Growth and Ideas. Romer (1990) and Jones (2005) Chad Jones Stanford GSB. Romer (1990) and Jones (2005) p. 1

Transcription

1 Growth and Ideas Romer (1990) and Jones (2005) Chad Jones Stanford GSB Romer (1990) and Jones (2005) p. 1

2 Key References Romer, Paul M Endogenous Technological Change Journal of Political Economy 98:S71-S102. Jones, Charles I Growth and Ideas in P. Aghion and S. Durlauf (eds.) Handbook of Economic Growth (Elsevier) Volume 1B, pp Jones, C The Facts of Economic Growth Handbook of Macroeconomics, forthcoming. See syllabus at Romer (1990) and Jones (2005) p. 2

3 Outline: How do we understand frontier growth? A Discussion of Ideas Simple Model Full Model, and various resource allocations Applications Romer (1990) and Jones (2005) p. 3

4 A Discussion of Ideas Romer (1990) and Jones (2005) p. 4

5 Solow and Romer Robert Solow (1950s) Capital versus Labor Cannot sustain long-run growth Paul Romer (1990s) Objects versus Ideas Sustains long-run growth Wide-ranging implications for intellectual property, antitrust policy, international trade, the limits to growth, sources of catch-up growth Romer s insight: Economic growth is sustained by discovering better and better ways to use the finite resources available to us Romer (1990) and Jones (2005) p. 5

6 The Idea Diagram Ideas Nonrivalry Increasing returns ր ց Long-run growth Problems with pure competition Romer (1990) and Jones (2005) p. 6

7 The Essence of Romer s Insight Let A denote Ideas Question: In generalizing from the neoclassical model to incorporate ideas, why do we not write the PF as Y = A α K β L 1 α β Romer (1990) and Jones (2005) p. 7

8 IRS and the Standard Replication Argument Familiar notation, but now let A t denote the stock of knowledge or ideas: Y t = F(K t,l t,a t ) = A t K α t L 1 α t Constant returns to scale in K and L holding knowledge fixed. Why? F(λK,λL,A) = λ F(K,L,A) But therefore increasing returns in K, L, and A together! F(λK,λL,λA) > F(λK,λL,A) Economics is quite straightforward: Replication argument implies CRS to objects Therefore there must be IRS to objects and ideas Romer (1990) and Jones (2005) p. 8

9 A Simple Model Romer (1990) and Jones (2005) p. 9

10 The Simple Model Production of final good Y t = A σ tl Yt Production of ideas A t = ν(a t )L At = νl At A φ t Resource constraint L Yt +L At = L t = L 0 e nt Allocation of labor L At = sl t, 0 < s < 1 φ > 0: Standing on shoulders φ < 0: Fishing out Romer (1990) and Jones (2005) p. 10

11 Solving (1) Y t = A σ t L Y t (2) A t = ν(a t )L At = νl At A φ t (3) L Y t +L At = L t = L 0 e nt (4) L At = sl t, 0 < s < 1 Romer (1990) and Jones (2005) p. 11

12 Discussion: g y = σn 1 φ Growth rate is the product of (1) degree of increasing returns and (2) rate at which scale is rising. More people more ideas more income per capita. But China is huge while Hong Kong is tiny? But Africa has fast population growth while Europe has slow? What happens if s permanently increases? Romer (1990) and Jones (2005) p. 12

13 From IRS to Growth Objects: Add one computer make one worker more productive. Output per worker # of computers per worker Ideas: Add one new idea make everyone better off. E.g. computer code for 1st spreadsheet or the software protocols for the internet itself Income per person the aggregate stock of knowledge, not on the number of ideas per person. But it is easy to make aggregates grow: population growth! IRS bigger is better. Romer (1990) and Jones (2005) p. 13

14 Romer (1990) and Scale Effects Romer (1990) / Aghion-Howitt (1992) / Grossman-Helpman (1991) have φ = 1 A t A t = νl At = ν sl t Policy effect: s raises long-run growth. Problem (Jones 1995): Growth in number of researchers (or population) implies accelerating growth True over the very long run (Kremer 1993) But not in the 20th century for the United States. Romer (1990) and Jones (2005) p. 14

15 Researchers in Advanced Countries Romer (1990) and Jones (2005) p. 15

16 U.S. GDP per Person Romer (1990) and Jones (2005) p. 16

17 Scale Effects Strong versus weak (versus none) Strong: φ = 1 scale affects the growth rate in LR Weak: φ < 1 scale affects the level in LR Literature Young (1998), Peretto (1998), Dinopoulos-Thompson (1998), Howitt (1999); discussed in Jones (1999 AEAPP) No role for scale? What about nonrivalry? Romer (1990) and Jones (2005) p. 17

