From Smith to Schumpeter: A Theory of Take-o¤ and Convergence to Sustained Growth Pietro F. Peretto Duke University September 2012 Peretto (Duke University) Take-o¤ to sustained growth September 2012 1 / 31
Motivation For much of human history, innovation had been primarily a byproduct of normal economic activity, punctuated by periodical ‡ashing insight that produced a macroinvention, such as water mills or the printing press. But sustained and continuous innovation resulting from systematic R&D carried out by professional experts was simply unheard of until the Industrial Revolution. (Mokyr 2010, p. 37) The Industrial Revolution, then, can be regarded not as the beginnings of growth altogether but as the time at which technology began to assume an ever-increasing weight in the generation of growth and when economic growth accelerated dramatically. (Mokyr 2005, p. 1118) But the exact connection between institutional change and the rate of innovation seems worth exploring, precisely because the Industrial Revolution marked the end of the old regime in which economic expansion was driven by commerce and the beginning of a new Schumpeterian world of innovation . (Mokyr 2010, pp. 37-38) Peretto (Duke University) Take-o¤ to sustained growth September 2012 2 / 31
Main actors Final producers: Homogeneous good that is consumed, used to produce intermediate goods, or invested in R&D. (Basically, one-sector structure.) This good is the numeraire , so P Y � 1. Intermediate producers: Develop new goods and set up operations to serve market (variety innovation or entry) and, when already in operation, invest in R&D internal to …rm (quality innovation). Households: Consume, save and, in extensions I’m working on, set path of population growth and resource use. Peretto (Duke University) Take-o¤ to sustained growth September 2012 3 / 31
Final producers (i) Technology: � 1 � γ ! 1 � θ � L � γ � R Z N Z N X θ Z α i Z 1 � α Y = , Z � ( Z j / N ) dj i N η L N η R 0 0 where: 0 < θ , γ < 1 standard parameters that map into factor shares; 0 < η L , η R < 1 congestion/rivarly parameters; i Z 1 � α vertical technology index, with α 2 [ 0 , 1 ) measure of private Z α returns to quality and 1 � α measure of social returns to quality; N is horizontal technology index, with social returns to variety given by σ � 1 � γη L � ( 1 � γ ) η R 2 [ 0 , 1 ) . Peretto (Duke University) Take-o¤ to sustained growth September 2012 4 / 31
Final producers (ii) Demand for product i : � θ � 1 1 � θ X i = N σ � 1 i Z 1 � α L γ R 1 � γ . Z α P i Factor payments: Z N N � PX = P i X i di = θ Y ; 0 Z N wL = w L i di = γ ( 1 � θ ) Y ; 0 Z N pR = p R i di = ( 1 � γ ) ( 1 � θ ) Y . 0 Peretto (Duke University) Take-o¤ to sustained growth September 2012 5 / 31
Intermediate producers Technologies: � i Z 1 � α � X i + φ Z α Cost i = 1 � ; ˙ Z i = I i . Firm’s objective: Z ∞ e � R t 0 r ( s ) ds � � i ( t ) Z 1 � α ( t ) � I i ( t ) X i ( t ) ( P i ( t ) � 1 ) � φ Z α V i ( 0 ) = dt . 0 In symmetric equilibrium : � 1 � P i , I i V i ) r = α X θ � 1 max � r Z ; Z � 1 � � φ Z � I ˙ θ � 1 = β X i ) r = X X V max + X � r N . i β X Peretto (Duke University) Take-o¤ to sustained growth September 2012 6 / 31
Households (i) Representative household chooses C ( t ) , L ( t ) , R ( t ) to maximize � C ( t ) � Z ∞ e � ( ρ � m ) t log U ( 0 ) = dt , ρ > m > 0 M ( t ) 0 subject to: ˙ A = rA + wL + pR � C ; M = M 0 e mt , m > 0 ; M � L � 0 , Ω � R � 0 , Ω > 0 . Two simpli…cations: M evolves according to exogenous (exponential) process; Ω is endowment of non-exhaustible resource (e.g., land). Peretto (Duke University) Take-o¤ to sustained growth September 2012 7 / 31
Households (ii) Factors supply: L = M ; R = Ω . Consumption/saving: ˙ C r = ρ + C � m . Peretto (Duke University) Take-o¤ to sustained growth September 2012 8 / 31
Equilibrium output Symmetry plus factor markets equilibrium yields 2 θ 1 � θ � N σ ZM γ Ω 1 � γ . Y = θ Accordingly, output per capita is � Ω � 1 � γ Y 2 θ 1 � θ � N σ Z M = θ . M Let y � ˙ Y / Y , n � ˙ N / N and z � ˙ Z / Z . Then, y � m = σ n + z � ( 1 � γ ) m . | {z } | {z } | {z } Hicks neutral output per capita growth drag TFP growth growth due to land Peretto (Duke University) Take-o¤ to sustained growth September 2012 9 / 31
