Schumpeterian quality ladder model: a synthesis (Aghion and Howitt, 2005, 2009)

1. Schumpeterian quality ladder model: a synthesis(Aghion and Howitt, 2005, 2009) But confront with Grossman Helpman (1991) and Acemoglu (2009, ch.14)

2. technology

3. Instantaneous production • A continuum [0, m] of intermediate goods varieties • 1 unit of x produced with 1 unit of Y • Elasticity of substit. between varieties:  1/(1  ) • Y sector is competitive: • input price = value marginal product of variety i

4. Intermediate good production • xit Produced by the innovator • Or by a competitive fringe of imitators that can produce the same good at a higher marginal cost χ . If 1 < χ ≤ monopoly price. → pit = χ • The innovator’s panning horizon is 1 period • the unconstrained monoploy price pit = 1/ results from: • Max: πit = (pit  1) xit = xitαα[Ait(L/m)] 1-α xit • dπit / dxit = 0 yields:

5. Monopoly power • If χ < 1/α the monopoly price is constrained. If χ increases, the constrained price increases, competition falls, monopoly output xit falls, but profit πit increases. • Assume full monopoly: χ ≥ 1/α = pit • πit = (p  1) xit = [(1   ) / ] xit = π Ait L/m • πL/m = πit / Ait = profit per unit of efficiency • π = (1   )(1 + ) /(  1) • log (χ) = log (χ  1) + 1/(  1) log (χ /) • χdlog (χ) / dχ = χ /(χ  1)  1/(1  ) > 0 • (1  ) = (1/  1)/(1/) = profit / price ratio (market power) of the monopolist

6. innovations • In each sector i the expected productivity growth rate is g = E (Ait /Ai, t-1 1) 1 • Conditional on R&D investment of Rit units of Y at time t in sector i, 1 innovation arrives at t with probability μit = λf(nit) • The time period is so short that at most 1 innovation arrives within the period. • where: nit = Rit/γAi,t-1f(0) = 0;f’ > 0 ; f’’ < 0 • f’(0) = +∞ Inada like condition which makes it sure that in equilibrium n > 0.

7. Productivity growth • Ait = γ Ai, t  1 with probability μit = λf(nit) • Ait = Ai, t  1 with probability 1  μit = 1  λf(nit) • in each i: E(Ait Ai, t  1) = λf(n)(γ 1)Ai, t  1 • Production function of technology linear in Ai, non-linear in ni; nidoes not depend on Ai. • In steady state symmetric equilibrium nit = n • No transition dynamics !! • g = E(Ait/ Ai, t  1)  1 = λf(n)(γ 1) • Where n = n(χ, λ)

8. R&D Expected sector profit from 1 extra unit of Y invested in R&D in sector i at time t is: • Bit = πitdμit /dNit = λf’(n)[πit / (γAi, t  1)] • Monopoly profit maximizing effort n when: Bit = 1 = output price = marginal cost • Innov. profit πit = (χ  1)xit = (χ)Ait L/m = (χ)γAi, t  1 L/m • R&D arbitrage equation: λf’(n) (χ)L/m = 1 • with f(n) = (2n)1/2, f’(n) = 1/ (2n)1/2 • f(n*) = (2n*)1/2 = λ(χ)L/m • Δ Ait = λf(n*)(γ 1) Ai, t  1n* depends on λ, π, L/m, not on Ai, t  1. • g = λf(n)(γ 1) = λ2(χ) (L/m) (γ 1)

9. Per capita GDP growth • GDP = Yt ∫i xitdi. = α2α/(1-α) At(L/m) (with χ = 1/α) • Per capita GDP is proportional to: • aver. technol. level = At = (1/m) ∫iAit di • Distribution of innovation statistically independent across sectors implies: at (t – 1) average techn. level of innovating and non innovating sectors is At1. • Fraction of innovating sectors = μ = λf(n) • At = μγAt1 + (1 – μ) At1 • At  At1 = μ (γ 1)At1 = gAt1 • g = λf(n)(γ 1) = λ2(χ) (L/m) (γ 1) • where (χ) (L/m) = productiv. adjusted sector profit

