Catch-up and Fall-back

Jess Benhabib Jesse Perla Christopher TonettiNYU NYU NYU

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Introduction-Questions

This paper was motivated by some questions raised by thiscourse last year and the papers of Perla-Tonetti andLucas-Moll, among others.

How to simultaneously model innovation, the technologyfrontier and technology diffusion when agents/countriesoptimally invest in fostering innovation and diffusion.

In equilibrium will some agents/countries choose to invest intechnology diffusion and catch-up, while others choose toinvest in innovation? What does the BGP look like?

Will some agents optimally choose instead to fall back relativeto the moving frontier?

Thus our title: Catch-up and Fall-back through Diffusion andInnovation.

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Motivation: Catch-up

Rapid, sustained growth of some emerging economies: often5− 10% over the last decade.

Growth can’t all be from factor accumulation and innovation.Some must be from technology diffusion and catch-up.

In 1977, after visiting a Samsung research lab in Korea as aconsutant for the British government, Ira Magaziner wrote:“[It] reminded me of a dilapidated high-school scienceroom. . .They’d gathered color televisions from every majorcompany in the world– RCA, GE, Hitachi– and were usingthem to design a model of their own.”Fifteen years laterthere would not be a UK owned TV manufacturer.

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Motivation: Fall-back?

Rapid growth of European countries in 1970s and 1980sslowing to stable ratio of GDP relative to the US

Since mid 1990s, low innovation middle-upper incomedeveloped countries decreasing GDP ratio relative to US.

"A growing productivity shortfall in the EU15’s leadingeconomies relative to the world’s innovation leader– theUnited States– was also recorded. While in 1985, Europeanproductivity was 95% of the US levels, this figure fell to 85%in 1995. The growth slowdown in Europe can also beexplained in terms of lagging innovation. ."

Golden Growth: Restoring the Lustre of the European EconomicModel, Gill and Raiser, World Bankhttp://www.bruegel.org/fileadmin/bruegel_files/Events/Write_ups/2012/120124_write-up_golden_growth.pdf

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Related Literature

There is some empirical evidence suggesting rate ofproductivity growth depends on distance to the frontier (e.g.Benhabib and Spiegel (1994, 2005)).

Early papers had some shortcomings in that economicvariables related to diffusion were not subject to choice (e.g.Nelson and Phelps (1966))

Later literature introduced models with investment decisionson innovation and "imitation" (e.g. Barro and Sala-i-Martin(1997), Aghion, Howitt, and Mayer-Foulkes (2005)) but hadexogenous permanent types of innovators and "imitators"

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Contribution

Highly Stylized model with a distribution of agentsendogenously choosing a portfolio of both diffusion andinnovation investments. Agents are identical other than initialproductivity.

How precisely does the initial distribution matter?

Catch-up or diffusion model that nests Nelson-Phelps,Gompertz, Logistic growth.

Productivity distribution influences the returns to diffusionthrough the distance to the frontier

Two competing forces for the choice of investment intensity:

Agents have an incentive to catch-up in order to have higherproductivity and hence consumption.Agents may also have an incentive to fall-back in order toexploit the higher productivity of diffusion from being furtherbehind the frontier

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Summary of Model

Log utility, linear production

Continuum of agents heterogeneous only over productivity:z(t), with CDF Φ(t, z)Frontier, F (t) ∈ (0,∞), is highest productivity among agentsAgents choose optimal portfolio in CRS growth technologies

Investments deterministically increase z(t)Investment costs paid in forgone consumptionInvest s in diffusion and/or γ in innovationReturns to innovation: geometric and autarkicReturns to diffusion: geometric and depend on distance (sortof) to the frontier through catch-up function

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Model: Growth Technologies

Technologies to produce goods could consist of variousprocesses, not all of which are instantly adopted. For example,production may start with an assembly plant and themanufacturing of parts. Associated processes may begradually adopted as know-how develops and accumulates.The productivity of innovation is: σThe productivity of diffusion is: D (t, z ;m)

D (t, z ;m) ≡ cm

(1−

(z (t)F (t)

)m)m = 1: logistic, m = −1: Nelson-Phelps, m = 0: Gompertz

D (t, z ;m) ≡ c(F (t)z (t)

− 1)if m = −1

D (t, z ;m) ≡ c(1− z (t)

F (t)

)= c z (t)F (t)

(F (t)z (t)

− 1)

if m = 1

Note: D (t,F (t);m) = 0 for all t,m.8 / 31

Agent’s Problem

The law of motion for productivity is a flow :

zz= σγ+ D(t, z ;m)s

Given discount rate r , and production parameter B:

maxs ,γ,z

∫ ∞

0(ln (Bz − sz − γz)) e−rtdt

s.t.zz= σγ+ D(t, z)s

s ≥ 0, γ ≥ 0

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The Technology Frontier

Agent’s at the frontier: z(t) = F (t)

D(t,F (t)) = 0 for all tHence, they choose s = 0 and γ ≥ 0

Assumption 1: σB − r > 0Don’t discount future so much that don’t want to invest ingrowing.

