# Quality Ladders, Competition and Endogenous Growth Michele Boldrin and David K. Levine

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1 Quality Ladders, Competition and Endogenous Growth Michele Boldrin and David K. Levine 1

2 The Standard Schumpeterian Competition Monopolistic Competition innovation modeled as endogenous rate of movement up a quality ladder incentive to innovate comes from short-run monopoly at each rung of the ladder Romer, Aghion-Howitt, Grossman-Helpman 2

3 The Questions Does imperfect competition have anything to do with this? Does fixed cost of innovation have anything to do with this? Do models where the incentive to innovate are a short-term monopoly have a better claim to fit the data well? 3

4 Benchmark Environment: Grossman&Helpman d the consumption (demand) for goods of quality ρ be the subective interest rate λ > 1 a constant = increase in quality each step up quality ladder consumer utility ρt U = e log λ d t dt 0 One unit of output requires a unit of labor to obtain 4

5 The first to reach has monopoly until + 1 is reached R&D intensity is ι, probability of innovating is dt One unit of labor, E steady state expenditure Wage rate is numeraire and price is λ The resource constraint is a ι + E/ λ = 1 I ι at a cost of I ι adt. 5

6 Monopolist gets a share (1 1/ λ) of expenditure Cost of innovation is I a, rate of return is (1 1/ λ) E/ a I. There is a chance ι of losing the monopoly, reducing the rate of return to the interest rate (1 1/ λ) E a I ι = ρ This and the resource constraint solve for R&D intensity ι (1 1/ λ) ρ =. λ a I 6

7 The Story moving up the capital ladder is unambiguously good the limitation on the rate at which you move up the ladder is the increasing marginal cost of labor used for innovation here the increasing marginal cost of labor is because it is drawn out of the production of output this is a trick to keep the model stationary 7

8 How industries walk up a quality ladder. 8

9 Key Feature gradual switching from one technology to the next suggests that there is a trade-off between increasing use (ramping up) an old technology and introducing a new one the fact suggest an alternative theory of why there is gradual movement up the quality ladder introduce a new technology when the benefits of the old one are exhausted quite different than the Romer, AH, GH story in their setup the new technology is always better, it is ust costly to go there right away 9

10 Innovation with Knowledge Capital Retain preferences and ladder structure Ladder corresponds to qualities of knowledge capital k and consumption d One unit of labor Consumption needs labor and knowledge capital Knowledge capital can be used to produce consumption, more knowledge of the same quality (widening, imitation), new knowledge (deepening, innovation) 10

11 Widening, imitation: use knowledge capital to produce knowledge capital of the same quality level at rate b > ρ Deepening, innovation: use knowledge capital h to produce knowledge capital of the next high quality level: one unit of quality + 1 needs a > 1 units of quality Deepening is costlier than widening, λ / a < 1. Law of motion: h a 1 = ( ) +. k b k d h k k a Allow 1 = / + 11

12 This is an ordinary diminishing return economy, CE is efficient Proposition: production uses at most two adacent qualities of capital 1, Proposition: after some initial period labor is fully employed: d d =. Proposition: Consumption grows at a rate b ρ or not at all Proposition: You innovate only when necessary, that is when d = 1 Proposition: Equilibrium paths cycle between widening and deepening 12

13 Widening At the beginning of this phase 1 d = and d + 1 = 0. Consumption during widening is 1 λ d + λ + d + 1 It increases as labor shifts from old to new capital Since d (0) = 1 and c(0) = λ + 1 ( b ρ) t d() t d+ 1() t e λ + λ = λ. 13

14 Using full employment condition d () t = e ( b ρ) t 1 λ λ. This continues until 1 d ( τ ) = 0 and d + 1( τ1) = 1 At which point widening ends. Solving 1 d ( τ ) = 0 we find the length of widening τ 1 logλ =. b ρ 14

15 Deepening Step back at the end of the previous widening phase, when only old capital was used to produce consumption. How much capital of new quality should we pile up before starting the new widening phase? Until we do so, full employment implies that consumption is constant d () t = 1, 15

