Endogenous Growth Theory

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1 Chapter 3 Endogenous Growth Theory 3.1 One-Sector Endogenous Growth Models 3.2 Two-sector Endogenous Growth Model 3.3 Technological Change: Horizontal Innovations References: Aghion, P./ Howitt, P. (1992), A Model of Growth Through Creative Destructio, Econometrica 60, Barro, R. J. (1990), Government Spending in a Simple Model of Endogenous Growth, Journal of Political Economy 98(5), Barro, R. J./Sala-I-Martin, X. (2004), Economic Growth, MIT Press, Cambridge, MA, 2nd ed. Rebelo, S. (1991), Long Run Policy Analysis and Long Run Growth, Journal of Political Economy 99, Romer, P. M. (1986), Increasing Returns and Long Run Growth, Journal of Political Economy 94, Romer, P.M. (1990), Endogenous technological change, Journal of Political Economy 98, ST2011 Growth and Natural Resources

2 Limits of Neoclassical Growth Models Growth peters out in the absence of technological progress due to decreasing marginal productivity of capital Policy ineffectiveness: no impact of economic policy on growth as growth rate exogenously determined Solow model: no impact of policy on savings rate»from the point of view of policy advice, growth theory had little to offer. In models with exogenous technological change and exogenous population growth, it never really mattered what the government did.«(romer, 1989, p. 51) 2 ST2011 Growth and Natural Resources

3 Endogenizing Economic Growth Growth driven by mechanisms that are endogenous to the economy and do not rely on exogenous forces. Two general types of models: 1. Accumulation-based models: accumulation of human and physical capital drive growth 2. Innovation-based models: investments in research and development (R&D) and technological development as growth engine Neoclassical growth models: growth peters out in the absence of technological progress due to decreasing marginal productivity of capital 3 ST2011 Growth and Natural Resources

4 Decreasing Marginal Productivity in Neoclassical Models Solow-Modell: Ramsey-Modell: 0 0 k0 c0 c * c Capital accumulation stops as net return to savings falls to zero due to the decreasing marginal product of capital. Decreasing net marginal product of capital adjusts to rate of time preference, such that endogenous incentives to accumulate vanish. 4 ST2011 Growth and Natural Resources

5 The Problem prerequisite for endogenous growth: non-decreasing returns to accumulating factors, resp. lower limit to productivity (lim 0) But empirical evidence: marginal product of physical capital decreasing also: perfect competition only compatible with constant returns to scale technology increasing returns (e.g. competitive equilibrium not sustainable, Euler s adding-up theorem) alternative: technological progress driven by R&D activities however: private R&D has to be financed by profits but constant returns to scale are associated with zero profits 5 ST2011 Growth and Natural Resources

6 3.1 One-Sector Endogenous Growth Models Barro/Sala-i-Martin (2004), chapter 4 one sector model: models that comprise only one production sector (mostly production of final ouput) (in contrast to models withdifferent sectors producing, e.g., final and intermediate goods or innovations) Three types of models considered here: The AK model Learning by doing model Productive government expenditure model 6 ST2011 Growth and Natural Resources

7 3.1.1 The AK model Rebelo (1991) Assumes constant returns to capital (employing a broadly defined measure of capital including, e.g., human capital), but neglects of input of labor. Households: utility maximization as in Ramsey model gives the Keynes-Ramsey rule: 1 Firms: Profit maximization FOC: Simultaneous optimum: 1 7 ST2011 Growth and Natural Resources

8 The AK model II Balanced growth path: 0 constant marginal product of capital sustains growth indefinitely if potential for long-run growth depends on productivity of capital, depreciation and preferences of households in contrast to empirical evidence: growth rate is constant over time, no transitory adjustment to the steady state (immediate jump), no convergence 8 ST2011 Growth and Natural Resources