General equilibrium: Role of …rm size (i) Theory draws distinction between per capita and per …rm variables: Per capita ) Household decisions (consumption/saving, labor/leisure, fertility); Per …rm ) Firms’ decisions (investment in vertical and horizontal innovation). Speci…cally, returns to innovation are functions of x � X ( P � 1 ) = gross cash ‡ow = “…rm size” , Z quality where in equilibrium 1 � θ � M γ Ω 1 � γ x = θ ( 1 � θ ) � Y 2 θ NZ = θ ( 1 � θ ) θ . N 1 � σ Peretto (Duke University) Take-o¤ to sustained growth September 2012 10 / 31
General equilibrium: Role of …rm size (ii) Expressions for returns are: r = α ( x � φ ) ; � � r = x � φ 1 � 1 + ˙ x + z x , π x π x where to simplify notation βθ β X entry cost π � 1 � θ = X ( P � 1 ) = gross cash ‡ow . Important : given mass of …rms N , …rm size x is increasing in aggregate market size Y and thus in population M . Peretto (Duke University) Take-o¤ to sustained growth September 2012 11 / 31
General equilibrium: Role of population De…ne market growth factor γ m > 0. Suppose initially n = z = 0. Then y � m < 0 but γ m > 0. Intuition: Population growth drives growth of aggregate market for intermediate goods and thus decisions to invest in variety and quality innovation. These decisions support positive output per capita growth i¤ resulting TFP growth rate is larger than the growth drag. Question : Does aggregate market growth drive transition from zero TFP growth to positive and su¢ciently strong TFP growth? If so, how? Peretto (Duke University) Take-o¤ to sustained growth September 2012 12 / 31
Transition dynamics: Role of corner solutions (i) Let x N be threshold of …rm size that triggers variety innovation and x Z threshold of …rm size that triggers quality innovation. There exists a condition on parameters such that thresholds are identical. Accordingly, we identify two cases: Dominant incentives to variety innovation x N < x Z . Dominant incentives to quality innovation x N > x Z . Most interesting consequence of this feature for the economy’s dynamics is that the sequence in which society turns on the two innovation engines determines the shape of the transition path. Peretto (Duke University) Take-o¤ to sustained growth September 2012 13 / 31
Transition dynamics: Role of corner solutions (ii) Speci…c values of x N and x Z di¤er in the two cases. Why? If x N < x Z , x Z is threshold for quality R&D given that market already supports entry of new …rms; if x N > x Z , x Z is threshold for quality R&D given that market does not yet support entry of new …rms. Intuition : in …rst case …rms undertaking quality R&D compete for resources with entrepreneurs that are setting up new …rms, in the second they do not. Similar reasoning applies to x N : If x N < x Z , x N is threshold for entry of new …rms given that market does not yet support quality R&D; if x N > x Z , x N is threshold for entry of new …rms given that market already supports quality R&D. Peretto (Duke University) Take-o¤ to sustained growth September 2012 14 / 31
The modern-growth steady state In the region x > max f x Z , x N g , there exists the steady state: � � ρ � m + γ m x � = ( 1 � α ) φ � 1 � σ � � > 0 ; ρ � m + γ m 1 � α � π 1 � σ γ m n � = 1 � σ > 0 ; " # � � α ( φπ � 1 ) ρ � m + γ m z � = � � � 1 > 0 . ρ � m + γ m 1 � σ 1 � α � π 1 � σ Exhibits growth of …nal output per capita y � � m = α ( x � � φ ) � ρ � m . Peretto (Duke University) Take-o¤ to sustained growth September 2012 15 / 31
The variety-…rst path to modern growth: equation Let σ / x ! 0 for x > max f x N , x Z g . Thresholds identify three regimes. In each one …rm size evolves according to linear di¤erential equation: 8 γ mx x � x N < x � � x ) x = ˙ ν ( ¯ ¯ x N < x � x Z , : ν ( x � � x ) x > x Z where: ( ) � � σ ( x � φ ) α � φ π x x N � 1 � ( ρ � m ) π ; x Z � arg solve ( 1 � σ ) ( ρ � m ) + γ m = 1 ; ν � ( 1 � σ ) φ φ x � = � � ; ¯ ¯ ; ρ � m + γ m x � π ¯ 1 � π 1 � σ � � ρ � m + γ m x � = ( 1 � α ) φ � ν � ( 1 � σ ) φ 1 � σ � � . ; ρ � m + γ m π x � 1 � α � π 1 � σ Peretto (Duke University) Take-o¤ to sustained growth September 2012 16 / 31
The variety-…rst path to modern growth: evolution of …rm size Peretto (Duke University) Take-o¤ to sustained growth September 2012 17 / 31
The variety-…rst path to modern growth: time of events � x N � 1 T N = γ m log ; x 0 � ¯ � x � � x N T Z = T N + 1 ν log . x � � x Z ¯ ¯ Peretto (Duke University) Take-o¤ to sustained growth September 2012 18 / 31
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