Consumption-led Growth

Transcription

1 Consumption-led Growth (preliminary work in progress) Markus Brunnermeier Pierre-Olivier Gourinchas Oleg Itskhoki Princeton University November / 25

2 Motivation 1 What is the relationship between openness and growth? trade openness financial openness 1 / 25

3 Motivation 1 What is the relationship between openness and growth? trade openness financial openness 2 Is it possible to borrow like Argentina or Spain and grow like China? (i) What is wrong with Spanish-style (consumption-led) growth? (ii) What is special about Chinese-style (export-led) growth? 1 / 25

4 Motivation 1 What is the relationship between openness and growth? trade openness financial openness 2 Is it possible to borrow like Argentina or Spain and grow like China? (i) What is wrong with Spanish-style (consumption-led) growth? (ii) What is special about Chinese-style (export-led) growth? A model of endogenous convergence growth to open the blackbox of productivity evolution under different regimes openness to international trade and capital flows a neoclassical (DRS) environment with endogenous innovation decisions by entrepreneurs emphasis on the feedback from international borrowing into the pace and composition (T vs NT) of converegence 1 / 25

5 Empirical Motivation I Figure: CA imbalances in the Euro Zone 2 / 25

6 Empirical Motivation II Figure: Sectoral reallocation in the Euro Zone 2 / 25

7 Main Insights Openness has two effects: (i) change in the relative size of the market (ii) increase in both foreign competition and in domestic cost of production (unit labor costs) 3 / 25

8 Main Insights Openness has two effects: (i) change in the relative size of the market (ii) increase in both foreign competition and in domestic cost of production (unit labor costs) With balanced trade, it s a wash: trade openness does not affect the pace and direction of productivity growth 3 / 25

9 Main Insights Openness has two effects: (i) change in the relative size of the market (ii) increase in both foreign competition and in domestic cost of production (unit labor costs) With balanced trade, it s a wash: trade openness does not affect the pace and direction of productivity growth Trade deficits unambiguously favor the non-tradable sector and tend to reduce the pace of innovation a reduced-form relationship between NX and sectoral growth furthermore, NX /Y is a sufficient statistic 3 / 25

10 Main Insights Openness has two effects: (i) change in the relative size of the market (ii) increase in both foreign competition and in domestic cost of production (unit labor costs) With balanced trade, it s a wash: trade openness does not affect the pace and direction of productivity growth Trade deficits unambiguously favor the non-tradable sector and tend to reduce the pace of innovation a reduced-form relationship between NX and sectoral growth furthermore, NX /Y is a sufficient statistic Sudden stops in financial flows are followed by both recessions and fast tradable productivity growth take off due to structural productivity imbalance, without sticky wages wage flexibility essential for a sharp productivity rebound 3 / 25

11 Main Insights Openness has two effects: (i) change in the relative size of the market (ii) increase in both foreign competition and in domestic cost of production (unit labor costs) With balanced trade, it s a wash: trade openness does not affect the pace and direction of productivity growth Trade deficits unambiguously favor the non-tradable sector and tend to reduce the pace of innovation a reduced-form relationship between NX and sectoral growth furthermore, NX /Y is a sufficient statistic Sudden stops in financial flows are followed by both recessions and fast tradable productivity growth take off due to structural productivity imbalance, without sticky wages wage flexibility essential for a sharp productivity rebound Laissez-faire productivity growth is in general suboptimal capital controls may improve upon market allocation 3 / 25

12 Learning-by-doing and dutch disease Literature Corden and Neary (1982), Krugman (1987), Young (1991)... Export-led growth: Rajan and Subramanian (2005) Trade and growth: Ventura (1997), Acemoglu and Ventura (2002) Technology transfer: Parente and Prescott (2002) Empirics: Frankel and Romer (1999) Transition growth after financial liberalization Aioke, Benigno and Kiyotaki (2009) Growth and trade with Frechet distribution: Kortum (1997), EK (2001, 2002), Klette and Kortum (2004) 4 / 25

