = α K K + (1 Ẏ = s α Y α) Ḣ H K s1 H α

13 Barro s model of Endogenous Growth with Government Spending and Taxation Barro (1990) suggests a simple endogenous growth model with government. In the Barro model public spending goes for public investment (infrastructures, schools, sanitation etc.). Public investments, which are nanced through income taxes, complement private investments. Since public investments raise the productivity of private investments, higher taxes can be associated with an increase or a decrease in overall growth.

14 Barro s model of Endogenous Growth with Government Spending and Taxation The model. Barro (1990) adds public spending to the Romer AK model. Y = BK 1 α G α where G = τy substituting into the production function Y = K 1 α (τy ) α solving for Y where B (τ) = B 1 1 α Y = B 1 1 α τ 1 α α τ α 1 α K = B (τ) K

15 Barro s model of Endogenous Growth with Government Spending and Taxation The low of motion of capital Then, K = s (1 τ) Y dk K K = s (1 τ) B (τ) d = s (1 τ) B 1 1 α τ 1 α α thus if s (1 τ) B (τ) > d =) K K > 0 Which is the e ect of taxation on growth? The economy faces a La er Curve Which is the optimal tax rate, i.e. the tax rate maximizing growth? We consider two models. 1) a model with exogenous savings; 2) A model with endogenous savings (Ramsey approach) d

16 Barro s model and the La er curve Optimal taxation in a model with exogenous savings It is su cient to take the derivative of solving for K K wrt τ and set equal to zero. K K τ = 0 : sb 1 1 α α 1 1+2α α τ 1 α + s (1 τ) B 1 α τ 1 α = 0 1 α τ = α which is the optimal tax rate, i.e. the tax rate that maximizes growth.

17 Optimal taxation in a model with exogenous savings

18 The Barro model with endogenous savings Optimal taxation in a model with endogenous savings For simplicity, and without loss of generality, we assume that population is constant and equal to L = 1, and that capital depreciation rate is d = 0. Given that L = 1 and constant, this means that per capita variables are identical to variables in level, C = c, Y = y, K = k. Then, the decentralized Ramsey problem is max fc,k g C 1 θ 1 θ e ρt s.t. K = (1 τ) Y C Y = BK 1 α G α

19 The Barro model with endogenous savings The present value Hamiltonian associated is H = C 1 θ 1 θ e ρt µ (1 τ) BK 1 α G α C FOCs wrt. consumption, capital and the costate variable are: H C = 0 : C θ e ρt µ = 0 H K = µ : µ (1 τ) (1 α) BK α G α µ = 0 H µ = K : (1 τ) BK 1 α G α C = K notice that G α = B α 1 α τ α 1 α K α.

20 The Barro model with endogenous savings Combining FOCs 1. and 2. 2 Ċ C = 1 4(1 τ) (1 α) BK α G α θ {z } MPK 2 = 1 θ 4(1 τ) (1 α) B 1 1 α α τ 1 α {z } MPK 3 ρ5 3 ρ5 where MPK states for Marginal Product of Capital.

21 The Barro model with endogenous savings Notice that the MPK is MPK = (1 τ) {z } negative e ect of taxation (1 α) BK α {z} G α positive e ect of public investment Growth in consumption depends on: i) the gap between the MPK and the rate of time preference ρ; ii) the intertemporal elasticity of substitution θ. Thus, Government a ects the MPK through two channels: i) increase in G raises the MPK to a point; ii) taxes always reduces the private return of capital. The main objective of a good Government is to balance these two e ects.

22 The Barro model with endogenous savings The tax rate maximizing consumption is obtained by di erentiating Ċ C w.r.t. τ. (Ċ /C) 0 τ = 1 θ (1 τ) α 1 α 1 2α 1 1 α (1 α) B 1 α τ 1 α 1 θ (1 α) B 1 α τ 1 α = simplifying and solving for τ the same value we found for K K τ τ GR = α

23 The Barro model with endogenous savings Is the Decentralized solution also the rst best solution? It is important to compare the decentralized solution with the Social Planner one. Which is the Social Planner solution? The Social Planner internalizes the e ect of G and thus the optimal problem becomes C 1 θ max fc,k,g g 1 θ e ρt s.t. Resource Constraint i.e. : Y = C + I + G or : K = Y C G = BK 1 α G α C G

24 The Barro model with endogenous savings The present value Hamiltonian of the Social Planner is H = C 1 θ 1 θ e ρt µ BK 1 α G α C G The Social Planner FOCs wrt. consumption, capital and the costate variable are: 1s. 2s. 3s. 4s. H C = 0 : C θ e ρt µ = 0 H G = 0 : αbk 1 α G α 1 = 1 =) Y G = 1 H K = µ : µ (1 α) K α G α µ = 0 H µ = K : BK 1 α G α C G = K

25 The Barro model with endogenous savings Combining FOCs 1s. and 2s. Ċ C = 1 i h(1 α) B 1 1 α α τ 1 α ρ θ Notice that (1 α) B 1 1 α 1 α α τ 1 α > (1 τ) (1 α) B 1 α τ 1 α, hence the MPK in the decentralized solution is (1 τ) Y K, which is smaller than what we get from the Social Planner solution, i.e. the social marginal product Y K, because of the tax rate. This gap between social and private returns leads to a lower growth rate in the decentralized solution.