Solution: Given Assumption 1, the agent chooses:

γ(t) = B − rσ , for all t

zz = Bσ− r ≡ g , for all tF (t) = F (0)egt

Thus, the frontier grows at a constant rate independent ofdiffusion technology

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Followers

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Agent’s Problem in Relative Productivity

With F (t) and g , do change of variables to x = zF :

x(t)x(t)

= σγ(t) +D(x(t))s(t)− F (t)F (t)

Problem of an agent not at the frontier becomes:

maxs ,γ,x

∫ ∞

0(ln x + ln(B − s − γ)) e−rtdt +

g + r ln(F (0))r2

xx= σγ+D(x)s − g

s ≥ 0, γ ≥ 0

Productivity of innovation is constant, while productivity ofdiffusion is constant conditional on x

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The Diffusion Technology

D(·) is autonomous in relative productivity x(t) ≡ z (t)F (t)

D(x) = cm (1− x

m) , for x ∈ (0, 1)

D ′(x) ≡ ddxD(x) < 0, diffusion more productive the further behind

Nelson-Phelps(m = −1), Gompertz(m→ 0), Logistic(m = 1)ddmD(x) < 0 =⇒ ↑ m means less effi cient diffusion

Figure: D (x) as a function of x for various m13 / 31

Innovation and Imitation

Simultaneous innovation and diffusion, s > 0,γ > 0, occurs onlyat a unique point, x∗ (via the bang-bang solution). Equate FOCs:

λx∗σ = λx∗cm(1− x∗m)

x∗ ≡ (1− σmc )

1/m

Assumption 2: σmc < 1

This assumption assures that x∗ > 0 : the effi ciency parameter fortechnology diffusion, cm (that also depends on the distance to thetechnology frontier) is larger than the effi ciency parameter forinnovation, σ.

x∗ =(1− σm

c

)1/m

x < x∗ =⇒ s(x) ≥ 0, γ(x) = 0

x > x∗ =⇒ s(x) = 0, γ(x) > 0

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Innovation Only

s = 0,γ > 0 occurs for x∗ < x < 1Dynamic and Stationary Solution:

γ(t) = Bσ−rσ for all t

xx = 0 =⇒

zz = g

Hence, all innovating agents invest the same amount ininnovation and grow at the same rate as the frontier

λ(t) = 1x1r

Constant shadow price of x , no future value from diffusion

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Imitation Only

s ≥ 0,γ = 0 occurs for 0 < x < x∗Unique Stationary Solution: (x , s, λ)

γ = B − rσ

λ =1x

1r + csxm

=1x

(1

r − s xD ′(x)

)<1r

s = B − 1c/mλx(1− xm) = B −

1λxD(x)

> 0

x =

12

(2−mσc +

m2c (σ− r

B ))−sign(m)

((2−mσ

c +m2c (σ− r

B ))2−4(1−mσ

c )) 12

1m

Assumption 3: m ≥ −1 and cσ < m+ 2

Assumption 3 ensures concavity of the Hamiltonian. Note that

both conditions in Assumption 3 restrict the effi ciency oftechnology diffusion. Concavity may be lost as technology diffusionbecomes too easy, with high c or low m.

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Dynamic Solution

Let Assumptions 1, 2, and 3 hold, then for arbitrary Φ(0, x)

1 There exists a threshold ratio x∗ =(1− mσ

c

)1/m such that allagents with x > x∗, including the leader, do only research.They set s = 0, γ = B − r

σ > 0, and grow at the rate σB − r .For x > x∗, the distribution Φ(t, x) = Φ(0, x) for all t.

2 All agents with initial conditions x < x∗ invest only intechnology adoption with s(t) ≥ 0 and γ(t) = 0. Thereexists a strictly positive BGP productivity ratio 0 < x < x∗

such that all initial productivity ratios x < x∗ converge to x .

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Example Dynamics Given D(x;m) and σ

Nelson-Phelps (m = -1) Gompertz (m = 0) Logistic (m = 1)

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Balanced Growth Path

Corollary (2)

A balanced growth path equilibrium is a growth rate g = σB − r ,a steady state imitator productivity ratio x , an innovator thresholdx∗, and an asymptotic distribution Φ(∞, x) such that

1 The growth rate of all agents is zz = g = σB − r , so thegrowth rate x

x = 0 for all x , t.2 For arbitrary Φ(0, x),

Φ(∞, x) =

0 for 0 ≤ x < xΦ(0, x∗) for x ≤ x < x∗

Φ(0, x) for x∗ ≤ x ≤ 1

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Balanced Growth Path Example

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Summary of Comparative Dynamics of m

Reminder: As m ↑ diffusion becomes less effi cient:ddmD(x) < 0

As technology diffusion becomes less effi cient:

A larger mass of agents become innovators and fewer chooseto be imitators: dx

∗dm < 0.