13 MODEL SETUP 5 / 25

14 Model Setup Real small open economy in continuous time exogenous world interest rate r in terms of world good Two sectors: tradable (exportable) and non-tradable (non-exportable) Rest of the world (ROW) in steady state: W = A T = A N = A and P F = P N = P = 1 We study convergence growth trajectories starting from A T (0), A N (0) < A 5 / 25

15 Households Representative household: max {C(t),L(t)} 0 e ϑt U(t)dt, U = 1 1 σ C 1 σ 1 1+ϕ L1+ϕ s.t. Ḃ = r B+NX, NX = WL + Π PC Results in labor supply: C σ t L ϕ t = W t P t w t Aggregate GDP and absorbtion: GDP = WL + Π and Y = PC GDP = Y + NX Special cases: σ 1 and ϕ (L = L) 6 / 25

16 Two sectors: Demand where Y = PC = γp T C T + (1 γ)p N C N C = C γ T C 1 γ N and C T = [ κ 1 ρ 1 ρ C ρ F ] ρ + (1 κ) 1 ρ 1 ρ 1 ρ C ρ H, ρ > 1 7 / 25

17 Two sectors: Demand where Y = PC = γp T C T + (1 γ)p N C N C = C γ T C 1 γ N and C T = [ κ 1 ρ 1 ρ C ρ F Aggregators of individual varieties: [ ] ρ 1 ΛT C H = C H (i) ρ 1 ρ 1 ρ di and C N = γ 0 ] ρ + (1 κ) 1 ρ 1 ρ 1 ρ C ρ H, ρ > 1 [ 1 1 γ ΛN 0 ] ρ C N (i) ρ 1 ρ 1 ρ di 7 / 25

18 Two sectors: Demand where Y = PC = γp T C T + (1 γ)p N C N C = C γ T C 1 γ N and C T = [ κ 1 ρ 1 ρ C ρ F Aggregators of individual varieties: [ ] ρ 1 ΛT C H = C H (i) ρ 1 ρ 1 ρ di and C N = γ 0 P T ] ρ + (1 κ) 1 ρ 1 ρ 1 ρ C ρ H, ρ > 1 [ 1 1 γ ΛN 0 ] ρ C N (i) ρ 1 ρ 1 ρ di Demand: ( ) ρ ( ) ρ PH (i) Y PN (i) Y C H (i) = (1 κ) and C N (i) = P T Price indexes P N P N 7 / 25

19 Tradable expenditure: Exports and Imports Aggregate imports: γp T C T = γp F C F + X = γp F C F = γκ Aggregate exports: Net exports: ( PF P T ΛT 0 P H (i)c H (i)di ) 1 ρ Y, P F = τp F = τ X = γp H C H = γκ(τp H) 1 ρ Y [ ] NX = X X = γκτ 1 ρ P 1 ρ H Y P ρ 1 T Y 8 / 25

20 Tradable expenditure: Exports and Imports Aggregate imports: γp T C T = γp F C F + X = γp F C F = γκ Aggregate exports: Net exports: ( PF P T ΛT 0 P H (i)c H (i)di ) 1 ρ Y, P F = τp F = τ X = γp H C H = γκ(τp H) 1 ρ Y [ NX = X X = γκτ 1 ρ P 1 ρ H Y P ρ 1 T Y ] = γκτ 1 ρ [ S ρ 1 Y 1 κτ 1 ρ +(1 κ)s ρ 1 Y ], S = P 1 H 8 / 25

21 Technology: Technology and Revenues Y J (i) = A J (i)l J (i), i [0, Λ J ], J {T, N} Marginal cost pricing if technology is non-excludable (same in N): P H (i) = W P H = W [, A T = 1 ΛT A T (i) A γ 0 A T (i) ρ 1 di T ] 1 ρ 1 9 / 25

