## Separability

## Separability

# Separability

Separability is a pivotal economic concept introduced independently by Masazo Sono (1945) and Wassily Leontief (1947) in order to deal with aggregation problems in both utility and production theory. Specifically, this concept was employed by Robert Strotz (1957) to analyze two-stage optimization in utility theory: Given separability, the first stage involves partitioning commodities into subsets and optimizing intensities within each subset. While holding the within-subset intensities fixed, the between-subset intensities are optimized in the second stage.

In formal terms, Leontief and Sono defined two factors *i* and *j* to be *separable* from factor *k* if and only if

where C(*Y, p _{1}, p_{2}, …, p_{n}* ) is a twice differentiable cost function with nonvanishing first and second partial derivatives with respect to input prices

*p*

_{1},…,

*p*

_{n}(Blackorby and Russell 1976, p. 286). A similar definition also applies to any twice differentiable production function F(x

_{1}, x

_{2},…, x

_{n})(see, e.g., Berndt and Christensen 1973, p. 404).

For aggregation purposes, the economic literature commonly distinguishes *strong* from *weak* separability, a terminology coined by Strotz (1957). A cost function C(*Y, p _{1}, p_{2}, …, p_{n}* ) or a production function

*F(x*) is said to be

_{1}, x_{2}, …, x_{n}*weakly separable*if condition (1) holds for all

*i*∈

*N*, and

_{s}, j∈E N_{s}*k ∈. N*where

_{s},*N*is a subset of the set of input factors

_{s}*N = {1, 2, …, n*}. Accordingly, cost function is said to be strongly separable if condition (1) holds for all

*i*,

**∈E**N_{s}*j ∈E N*and

_{s}j∈N_{t}*k ∈.N*, where N

_{s}U N_{t}_{t}is another subset of

*N*.

Steven Goldman and Hirofumi Uzawa (1964) prove that weak and strong separability impose specific restrictions on the underlying cost and production functions. Weak separability with respect to any partition *R* of *r* mutually exclusive and exhaustive subsets N_{1}, N_{2}, …, N_{r} is necessary and sufficient for a production function to be of the form *F(X _{1}*, X

_{2}, …,

*X*where the aggregate

_{r}),*X*is a function of only the elements of N

_{s}_{s}. Similarly, strong separability implies that the production function is of the form

*F(X*+

_{1}*X*2 + … +

_{2}*X*).

_{r}In empirical applications it is often indispensable to invoke separability assumptions if data on some production factors are lacking. For instance, numerous empirical studies investigating the substitutability of capital *(K)* and labor *(L)* have assumed that both inputs are weakly separable from energy *(E)* and materials *(M),* implying the so-called “value-added” specification of production technologies, *F(V = G(K, L), E, M).* This assumption, it is commonly asserted, allows for focusing on the capital-labor aggregate *V= G(K, L)* when estimating the degree of substitution between capital and labor, whereas energy and materials inputs can be safely ignored (Berndt and Wood 1975, p. 265).

This example illustrates that the concept of separability is intimately associated with the notion of substitutability of production factors and consumer goods (see, e.g., Berndt and Christensen 1973 and Blackorby and Russell 1976). The natural intuition about the separability of a nonnegligible production factor, such as energy after the oil crises, from all other factors is that the substitution elasticity estimates of nonenergy inputs remain the same, irrespective of whether energy is included in any substitution analysis (Hamermesh 1993, p. 34).

This intuition, however, does not apply to the typically employed substitution measures, such as the Allen and Morishima elasticities of substitution, or cross-price elasticities, because the classical LeontiefSono condition(1) only implies that the *input proportion x _{i}/x_{j}* remains unaffected by changes in the price

*p*of factor

_{k}*k:*

where Shephard’s Lemma , has been inserted into condition (1). When replacing the cost function by its dual production function *F(x _{1}, x_{2}, …, x_{n})*, the primal analogon of condition (1) suggests that the

*marginal rate of substitution*between any two 2 inputs

*i*and

*j,*is independent from the input x

_{k}of factor

*k*(Berndt and Christensen 1973, p. 404).

Yet, virtually no empirical study measures the ease of substitution between *i* and *j* in terms of either their input proportion or their marginal rate of substitution. Manuel Frondel and Christoph Schmidt (2004) therefore suggest an empirically oriented definition of separability that guarantees the invariance of cross-price elasticities and due to changes in the price *p _{k}* :

It can be shown that this definition represents a stronger requirement than the classical Leontief-Sono separability condition (1). In other words, the Leontief-Sono condition (1) is necessary, but not sufficient, for the separability definition (3) to hold (Frondel and Schmidt 2004, p. 221). Frondel and Schmidt (2004, p. 220) argue, however, that only these stronger conditions capture a separability notion coming close to Hamermesh’s (1993) intuition.

**SEE ALSO** *Elasticity; Leontief, Wassily; Optimizing Behavior; Production Function; Utility Function; Uzawa, Hirofumi*

## BIBLIOGRAPHY

Allen, R. G. D. 1938. *Mathematical Analysis for Economists.* London: Macmillan.

Berndt, Ernst R., and Laurits R. Christensen. 1973. The Internal Structure of Functional Relationships: Separability, Substitution and Aggregation. *Review of Economic Studies* 40: 403–410.

Berndt, Ernst R., and David O. Wood. 1975. Technology, Prices, and the Derived Demand for Energy. *Review of Economics and Statistics* 57: 259–268.

Blackorby, Charles, and R. Robert Russell. 1976. Functional Structure and the Allen Partial Elasticities of Substitution: An Application of Duality Theory. *Review of Economic Studies* 43: 285–292.

Frondel, Manuel, and Christoph M. Schmidt. 2004. Facing the Truth about Separability: Nothing Works without Energy. *Ecological Economics* 51 (3–4): 217–223.

Goldman, Steven M., and Hirofumi Uzawa. 1964. A Note on Separability in Demand Analysis. *Econometrica* 32: 387–398.

Hamermesh, Daniel S. 1993. *Labor Demand*. Princeton, NJ: Princeton University Press.

Leontief, Wassily W. 1947. Introduction to a Theory of the Internal Structure of Functional Relationships. *Econometrica* 15: 361–373.

Morishima, M. 1967. A Few Suggestions on the Theory of Elasticity. *Keizai Hyoron* (Economic Review) 16:145–150.

Sono, Masazo. 1945. The Effect of Price Changes on the Demand and Supply of Separable Goods. *International Economic Review* 2: 1–51.

Strotz, Robert H. 1957. The Empirical Implications of a Utility Tree. *Econometrica* 25: 269–280.

*Manuel Frondel*

*Christoph M. Schmidt*