And under some additional restrictions:

Imitating agents optimally let the ratio of their productivity tothe technology frontier slip lower and they benefit from thehigher diffusion rate that comes from falling behind thefrontier: d xdm < 0.Since s is increasing in m, imitators increase expenditures thataid technology diffusion even as they fall further behind thetechnology frontier: d s

dm > 0

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Numerical Example of Equilibrium Given m

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Catch-up, Fall-back, and the Shadow Value of x

From the Dynamic Euler equation:

r =λ

λ+1x+ (σγ+ sD(x)− g) + sxD ′(x)

Euler equation: total benefit of a marginal x = discount rateλλ : Appreciation in the shadow price1x : Marginal utility of x

(σγ+ sD(x)− g): Marginal product of xsxD ′(x) < 0: Fall-back incentive

Equilibrium shadow price:

λ =1x

(1

r − s xD ′(x)

)<1r

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The Ratio of Marginal Productivities

Define θ ≡ σc :

x∗ = (1−mθ)1/m

x =

12

2+(m2−m)θ−m2rcB −sign(m)

((2+(m2−m)θ−m

2rcB

)2−4(1−mθ)

) 12

1m

Note that:

The innovator threshold only depends on the ratio of marginalproductivities

The equilibrium imitator productivity depends on c separately∂D ′(x ;c )

∂c = ∂2D (x ;c )∂x∂c < 0, so as c ↑, the fall-back incentive

sxD ′(x) ↑

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Comparative Statics with “Hicks-Nutral”TechnologyChange

Uniformly change the productivity of growth technologies

Fix θ = σc

Multiply both by a common A: σ = Aσ and c = Ac

This seemingly Hicks-neutral change distorts the equilibrium

1 x∗(A) = x∗(1) =(1−mθ

)1/m , no distortion

2d x (A)dA < 0 =⇒ x(A) is decreasing in A

3d s(A)dA > 0 =⇒ s(A) is increasing in A

4 limA→∞ x(A) > 0

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Equilibrium with “Hicks Neutral”A

Increasing Hicks-neutral progress causes fall backA ↑ =⇒ g = σAB − r ↑, so the imitators have to investmore, s ↑, to keep up with the frontier. 26 / 31

Conclusion:Key Results

The technology frontier, and the BGP productivity growthrate, depend only on the innovation technology

Agents endogenously segment into innovators and imitators:

The proportion of innovators depends on the relativeproductivity of diffusion vs. innovation

Agents may endogenously choose to “fall-back” to a lowerproductivity relative to the frontier

As diffusion becomes less effi cient imitators fall further behindthe frontier on the BGP

A seemingly Hicks-neutral technology increase:

Has no effect on the proportion of innovatorsDecreases the equilibrium relative productivity of imitators

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Motivation Revisited

Rapid, sustained growth of some developing countries

“Catch-up”not surprising– we’ve seen it beforeHistorically, growth rates depend on distance to frontier

European convergence to GDP ratio below unity and lowerinnovation

– Technology diffusion effi ciency incentives

“Fall-back ”

Not neccesarily sub-optimal–

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The Lucas-Moll Model-Baseline Model Described: Poissonarrivals

What happens in a meeting?– Agent (a draw z from F (z , t)) meets another, z ′.– Not symmetric: z is active, learns from meeting, but z ′ ispassive.– If z ′ ≥ z , nothing happens; if z ′ ≤ z agent z adopts z ′– Poisson Idea Arrivals: over time interval (t, t + ∆) meetanother with probability α∆– More than one meeting with with probability o (∆) (where thefunction o (∆) satisfies lim→∞

o(∆)∆ = 0)

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Now motivate differential equation for right cdfF (z , t) = Pr {z ≥ z at date t}

F (z , t + ∆) = F (z , t)×Pr{no lower cost idea darrives in (t, t+∆)}

Pr{no lower cost arrives in(t, t + ∆) = Pr{no ideas arrive in (t, t + ∆)}+Pr{one idea > z arrives from F (z , t) in (t, t + ∆)}

+Pr{more than one idea > z arrives in (t, t + ∆)}= 1− α+ α∆F (t, z) + o (∆)

Combine to get

F (z , t + ∆) = F (z , t) [1− α∆+ α∆F (t, z) + o (∆)]

F (z , t + ∆)− F (z , t)∆

= −F (z , t)[

α− αF (z , t)− o (∆)∆

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F (z , t + ∆)− F (z , t)∆

= −F (z , t)[

α− αF (z , t)− o (∆)∆

]Let h→ 0 to get

∂F (z , t)∂t

= −αF (z , t) [1− F (z , t)]

Now fix z , treat as ODE, solve given initial F (z , 0) = G (z , ) andcharacterize.For fixed z solution is:

F (z , t) =G (z)

G (z) + eαt (1− G (z))

and if the initial distribution is bounded below at z so G (z) = 1,then all mass converges down to z .

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