18 What happens ifφ > 1? Romer (1990) and Jones (2005) p. 18

19 What happens ifφ > 1? Let φ = 1 + ǫ and assume population is constant. Ȧ t = νla 1+ǫ t Integrating this differential equation from 0 to t gives A t = ( 1 A ǫ 0 ǫνlt ) 1/ǫ There exists a finite date t at which point A becomes infinite! Romer (1990) and Jones (2005) p. 19

20 The Full Romer Model Romer (1990) and Jones (2005) p. 20

21 Overview of Romer Model All of the key insights regarding nonrivalry and increasing returns. In addition, a significant troubling question was how to decentralize the allocation of resources With increasing returns, why doesn t one firm come to dominate? Imperfect competition, a la Spence (1976), Dixit-Stiglitz (1977), and Ethier (1982, production side) Romer (1990) and Jones (2005) p. 21

22 The Economic Environment Final good Y t = ( At 0 x θ it di ) α/θl 1 α Yt, 0 < θ < 1 Capital K t = I t δk t, C t +I t = Y t Production of ideas A t = νl λ At Aφ t, φ < 1 Resource constraint (capital) Resource constraint (labor) At 0 x it di = K t L Yt +L At = L t = L 0 e nt Preferences U t = t L s u(c s )e ρ(s t) ds, u(c) = c1 ζ 1 1 ζ Romer (1990) and Jones (2005) p. 22

23 Allocating Resources Environment features 9 unknowns: Y,A,{x i },L Y,L A,K,C,I,L 6 1/2 equations (use x it di = K to get x t below) Need 2 1/2 more equations to complete A rule of thumb allocation in this economy features I t /Y t = s K (0,1) L At /L t = s A (0,1) x it = x t for all i [0,A t ]. Romer (1990) and Jones (2005) p. 23

24 Balanced Growth Path A balanced growth path in this economy is a situation in which all variables grow at constant exponential rates (possibly zero) forever. Romer (1990) and Jones (2005) p. 24

25 Result 1 (Rule of Thumb) (a) Symmetry of capital goods implies Y t = A σ tk α t L 1 α Yt, σ α( 1 θ 1) (b) Along BGP, y depends on the total stock of ideas: y t = ( s K n+g k +δ ) α 1 α A σ 1 α t. (c) Along BGP, the stock of ideas depends on the number of researchers ( ) 1 ν A 1 φ t = L λ 1 φ At g A Romer (1990) and Jones (2005) p. 25

26 Result 1 (continued) (d) Combining these last two results(b) and (c), y t L γ At = ( s AL t ) γ, γ σ 1 α (e) Finally, TLAD gives the growth rates λ 1 φ g y = g k = σ 1 α g A = γg LA = g γn. Romer (1990) and Jones (2005) p. 26

27 The Optimal Allocation The optimal allocation features time paths {c t,s At,{x it }} t=0 that maximize utility U t at each point in time given the economic environment. Using symmetry of x it : subject to max {c t,s At } 0 L t u(c t )e ρt dt y t = A σ tk α t (1 s At ) 1 α A t = ν(s At L t ) λ A φ t k t = y t c t (n+δ)k t Romer (1990) and Jones (2005) p. 27

28 The Maximum Principle in the Romer Model The Hamiltonian for the optimal allocation is H t = u(c t )+µ 1t (y t c t (n+δ)k t )+µ 2t νs λ AtNt λ A φ t, where y t = A σ tkt α (1 s At ) 1 α. First order necessary conditions H t / c t = 0, H t / s At = 0 ρ = H t/ k t µ 1t + µ 1t µ 1t, ρ = H t/ A t µ 2t + µ 2t µ 2t lim µ 1te ρt k t = 0, lim µ 2t e ρt A t = 0 t t Romer (1990) and Jones (2005) p. 28

29 Result 2 (Optimal Allocation) (a) All of Result 1 continues to hold. Same long-run growth rate! (b) Optimal consumption satisfies the Euler equation ċ t c t = 1 ζ (c) Optimal saving along BGP ( ) yt δ ρ k t s op K = α(n+g +δ) ρ+δ +ζg Romer (1990) and Jones (2005) p. 29

30 Result 2 (Optimal Allocation continued) (d) Optimal labor allocation equates marginal products s op At 1 s op At Along the BGP, this implies = µ 2t µ 1t λȧt (1 α)y t s op A 1 s op A = σy t /A t r (g Y g A ) φg A λa t (1 α)y t Romer (1990) and Jones (2005) p. 30