26 Endogenous Growth and R&D Sector The Romer model try to explain why and how advanced countries of the world exhibit sustained growth. Technological progress is driven by R&D sector in advanced world. Romer endogenizes technological progress by introducing an R&D sector, i.e. search of new ideas by researcher interested in pro ting from their invention. The aggregate production function in the Romer model is Y = K α (AL Y ) 1 α Capital accumulation is K = s K Y dk population growth is L L = n.

27 Endogenous Growth and R&D Sector The key equation of the Romer model is the one describing the R&D sector. According to Romer A is the number of ideas, or the stock of knowledge accumulated up until time t. The number of new ideas Ȧ is equal to the number of people devoting their time in discovering new ideas L A, multiplied by the rate at which they discover new ideas, i.e. δ. Thus, Ȧ = δl A Labor is used either to produce good, L Y, or to produce new ideas L A. So the economy faces the following resource constraint: L = L Y + L A

28 Endogenous Growth and R&D Sector The rate at which new ideas are discovered, δ, might be constant, or an increasing function of A where δ and φ are constants. δ = δa φ Notice that with φ > 0 the productivity of research increases with the stock of ideas that have already been discovered. On the contrary with φ < 0, discovering new ideas becomes harder over time. With φ = 0 the discovery rate is independent from the stock of knowledge.

29 Endogenous Growth and R&D Sector It is possible that new ideas are more likely when there are more persons engaged in research. Thus, the e ect of L A is not proportional. Hence, it can be assumed that it is LA λ that enter in the production function of new ideas, with 0 < λ < 1. The general production function of new ideas is Ȧ = δl λ A Aφ Assuming that 0 < φ < 1. Dividing by A Ȧ A = δ Lλ A A 1 which is the rate of growth along the BGP? φ

30 Endogenous Growth and R&D Sector Along the BGP Ȧ A = g A = constant. Thus, the numerator and the denominator should growth at the same rate, which means along the BGP L A L A λ L A L A (1 φ) Ȧ A = 0 = n and thus Ȧ A = g A = nλ 1 φ In this model, as in the Neoclassical model, even if growth is an endogenous process, policy maker cannot do nothing to increase the long-run growth rate. Indeed bot λ and φ are parameters independent on policies, such as subsidies to R&D

31 Endogenous Growth and R&D Sector Introducing Microfoundation. Romer (1990 JPE) Romer (1990) explains how to construct an economy of pro ts-maximizing agents that endogenize technological progress. The economy consists of three sectors: 1 A nal good-producing sector 2 An intermediate good-producing sector: producing capital goods 3 A research sector The research sector sells the exclusive right to produce a speci c capital good to an intermediate-good rm. The intermediate-good rm, is monopolist, manufactures the capital good and sells it to the nal good sector which produces output.

32 Endogenous Growth and R&D Sector The nal-good sector is composed by a large number of perfectly competitive rms that combine labor and capital to produce the nal good, Y. There is more than one type of capital in the production function, thus it is speci ed as follows Y = L 1 Y α N j=1 x α j where the capital goods x j, come from the intermediate good-producing sector. Inventions, or new ideas correspond to the creation of new capital that can be used by the nal-good sector to produce the nal output.

33 Endogenous Growth and R&D Sector The nal-good sector If A is the number of capital goods. Then N = A and the production can be rewritten as Y = L 1 if the number of goods is continuos Y = L 1 Y α A j=1 Z A Y α 0 x α j x α j dj For simplicity we will use the second de nition. Notice that, whether we use a discrete number of goods or a continuos number, results remain unchanged.

34 Endogenous Growth and R&D Sector Final good price P is normalized to 1. Firms in the nal-good sector, choose labor and capital to maximize pro ts, Z A max fl Y,x J g L1 Y α 0 x α j dj wl Y Z A 0 p j x j dj where p j is the rental price for capital-goods and w the wage paid for labor. The FOCs imply: w = (1 α) Y L Y p j = αl 1 Y α x j α 1 for each j As usual prices of inputs equate their marginal product.