22 Technology: Technology and Revenues Y J (i) = A J (i)l J (i), i [0, Λ J ], J {T, N} Marginal cost pricing if technology is non-excludable (same in N): P H (i) = W P H = W [, A T = 1 ΛT A T (i) A γ 0 A T (i) ρ 1 di T Revenues: ( ) PN (i) 1 ρ R N (i) = P N (i)c N (i) = R N, where P N R T (i) = P H (i)c H (i) + P H (i)c H (i) = ( PH (i) P H R N = Y and R T = (1 κ) ( PH P T ) 1 ρ R T ) 1 ρ Y + κ(τp H ) 1 ρ Y ] 1 ρ 1 9 / 25

23 Technology Draws An entrepreneur has n 1 possible ideas (projects): Z J(l) (l) iid Frechet(z, θ), l = 1..n, θ > ρ 1 A fraction γ of ideas are tradable, J(l) = T 10 / 25

24 Technology Draws An entrepreneur has n 1 possible ideas (projects): Z J(l) (l) iid Frechet(z, θ), l = 1..n, θ > ρ 1 A fraction γ of ideas are tradable, J(l) = T An entrepreneur can adopt only one project The technology is privately owned for one period, then rival 10 / 25

25 Technology Draws An entrepreneur has n 1 possible ideas (projects): Z J(l) (l) iid Frechet(z, θ), l = 1..n, θ > ρ 1 A fraction γ of ideas are tradable, J(l) = T An entrepreneur can adopt only one project The technology is privately owned for one period, then rival Period profits: Π T (l) = 1 ( ) ρ W 1 1 ρ R T ρ ρ 1 Z T (l) Π N (l) = 1 ( ρ W ρ ρ 1 Z N (l) P H ) 1 1 ρ R N P N 10 / 25

26 Technology Draws An entrepreneur has n 1 possible ideas (projects): Z J(l) (l) iid Frechet(z, θ), l = 1..n, θ > ρ 1 A fraction γ of ideas are tradable, J(l) = T An entrepreneur can adopt only one project The technology is privately owned for one period, then rival Period profits: Π T (l) = 1 ( ρ W ρ ρ 1 Z T (l) Π N (l) = 1 ( ρ W ρ ρ 1 Z N (l) ) 1 1 ρ R T = ϱ R T A ρ 1 Z T (l) ρ 1 T ) 1 1 ρ R N = ϱ R N A ρ 1 Z N (l) ρ 1 N P H P N 10 / 25

27 Project choice: Technology Adoption ˆl = arg max l=1..n Π J(l)(l) and we define (Ẑ T, Ẑ N, Ẑ) and (ˆΠ T, ˆΠ N, ˆΠ) 11 / 25

28 Project choice: Technology Adoption ˆl = arg max l=1..n Π J(l)(l) and we define (Ẑ T, Ẑ N, Ẑ) and (ˆΠ T, ˆΠ N, ˆΠ) Lemma 1 (i) The probability to adopt a tradable project: π T P{ˆΠ T ˆΠ N } = γ χ θ ρ 1 γ χ θ ρ γ, χ ( PH P N ) ρ 1 R T R N. 11 / 25

29 Project choice: Technology Adoption ˆl = arg max l=1..n Π J(l)(l) and we define (Ẑ T, Ẑ N, Ẑ) and (ˆΠ T, ˆΠ N, ˆΠ) Lemma 1 (i) The probability to adopt a tradable project: π T P{ˆΠ T ˆΠ N } = γ χ θ ρ 1 γ χ θ ρ γ, χ ( AN A T ) ρ 1 R T R N. 11 / 25

