This process was named creative destruction by Joseph Schumpeter. Aghion and Howitt (1992) formalized Schumpeter s ideas in a growth model of the type we have been studying. A simple version of their model is contained in their textbook, Endogenous Growth Theory (MIT Press, 1998), and I have put together a summary of the model which is linked to the course outline. We will use this model to illustrate: 1. the potentially adverse incentive effects of the process of creative destruction 2. that when growth is suboptimal it may be either to low or too high. 3. that beliefs are an essential component of the process of technological innovation. 285
The Aghion and Howitt model of growth through creative destruction I. The Environment Similar to the version of the Romer model which we studied earlier. Focus on the market equilibrium of an economy with three sectors: 1. a perfectly competitive final output (manufacturing) sector 2. a monopolistic sector producing a single intermediate good 3. a research and development sector comprised of many identical researchers. The allocation of labour across sectors (effort into R&D) will again be the key margin. 286
Some preliminaries... Time is continuous and indexed by τ 0. There are two goods, a final consumption good and an intermediate good There is one basic factor of production, labour. There are a large number (L) of identical consumers who maximize utility over an infinite time horizon. Preferences: U (c(τ)) = 0 e rτ c(τ)dτ (213) where r is both the rate of time preference and interest rate. 287
Technologies The final consumption good is produced using only the intermediate good: c(τ) y(τ) = A t x(τ) α 0 < α < 1 (214) Here the productivity parameter, A t, given by A t = A 0 γ t γ > 1, (215) is the productivity of the t th generation of intermediate good x is the quantity of the intermediate good employed. The intermediate good is produced one-for-one using labour. So x denotes both the quantity of labour employed in production and the quantity of intermediate good produced. 288
If the t th innovation has occurred previously, productivity is given by A t. Then the t + 1 st innovation raises productivity by the factor γ, to A t+1 = A 0 γ t+1 = γa t. (216) These increases in productivity occur randomly at a rate which depends on the amount of effort devoted to R&D. At each point in time, the approximate probability that an innovation occurs right then is given by λn, where i. λ > 0 is a parameter ii. n is the amount of labour employed in research and development. λn is the expected (or average) number of innovations that occur within a single unit of time. 289
Aside... Poisson processes: Random events are said to follow a Poisson process with arrival rate µ in the following circumstances: F(T) Prob{event occuring before T } = 1 e µt (217) Then, the density: f(t) = F (T) = µe µt (218) tells us the probability that the event will occur between T and a time a little later: T = T + dτ. With T = 0 (now) the probability that it will occur a short time from now is µdt. Thus µ (= λn in the Aghion-Howitt model) is the flow probability or the number of times the event is expected to occur in a unit interval of time. 290
Endowments: The total amount of labour available is L: x + n L (219) Market Structure: The final goods sector is perfectly competitive the (single) intermediate good is supplied by a monopolist. These sectors are similar to their counterparts in the Romer model. The models differ, however with regard to their research and development sectors 291
The R&D sector is characterized by a patent race. 1. Fraction n of the labour force, L, devotes its time (labour) to trying to invent the next generation of intermediate good. 2. Eventually, one of them succeeds. This lucky researcher can then either: i. become the monopolistic supplier of the intermediate good to the final goods sector ii. sell an infinitely lived patent to another firm to produce it monopolistically. Note it is assumed that the innovator will be a monopolist because he/she will have a sufficiently higher productivity than the previous intermediate goods producer to limit price. (Innovations are drastic ) 292
II. Equilibria We will use Period t to refer to the time between the introduction of the t th and t + 1 st innovations. Key issue: the equilibrium division of labour between manufacturing and R&D. Since all individuals are identical, in equilibrium the returns to labour in its two uses must be equalized: where, w t = λv t+1 (220) w t is the return to working in manufacturing during period t, i.e. the manufacturing wage. 293
λv t+1 is the expected return to working in the patent race. This is equal to the product of i. the value of t + 1 st innovation to the researcher who wins the patent race, V t+1 ii. the expected length of time until the t + 1 st innovation appears, (λn) iii. the probability that a given individual researcher wins the race (1/n). If w t λv t+1 (221) then the returns to working in one sector or the other is higher. This is will lead to a change in the allocation of labour and is not consistent with equilibrium. 294