9 Alternative One-Sector Approaches Decreasing marginal product of capital compensated by increases in efficiency through learning by doing (Romer 1986) productive Government Expenditures (Barro 1990) Instead of postulating constant returns to scale with respect to private capital, these models attribute non-decreasing returns to reproducible factors partly to private capital accumulation and partly to positive productivity effects of public goods. 9 ST2011 Growth and Natural Resources

10 3.1.2 Learning By Doing based on Romer (1986) Individual investment in capital increases experience. Increased experience enhances labor productivity by generating knowledge (human capital) that can also be used costlessly by other firms (non-rival character of knowledge public good). no enforcement of property rights possible knowledge as a byproduct of investment activities knowledge exerts productivity effect on privately owned factors of production Knowledge stock is approximated by the aggregate capital stock increases with individual investment Important: positive productivity effect (learning effect) external to the individual firm 10 ST2011 Growth and Natural Resources

11 Learning By Doing II Production function for the individual firm (assumption: identical firms) = aggregate capital stock (with, aggregates equal averages if number of firms is normalized to unity, i.e. defined on continuum [0,1]) constant population ( 0) Constant returns to scale in privately owned factors consistent with perfect competition Increasing returns to scale over all factor inputs Constant social return to capital (i.e. social return to capital exceeds private return) Implication: market equilibrium is not pareto-optimal as external effects of knowledge accumulation are not internalized by the individual firm 11 ST2011 Growth and Natural Resources

12 Learning By Doing III Market solution Households: Keynes-Ramsey rule Firms: Profit maximization over privately owned factors of production 1 1 constant marginal product of capital for constant population note: scale effect with respect to population size (not supported by empirical data) Simultaneous optimum: 12 ST2011 Growth and Natural Resources

13 Learning By Doing IV Optimal Solution I Assumption: social planner who has perfect information maximizes household welfare by internalizing externality and thereby remunerates capital its full return Optimization problem: maximization of household utility subject to aggregate technology s.t. max lim 0 13 ST2011 Growth and Natural Resources

14 Learning By Doing V Optimal Solution II Hamiltonian: FOCs: Balanced growth rate in social optimum: 1 Comparison to market growth rate ( ) shows that growth rate rises due to internalization of knowledge externality Scope for economic policy to enhance growth: e.g. subsidy on capital input 14 ST2011 Growth and Natural Resources

15 Learning By Doing VI Capital Subsidization Subsidy rate z on capital input (financed by lump-sum tax). Modified profit function: FOC for capital: 1 Optimal subsidy rate: equalizes the market growth rate to the optimal growth rate 1 15 ST2011 Growth and Natural Resources

16 3.1.3 Productive Government Expenditures I based on Barro (1990) government provides public goods e.g. infrastructure, education, public research, legal system firms can use public goods costlessly in production (no rivalry, no exclusion possible) productivity enhancing effect on private factors of production but need to finance public expenditures Production function for the individual firm (assumption: identical firms) with = provision of public goods (external to individual firm) Provision of public goods financed by income tax (tax rate ) government budget constraint: household income constraint: 1 16 ST2011 Growth and Natural Resources

17 Productive Government Expenditures II Decentralized solution as function of : 1 α δρ) (inserting for ) 1 α δρ Optimal tax policy tax rate that maximizes growth rate also maximizes welfare: max 1 (= production elasticity with respect to the public input) 1 δρ Comparison to optimal growth: 1 δρ Decentralized growth rate lower despite optimal policy as households neglect the productivity increasing effect of (second-best policy) 17 ST2011 Growth and Natural Resources

18 Lesson from government policies in the two preceding models: Efficiency of the decentralized solutions depends strongly on the policy instruments chosen capital income taxation: first-best policy in learning-by-doing model (welfare in decentralized solution with policy = welfare in social planner solution) income taxation: second-best policy in government expenditure model (welfare in decentralized solution with policy = welfare in social planner solution) 18 ST2011 Growth and Natural Resources