30 Project choice: Technology Adoption ˆl = arg max l=1..n Π J(l)(l) and we define (Ẑ T, Ẑ N, Ẑ) and (ˆΠ T, ˆΠ N, ˆΠ) Lemma 1 (i) The probability to adopt a tradable project: π T P{ˆΠ T ˆΠ N } = γ χ θ ρ 1 γ χ θ ρ γ, χ ( AN A T (ii) The productivity conditional on adoption: ρ 1 E {Ẑ T ˆΠT ˆΠ } ( ) ν 1 πt N = A ρ 1, γ ) ρ 1 R T R N. where A EẐ = (nz) 1/θ Γ(ν) 1 ρ 1 and ν 1 ρ 1 θ (0, 1). 11 / 25

31 Productivity Dynamics λ is the innovation rate and δ is the rate at which technologies become obsolete: Λ T = λπ T δλ T Assume λ is country-specific and λ δ 12 / 25

32 Productivity Dynamics λ is the innovation rate and δ is the rate at which technologies become obsolete: Λ T = λπ T δλ T Assume λ is country-specific and λ δ Lemma 2 The sectoral productivity dynamics is given by: [ A ( T = δ ) ρ 1 ( ) Ā ν πt 1] A T ρ 1 A T γ where Ā A ( λ δ ) 1 ρ / 25

33 CLOSED ECONOMY 13 / 25

34 Closed Economy, κ 0 In closed economy R T = R N = Y, and therefore: ( ) ρ 1 ( ) ρ 1 PH AN χ = = P N The project choice is, thus: π T (t) 1 π T (t) = γ 1 γ A T ( ) AN (t) θ A T (t) 13 / 25

35 Closed Economy, κ 0 In closed economy R T = R N = Y, and therefore: ( ) ρ 1 ( ) ρ 1 PH AN χ = = P N The project choice is, thus: π T (t) 1 π T (t) = γ 1 γ A T ( ) AN (t) θ A T (t) Proposition 1 (i) Starting from A T (0) = A N (0), equilibrium project choice in the closed economy is π T (t) γ, A T (t) = [e δt A T (0) ρ 1 + ( 1 e δt) Ā ρ 1] 1 ρ 1 and Λ T = γ λ δ. 13 / 25

36 Closed Economy, κ 0 In closed economy R T = R N = Y, and therefore: ( ) ρ 1 ( ) ρ 1 PH AN χ = = P N The project choice is, thus: π T (t) 1 π T (t) = γ 1 γ A T ( ) AN (t) θ A T (t) Proposition 1 (i) Starting from A T (0) = A N (0), equilibrium project choice in the closed economy is π T (t) γ, A T (t) = [e δt A T (0) ρ 1 + ( 1 e δt) Ā ρ 1] 1 ρ 1 and Λ T = γ λ δ. (ii) Equilibrium allocation C = w 1+ϕ σ+ϕ, L = w 1 σ σ+ϕ, w = A. (iii) Efficiency: / 25

37 OPEN ECONOMY I BALANCED TRADE 14 / 25

38 Balanced Trade Consider open economy with κ > 0 and τ 1 Lemma 3 (i) The relative revenue shifter is given by: ( ) 1 ρ R T PH = (1 κ) + κ(τp H ) 1 ρ Y R N Y = 1 + NX γy. P T (ii) Under balanced trade, χ = (A N /A T ) ρ 1, and hence π T (t) and (A T (t), A N (t)) follow the same path as in autarky. 14 / 25

39 Balanced Trade Consider open economy with κ > 0 and τ 1 Lemma 3 (i) The relative revenue shifter is given by: ( ) 1 ρ R T PH = (1 κ) + κ(τp H ) 1 ρ Y R N Y = 1 + NX γy. P T (ii) Under balanced trade, χ = (A N /A T ) ρ 1, and hence π T (t) and (A T (t), A N (t)) follow the same path as in autarky. Equilibrium allocation is nonetheless different from autarkic. For σ = 1: ( 1 A ) κγ 1+(2 κ)(ρ 1) w = C = A τ 2ρ 1 A T 14 / 25

