Economic Growth and the Elasticity of Substitution: Two Theorems and Some Suggestions

By R. Klump, O. Grandville. 2000

368 citations recorded by the source corpus.

At the beginning of the twentieth century, the work of the Swedish economist Knut Wicksell (1901) marked an important advance in our understanding of the relationship between the structure of production functions and income distribution. Wicksell had asked the following question, among many others: Suppose that the marginal productivities are equal to the remuneration rates of each factor. What then should be the form of the corresponding production function? He pointed out (p. 125) that the general integral of the partial differential equation implied by this relationship was of the form Y = Lf(KIL), which meant in turn that the product Y was necessarily a homogeneous of degree one function of capital (K) and labor (L). As an example, he gave a function where the shares of K and L in Y were constant and wrote down the function which would be rediscovered decades later by Charles W. Cobb and Paul H. Douglas (1928).' However, it is only half a century later, in Robert M. Solow's Nobel Prize-winning 1956 essay, that implications between the structure of the production function and long-run equilibrium or disequilibrium of the economy were drawn. In his classic article " A Contribution to the Theory of Economic Growth," Solow had considered three cases analytically (other cases were also discussed geometrically, using a phase diagram). He had labeled the two first cases "1Hlarrod-Domar" and "Cobb-Douglas," respectively, and he referred to the third type of production function as "offering a bit of variety"; it was Y (aK"/2 + L/2)2, a particular case of Y (aKP + LP)"1P. In those early days, Solow had not pointed out to the reader that his cases could have been numbered 0, 1, and 2, following the elasticity of substitution of the corresponding production function. It would be only five years later, in 1961, that with his coauthors Kenneth J. Arrow et al., he would find the general form of the two-factor constant-elasticity-of-substitution (CES) production function by integrating the differential equation corresponding to the observed relationship between output per capita and the wage rate. Solow's third case was indeed highly interesting since it allowed the possibility of everincreasing income per head, even without any technical progress. Indeed, Solow showed that due to the efficiency of the system entailed by the particular production function he had chosen, there existed a threshold for the savings rates s to generate investment per head large enough for income per head to grow forever. That threshold was s nla2 where n is the growth rate of the population. It was then natural to ask whether one could express that threshold in terms of the elasticity of substitution a. The value of that threshold was determined by de La Grandville (1989) to be equal to s = no3(or"' ), where (a) is K's coefficient in a normalized CES function (equal to a in Solow's third case, where u -2); for any s and n values, this relationship gives also the threshold value (& in implicit form by the solution of ao1/[1 log 3Q(or)Ilog(nls)]. By totally differentiating the last relationship, one could observe that the threshold for the elasticity of substitution to entail ever-sustained growth was an in* Klump: Faculty of Mathematics and Economics, University of Ulm, Helmholtzstrasse 20, D-89081 Ulm, Germany; de La Grandville: Department of Economics, University of Geneva, 102 Boulevard Carl-Vogt, CH-1211 Geneva 4, Switzerland. We gratefully acknowledge the comments of seminar participants at Stanford University and at the Universities of Stuttgart-Hohenheim and Munich, where parts of the paper were presented. Also, we would like to thank, without implicating in any way, Robert Solow for his insightful remarks, and an anonymous referee for very helnfuil supgestions. ' Wicksell, Cobb, and Douglas would no doubt burst with pride if they knew that one of our first-year students wrote, in a paper of his: "Of course, we recognize here a Kirk Douglas production function."

Subfields: Economics and Econometrics

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Citations in the corpus

The Solow model with CES technology: nonlinearities and parameter heterogeneity (2004, background): “tura (1997), Klump and de La Grandville (2000) , Azariadis (2001) and Azariadis and de la Croix”
The Context of Entrepreneurial Judgment: Organizations, Markets, and Institutions (2019, background): “They focus on the flexibility with which resources can be combined and recombined, corresponding to the production theory notion of the ‘elasticity of factor substitution’ (Klump and de la Grandville, 2000).8 If this elasticity is high, then resources are more easily allocated to highly valued uses.”
Getting Normalization Right: Dealing with ‘Dimensional Constants’ in Macroeconomics (2012, methodology): “Indeed Klump and Preissler (2000) and Klump and de La Grandville (2000) explain that the normalized CES production function employed permits one to compare results with steady-state allocations and factor income shares that are constant as the elasticity of substitution is changed.”