Endogenous Growth Theory - LMU München · Barro, R. J. (1990), Government ... Households: Keynes-Ramsey rule C ... Endogenous Growth Theory 20 ST2011 Growth and Natural Resources

Endogenous Growth Theory

1 ST2011 Growth and Natural Resources

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, 323-351. Barro, R. J. (1990), Government Spending in a Simple Model of Endogenous Growth, Journal of Political Economy 98(5), 103–125. 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, 500–521. Romer, P. M. (1986), Increasing Returns and Long–Run Growth, Journal of Political Economy 94, 1002–1037. Romer, P.M. (1990), Endogenous technological change,’ Journal of Political Economy 98, 71–102.



Limits of Neoclassical Growth Models

Growth peters out in the absence of technological progress due to decreasing marginal produc-tivity 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)



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 technologi-cal development as growth engine

Neoclassical growth models: growth peters out in the absence of technological progress due to decreasing marginal productivity of capital



Decreasing Marginal Productivity in Neoclassical Models Solow-Modell: Ramsey-Modell:

c

c*∗

0

k0 c0

0

Decreasing net marginal product of capital ad-justs to rate of time preference, such that en-dogenous incentives to accumulate vanish.

Capital accumulation stops as net return to savings falls to zero due to the de-creasing marginal product of capital.



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, Eu-ler’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



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 fi-nal ouput)

(in contrast to models withdifferent sectors producing, e.g., final and intermediate goods or in-novations)

Three types of models considered here:

3.1.1 The AK model

3.1.2 Learning by doing model

3.1.3 Productive government expenditure model



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



The AK model II Balanced growth path:

→ 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

0



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 at-

tribute non-decreasing returns to reproducible factors partly to private capital accumulation and partly to positive productivity effects of public goods.



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



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 accumula-tion are not internalized by the individual firm



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:



Learning By Doing IV

Optimal Solution I

Assumption: social planner who has perfect information maximizes household welfare by inter-

nalizing externality and thereby remunerates capital its full return Optimization problem: maximization of household utility subject to aggregate technology

max 0 1

s.t.

0 0

lim→

0



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



Learning By Doing VI

Capital Subsidization

Subsidy rate zon 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

⟺



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 expendi-

tures

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



Productive Government Expenditures II Decentralized solution as function of : 1 α δ ρ)

(inserting for ) → 1 α δ ρ

Optimal tax policy → tax rate that maximizes growth rate also maximizes welfare:

max → 1 (= production elasticity with re-spect to the public input)

→ 1 δ ρ

Comparison to optimal growth: 1 δ ρ

→ Decentralized growth rate lower despite optimal policy as households neglect the produc-tivity increasing effect of (second-best policy)



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)



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



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



Representative Consumer I

Utility derived from consumption; no disutility from labor

Household allocates time optimally between working and schooling

Household optimization problem:

max,

0 1

s.t.

1

0 0, 0 0



Household Optimization I

Hamiltonian:

Η 1 1

→ two controls ( , ) and two state variables ( , )

FOCs:

Η0 ⇔ C e μ

Η0 ⇔ 1

Η ⇔

Η ⇔ μ 1 α K u H μ B 1 u μ



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



Solving for the Long-Run Equilibrium I From the FOCs and equations of motion for and we get

→ 1

→ Growth rates of composite variables

1

1

→ 2 equations of motion (for , ) and 3 unknowns ( , , ) → equation of motion for has to be derived using FOCs



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

1 1

⟺ 1



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:

and for the equilibrium growth rate:

1

time spent working 0 if 1

1

1

1



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 hu-

man 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



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 corre-sponding patent (vertical differentiation)



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 substi-

tutes 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



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 Nvarieties of a continuum of differentiated intermediate goods X(i)

,0 1

Constant population

Diminishing marginal productivity of each input and ,

Constant returns to scale in all inputs together



Market Solution II

Final Goods Sector II:

Profits maximization:

max,

Π

FOC for labor:

1

FOC for intermediate product i:

⟺

demand for



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



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:



Market Solution V

Intermediate Goods Sector III:

1

1

Demand:

. ..



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 mo-

nopoly 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 aunits of the final good: &

market entry condition (no entry barriers for new firms):



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:



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



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:

max,

0 1

s.t. 1

Optimal equilibrium growth rate: 1

1

&

⇔1



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 inven-tors to create new types of products

subsidies to purchases of intermediate goods

subsidies to final good

subsidies to research



The Original Romer (1990) Model Alternative specification of R&D production technology with labor rather than final product as in-

put 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

Documents

Endogenous Growth Theory - LMU München · Barro, R. J. (1990), Government ... Households: Keynes-Ramsey rule C ... Endogenous Growth Theory 20 ST2011 Growth and Natural Resources