40 Balanced Trade Consider open economy with κ > 0 and τ 1 Lemma 3 (i) The relative revenue shifter is given by: ( ) 1 ρ R T PH = (1 κ) + κ(τp H ) 1 ρ Y R N Y = 1 + NX γy. P T (ii) Under balanced trade, χ = (A N /A T ) ρ 1, and hence π T (t) and (A T (t), A N (t)) follow the same path as in autarky. Equilibrium allocation is nonetheless different from autarkic. For σ = 1: ( 1 A ) κγ 1+(2 κ)(ρ 1) w = C = A τ 2ρ 1 A T Laisser-faire productivity dynamics is suboptimal. The planner would choose π T (t) < γ for all t / 25

41 OPEN ECONOMY II FINANCIAL OPENNESS 15 / 25

42 Financial Openness With open current account: π T = γ 1 π T 1 γ ( AN A T ) θ [ 1 + NX ] θ ρ 1 γy 15 / 25

43 Financial Openness With open current account: π T = γ 1 π T 1 γ ( AN A T ) θ [ 1 + NX ] θ ρ 1 γy Lemma 4 NX (t)<0 and A T (t) A N (t) A T (t)< A N (t). 15 / 25

44 Financial Openness With open current account: π T = γ 1 π T 1 γ ( AN A T ) θ [ 1 + NX ] θ ρ 1 γy Lemma 4 NX (t)<0 and A T (t) A N (t) A T (t)< A N (t). Proposition 5 In st.st. with NX = r B > 0: Ā T > Ā > Ā N. 15 / 25

45 Financial Openness With open current account: π T = γ 1 π T 1 γ ( AN A T ) θ [ 1 + NX ] θ ρ 1 γy Lemma 4 NX (t)<0 and A T (t) A N (t) A T (t)< A N (t). Proposition 5 In st.st. with NX = r B > 0: Ā T > Ā > Ā N. Proposition 6 Starting from A T (0) = A N (0) < Ā, there exist two cutoffs 0 < t 1 < t 2 < : NX (t) < 0 for t [0, t 1 ) and NX (t) > 0 for t > t 1, and A T (t) < A N (t) for t (0, t 2 ) and A T (t) > A N (t) for t > t 2. At t = t 2, A T (t) = A N (t) = A(t) < A a (t). 15 / 25

46 Convergence Path Figure: Productivity convergence in closed and open economies 16 / 25

47 0.5 Impact of Openness Two effects of openness: 1 Relative size of the market: Y /Y 2 Competition: P T /P H < 1 =X /X 1 + NX ( ) [ {}}{ 1 ρ ( ) γy = PH τ 1 ρ P 1 ρ H (1 κ) + κ Y ] P T P H P ρ 1 T Y 17 / 25

48 ENDOGENOUS INNOVATION 18 / 25

49 Endogenous Innovation Rate Endogenous participation of entrepreneurs if E ˆΠ φw : ( ) EˆΠ EˆΠ λ = Φ and W W = ϱr { } N/W A ρ 1 E max χẑ ρ 1 T, Ẑ ρ 1 N N 18 / 25

50 Endogenous Innovation Rate Endogenous participation of entrepreneurs if E ˆΠ φw : ( ) EˆΠ EˆΠ λ = Φ and W W = ϱr { } N/W A ρ 1 E max χẑ ρ 1 T, Ẑ ρ 1 N N Lemma 5 EˆΠ ( A W = ϱ A ) ρ 1 A Ψ (1 + nx) Â θ 18 / 25

51 Endogenous Innovation Rate Endogenous participation of entrepreneurs if E ˆΠ φw : ( ) EˆΠ EˆΠ λ = Φ and W W = ϱr { } N/W A ρ 1 E max χẑ ρ 1 T, Ẑ ρ 1 N N Lemma 5 EˆΠ ( A W = ϱ A ) ρ 1 A Ψ (1 + nx) Â θ Proposition 8 (i) λ is increasing in A /A and in A/Âθ 1. (ii) λ increases with trade openness iff σ < 1 and ϕ <. ] nx, [ ( ) 1 γ (iii) When σ = 1, Ψ 1 + AN A T and λ increases with NX when A N A T. ϕ 1+ϕ 18 / 25