16 bτ A reduction in consumption now give e 0 / the end deepening. a units of + 1 capital by e ρτ 0 This future capital gives consumption worth consumption. units of current Consumption of + 1 quality is worth λ time consumption of quality At the social optimum, this shift must be neutral, λ ρτ bτ e e a ( / ) τ 0 =. = log a b logλ ρ, The same flow of consumption service can be obtained through a smooth innovation process 16

17 Solve µ τ 0 0 b( τ t) bτ e dt = e 0 0 for µ, to get µ e bτ 0 = b/(1 ). Hence there is a continuum of payoff equivalent equilibria. Focus on the one in which innovation is done at end of deepening 17

18 Intensity of innovation This is ust the inverse of the length of the cycle, i.e. of the sum τ + τ of the two parts of the cycle * 0 1 b ρ loga =. Length of cycle is endogenous but does not depend on how high the step ladder is 18

19 Remarks New knowledge is costlier than old New knowledge loses the productive capacity of the old Conversion is instantaneous use the cost of conversion b ae time delay is capitalized into 19

20 Evolution of the stocks Deepening: growth rate of consumption is zero t value at 0 = τ of old capital converted to new is F conversion takes place at t = 0, hence k 0 τ ( τ ) = 1+ F b( τ t) k () t = 1+ Fe and d 1( t) 0 so 0 + = during deepening 20

21 Widening: k () t and d () t shrink from 1 at t τ 0 d = d + and c/ c = b ρ from 1 d ( b ρλ ) = + d( b ρ) 1 λ, derive which has the solution given earlier, i.e. d () t ( b ρ )( t τ ) 0 e λ = 1 λ 1 λ =, to 0 at 0 1 New capital producing consumption expands as d + 1() t = ( b ρ)( t τ ) 1 e 1 λ 1 λ 0, t = τ + τ. 21

22 Plugging the results in the law of motion for k+ 1 () t, the expanding stock, we have ( b ρ )( t τ ) 0 b e k + 1() t = bk+ 1() t + b( a 1) + 1 λ a(1 λ) t ( τ, τ + τ ]. for [ ρ] 22

23 Solving this we find that ( b ρ )( t τ ) 0 ba ( 1) + ρ e 1 k+ 1() t = + + C aρ 1 λ 1 λ k ( τ ) ( F/ a) The initial condition = can be used to eliminate the constant of integration to get ba ( 1) + ρ ( b ρ)( t τ0 ) F k+ 1() t = [1 e ] + aρ(1 λ) a. 23

24 For the cycle to repeat itself, at the end of the widening period the stock b 0 of capital of quality + 1 must equal 1+ Fe τ again. Use this to * compute F, the (pseudo) fixed cost invested in innovation along the competitive equilibrium path 24

25 Comparison of the Models Ignore technical differences, stick to substantive First, the parameterλ two offsetting effects during widening and deepening respectively Second, our model has the extra widening parameter b. In a certain sense the Grossman-Helpman model assumes that b =. Third, our model does not require a fixed cost to innovate. Move on to this issue Fixed Cost 25

26 Assume that there is a technologically determined fixed cost F that gets you k = F/ a units of new capital. Once the fixed cost is incurred, it is possible to convert additional units of old capital to new capital at the same rate a. t then + 1 cannot also be If is introduced for the first time at introduced at time t, hence the distance in time between t and t + 1 is either constant or infinity. We are interested in F * F small fixed cost, and F > F * large fixed cost. 26

27 Behavioral economics: who innovates? Competitive means no one has monopoly power. Can someone affect prices by innovating? Can he/she take this into account when choosing action? When everyone believes that nothing affects equilibrium prices competitive equilibrium with non-atomic innovators, When someone feels powerful, we have its schumpeterian perturbation: entrepreneurial competitive equilibrium 27

28 Fixed Cost: Non- Atomic Innovators viable initial stocks of knowledge capital k = ( k, k,, k ) J kj k 0 1 J feasible paths kt () = [ k(), t k(),..., t k(),...] t, t [0, ) 0 1 kt ( ) = [ k ( t ), k ( t ),..., k ( t ),...] J+ 1 J+ 1 J+ 2 J+ 2 J+ n J+ n 28