19 3.2 Two-sector Endogenous Growth Model Barro/Sala-i-Martin (2004), chapter 5 Assumptions Two sector model with two accumulating factors: real capital & human capital Human capital is labor augmenting and has a positive impact on factor productivity. Therefore, human capital is an important determinant of the growth rate of the economy. Human capital is a private good and tied to labor inputs. Individuals invest in human capital, but accumulation of knowledge has opportunity costs in terms of forgone income. time spent for schooling cannot be used for working the individual weighs costs of education against benefits 19 ST2011 Growth and Natural Resources

20 Set-up Robinson-Crusoe economy (household as consumer and producer) effective labor input: = working time = individual human capital stock no population growth and labor force normalized to 1 aggregate output (net of depreciation): simplifying assumption: no physical capital used in the production of human capital production of human capital: 1 1 = schooling time ( 0,1) generation of new human capital profits from existing human capital human capital does not depreciate 20 ST2011 Growth and Natural Resources

21 Representative Consumer I Utility derived from consumption; no disutility from labor Household allocates time optimally between working and schooling Household optimization problem: s.t. max, , ST2011 Growth and Natural Resources

22 Household Optimization I Hamiltonian: Η 1 1 two controls (,) and two state variables (, ) FOCs: Η 0 C e μ Η 0 1 Η Η μ 1 αk u H μ B1u μ 22 ST2011 Growth and Natural Resources

23 Household Optimization II goods market clearing: conditions for long-run equilibrium:,, and grow at common rate: allocation of time constant over time ( and 1 = constant) helpful to express variables in ratios: and 23 ST2011 Growth and Natural Resources

24 Solving for the Long-Run Equilibrium I From the FOCs and equations of motion for and we get 1 Growth rates of composite variables equations of motion (for, ) and 3 unknowns (,, ) equation of motion for has to be derived using FOCs 24 ST2011 Growth and Natural Resources

25 Solving for the Long-Run Equilibrium II Deriving From the FOC for u we get Taking the time derivative and expression the resulting expression in growth rates gives Substituting for and from the FOCs for and as well as for gives ST2011 Growth and Natural Resources

26 Solving for the Long-Run Equilibrium III Taking into account that in the long-run equilibrium 0 and 0 have to hold and solving for, and gives: 1 time spent working 0 if and for the equilibrium growth rate: 1 26 ST2011 Growth and Natural Resources

27 Intuition for Equilibrium Values Productivity in schooling drives endogenous growth Optimal schooling time, 1, decreases with impatience and the intertemporal elasticity of substitution 1/. Ratio of physical to human capital ( /) decreases in schooling time Long-run equilibrium is Pareto-optimal as no market failures arise (e.g. no externalities, ) Model Extension Assumption: Externalities arise from aggregate human capital on individual accumulation of human capital: with 0 In the long-run equilibrium, the human capital stock grows faster than the physical capital stock due to the externality from Also due to externality: market solution is not Pareto-optimal scope for educational policy 27 ST2011 Growth and Natural Resources

28 3.3 Technological Change Barro/Sala-i-Martin (2004), chapter 6 General problems of endogenizing technological change: public good property of knowledge is a barrier to innovation given insufficient property rights enforcement only constant returns to scale (CRS) technologies compatible with competitive equilibrium but: zero profits under CRS, i.e. no compensation of investment in R&D development of imperfect competition models Pareto inefficiency of because of monopoly profits Romer (1990): monopolistic competition in the production of differentiated intermediates that are imperfect substitutes (horizontal differentiation) Aghion/Howitt (1992): qualitative improvement of existing products that replace old products (quality ladder) and are produced by monopolists holding the corresponding patent (vertical differentiation) 28 ST2011 Growth and Natural Resources

29 Romer 1990 type Model Multi sectoral model: final goods sector intermediate goods sector research sector Intellectual property rights are protected by perfectly enforceable patents. Knowledge creation (innovation) is result of deliberate research efforts (not merely a by product of investment). firms weigh the costs of R&D against the benefits. Innovation leads to the development of new (intermediate) products which are imperfect substitutes of existing ones monopolistic competition in the intermediates sector profits compensate for R&D costs Expanding product variety generates ongoing growth of per capita incomes 29 ST2011 Growth and Natural Resources