31 Equilibrium in the Romer Model Romer (1990) and Jones (2005) p. 31

32 Market Equilibrium with Imperfect Competition Increasing returns means a perfectly competitive equilibrium does not exist. Romer (1990) adds infinitely lived patents on ideas to set up an equilibrium with imperfect competition as in Spence (1976), Dixit-Stiglitz (1977), Ethier (1982). Partial excludability associated with patents leads individuals to exert effort to discover new ideas. We define the decision problems first and then the equilibrium. Romer (1990) and Jones (2005) p. 32

33 Households: Problem (HH) Taking the time path of {w t,r t } as given, HHs solve max {c t } 0 u(c t )e ρt dt subject to v t = (r t n)v t +w t c t, v 0 given lim v te t (rs n)ds 0 0 t where v t is financial wealth, w t is the wage, and r t is the interest rate. Romer (1990) and Jones (2005) p. 33

34 Final Goods Producers: Problem (FG) Perfectly competitive At each t, taking w t, A t, and {p it } as given, the representative firm chooses L Yt and {x it } to solve max {x it },L Y t ( At 0 ) α/θ x θ it di L 1 α Yt w t L Yt At 0 p it x it di. Romer (1990) and Jones (2005) p. 34

35 Capital Goods Producers: Problem (CG) Monopolistic competition a patent gives a single firm the exclusive right to produce each variety. At each t and for each capital good i, taking r t and x( ) as given, a monopolist solves max p it π it (p it r t δ)x(p it ) where x(p it ) is the (constant elasticity) demand from the final goods sector for variety i from the FOC in Problem (FG). Romer (1990) and Jones (2005) p. 35

36 Idea Producers: Problem (R&D) Perfectly competitive, seeing the idea production function as A t = ν t L At The representative research firm solves max L At P At ν t L At w t L At taking the price of an idea P At, research productivity ν t, and the wage rate w t as given. Romer (1990) and Jones (2005) p. 36

37 Market Equilibrium An equilibrium with imperfect competition is {c t,{x it },Y t,k t, I t,v t,{π it },L Yt,L At,L t,a t, ν t } t=0 and prices {w t,r t, {p it },P At } t=0 such that for all t: 1. c t, v t solve Problem (HH). 2. {x it } and L Y t solve Problem (FG). 3. p it and π it solve Problem (CG) for all i [0,A t ]. 4. L At solves Problem (R&D). 5. (r t ) The capital market clears: V t v t L t = K t +P At A t. 6. (w t ) The labor market clears: L Y t +L At = L t. 7. ( ν t ) The idea production function is satisfied: ν t = νl λ 1 At A φ t. 8. (K t ) The capital resource constraint is satisfied: A t 0 x it di = K t. 9. (P At ) Assets have equal returns: r t = π it P + At. P At P At 10. Y t, A t,l t, I t are determined by their production functions (16 equations, 16 unknowns. Goods market clears by Walras Law: C t +I t = Y t ) Romer (1990) and Jones (2005) p. 37

38 Result 3 (Market Equilibrium Allocation) (a) All of Result 1 continues to hold: Same long-run growth rate! (b) The same Euler equation characterizes consumption ċ t c t = 1 ζ (req t ρ) Note: Since the growth rate is the same and the Euler equation is the same, r eq = r (steady state). Romer (1990) and Jones (2005) p. 38

39 Result 3 (Market Equilibrium continued) (c) Capital goods are priced with the usual monopoly markup: p eq it = peq t 1 θ (req t +δ) As a result, capital is paid less than its marginal product: r eq t = αθ Y t K t δ. Same growth rate same interest rate. Underpaying capital leads to low K/Y : s eq K = αθ(n+g +δ) ρ+δ +ζg = θs op K Romer (1990) and Jones (2005) p. 39

40 Result 3 (Market Equilibrium continued) (d) Free flow of labor equates marginal products s eq At 1 s eq At = P AtȦt (1 α)y t Along BGP s eq A 1 s eq A = σθy t /A t r eq (g Y g A ) A t (1 α)y t Romer (1990) and Jones (2005) p. 40

41 Compares op A andseq A s eq A 1 s eq A = σθy t /A t r eq (g Y g A ) A t (1 α)y t, s op A 1 s op A = σy t /A t r (g Y g A ) φg A λa t (1 α)y t Three differences: λ < 1: Duplication externality φ 0: Knowledge spillover externality θ < 1: Appropriability effect Equilibrium may involve either too much or too little research. Romer (1990) and Jones (2005) p. 41