29 Definition 1. A competitive equilibrium with atomic innovators E with respect to a viable k J consists of: (i) a non-decreasing sequence of times 0 1 J and, for > J, either t > t 1, or t ( t, t, ), with t = 0 for = ; (ii) a path of capital k() t 0and capital prices q() t 0for t t, and a path of consumption ct () 0and consumption prices pt () 0 that satisfy the conditions (1) [Consumer Optimality] ct () max ρ t e log( c ( t)) dt subect to 0 ρt e ptctdt ()() e ptctdt ()() 0 0 ρt. 29

30 (2) [Optimal Production Plans at t ] k ( t ) = k ( t )/ a k 1 q ( t ) = aq ( t ) 1 (3) [Optimal Production Plans for t > t ] h 1( t) k () t = b( k() t d()) t h() t +, a q () t aq () t k () t max{ d (), t h ()} t, 1 and 1 q () t = aq () t if h 1( t) > 0, k (), t d () t maximize profits 30

31 (4) [Social Feasibility] ct () = λ d () t t (5) [Boundedness] for some number K > 0 k() t < K,b at all t. = 0 31

32 c( τ0) c(0) = e ( b ρτ ) a Implications of the zero profit condition 0 λ. ( k, τ, F) 0 bτ0 is a candidate for equilibrium if zero profits and ke F hold. Also ( )( ) () ( ) b ρ ct c e t τ 0 = τ. 0 1 When t = τ0 + τ1 ct () = λ +. This gives 1 ( b ρτ ) c( τ0) λ + e 1 =, and, because c(0) e ρ τ0+ τ1 =. ( b )( ) a = λ, the zero profit condition simplifies to 32

33 The Case of Small Fixed Cost Theorem 1: In the economy with a small fixed cost, for given initial conditions, there exists a unique competitive equilibrium with non-atomic innovators. This equilibrium is efficient. 33

34 The Case of Large Fixed Cost F > F * Assume now that The dates innovations take place less easy to pin down. First, the competitive equilibrium of the economy without fixed cost is no longer feasible. Second, the new competitive equilibrium is not efficient. Equilibrium exists; in fact: quite a few equilibria exist ( k, τ ) The set of equilibria is parameterized by,0 34

35 Theorem 2: - The earliest competitive equilibrium with non-atomic innovators pareto dominates all other steady state competitive equilibria with non-atomic innovators, but is not first best. - Given k F/ a least one cyclical equilibrium of period two., there is at most onestationary equilibrium and at - There are also equilibria with k = 0 for * > J, and τ * + =+ 1,0 J 35

36 Fixed Cost: Entrepreneurial Innovators Fix one competitive equilibrium Ê. A -innovation is a pair ( t, k ()) t composed of the time tˆ ˆ 1 < t < t at which a single agent purchases F units of capital of F a units of capital of quality. quality 1 and turns them into / We say that a competitive equilibrium Ẽ is a feasible continuation for if the -innovation ( tf, ) k () t = kˆ () t F, k ˆ 1() t = k 1( t'), for t < t ; (a) 1 1 k () t = kˆ () t for ' < 1 and all t. (b) ' ' 36

37 kt ( ) = kt ˆ( ˆ) and kt ( ) = kt ˆ( ˆ ). It is Markov if We say that a -innovation ( t, F ) is profitable with respect to a feasible continuation Ẽ if q ( t ) aq ( t ), and 1. q ( t ) > aqˆ ( t ) 1 37

38 Definition 2. An entrepreneurial competitive equilibrium is defined as a competitive equilibrium with non-atomic innovators that (1) does not admit innovations that are profitable with respect to feasible Markov continuations, (2) does not stop innovating, that is, t < for all. Theorem 3: There is a unique entrepreneurial competitive equilibrium: it is the earliest competitive equilibrium with non-atomic innovators. 38

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