Schumpeterian Growth Models

Transcription

1 Schumpeterian Growth Models Yin-Chi Wang The Chinese University of Hong Kong November, 2012

2 References: Acemoglu (2009) ch14

3 Introduction Most process innovations either increase the quality of an existing product or reduce the costs of production Competitive aspect of innovations: a newly-invented superior computer often replaces existing vintages Schumpeterian creative destruction 1. Direct price competition between producers with different vintages of quality or different costs of producing 2. Competition between incumbents and entrants: business stealing effect

4 Baseline Model Preferences and Technology 1 Continuous time. Representative household with standard CRRA preferences 0 exp ( ρt) C (t)1 θ 1 1 θ Constant population L: labor supplied inelastically. Resource constraint: dt (1) C (t) + X (t) + Z (t) Y (t) (2) Normalize the measure of inputs to 1, and denote each machine line by ν [0, 1].

5 Baseline Model Preferences and Technology 2 Engine of economic growth: quality improvement. q(ν, t) = quality of machine line ν at time t. Quality ladder for each machine type: where q(ν, t) = λ n(ν,t) q(ν, 0) for all ν and t (3) λ > 1 n(ν, t) = innovations on this machine line between 0 and t. Production function of the final good: Y (t) = 1 [ ] 1 1 β 0 q(ν, t)x (ν, t q)1 β dν L β (4) where x (ν, t q) = quantity of machine of type ν quality q.

6 Baseline Model Preferences and Technology 3 Implicit assumption in (4): at any point in time only one quality of any specific machine is used. Creative destruction: when a higher-quality machine is invented it will replace ( destroy ) the previous vintage of machines.

7 Baseline Model Innovation Technology 1 R&D technology: flow rate of innovation (cumulative R&D) where ηz (ν, t) /q (ν, t) Z (ν, t) : units of the final goods spent on research on machine line ν q(ν, t) : current quality of machine line ν Note that one unit of R&D spending is proportionately less effective when applied to a more advanced machine Assume free entry into research The firm that makes an innovation has a perpetual patent But other firms can undertake research based on the product invented by this firm

8 Baseline Model Innovation Technology 2 Once a machine of quality q(ν, t) has been invented, any quantity can be produced at the marginal cost ψq(ν, t). New entrants undertake the R&D and innovation: The incumbent has weaker incentives to innovate, since it would be replacing its own machine, and thus destroying the profits that it is already making (Arrow s replacement effect).

9 Equilibrium Allocation: time paths of consumption levels, aggregate spending on machines, and aggregate R&D expenditure [C (t), X (t), Z (t)] t=0 machine qualities [q (ν, t)] ν [0,1],t=0 prices and quantities of each machine and the net present discounted value of profits from that machine,[p x (ν, t q), x (ν, t q), V (ν, t q)] ν [0,1],t=0 interest rates and wage rates, [r (t), w (t)] t=0

10 Equilibrium: Innovations Regimes The demand for machines can be derived from final goods producer s maximization problem ( ) 1 q (ν, t) β x (ν, t q) = L (5) p x (ν, t q) for all ν [0, 1] and all t. p x (ν, t q) refers to the price of machine type ν of quality q(ν, t) at time t. Two regimes: 1. innovation is drastic and each firm can charge the unconstrained monopoly price 2. limit prices have to be used Assume drastic innovations regime: λ is suffi ciently large ( ) 1 β 1 β λ 1 β Normalize ψ 1 β (6)

11 Monopoly Profits Profit-maximizing monopoly price p x (ν, t q) = Combining with (5) Thus, flow profits of monopolist: ψ q(ν, t) = q(ν, t) (7) 1 β x (ν, t q) = L. (8) π (ν, t q) = p x (ν, t q)x (ν, t q) ψq(ν, t)x (ν, t q) = βq(ν, t)l.

