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

Daniele Tavani; Luca Zamparelli

doi:10.1515/snde-2013-0117

Published by De Gruyter July 4, 2014

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

Daniele Tavani and Luca Zamparelli

From the journal Studies in Nonlinear Dynamics & Econometrics

https://doi.org/10.1515/snde-2013-0117

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Abstract

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.

Keywords: endogenous technical change; Goodwin model; income shares; employment.

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: daniele.tavani@colostate.edu

Acknowledgments

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.

Appendix

A Dynamic optimization

The current-value Hamiltonian is:

ℋ=lnc+μ(B(1−wA)K−δK−c(1+τ)−R(1−s))+λϕ[n]A,

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

(12)

c−1=μ(1+τ) (12)

(13)

μ(1−s)=λϕ′[n] (13)

(14)

ρμ−μ˙=μ(B(1−ω)−δ) (14)

(15)

ρλ−λ˙=μBKw(1A2)+λ(ϕ[n]−ϕ′[n]n)=μωL+λ(ϕ[n]−ϕ′[n]n) (15)

plus two transversality conditions: limt→∞e−ρtμ(t)K(t)=0=limt→∞e−ρtλ(t)A(t). From (14), we get ρ−μ˙/μ=B(1−ω)−δ, which, given (12), yields c˙/c=B(1−ω)−(ρ+δ). Next, differentiate log of (13) to obtain:

λ˙λ−μ˙μ=(1−χ)n˙n

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

ρ−λ˙λ=(ϕ[n]+ϕ′[n](ωv1−s−n)),

and then λ˙λ−μ˙μ=ρ−μ˙μ−(ρ−λ˙λ)=B(1−ω)−δ−(ϕ[n]−ϕ′[n](n−ωv1−s)), 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:

dρ(1+h[nss]ϕ′[nss](1−s)B)=ϕ″[nss](h[nss]1−s(1−ρ+δ+ϕ[nss])B)−nss)dn+ϕ′[nss](h′[nss]1−s(1−ρ+δ+ϕ[nss])B)−h[nss]ϕ′[nss](1−s)B−1)dn.

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

(16)

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

From (8), h[nss]/nss(1−s)>(1−ρ+δ+ϕ[nss])B)−1, therefore h[nss]ϕ′[nss](1−s)B(1−ρ+δ+ϕ[nss])B)−1+(1−ρ+δ+ϕ[nss])B)−1=(1−ρ+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′[n_ss]<1–s is a sufficient condition for dn_ss/dρ<0.

B.2 Labor market conditions

Rewrite (R) as

ρ=ϕ′[nss]h[nss, σ](1−B−1(ρ+δ+ϕ[nss]))1−s−ϕ′[nss]nss

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

dσ(−dhdσϕ′[nss])1−s(1−ρ+δ+ϕ[nss])B))=[ϕ″[nss](h[nss, σ]1−s(1−ρ+δ+ϕ[nss])B)−nss)+ϕ′[nss](dhdn(1−ρ+δ+ϕ[nss])B)1−s−h[nss, σ]ϕ′[nss](1−s)B−1)]dn.

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

B.3 R&D subsidy

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

−(ϕ′[nss]h[nss](1−s)2(1−ρ+δ+ϕ[nss])B))ds=[ϕ″[nss](h[nss]1−s(1−ρ+δ+ϕ[nss])B)−nss)+ϕ′[nss](h′[nss]1−s(1−ρ+δ+ϕ[nss])B)−h[nss]ϕ′[nss](1−s)B−1)]dn.

Since ϕ′[nss]h[nss](1−ρ−(γ−1)ϕ[nss])(1−s)2>0, proceeding similarly to B.1 and B.2 yields h′[n_ss]<1–s as a sufficient condition for dn_ss/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]=an^{^χ}, differentiate (N) w.r.t. ω and n to find:

−dω(B+11−sϕ′[nss]h[nss])=dn[ω(h′[nss](1−s)ϕ′[nss]+h[nss](1−s)ϕ″[nss])+(1−χ)ϕ′[nss]].

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

−dω(Bϕ′[nss]+11−sh[nss])=dn[ω(h′[nss](1−s)−(1−χ)h[nss](1−s)nss)+(1−χ)].

Accordingly, h′[nss]>(1−χ)h[nss]nss⇒dωdn<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.