35 Endogenous Growth and R&D Sector The intermediate good sector consists of monopolists who produce the capital goods to sell to the nal sector. Firms gain their monopoly power by purchasing the design for a speci c capital good from the R&D sector. Because of patent protection only one rm manufactures each capital good. Each rm uses a very simple production function. One unit of raw capital (purchased in the R&D sector) translates into one unit of manufactured capital. The pro t maximization problem of the representative intermediate-good rm is max x j p j (x j ) x j rx j where p j (x j ) is the demand function of the capital good, corresponding to p j = αl 1 Y α x j α 1 and r is the interest rate, or the rental rate of capital.

36 Endogenous Growth and R&D Sector The FOC of the intermediate-good rm is. pj 0 (x j ) x j + p j (x j ) r = 0 α 2 L 1 Y α j α 1 {z } r = 0 αp j Imposing symmetry and solving for p p = p0 (x )x p r = 1 α r. which is the optimal price set in the intermediate-good sector.

37 Endogenous Growth and R&D Sector Equilibrium and Aggregation The total demand for capital from the intermediate good sector must equal the total capital stock in the economy. Thus, Z A x j dj = K 0 Since the capital goods are each used in the same amount, x, the previous equation can be used to determine x x = K A The nal good production function can be rewritten as Y = L 1 Z A Y α 0 x α dj = L 1 Y α Ax α substituting for x = K A Y = K α (AL Y ) 1 α

38 Endogenous Growth and R&D Sector In the Research Sector new design are discovered according to Ȧ = δl λ A Aφ When a design is discovered, the inventor receives a patent from the Government for the exclusive right to produce the new capital good. The patent last forever. The inventor sells the patent to an intermediate good rm and uses the proceeds to consume and save. What is the price of a new patent? Anyone can bid for a patent. The potential bidder will be willing to pay the discounted value of the pro ts earned by an intermediate-good rm. Let the discounted value of pro ts earned by an intermediate-good rm be P A, where pro ts are: π = α (1 α) Y A

39 Endogenous Growth and R&D Sector The research sector How does P A change over time? Firms can put money (an amount equivalent to the value of a patent, P A ), in a bank, earning the interest rate r. Alternatively, they can purchase patent for one period, manufacture capital, earn pro ts and then sell the patent. In equilibrium the return of these two alternatives must be the same. Thus, rp A = π + Ṗ A Which gives r = π + ṖA P A P A Along the BGP r is constant and thus π and P A must grow at the same rate, which is the population growth rate n (when λ = 1 and φ = 0). Thus, along the BGP P A = π r n

40 Endogenous Growth and R&D Sector Share of population working in the R&D and good producing sector Once again we can use the arbitrage concept. It must be the case that at the margin, individual are indi erent between working in the nal-good sector or the R&D sector. We know that in the nal-good sector w Y = (1 α) Y L Y in the R&D sector, real wages are equal to the marginal product of labor δ, multiplied by the value of new ideas created, i.e. P A, thus w R = δp A

41 Endogenous Growth and R&D Sector Because there is free entry in the two labor markets it must be that w Y = w R, then then (1 α) Y = δp A = δπ L Y r n = δα (1 α) Y A r n 1 = δα (1 α) Y (1 α) = α δ L Y r n A Y r n A Rearranging and considering that Ȧ = δl A =) Ȧ the BGP, then 1 = α g A L Y r n L A L A L Y = αg A r n = s R 1 s R and s R = L A L s R = is r n. αg A A = δl A A = g A along

42 Endogenous Growth and R&D Sector OPTIMAL R&D. Is the share of population involved in R&D sector optimal? The answer is no. Why? The economy is characterized by three distortions 1 The market does not endogenize the fact that new research may a ect the productivity of future research. φ > 0, implies that productivity of research increases with the stock of ideas. Researcher are not compensated for their contribution toward improving the productivity of future researcher. Thus, with φ > 0 the market provides too little research and the fraction of population hired by R&S is too low. This e ect is called spillover e ect or "standing on the shoulders e ect". 2 With λ < 1 research productivity is lower because of duplications. Thus, too many people are hired by the research sector. This e ect is called "stepping on toes e ect". 3 Consumer surplus e ect. The monopoly pro ts are less than the consumer surplus. This e ect tends to generate too little innovations.

43 Endogenous Growth and R&D Sector OPTIMAL R&D Classical economic theory: imperfect competition and monopoly are bad for welfare and e ciency because they generate a deathweight-loss in the economy. This happens because prices are higher than marginal costs. However, the literature on the economic of ideas suggests that it is the possibility to make pro ts, and thus to set a markup over marginal costs, that incentives rms, or the R&D sector, to produce more ideas. This means, that there is a trade-o between short-run losses and long-run gains. Concluding. In deciding antitrust policies, the regulator has to weight the deathweight losses against the incentive to innovate.

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