52 Endogenous Innovation Rate Endogenous participation of entrepreneurs if E ˆΠ φw : ( ) EˆΠ EˆΠ λ = Φ and W W = ϱr { } N/W A ρ 1 E max χẑ ρ 1 T, Ẑ ρ 1 N N Lemma 5 EˆΠ ( A W = ϱ A ) ρ 1 A Ψ (1 + nx) Â θ Proposition 8 (i) λ is increasing in A /A and in A/Âθ 1. (ii) λ increases with trade openness iff σ < 1 and ϕ <. ] nx, [ ( ) 1 γ (iii) When σ = 1, Ψ 1 + AN A T and λ increases with NX when A N A T. ϕ 1+ϕ Endogenous non-tradable tilt reinforces the negative effect of trade deficits on innovation rate 18 / 25

53 Endogenous Innovation Rate Endogenous participation of entrepreneurs if E ˆΠ φw : ( ) EˆΠ EˆΠ λ = Φ and W W = ϱr { } N/W A ρ 1 E max χẑ ρ 1 T, Ẑ ρ 1 N N Lemma 5 EˆΠ ( A W = ϱ A ) ρ 1 A Ψ (1 + nx) Â θ Proposition 8 (i) λ is increasing in A /A and in A/Âθ 1. (ii) λ increases with trade openness iff σ < 1 and ϕ <. ] nx, [ ( ) 1 γ (iii) When σ = 1, Ψ 1 + AN A T and λ increases with NX when A N A T. ϕ 1+ϕ Endogenous non-tradable tilt reinforces the negative effect of trade deficits on innovation rate Induce NX > 0 with policy if the goal is max growth rate 18 / 25

54 Comparison with Learning-by-Doing General learning-by-doing formulation: Y T (t) = F ( A T (t), L T (t) ), A T (t) = G ( A T (t), A N (t), L T (t), L N (t) ) 19 / 25

55 Comparison with Learning-by-Doing General learning-by-doing formulation: Y T (t) = F ( A T (t), L T (t) ), A T (t) = G ( A T (t), A N (t), L T (t), L N (t) ) Mapping of the baseline model into learning-by-doing: F (A T, L T ) = AL, G(A T, A N, L T, L N ) = G ( A T, π T (A T, A N, L T, L N ) ), [ ( G(A T, π T ) = δ ) ρ 1 ( ) Ā ν πt 1], ρ 1 γ A T π T 1 γ 1 π T γ = ( AN A T ) θ ( RT R N ) θ ρ 1 and R T R N = L T L N 19 / 25

56 EMPIRICAL IMPLICATIONS 20 / 25

57 Empirical Implications Reduced-form relationship between NX and sectoral growth: A T (t) A T (t) A [ N (t) A N (t) = g 0 (ρ 1) log A T (t) A N (t) + ν ( ] πt (t) γ) γ(1 γ) [ = g 0 (ρ 1) (1 + µ) log A T (t) A N (t) + µ ] NX (t), γ Y (0) with g 0 ( ) δ λ A ρ 1, ρ 1 δ A 0 which is also the base growth rate holds whether NX 0 are market outcomes or policy-induced i.e., applies equally for NX < 0 in Spain and NX > 0 in China NX /Y is a sufficient statistic for the feedback effect from equilibrium allocation to sectoral productivity growth 20 / 25

58 Empirical Implications Reduced-form relationship between NX and sectoral growth: A T (t) A T (t) A [ N (t) A N (t) = g 0 (ρ 1) log A T (t) A N (t) + ν ( ] πt (t) γ) γ(1 γ) [ = g 0 (ρ 1) (1 + µ) log A T (t) A N (t) + µ ] NX (t), γ Y (0) with g 0 ( ) δ λ A ρ 1, ρ 1 δ A 0 which is also the base growth rate holds whether NX 0 are market outcomes or policy-induced i.e., applies equally for NX < 0 in Spain and NX > 0 in China NX /Y is a sufficient statistic for the feedback effect from equilibrium allocation to sectoral productivity growth Preliminary evidence of this effect in the KLEMS data CA/Y interacted with sector i tradability predicts sector i productivity growth rate in the panel of country-sectors 20 / 25