42 Summary More people more ideas more per capita income (nonrivalry) Log-difference per capita growth is proportional to population growth, where proportionality measures increasing returns Long-run growth rate is independent of the allocation of resources Subsides or taxes on research affect growth along the transition path and have long-run level effects (like Solow) In contrast, in Romer (1990), they lead to long-run growth effects Romer (1990) and Jones (2005) p. 42

43 Applications Romer (1990) and Jones (2005) p. 43

44 Several Applications of this Framework 1. Endogenizing population growth 2. Growth over the very long run 3. The linearity critique 4. Growth accounting Romer (1990) and Jones (2005) p. 44

45 1. Endogenizing Population Growth (Jones 2003) Let fertility be a choice variable in a Barro and Becker (1989) kind of framework. Converts this to a fully endogenous growth model But policy can have odd effects: a subsidy to research shifts labor toward producing ideas but away from producing kids lower long-run growth! A recent reference on some intriguing issues that arise along this line of research is Cordoba and Ripoll, The Elasticity of Intergenerational Substitution, Parental Altruism, and Fertility Choice 2014 working paper. Romer (1990) and Jones (2005) p. 45

46 2. Growth over the Very Long Run Malthus: c = y = AL α, α < 1 Fixed supply of land: L c holding A fixed Story (Lee 1988, Kremer 1993): 100,000 BC: small population ideas come very slowly New ideas temporary blip in consumption, but permanently higher population This means ideas come more frequently Eventually, ideas arrive faster than Malthus can reduce consumption! People produce ideas and Ideas produce people If nonrivarly > Malthus, this leads to the hockey stick Romer (1990) and Jones (2005) p. 46

47 Population and Per Capita GDP: the Very Long Run INDEX (1.0 IN INITIAL YEAR) Per capita GDP Population YEAR Romer (1990) and Jones (2005) p. 47

48 3. The Linearity Critique The result of any successful growth model is an equation like ẏ t = ḡy t, with a story about ḡ. An essential ingredient to getting this result is (essentially) some linear differential equation somewhere in the economic environment: Ẋ t = X t Growth models differ according to what they call the X t variable and how they fill in the blank. Romer (1990) and Jones (2005) p. 48

49 Catalog of Growth Models: What isx t? Solow y t = k α, kt = k α t c t δk t Solow Ȧ t = ḡa t AK model Y t = ĀK t, K t = AK t C t δk t Lucas Y t = K α t (u t h t L) 1 α, ḣ t = (1 u t )h t Romer (φ = 1) Ȧ t = νl at A t Romer (φ < 1) Lt = nl t Romer (1990) and Jones (2005) p. 49

50 4. Growth Accounting Jones (2002): Sources of U.S. Economic Growth in a World of Ideas Puzzle (earlier graphs) A straight line fits log U.S. GDP per person quite well But human capital and R&D investment rates appear to be rising Human capital: Completion rates for adult population 1940: 25% high school, 5% college 1993: 80% high school, 20% college U.S. science/eng researchers as a fraction of labor force: 1950: 1/4 of 1% 1993: 3/4 of 1% Romer (1990) and Jones (2005) p. 50

51 How to reconcile? Imagine a Solow model What happens if the investment rate grows over time? Same thing could be going on in an idea model with φ < 1 Only it is human capital and R&D investment rates that rise Implication for the long run... Romer (1990) and Jones (2005) p. 51

52 Key Equations of the Model Production of final good Production of ideas Y t = A σ tk α t H 1 α yt A t = νh λ ata φ t Efficiency units of labor H it = h t L it, h t = e ψl ht Resource constraint L yt +L at = (1 l ht )L t Rewriting the production function y t = ( Kt Y t ) α 1 α lyt h t A σ 1 α t Balanced growth path: g y = γn where γ σ 1 α λ 1 φ Romer (1990) and Jones (2005) p. 52

53 Accounting for Growth (γ = 1/3), Educational attainment rises 1 year per decade. With ψ =.06 about 0.6 percentage points of growth per year. Transition dynamics are 80 percent of growth. Steady state growth is only 20 percent of recent growth! Numbers from The Future of U.S. Economic Growth (with Fernald) Romer (1990) and Jones (2005) p. 53

54 Alternative Futures? The shape of the idea production function, f(a) Increasing returns The past GPT "Waves" Today Run out of ideas The stock of ideas, A Romer (1990) and Jones (2005) p. 54