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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)

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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 PY 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.

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Final producers (i)

Technology: Y =

Z N

X θ

i

Z α

i Z 1α

L NηL γ R NηR 1γ!1θ , Z

Z N

(Zj/N) dj where: 0 < θ, γ < 1 standard parameters that map into factor shares; 0 < ηL, ηR < 1 congestion/rivarly parameters; Z α

i Z 1α vertical technology index, with α 2 [0, 1) measure of private

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 γηL (1 γ) ηR 2 [0, 1).

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Final producers (ii)

Demand for product i: Xi = Nσ1 θ Pi

• 1

1θ

Z α

i Z 1αLγR1γ.

Factor payments: N PX =

Z N

PiXidi = θY ; wL = w

Z N

Lidi = γ (1 θ) Y ; pR = p

Z N

Ridi = (1 γ) (1 θ) Y .

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Intermediate producers

Technologies: Costi = 1

• Xi + φZ α

i Z 1α

; ˙ Zi = Ii. Firm’s objective: Vi (0) =

Z ∞

e R t

0 r(s)ds

Xi (t) (Pi (t) 1) φZ α

i (t) Z 1α (t) Ii (t)

• dt.

In symmetric equilibrium: max

Pi,Ii Vi ) r = αX

1

θ 1

• Z

rZ ; V max

i

= βXi ) r = X 1

θ 1

φZ I βX + ˙ X X rN.

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Households (i)

Representative household chooses C (t), L (t), R (t) to maximize U (0) =

Z ∞

e(ρm)t log C (t) M (t)

• dt,

ρ > m > 0 subject to: ˙ A = rA + wL + pR C; M L 0, M = M0emt, m > 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).

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Households (ii)

Factors supply: L = M; R = Ω. Consumption/saving: r = ρ + ˙ C C m.

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Equilibrium output

Symmetry plus factor markets equilibrium yields Y = θ

2θ 1θ NσZMγΩ1γ.

Accordingly, output per capita is Y M = θ

2θ 1θ NσZ

Ω M 1γ . Let y ˙ Y /Y , n ˙ N/N and z ˙ Z/Z. Then, y m | {z }

• utput per capita

growth = σn + z | {z } Hicks neutral TFP growth

• (1 γ) m

| {z } growth drag due to land .

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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) Z = gross cash ‡ow quality = “…rm size”, where in equilibrium x = θ (1 θ) Y NZ = θ (1 θ) θ

2θ 1θ MγΩ1γ

N1σ .

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General equilibrium: Role of …rm size (ii)

Expressions for returns are: r = α (x φ) ; r = x φ πx + z

• 1 1

πx

• + ˙

x x , where to simplify notation π βθ 1 θ = βX X (P 1) = entry cost gross cash ‡ow. Important: given mass of …rms N, …rm size x is increasing in aggregate market size Y and thus in population M.

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General equilibrium: Role of population

De…ne market growth factor γm > 0. Suppose initially n = z = 0. Then y m < 0 but γm > 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?

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Transition dynamics: Role of corner solutions (i)

Let xN be threshold of …rm size that triggers variety innovation and xZ 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 xN < xZ . Dominant incentives to quality innovation xN > xZ . 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.

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Transition dynamics: Role of corner solutions (ii)

Speci…c values of xN and xZ di¤er in the two cases. Why? If xN < xZ , xZ is threshold for quality R&D given that market already supports entry of new …rms; if xN > xZ , xZ 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 xN: If xN < xZ , xN is threshold for entry of new …rms given that market does not yet support quality R&D; if xN > xZ , xN is threshold for entry of new …rms given that market already supports quality R&D.

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The modern-growth steady state

In the region x > max fxZ , xNg , there exists the steady state: x = (1 α) φ

• ρ m + γm

1σ

• 1 α π
• ρ m + γm

1σ

> 0; n = γm 1 σ > 0; z = " α (φπ 1) 1 α π

• ρ m + γm

1σ

1 # ρ m + γm 1 σ

• > 0.

Exhibits growth of …nal output per capita y m = α (x φ) ρ m.

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The variety-…rst path to modern growth: equation

Let σ/x ! 0 for x > max fxN, xZ g. Thresholds identify three regimes. In each one …rm size evolves according to linear di¤erential equation: ˙ x = 8 < : γmx x xN ¯ ν (¯ x x) xN < x xZ ν (x x) x > xZ , where: xN φ 1 (ρ m) π; xZ arg solve ( (x φ)

• α

σ πx

• (1 σ) (ρ m) + γm = 1

) ; ¯ ν (1 σ) φ π¯ x ; ¯ x = φ 1 π

• ρ m + γm

1σ

; ν (1 σ) φ πx ; x = (1 α) φ

• ρ m + γm

1σ

• 1 α π
• ρ m + γm

1σ

.