References

Aghion, P., and P. Howitt. 2010. The Economics of Growth. Cambridge, MA: MIT Press.Search in Google Scholar

Benhabib, J., and K. Nishimura. 1979. “The Hopf Bifurcation and Existence and Stability of Closed Orbits in Multisector Models of Optimal Economic Growth.” Journal of Economic Theory 21: 421–444.10.1016/0022-0531(79)90050-4Search in Google Scholar

Blanchard, O., and C. Kahn. 1980. “The Solution of Linear Difference Models under Rational Expectations.” Econometrica 48: 1303–1311.10.2307/1912186Search in Google Scholar

Desai, M., B. Henry, A. Mosley and M. Penderton. 2006. “A Clarification of The Goodwin Model of The Growth Cycle.” Journal of Economic Dynamics and Control 30: 2661–2670.10.1016/j.jedc.2005.08.006Search in Google Scholar

Fiorio, C., S. Mohun, and R. Veneziani. 2013. “Social Democracy and Distributive Conflict in the UK, 1950-2010.” Working Paper No. 705, School of Economics and Finance, Queen Mary University of London. ISSN: 1473-0278.Search in Google Scholar

Foley, D. K. 2003. “Endogenous Technical Change with Externalities in a Classical Growth Model.” Journal of Economic Behavior and Organization 52: 167–189.10.1016/S0167-2681(03)00020-9Search in Google Scholar

Foley, D. K., and T. Michl. 1999. Growth and Distribution. Cambridge, MA: Harvard University Press.Search in Google Scholar

Gandolfo, G. 1997. Economic Dynamics. Berlin: Springer-Verlag.10.1007/978-3-662-06822-9Search in Google Scholar

Goodwin, R. 1967. “A Growth Cycle.” In Socialism, Capitalism, and Economic Growth, edited by Carl Feinstein, Cambridge, UK: Cambridge University Press.Search in Google Scholar

Harvie, D. 2000. “Testing Goodwin: Growth Cycles in OECD Countries.” Cambridge Journal of Economics 24: 349–376.10.1093/cje/24.3.349Search in Google Scholar

Harvie, D., M. A. Kelmanson, and D. G. Knapp. 2007. “A Dynamical Model of Business–Cycle Asymmetries: Extending Goodwin.” Economic Issues 12: 53–92.Search in Google Scholar

Impullitti, G. 2010. “International Competition and U.S. R&D Subsidies: A Quantitative Welfare Analysis.” International Economic Review 51: 1127–1158.10.1111/j.1468-2354.2010.00613.xSearch in Google Scholar

Julius, A. J. 2005. “Steady-State Growth and Distribution with An Endogenous Direction of Technical Change.” Metroeconomica 56: 101–125.10.1111/j.1467-999X.2005.00209.xSearch in Google Scholar

Kennedy, C. 1964. “Induced Bias in Innovation and the Theory of Distribution.” Economic Journal 74: 541–547.10.2307/2228295Search in Google Scholar

Kortum, S. 1993. “Equilibrium R&D Ratio and the Patent-R&D Ratio: U.S. Evidence.” American Economic Review, Papers and Proceedings 83: 450–457.Search in Google Scholar

Mehra, R., and E. C. Prescott. 1985. “The Equity Premium: A Puzzle.” Journal of Monetary Economics 15: 145–161.10.1016/0304-3932(85)90061-3Search in Google Scholar

Montrucchio, L. 1992. “Dynamical Systems that Solve Continuous-Time Concave Optimization Problem: Anything Goes.” In Cycles and Chaos in Economic Equilibrium, edited by J. Benhabib, 277–288. Princeton, NJ: Princeton University Press.10.2307/j.ctv19fvxt1.15Search in Google Scholar

Shah, A., and M. Desai. 1981. “Growth Cycles with Induced Technical Change.” Economic Journal 91: 1006–1010.10.2307/2232506Search in Google Scholar

Smulders, S., and T. van de Klundert. 1995. “Imperfect Competion, Concentration and Growth with Firm-specific R&D.” European Economic Review 39: 139–160.10.1016/0014-2921(94)E0072-7Search in Google Scholar

van der Ploeg, F. 1987. “Growth Cycles, Induced Technical Change, and Perpetual Conflict over the Distribution of Income.” Journal of Macroeconomics 9: 1–12.10.1016/S0164-0704(87)80002-2Search in Google Scholar

Published Online: 2014-7-4

Published in Print: 2015-4-1

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

Abstract

Acknowledgments

Appendix

A Dynamic optimization

B Comparative steady states

B.1 Discount rate

B.2 Labor market conditions

B.3 R&D subsidy

B.4 On the slope of the N isocline

References

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