10. conclusions • Δ At = λf(n*)(γ 1) At  1n* = λ2π L/m is independent of At  1 (otherwise the model would be semi-endogenous!) • g = λf(n*)(γ 1) • L is the scale effect on growth; it can be avoided by imposing: limt→∞ L/m = constant • monopoly profit π is good for growth. π grows if competition within industry is lower, but the influence of competition between industries (α) is more ambiguous. • λ is the productivity of the innovation system

11. 4 (potential) sources of Pareto ineff. • Monopoly output of innovation goods (price > marginal cost) • Outsiders investing in R&D ignore that 1 innovation destroys current monopoly rents. • Innovators are only interested in private benefits, and ignore that the social benefits of innovations may outlast the private ones. • R&D investors may not correctly compute the marginal probability λ f’(n).

12. An inter-temporal compensation • The blue and green inefficiencies are avoided forcing: • each innovator pays the incumbent monopolist for her loss of rents: → current net benefit of innovation ! • The current innovator will be compensated in the future. She evaluates the future social benefits of the innovation outlasting the period in which it commands a demand in the market. • In steady state, the productivity adjusted undiscounted value of the two payments is the same: → • The productivity adjusted present value of the second (later) payment is lower.

13. Result: If • marginal effect of R&D effort on innovation probability is correctly evaluated • Demand price of x is lowered by a subsidy: monopoly price - subsidy = marg. cost of x • there is no compensation schema • Then Market R&D > Pareto optimal R&D

14. interpretation • In general intertemporal equilibrium, prices are present value prices: • The market value of the innovator’s gain for the missing payment to the incumbent monopolist is larger than the market value loss from the missing future recovery of rents (when her innovation will be displaced). • The market incentive to R&D is ‘too high’.

15. Welfare conclusions • If we still leave aside incorrect computation of the marginal probability λf’(n); • But leave in place the static inefficiency from monopoly. • Market R&D can be higher or lower than ‘Pareto efficient R&D’ depending on parameters.

16. Technology transfer • h countries • m sectors in each country • Cross country sector-specific spillovers • A*t1 cross country maximum sector productivity (subscript i omitted) at the beginnining of period t. If 1 innovation arrives in t: At = γ A*t  1γ > 1 • knowledge spillovers induce technology catch up • At = γ A* t  1 with probability μ • At = At  1 with probability 1  μ • μ = λf(n) n = nt = Rt/γA*t-1 • μ = country innovation rate in the given sector (sectors are ex ante identical)

17. Sector technology frontier • logA*t = logγ +logA* t  1 with probability μ* • logA*t = logA*t  1 with probability 1  μ* • μ* = ∑hj = 1λjf(nj) = probability that some innovation occurs in some country • g* = E[logA*t -logA* t  1] = • = μ*logγ≈ μ*(γ 1) for γ close to 1 • distance to frontier: dt  1 = log(A*t  1 /A t  1)

18. Expected sector growth in 1 country gt = E[logAt]  logA t  1 = • = μlogA*t  1 + μlogγ + (1μ)logAt  1 logAt  1 • = μ[logA*t  1 + logγ logAt  1] • = μ[logγ + log(A*t  1 /A t  1)] • dt  1 = log(A*t  1 /A t  1) • gt = μ[logγ + dt  1]

19. Expected sector growth in 1 country • gt = μ[logγ + dt  1] • dt = dt  1with prob. (1  μ*) (no sect. innov. world-wide) • dt = 0 with prob.μ(sector innovation in the country) • dt =logγ + dt  1with prob. (μ*  μ) = (sector innovation outside) • E(dt) =dt  1 (μ*  μ +1  μ*) + (μ*  μ) logγ • = dt  1 (1 μ) + (μ*  μ) logγ

20. Law of motion of d E(dt) = (logγ + dt  1)(μ*  μ) + dt  1 (1  μ*) • = dt  1 (μ*  μ +1  μ*) + (μ*  μ) logγ • = dt  1 (1 μ) + (μ*  μ) logγ