12 Characterization of Equilibrium 1 Substituting (8) into (4) where Aggregate spending on machines is thus Equilibrium wage rate: Y (t) = 1 Q (t) L (9) 1 β Q (t) = 1 q 0 (ν, t) dν. (10) X (t) = (1 β) Q (t) L (11) w(t) = β Q (t). (12) 1 β

13 Characterization of Equilibrium 2 Value function for monopolist of variety ν of quality q (ν, t) at time t r(t)v (ν, t q) V (ν, t q) = π(ν, t q) z(ν, t q)v (ν, t q) (13) where z(ν, t q) = rate at which new innovations occur in sector ν at time t π(ν, t q) = flow of profits Last term captures the essence of Schumpeterian growth: When innovation occurs, the monopolist loses its monopoly position and is replaced by the producer of the higher-quality machine. Once that happens, it receives zero profits, and thus has zero value. Because of Arrow s replacement effect, an entrant undertakes the innovation, thus z(ν, t q) is the flow rate at which the incumbent will be replaced.

14 Characterization of Equilibrium 3 Free entry conditions ηv (ν, t q) λ 1 q(ν, t) (14) ηv (ν, t q) = λ 1 q(ν, t) if Z (ν, t q) > 0 Though the q(ν, t) s are stochastic, as long as Z (ν, t q) s are nonstochastic, average quality Q(t), total output, Y (t), and total spending on machines, X (t), will be nonstochastic. Consumer maximization implies the Euler equation Transversality condition: [ ( t exp lim t for all q. Ċ (t) C (t) = 1 (r (t) ρ) (15) θ 0 ) t r (s) ds 0 ] V (ν, t q) dν = 0 (16)

15 Definition of Equilibrium V (ν, t q) is nonstochastic: either q is not the highest quality in this machine line and V (ν, t q) is equal to 0, or it is given by (13) An equilibrium can then be represented as time paths of [C (t), X (t), Z (t)] t=0 that satisfy (2), (11), (16) [Q (t)] t=0 and [V (ν, t q)] ν [0,1],t=0 consistent with (10), (13) and (14) [p x (ν, t q), x (ν, t q)] ν [0,1],t=0 given by (7) and (8), and [r (t), w (t)] t=0 that are consistent with (12) and (15) Balanced Growth Path defined similarly to before (constant growth of output, constant interest rate)

16 Balanced Growth Path 1 In BGP, consumption grows at the constant rate g C, that must be the same rate as output growth, g. From (15), r(t) = r for all t. If there is positive growth in BGP, there must be research at least in some sectors. Since profits and R&D costs are proportional to quality, whenever the free entry condition (14) holds as equality for one machine type, it will hold as equality for all of them. Thus, V (ν, t q) = q (ν, t) λη (17) Moreover, if it holds between t and t + t, V (ν, t q) = 0, because the right-hand side of equation (17) is constant over time q (ν, t) refers to the quality of the machine supplied by the incumbent, which does not change.

17 Balanced Growth Path 2 Since R&D for each machine type has the same productivity, constant in BGP Then (13) implies z(ν, t) = z(t) = z V (ν, t q) = βq (ν, t) L r + z (18) Effective discount rate: r + z Combining this with (17) r + z = ληβl (19) From the fact that gc = g and (15), g = (r ρ) /θ, or r = θg + ρ. (20)

18 Balanced Growth Path 3 To solve for the BGP equilibrium, we need a final equation relating g to z. From (9) Ẏ (t) Y (t) = Q (t) Q (t) Note that in an interval of time t, z(t) t sectors experience one innovation, and this will increase their productivity by λ. The measure of sectors experiencing more than one innovation within this time interval is o( t) i.e., it is second-order in t, so that Therefore, we have as t 0, o ( t) / t 0. Q (t + t) = λq (t) z (t) t + (1 z (t) t) Q (t) + o (t)

19 Balanced Growth Path 4 Now subtracting Q(t) from both sides, dividing by t and taking the limit as t 0, we obtain Therefore Q (t) = (λ 1) z (t) Q (t). Now combining (19)-(21), we obtain: g = (λ 1) z. (21) g = ληβl ρ θ + (λ 1) 1 (22)

20 Summary of Balanced Growth Path Proposition 14.1 Consider the model of Schumpeterian growth described above. Suppose that ληβl > ρ > (1 θ) ληβl ρ θ + (λ 1) 1 (23) Then, there exists a unique balanced growth path in which average quality of machines, output and consumption grow at rate g given by (22). The rate of innovation is g /(λ 1). Important: Scale effects and implicit knowledge spillovers are present. knowledge spillovers arise because innovation is cumulative.

