Quality Ladders, Competition and Endogenous Growth Michele Boldrin and David K. Levine 1
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
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
Benchmark Environment: Grossman&Helpman d the consumption (demand) for goods of quality j j ρ be the subjective interest rate a constant = increase in quality each step up quality ladder λ > 1 consumer utility ∑ ∞ − ∫ t j = ρ λ U e log d dt jt j 0 One unit of output requires a unit of labor to obtain 4
The first to reach j has monopoly until j + 1 is reached ɶ , probability of innovating is dt ɶ ɶ a dt R&D intensity is ι at a cost of . ι ι I One unit of labor, E steady state expenditure Wage rate is numeraire and price is λ ι λ a E / 1 The resource constraint is + = I 5
Monopolist gets a share (1 1/ λ ) of expenditure − a , rate of return is (1 1/ λ ) E / a Cost of innovation is I . − I There is a chance ι of losing the monopoly, reducing the rate of return to the interest rate (1 1/ λ ) E − ι ρ − = a I This and the resource constraint solve for R&D intensity (1 1/ λ ) ρ − . ι = − a λ I 6
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
How industries walk up a quality ladder . 8
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 just costly to go there right away 9
Innovation with Knowledge Capital Retain preferences and ladder structure k and Ladder corresponds to qualities of knowledge capital j d consumption j 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
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 j j + 1 capital of the next high quality level: one unit of quality needs a > 1 units of quality j Deepening is costlier than widening, / λ a 1 . < Law of motion: h j 1 ɺ k b k ( d ) h − . = − − + j j j j a ∆ = −∆ k k / a Allow j 1 j + 11
This is an ordinary diminishing return economy, CE is efficient Proposition: production uses at most two adjacent qualities of capital j 1, j − Proposition: after some initial period labor is fully employed: d d + 1 + = . j j 1 − Proposition: Consumption grows at a rate b or not at all ρ d = 1 Proposition: You innovate only when necessary, that is when j Proposition: Equilibrium paths cycle between widening and deepening 12
Widening d = 1 d + = 0 At the beginning of this phase and . j j 1 λ + λ + j j 1 d d Consumption during widening is + j j 1 It increases as labor shifts from old to new capital = λ j d (0) 1 and (0) c Since = j j j 1 j ( b ) t + − λ d t ( ) λ d ( ) t λ e ρ . + = j j 1 + 13
Using full employment condition ( b ) t ρ λ e − − d t ( ) . = j 1 λ − d τ ( ) 0 d ( ) 1 This continues until and τ = = j 1 j 1 1 + At which point widening ends. d τ ( ) 0 Solving we find the length of widening = j 1 log λ τ . = 1 b ρ − 14
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 , j 15
0 / b e τ a j + 1 A reduction in consumption now give units of capital by the end deepening. e ρτ This future capital gives consumption worth 0 units of current − consumption. j + 1 quality is worth λ time consumption of quality j Consumption of At the social optimum, this shift must be neutral, b 1 λ − e ρτ ( e τ / a ) . 0 0 = λ log a log − τ , = 0 b ρ − The same flow of consumption service can be obtained through a smooth innovation process 16
Solve τ 0 τ − τ µ = b ( t ) b e dt e 0 0 ∫ 0 − τ µ = − for µ , to get b b /(1 e ) 0 . Hence there is a continuum of payoff equivalent equilibria. Focus on the one in which innovation is done at end of deepening 17
Intensity of innovation This is just the inverse of the length of the cycle, i.e. of the sum of the two parts of the cycle + τ τ 0 1 b ρ − j * . = log a Length of cycle is endogenous but does not depend on how high the step ladder is 18
Remarks New knowledge is costlier than old New knowledge loses the productive capacity of the old ae ∆ time delay is capitalized into b Conversion is instantaneous – use the cost of conversion 19
Evolution of the stocks Deepening : growth rate of consumption is zero = τ t of old capital converted to new is F value at 0 = τ τ = + t k ( ) 1 F conversion takes place at , hence 0 j 0 − τ − = + b ( t ) k ( ) t 1 Fe d 1 ( ) t 0 so 0 and during deepening = j j + 20
Widening: = τ = τ + τ k ( ) t and d ( ) t shrink from 1 at t t , to 0 at . j j 0 0 1 = − ρ = − and / c c b ɺ ɺ ɺ d + d from derive j j 1 ρ λ ( b ) − ɺ d d b ( ρ ) , = + − j j 1 λ − which has the solution given earlier, i.e. − ρ − τ λ ( b )( t ) e 0 = − d ( ) t j − λ − λ 1 1 New capital producing consumption expands as − ρ − τ ( b )( t ) 1 e 0 = − d ( ) t , + j 1 − λ − λ 1 1 21
k 1 ( ) t Plugging the results in the law of motion for , the expanding + j stock, we have − ρ − τ ( b )( t ) b e 0 [ ] = − + − + ρ ɺ k ( ) t bk ( ) t b a ( 1) + + j 1 j 1 − λ − λ 1 a (1 ) ∈ τ τ + τ t ( , ] for . 0 0 1 22
Solving this we find that − ρ − τ − + ρ ( b )( t ) b a ( 1) e 1 0 = − + + k ( ) t C + j 1 ρ − λ − λ a 1 1 τ = k ( ) ( F a / ) The initial condition can be used to eliminate the + j 1 0 constant of integration to get − + ρ b a ( 1) F − ρ − τ = − + ( b )( t ) k ( ) t [1 e ] 0 . + j 1 ρ − λ a (1 ) a 23
For the cycle to repeat itself, at the end of the widening period the stock − τ j + must equal + b 1 1 Fe 0 of capital of quality again. Use this to * F , the (pseudo) fixed cost invested in innovation along the compute competitive equilibrium path 24
Comparison of the Models Ignore technical differences, stick to substantive First, the parameter λ – 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
Assume that there is a technologically determined fixed cost F that k F / a gets you 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 . If j is introduced for the first time at j t then j + 1 cannot also be t , hence the distance in time between j t and t + is introduced at time j j 1 either constant or infinity. We are interested in * F F ≤ small fixed cost , and * F F large fixed cost . > 26
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
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