21. z = expected distance from frontier • zt = E(dt) = (1  μ) dt  1 + (μ*  μ)logγ = • E(zt) = zt = (1  μ) zt  1 + (μ*  μ)logγ • The long run expected country distance z* from the growing frontier converges to: • z* = [(μ*  μ) / μ]logγ if μ > 0 • z* = +∞ if μ = 0 • Long-run expected country h growth: • gh* = g* = μ*logγif μ > 0 • gh* = 0 if μ = 0

22. Conditions for no R&D • Assume that the innovation arrival is ruled by productivity weighted innovation effort n according to f(n) such that f’(0) < +∞. • In this case the research arbitrage equation is: 1 ≥ λf’(n) πL/m • n = 0 if strict inequality holds. • If research productivity λ and/or monopoly profit π is too low, then → no R&D

23. A model of step by step innovations and knowledge spillovers • PRODUCTION: • A continuum of consumer-good sectors • Each sector j is a duopoly. firms A and B in j produce xAj = LAj xBj = LBj xj = xAj + xBj • Firm i has labour productivity Ai = γk(i) i = A, B k(i) = i’s technology level γ > 1 γ─ k(i) = labour input per unit of output • i’s unit cost is w γ─ k(i)

24. innovation • Tacit knowledge: before improving upon frontier-knowledge a follower must catch up with the leader • Knowledge spillovers are such that maximum technological distance is 1 step: →a sector j can belev or unlev. No R&D by the leader if j is unlevel • R&D expenditure ψ(n) = n2/2 by the leader moves technology 1 step ahead with probability n. • R&D expenditure 0 by the laggard moves technology 1 step ahead with probability h. • R&D expenditure ψ(n) = n2/2 by the laggard moves technology 1 step ahead with probability n+h

25. consumption • A unit mass of identical consumers, each with current utility u = ∫(log xj)dj • In equilibrium, for each j in [0, 1]: pjxj = E = 1 (expenditure is our numeraire) • The representative household chooses xAj , xBj to maximize log(xAj + xBj) subject to: pAjxAj + pBjxBj = 1 = E • She will choose the least expensive between xAj , xBj

26. Firm profit π1 in un-level sector 1 • If the leader’s unit cost is c, the laggard’s unit cost is γc > c • leader’s profit = π1 = p1x1 ─ cx1 = 1 ─ cx1 • Leader chooses maximum price p1 consistent with p1x1 = 1 • → p1= γc x1= 1/γc cx1= 1/γ • π1 = γc x1 ─ cx1 = 1 ─ cx1 = 1 ─ 1/γ

27. firm profit π0 in level sector • Bertrand price competition → π0 = 0 • Perfect collusion with potential imitators (one step behind) → π0 = (1/2) π1 • With degree of competition Δ in a level sector: • → π0 = (1 ─ Δ)π1 ½ ≤ Δ ≤ 1 • Δ = (π1 ─ π0) / π1 • Perfect collusion Δ = ½ • Bertrand competition Δ = 1

28. Innovation intensity n0 in a level sector • Planning horizon: 1 period • At most 1 innovation per period (1 firm succeeds) • Innovator’s gross profit: • π1 with probability n0 • π0 with probability 1 ─ n0 • Max: [π1 n0 + π0 (1 ─ n0)] ─ (n0)2 / 2 with respect to n0 • → n0 = π1 ─π0 = Δπ1 • Escape competition effect: dn0 / dΔ > 0

29. Innovation intensity in un-level sector • The laggard -1 chooses n ─1 to maximize: • (n-1 + h) π0 ─ (n-1)2 / 2 • → n-1 = π0 = (1 ─ Δ) π1 • Lower competition Δ in a level sector increases innovation intensity in un-level sector(‘Schumpeter’s effect’ according to Aghion and Howitt)

