Google’s logo doodle yesterday paid tribute to the late Sir Arthur Lewis, the first economist to be so honored. The accolade bookends the Nobel Prize he won in 1979. Lewis was an influential professor of mine at Princeton. Indeed, my PhD thesis used nonlinear difference equations to restate and expand his important two-sector model of economic growth in developing countries.

The irony of the Google recognition four decades after his Nobel is that his pioneering, still important work is no longer taught in our most prestigious graduate schools. One of the original goals of the GEM Project was the restoration of Sir Arthur to his rightful place in modern macro theory.

__Original Model __

Lewis’s two-sector growth theory (1954, 1958) quickly became the benchmark analysis of the transformation of subsistence economies. Sir Arthur used two-sector technical heterogeneity to motivate separate labor-pricing regimes. Rational behavior in each venue is governed by discrete decision rules, constraints, and mechanisms of exchange. Lewis heterogeneities, like those featured in the more general GEM modeling, prevent meaningful aggregation. Gustav Ranis aptly named Lewis’s aggregation “organizational dualism”.

Lewis macrodynamics identify the low-productivity venue as subsistence agriculture and its high-productivity counterpart as an industrial enclave. The former is characterized by small production units, primitive production techniques, and the absence of input specificities, all captured by positing near-subsistence productivity and the absence of saving and capital accumulation. Low-productivity farming produces total real output *X*_{S}: *X*_{S}(t)=*b*^{S}*H*_{S}(t), where *S* stands for the subsistence venue, *b*^{S} is constant labor productivity, and *H*_{S} denotes sector labor supply. Product pricing (*P*^{S}) is also constant.

Lewis posited, in the subsistence sector, marginal labor productivity to be zero and market institutions to be poorly developed. In place of the market, labor compensation and employment are determined by equity-based income-sharing arrangements: *W*_{S}(t)=*X*_{S}(t)/*H*_{S}(t). *H*_{S} is exogenously determined by subsistence-sector population growth. The real wage (*W*_{S}) varies as a result of exogenous factors, such as weather and disease. The core macrodynamics here, focusing on the interaction between labor productivity and the preference to procreate, were famously provided by Malthus.

The high-productivity venue, by contrast, exploits input specialization and scale, generating total real output *X*_{I}: *X*_{I}(t)=*b*^{I}(t)*H*_{I}(t), such that *b*^{I}>*b*^{S}, where *I *indicates the industrial venue and *b*^{I} is the sector’s labor-productivity, assumed to be constant along with the product price (*P*^{Ĭ}) and the labor-capital ratio.

Within his two-sector framework, Lewis constructed macrodynamics for saving, investment, and sectoral labor transfer that explain how economies break out of subsistence. Profits are posited to be the source of all saving, which is wholly invested in capital accumulation: Δ*K*_{I}(t)=*Π*_{I}(t−1)=*P*^{I}*X*_{I}(t-1)–*W*^{I}*H*_{I}(t-1), and Δ*K*_{I}(t)/*K*_{I}(t−1)=*Π*_{I}(t−1)/*K*_{I}(t−1). There is no depreciation, and the capital price is constant. It follows that the rate of growth of the homogeneous capital stock equals the rate of return on capital.

Reflecting Leibenstein and Stiglitz’s anticipatory work on efficiency wages, it is posited (although not explicitly by Lewis) that labor productivity is significantly increasing in nutrition and health, which in turn are increasing in the real wage paid, to motivate the rational payment of constant labor rents. The wage premium and (point-of-hire) labor homogeneity imply a horizontal labor supply for industrial establishments. Absorption of workers from subsistence farming is determined, given the constant capital-labor ratio, by the intertemporal path of the capital stock: Δ*H*_{I}(t)/*H*_{I}(t−1)=Δ*K*_{I}(t)/*K*_{I}(t–1)=*k*(t)=*П*_{I}(t−1)/*K*_{I}(t−1). The final source of dynamics in the Lewis model is total labor-force (*H*_{T}) growth, assumed to be a positive constant, *c*: *H*_{T}(t)=(1+*c*)*H*_{T}(t−1).

An initial condition of Lewis’s macrodynamics is that *H*_{I}(0)/*H*_{T}(0) is near zero. A turning point will eventually be reached iff: *П*_{I}/*K*_{I}>c. In Lewis’s *turning-point hypothesis*, once surplus workers have exited the subsistence sector, market forces assert control of all labor pricing. Homogeneous wage determination signals the economy’s consolidation into a single (market) venue. Lewis had little interest in post turning-point macrodynamics, believing that in such circumstances his two-sector model was no longer useful.

__GEM Project Generalization of the Lewis Model__

Familiar GEM modeling permits a richer specification of Lewis’s large-establishment, industrial venue: *X*_{J}(t)=*b*_{J}(t)*Ź*_{J}(t)*H*_{J}(t), such that *Ź*_{J}=*Έ*_{J}/*H*_{J}=*Ź*^{n}_{J}, and *W*_{J}=*W*^{n}_{J}>*W*^{m}. The variable *b*_{J}=*X*_{J}/*Έ*_{J} denotes the *J*th sector’s technical efficiency of labor. Worker productivity rises over time, driven by physical and human capital accumulation, scale economies, and technological advance. Meanwhile, production in the low-productivity, effective labor-supervision venue is: *X*_{K}(t)=*b*_{K}(t)*H*_{K}(t), such that *Ź*_{K}=*Έ*_{K}/*H*_{K}, *Ź*_{K}=*Ź*^{ɱ}_{K}, and *W*_{K}=*W*^{m}. Output per worker hour is denoted by *b*_{K}, moving in lockstep with technically-efficient labor productivity (*X*_{k}/*Έ*_{k}). The assumption of constant capital and technology in the *K*th venue simplifies the analysis. The sector is also assumed to generate zero savings.

Rational workplace exchange extends the life of high-productivity-venue wage rents beyond Lewis’s turning point, implying that labor supply to that venue continues is elastic. Moreover, generalizing the Lewis model significantly enriches the turning point. Once surplus workers have been eliminated in the subsistence sector, implying the introduction of more robust labor-transfer opportunity costs, market forces replace the underdeveloped market institutions, asserting control over labor pricing in *K*th-venue establishments. Nonmarket labor pricing, minimizing unit labor costs, persists in *J*th-venue firms. Post turning-point economies, with optimizing *Z*_{J} playing the critical role, rationally generate downward rigid nominal wage over stationary business cycles as well as chronic, time-varying labor rents. Adverse nominal-demand disturbances now play a central role, combining with meaningful wage rigidity to induce job-loss dynamics that are consistent with the evidence on high- and low-frequency employment instability. Such instability influences rational labor transfer between the two venues. Meanwhile, trend aggregate labor-productivity and living-standard advance continues to depend on technical change, capital accumulation and labor transfer to the high-productivity venue. The generalized Lewis model becomes a uniquely powerful platform for macrodynamic analysis.

__Conclusion__

I learned from the Google piece that Sir Arthur was born on the island of St. Lucia in the Caribbean. I knew he was from Caribbean but did not know that the location is one of my favorite sailing destinations. St. Lucia is flat-out beautiful.

Blog Type: Wonkish Saint Joseph, Michigan

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