21 Transitional Dynamics Proposition 14.2 In the model of Schumpeterian growth described above, starting with any average quality of machines Q(0) > 0, there are no transitional dynamics and the equilibrium path always involves constant growth at the rate g given by (22). Note only the average quality of machines, Q(t), matters for the allocation of resources. Moreover, the incentives to undertake research are identical for two machine types ν and ν, with different quality levels q(ν, t) and q(ν, t).

22 Pareto Optimality This equilibrium is typically Pareto suboptimal. But now distortions more complex than the expanding varieties model. monopolists are not able to capture the entire social gain created by an innovation. Business stealing effect. The equilibrium rate of innovation and growth can be too high or too low.

23 Social Planner s Problem 1 Quantities of machines used in the final good sector: no markup. Substituting into (4): x S (ν, t q) = ψ 1 β L = (1 β) 1 β L. Y S (t) = (1 β) 1 β Q S (t) L.

24 Social Planner s Problem 2 Maximization problem of the social planner: subject to C S (t) 1 θ 1 max exp ( ρt) dt 0 1 θ Q S (t) = η (λ 1) (1 β) 1 β βq S (t) L η (λ 1) C S (t) where (1 β) 1 β βq S (t) L is net output.

25 Social Planner s Problem 3 Current-value Hamiltonian ( ) H Q S, C S, µ S = C S (t) 1 θ 1 + µ S (t) [η (λ 1) (1 β) 1 β 1 θ η (λ 1) C S (t)]. Necessary conditions: C S (t) θ = µ S (t) η (λ 1) µ S (t) = ρµ S (t) µ S (t) η (λ 1) (1 β) 1 β βl [ ] exp ( ρt) µ S (t) Q S (t) = 0 lim t Keynes-Ramsey eq: Ċ S (t) C S (t) = g S = 1 [ ] η (λ 1) (1 β) 1 β βl ρ θ (24)

26 Summary of Social Planner s Problem Total output and average quality will also grow at the rate g S. Comparing g S to g, either could be greater. When λ is very large, g S > g. As λ, g S g (1 β) 1 β > 1. Proposition 14.3 In the model of Schumpeterian growth described above, the decentralized equilibrium is generally Pareto suboptimal, and may have a higher or lower rate of innovation and growth than the Pareto optimal allocation.

27 Policies 1 Creative destruction implies a natural conflict of interest, and certain types of policies may have a constituency. Suppose there is a tax τ imposed on R&D spending. This has no eect on the pro... ts of existing monopolists, and only influences their net present discounted value via replacement. Since taxes on R&D will discourage R&D, there will be replacement at a slower rate, i.e., z will fall. This increases the steady-state value of all monopolists given by (18): βql V (q) = r (τ) + z (τ) The free entry condition becomes V (q) = (1 + τ) q. λη

28 Policies 2 V (q) is clearly increasing in the tax rate on R&D, τ. Combining the previous two equations, we see that in response to a positive rate of taxation, r (τ) + z (τ) must adjust downward. Intuitively, when the costs of R&D are raised because of tax policy, the value of a successful innovation, V (q), must increase to satisfy the free entry condition. This can only happen through a decline the effective discount rate r (τ) + z (τ). A lower eective discount rate, in turn, is achieved by a decline in the equilibrium growth rate of the economy: g (τ) = (1 + τ) 1 ληβl ρ θ + (λ 1) 1. This growth rate is strictly decreasing in τ, but incumbent monopolists would be in favor of increasing τ.

29 Conclusion Schumpeterian models introduce the possibility of creative destruction. But in the baseline model this does not change the major implications of the endogenous input/product variety models (in particular, constant growth subject to scale effects; growth rate decreasing in market competitiveness; steady aggregate behavior; no interesting/realistic firm-level dynamics; little firm-level heterogeneity). Heterogenous firm: Melitz (2003): a dynamic industry model that incorporates firm productivity heterogeneity into the Krugman (1979) monopolistic competition framework, and focuses on steady state equilibrium only. Nevertheless, we will see that to go beyond the baseline model and study the dynamics of firm-level innovation and its aggregate implications, the Schumpeterian model will be a good starting point.