59 Two ULC measures: w/a and W /A T move together holding τ constant Unit Labor Costs Autarky (assume σ = 1): w a (t) = C a (t) = A(t) Balanced trade: ( A w b (t) = C b (t) = A(t) A T (t) Open financial account: w b (0) < w(0) < C(0) ) κγ 1+(2 κ)(ρ 1) ULC increase on impact and gradually fall along the convergence path > A(t) 21 / 25

60 APPLICATIONS 22 / 25

61 Application 1 Rollover crisis Sudden stop in capital flows during transition triggers a reversal in trade deficits and a recession in non-tradable sector Rapid take off in tradable productivity growth, provided labor market can flexibly adjust by a sharp decline in wages 2 Misallocation and growth policy 3 Physical capital and financial frictions 22 / 25

62 Rollover Crisis / 25

63 CONCLUSION 24 / 25

64 Conclusion Standard endogenous growth forces have a robust implication for the relationship between trade deficits and: 1 non-tradable tilt of innovation 2 overall lower speed of convergence growth Countries that borrow along the convergence growth trajectory are likely to experience asymmetric and slower convergence lagging tradable productivity high unit labor costs and depressed innovation rate particularly vulnerable to rollover crisis along such trajectories Countries that are concerned with GDP growth rather than welfare might find it optimal to subsidize exports 25 / 25

65 APPENDIX 26 / 25

66 Average sectoral prices: [ 1 ΛT P H = P H (i) 1 ρ di γ 0 Aggregate price indexes: ] 1 1 ρ P = P γ T P1 γ N where P T = Equilibrium sectoral prices: [ 1 and P N = 1 γ [ κp 1 ρ F Price Indexes back to slides ΛN 0 P N (i) 1 ρ di ] + (1 κ)p 1 ρ 1 1 ρ H P H = W A T, P N = W A N and P F = τ ] 1 1 ρ Real wage rate: w = W ( ) ] γ ρ 1 [1 P = A ρ 1 κ + κ W τa T, A A γ T A1 γ N 26 / 25

67 Solution for NX back to slides Equilibrium system: [ C = w 1+ϕ σ+ϕ 1 + NX ] ϕ σ+ϕ Y where ( W w = A τa T ) κγ and NX Y = γκ ( ) ρ κγ W τa T [ τ 1+ϕ 1 2ρ A σ+ϕ C ( ) ] A W (1 κγ)+(2 κ)(ρ 1) A T τa T 27 / 25

68 Efficiency in Closed Economy Proposition (i) If A T (0) = A N (0), then πt (t) = γ and A T (t) = A N (t) for all t maximizes A(t) for all t. (ii) If A N (t) > A T (t) at some t, then πt (t) ( γ, π T (t) ), and laissez-faire dynamics with π T (t) is suboptimal. Optimal policy satisfies (for J {T, N}): ( ) π 1 ν T 1 γ = ξ ( ) ρ 1 T AN, 1 πt γ ξ N A T where and b J (t)ξ T (t) ξ J (t) = a J (t), a J (t) ( ) η 1 ( ) ρ 1 ( AJ (t) A(t) ζ Ā πj (t), b J (t) ϑ + δ A(t) A J (t) γ J ) ν b J (t) plays the role of discount rate and a J (t) is the flow benefit ξ T /ξ N = R T /R N in the limit of ϑ (perfect impatience) Otherwise, ξ T /ξ T (1, R T /R N ) Patents with finite time-varying duration can decentralize π T (t) back to slides 28 / 25