A no arbitrage argument can also be used to derive an equation for V t+1 : Think of V t+1 as the price required to buy the patent for the t + 1 st innovation from the from the researcher who obtains it. Then, rv t+1 is the value per unit time of resources required to become the t + 1 st monopolist. This must satisfy rv t+1 = π t+1 λn t+1 V t+1 (222) π t+1 : is the flow of monopoly profits generated by the t + 1 st innovation. n t+1 : is future effort in R&D (that is the quantity of labour devoted, in period t+1, to producing the t + 2 nd innovation). 295
λn t+1 V t+1 : the capital loss which occurs when the t + 2 nd innovation makes the t + 1 st obsolete. Solving for V t+1 ; V t+1 = π t+1 r + λn t+1. (223) The value of the t + 1 st innovation is equal to the profit stream it generates, π t+1, discounted by the obsolescence-adjusted interest rate, r + λn t+1. Note that the discount rate here, however, is higher than the rate of time preference, whereas in the Romer model it was lower: r + λn t+1 > r > r n pop (224) In the Romer model innovations became more valuable over time (complementarities) whereas here they become less valuable (obsolescence) 296
As in our analysis of the Romer model, in order to go further we must solve the profit maximization problem of the successful innovator. π t = max x p t (x)x w t x (225) p t (x): is the inverse demand for the intermediate good. Since the final good sector is perfectly competitive the intermediate good will be employed until its price equals its marginal value product: p t (x) = αa t x α 1, (226) so that π t = max x αa t x α w t x (227) 297
The first-order condition is x = α 2 A t x α 1 = w t (228) [ α 2 A t w t ] 1 1 α x ( wt A t ) (229) x expresses employment in manufacturing as a function of the productivity adjusted wage, w t /A t, which will be denoted ω t. Solution of the profit maximization problem gives rise to the function x(ω t ), which can be shown to be decreasing in ω t : As the productivity adjusted wage rises, employment in manufacturing decreases. 298
Plugging (229) back into the equation for profit (228) we have π t = αa x α t w t x t = αa ( α 2 A t w t ) α ( ) 1 α α 2 1 1 α A t w t = α2 α w t w t A t ( α 2 A t w t ) α ( ) 1 α α 2 1 1 α A t wt w t = 1 α w t x t w t x t [ ] 1 α π t = w t x t (230) α Here the function π t (ω t ) is also decreasing in ω t. 299
Combining the labour market equilibrium condition: w t = λv t+1 with the equation for V t+1 : V t+1 = π t+1 r + λn t+1 we can write (leaving out some algebraic steps see the lecture notes): ω t = λ A t π t+1 r + λn t+1 = λ A t A t+1 π(ω t+1 ) r + λn t+1 = λγ π(ω t+1) r + λn t+1 (231) 300
Equilibria are characterized by the following two equations: w t A t = λγ π(ω t+1) r + λn t+1 (232) L = x t + n t (233) These equations, however, contain more than two unknowns. So, it is natural to expect that they have multiple solutions. These different solutions are associated with different equilibria. We will consider both a balanced growth path and equilibria with time-varying growth rates. 301
1. The Balanced Growth Path Consider the steady-state case in which both 1. the productivity-adjusted wage, ω t 2. employment in R&D, n t are constant: ω t = ω and n t = n (234) In this case equations (232) and (233) may be rewritten: ω = λγ π(ω) r + λn (235) This system can be solved for a unique ˆω and ˆn. n = x(ω) (236) 302
Balanced Growth in the Aghion-Howitt Model ω (236) (235) ^ n ^ L n
The average growth rate along the balanced growth path: In the steady-state, aggregate output is given by Since, A t+1 = γa t, we have y t+1 = γy t. y t = A tˆx α = A t (L ˆn) α. (237) In this case γ is the rate at which output changes as productivity advances due to innovation. It is not the rate at which output grows per unit of time. The average growth rate in the steady-state, ĝ, is given by the expected rate of change of output per unit time: ĝ = E [ln y(τ + 1)] ln y(τ) (238) 304
Since y t+1 = γy t, ln y(τ + 1) = lny(τ) + ln γǫ(τ), (239) where ǫ(τ) is the number of innovations that occur between τ and τ + 1. ǫ(τ) is a random number, but (as noted earlier) its expected value is given by the Poisson arrival rate of innovations: Eǫ(τ) = λˆn. (240) So the expected value of output one unit of time in the future is: 305
We can then write: E [ln y(τ + 1)] = E[ln y(τ)] + E[lnγǫ(τ)] = ln y(τ) + ln γe[ǫ(τ)] = ln y(τ) + ln γλˆn (241) ĝ = E [ln y(τ + 1)] ln y(τ) = ln y(τ) + ln γλˆn ln y(τ) ĝ = λˆn ln γ. (242) The average growth rate along the balanced growth path is determined by the steady-state level of employment in R&D, ˆn: 306
ln y t Balanced Growth in the Aghion-Howitt Model ln y4 ln y3 ln y2 ln y1 ln y0 t = 1 t = 2 t = 3 t = 4 t = 5 τ