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The variety-…rst path to modern growth: evolution of …rm size

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The variety-…rst path to modern growth: time of events

TN = 1 γm log xN x0

• ;

TZ = TN + 1 ¯ ν log ¯ x xN ¯ x xZ

• .

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The quality-…rst path to modern growth: equation

As before, let σ/x ! 0 for x > max fxN, xZ g. Thresholds identify three

• regimes. In each one …rm size evolves according to linear di¤erential

equation: ˙ x = 8 < : γmx x xZ γmx xZ < x xN ν (x x) x > xN , where: xN (1 α) φ ρ + m γm 1 α (ρ m) π ; xZ arg solve ( α (x φ) = 1 + 1/θ 1 φ

x + 1/θ

γm ) .

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The quality-…rst path to modern growth: evolution of …rm size

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The quality-…rst path to modern growth: time of events

TZ = 1 γm log xZ x0

• ;

TZ = TN + 1 γm log xN xZ

• .

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The anatomy of the transition: GDP

Let G denote this economy’s GDP. Subtracting the cost of intermediate production from the value of …nal production yields G = (1 θ)

• θ
• 1 φ

x

• + 1
• |

{z } unit cost of intermediate …rm falls as scale of operation rises Y . Taking logs and time derivatives, g ˙ G G = y + ξ (x) ˙ x x , ξ (x) θφ (1 + θ) x θφ, where ξ (x) is the elasticity of GDP with respect to …rm size (strictly positive since x φ).

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From Smith to Schumpeter: variety-…rst case (i)

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From Smith to Schumpeter: variety-…rst case (ii)

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From Smith to Schumpeter: variety-…rst case (iii)

The expressions for TN and TZ reveal the following pattern. The activation of horizontal innovation occurs earlier, i.e., TN is lower, in economies where the ratio xN/x0 is lower. The activation of vertical innovation occurs if and only if the steady-state …rm size ¯ x associated to the phase with horizontal innovation only is smaller than the threshold for quality innovation xZ . Given TN, and conditional on ¯ x > xZ , the activation of vertical innovation occurs earlier, i.e., TZ is lower, in economies where:

the steady-state …rm size ¯ x associated to the equilibrium with no quality innovation is larger; convergence in the variety-driven phase is faster, i.e., where ¯ ν is higher; the threshold for quality innovation xZ is smaller.

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From Smith to Schumpeter: quality-…rst case (i)

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From Smith to Schumpeter: quality-…rst case (ii)

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From Smith to Schumpeter: quality-…rst case (iii)

The expressions for TZ and TN in and yield the following pattern. The activation of vertical innovation occurs earlier, i.e., TZ is lower, in economies where the ratio xZ /x0 is lower. Given TZ , the activation of horizontal innovation occurs earlier, i.e., TN is lower, in economies where the threshold for variety innovation xN is smaller.

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Bringing it all together: When does the take-o¤ occur?

De…ne generic value xT min ( φ 1 (ρ m) π, arg solve ( α (x φ) = 1 + 1/θ 1 φ

x + 1/θ

γm )) and think of the take-o¤ date as T = 1 γm ln xT x0

• =

1 γm ln B @ xT θ (1 θ) θ

2θ 1θ M γ 0 Ω1γ

N 1σ

1 C A . Suppose x0 = xT /2, γ = 0.8, m = 0.1%. Then, using “rule of 70”, T = ln 2/ (0.08%) = (70%) / (0.08%) = 875. Note: This economy experiences an increase in population given by M (T) = 21/γ M0 = 2.38 M0.

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Remarks (i)

Prior to the onset of pro…t-driven systematic innovation, static economies

• f scale in intermediate production deliver income per capita growth in

periods of population expansion. Such Smithian growth, however, is not self-sustaining and eventually must vanish. Changes in fundamentals that result in an earlier take-o¤ date do not necessarily result in immediate take-o¤. When the economy turns on variety innovation …rst, it can fail to cross the threshold for quality innovation. When the economy turns on quality innovation …rst, it exhibits explosive growth that ends in …nite time because it cannot fail to cross the threshold for variety innovation.

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Remarks (ii)

Recall that in steady state x is constant. Hence, N =

• θ (1 θ) κMγΩ1γ

x

• 1

1σ

. Elimination of scale e¤ect through product proliferation does not require knife-edge assumption N = M{, { = 1. Rather, theory says { = γ 1 σ Q 1. To get { = 1 need to assume either (a) γ = 1 (no land) and σ = 0 (no love of variety) or (b) γ = 1 σ ) 1 γ = σ.

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