30. composition between lev. / unlev sectors • μ1 = steady state fraction of unlevel sectors • μ0 = 1 ─ μ1 = steady st. fraction of level sect.s • (n-1 + h) = prob. that a unlevel sector becomes level • (n-1 + h) μ1 = steady state expected number of sectors moving from un-level to level • n0 = prob. that a level sector becomes unlevel • n0 (1─ μ1) = steady state expected number of sectors moving from level to unlevel

31. Δ and the steady state composition • the steady state number of level/unlevel sectors is stationary • (n-1 + h) μ1 = n0 (1─ μ1) • μ1 = n0 / (n-1 + h + n0) • The aggregate innovation flow is: • I = (n-1 + h) μ1 + n0 (1─ μ1) = 2 (n-1 + h) μ1 = • I = [2 (n-1 + h)n0 ] / (n-1 + h + n0) • I = {2[(1 ─ Δ) π1 + h] Δπ1} / (π1 + h) • dI / dΔ = {2 π1[(1 ─ 2Δ) π1 + h]} / (π1 + h)

32. Relation between Δ and aggregate innovation • dI / dΔ = {2 π1[(1 ─ 2Δ) π1 + h]} / (π1 + h) • At Δ = ½ dI / dΔ = (2π1h) / (π1 + h) > 0 • d2I / (dΔ)2 < 0 • At Δ = 1: dI / dΔ = (- π1 + h) 2 π1 / (π1 + h) dI / dΔ < 0 if and only if π1 > h • → if π1 < h dI / dΔ > 0 for any Δ • → if π1 > h dI / dΔ < 0 if Δ suff. Large • Is this a re-conciliation between the Schumpeterian model and the empirical evidence? • Δ is degree of competition in level sectors only!!

33. Alternative approach: Competitive innovation (Introductory notes to Boldrin and Levine 2010)

34. IPR: Arrow, Romer, Aghion-Howitt … Knowledge (Kn) non rival K is transferred at zero (negligible) cost. Copying of ideas is negligibly affected by their material support number of copies per unit of time is unbounded Hellwig-Irmen (JET, 2002), Boldrin-Levine (JME, 2008), Quah (2002): (Kn) non rival Kn always embodied in goods or human capital Kn transfer requires access to copies of the innovative good number of copies per unit of time is finite IPR vs. competitive innovation

35. Romer (1986, 1990): The innovative good is produced under increasing returns to scale: Fixed cost (R&D) + small constant marginal cost = increasing returns to scale: price = marginal cost would imply: producer’s profit < 0 Technology set is non-convex R&D cost is related to the fact that the productivity of quality j input is lower in producing quality j+1 output than in producing quality j output. Technology set may be convex. Innovative good production

36. Three uses of Knowledge capital kj, t • Copying (widening): • flow output of quality j is b kj, t • Innovating (deepening): • flow output of quality j +1 is (1/a) kjt • where b > λ / a and λj is quality adjusted value of j in consumption • Producing consumption: flow output of quality j is cjt = min{kjt, Ljt} • kj, t = kwj, t+ kdj, t+ kcj, t

37. Pricing of knowledge capital • Zero profits on deepening:qj+1, t– aqj, t ≤ 0 • If innovation occurs at t: qj+1, t– aqj, t = 0 (1) • Zero profits on widening: • b + (dqj, t/dt) /qj, t = ρ (optimality condition on state variable kj) • In competitive equlibrium no quality j+1 can be produced before its price qj+1 rises relative to qj so that (1) holds. • This occurs when all labour L is fully employed in producing consumption with capital of quality j.

38. IPRs protect the excludability of the innovation good, not of the general knowledge embodied in patents, which is disclosed by patent publication. At monopoly price pm the innovator confers to the buyer the right to use the good in consumption or production in the ways that are specified by the owner of the IPR. Knowledge externalities: Pareto inefficiency of market economies. At competitive price qj the seller confers to the buyer the right to use the good in consumption or production, including production and sale of innovation copies to the market q confers different rights, compared to monopoly price. Demand, hence q are affected An idea is accessible only through the appropriation of its material support: there are not knowledge externalities Pareto efficient equilibrium What covers innovation price?