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Endogenous technical change, employment and distribution in the Goodwin model of the growth cycle

Daniele Tavani and Luca Zamparelli


In this paper, we introduce endogenous technological change through R&D expenditure on labor-augmenting innovation in the cyclical growth model by Goodwin (Goodwin, R. 1967. “A Growth Cycle.” In Socialism, Capitalism, and Economic Growth, edited by Carl Feinstein, Cambridge, UK: Cambridge University Press.). Innovation is a costly, forward-looking process financed out of profits, and pursued by owners of capital stock (capitalists) in order to foster labor productivity and save on labor requirements. Our main findings are: (i) Goodwin-type distributive cycles arise even with dynamic optimization, but (ii) endogenous technical change has a dampening effect on economic fluctuations; (iii) steady state per capita growth, income distribution and employment rate are endogenous, and depend on the capitalists’ discount rate, the institutional variables regulating the labor market, and policy variables such as subsidies to R&D activity. Implementing the model numerically to match long run data for the US, we show that: (iv) an increase in the capitalists’ discount rate lowers per-capita growth, the employment rate and the labor share; (v) an increase in workers’ bargaining strength moderately raises the labor share and moderately decreases per-capita growth, while sharply reducing employment: quarterly US fluctuations (1948–2006) in employment and the labor share seem to support this result; (vi) a balanced budget increase in the R&D subsidy also fosters per-capita growth at the expenses of the labor share, even though the corresponding variations might be small.

JEL codes:: E32; O33

Corresponding author: Daniele Tavani, Department of Economics, Colorado State University, 1771 Campus Delivery, Fort Collins, Colorado 80523-1771, USA, Phone: +1 970 491 6657, e-mail:


We thank Corrado Di Guilmi, Duncan Foley, Giancarlo Gandolfo, Codrina Rada, Peter Skott, Rick van der Ploeg, and participants at the EEA Conference 2013 and at the FMM Conference 2013 for very useful comments on earlier drafts. Comments from the Associate Editor and an anonymous referee greatly improved the paper. The usual disclaimer applies.


A Dynamic optimization

The current-value Hamiltonian is:


where μ, λ are the current-value costate variables. The first order conditions are:

(12)c1=μ(1+τ) (12)
(13)μ(1s)=λϕ[n] (13)
(14)ρμμ˙=μ(B(1ω)δ) (14)
(15)ρλλ˙=μBKw(1A2)+λ(ϕ[n]ϕ[n]n)=μωL+λ(ϕ[n]ϕ[n]n) (15)

plus two transversality conditions: limteρtμ(t)K(t)=0=limteρtλ(t)A(t). From (14), we get ρμ˙/μ=B(1ω)δ, which, given (12), yields c˙/c=B(1ω)(ρ+δ). Next, differentiate log of (13) to obtain:


and use (13), (14) and (15) to find, first:


and then λ˙λμ˙μ=ρμ˙μ(ρλ˙λ)=B(1ω)δ(ϕ[n]ϕ[n](nωv1s)), from which equation (4) follows.

B Comparative steady states

B.1 Discount rate

Start with (R), and totally differentiate it w.r.t. n and ρ to find:


Given φ″[nss]<0 and (8), h[nss](1ρ+δ+ϕ[nss])B)/(1s)h[nss]ϕ[nss]/((1s)B)1<0 implies dnss/<0 Using the definition of φ[nss] and rearranging the previous condition yields

(16)h[nss]1s<h[nss]ϕ[nss](1s)B(1ρ+δ+ϕ[nss])B)1+(1ρ+δ+ϕ[nss])B)1. (16)

From (8), h[nss]/nss(1s)>(1ρ+δ+ϕ[nss])B)1, therefore h[nss]ϕ[nss](1s)B(1ρ+δ+ϕ[nss])B)1+(1ρ+δ+ϕ[nss])B)1=(1ρ+δ+ϕ[nss])B)1(1+ζ)>1, where ζ is an unknown positive scalar. The right hand side of (16) is strictly larger than one so that h′[nss]<1–s is a sufficient condition for dnss/<0.

B.2 Labor market conditions

Rewrite (R) as


to emphasize the role of the conflict parameter σ. Totally differentiate it w.r.t. n and σ to find:


After noting that dhdσ(ϕ[nss](1ρϕ[nss])1s)>0, proceed as in B.1 to find that h′[nss]<1–s is a sufficient condition for dnss/<0.

B.3 R&D subsidy

Totally differentiate (R) w.r.t. n and s to find:


Since ϕ[nss]h[nss](1ρ(γ1)ϕ[nss])(1s)2>0, proceeding similarly to B.1 and B.2 yields h′[nss]<1–s as a sufficient condition for dnss/ds>0.

B.4 On the slope of the N isocline

Our calibrated economy already proves the possibility that the (N) isocline be upward sloping. Here we provide a sufficient condition for the isocline to be downward sloping. Keeping in mind that φ[n]=anχ, differentiate (N) w.r.t. ω and n to find:


Divide the both sides of the previous equation by φ′[nss] to find


Accordingly, h[nss]>(1χ)h[nss]nssdωdn<0. In other words, a sufficient condition for a negative slope of the (N) isocline is that the elasticity of the employment rate to R&D share is larger than 1–χ.


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Published Online: 2014-7-4
Published in Print: 2015-4-1

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