30 Market Solution I Households: Keynes-Ramsey rule Final Goods Sector I Aggregate output Y is produced on a perfectly competitive market by using labor input L(t) and N varieties of a continuum of differentiated intermediate goods X(i) Constant population, 0 1 Diminishing marginal productivity of each input and, Constant returns to scale in all inputs together 30 ST2011 Growth and Natural Resources

31 Market Solution II Final Goods Sector II: Profits maximization: FOC for labor: max, Π 1 FOC for intermediate product i: demand for 31 ST2011 Growth and Natural Resources

32 Market Solution III Intermediate Goods Sector I: monopolistic competition large number of small firms firms produce heterogeneous intermediate goods with identical technology goods are close but imperfect substitutes each firm is a monopolist and identified with one product variety i price setting behavior, i.e. firms extract monopoly rents allocation is not Pareto efficient: smaller quantities are traded at higher prices than under perfect competition 32 ST2011 Growth and Natural Resources

33 Market Solution IV Intermediate Goods Sector II: production of intermediate good at constant marginal costs (normalized to unity) profit maximization of an individual monopolist i: max 1 1 FOC: 1 0 monopoly price: markup of over the marginal cost of production constant over time and identical for all firms quantity produced: 33 ST2011 Growth and Natural Resources

34 Market Solution V Intermediate Goods Sector III: Demand: 34 ST2011 Growth and Natural Resources

35 Market Solution VI Research Sector I: Innovation in form of the development of a new product requires R&D effort Firm i retains a perpetual monopoly right on the production and sale of good i. The flow of monopoly rentals provide the incentive for innovation. Present value of future profits from inventing intermediate at : Assumption: cost of inventing a new intermediate are constant (in terms of the final product): R&D requires a units of the final good: & market entry condition (no entry barriers for new firms): 35 ST2011 Growth and Natural Resources

36 Market Solution VI Long-run Equilibrium I: Equilibrium entry decision: infinite amount of resources channeled into R&D not consistent with equilibrium no resources spent on R&D number of intermediates remains constant over time positive R&D at all points of time number of intermediate goods grows over time Taking the time derivative of the market entry condition (Leibniz rule) gives stability condition: 0 where and the equilibrium interest rate: 36 ST2011 Growth and Natural Resources

37 Market Solution VII Long-run Equilibrium II: Due to identical prices for all, all intermediates are used in the same quantity in -production. aggregate quantity of intermediates: aggregate output: Profit of each firm i : From equilibrium interest rate, intermediates firms profits and the Keynes-Ramsey rule we get: 1 1 a decrease in the costs of innovation, a, increases the real return and raises growth again scale effect with respect to labor 37 ST2011 Growth and Natural Resources

38 Optimal Solution I Market solution not pareto-optimal due to monopolistic price setting in intermediates sector Social planner takes into account that intermediate firms produce identical quantities ( ) Optimization problem: s.t. max, & 1 Optimal equilibrium growth rate: ST2011 Growth and Natural Resources

39 Economic Policies (1) Make existing innovations freely available to the public ex post but: this policy destroys ex ante incentives for further inventions (one shot game) (2) tax subsidy policy that induces marginal cost pricing without eliminating the incentive for inventors to create new types of products subsidies to purchases of intermediate goods subsidies to final good subsidies to research 39 ST2011 Growth and Natural Resources

40 The Original Romer (1990) Model Alternative specification of R&D production technology with labor rather than final product as input to R&D Problem: as the population is assumed to be constant, growth would cease in the long run due to the limitedness of labor input. Therefore additional assumption: R&D productivity depends positively on the accumulated knowledge of past research cost of additional innovations decline with increasing knowledge Accumulated knowledge stock: approximated by number of past innovations () Modified R&D production function: & with & = input of labor in R&D sector Additional market failure due to spillovers of past research onto present research 40 ST2011 Growth and Natural Resources

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