Homogeneity and supply.

Economists' motivations for choice of functional form include ease of implementation and interpretation, needed flexibility, and aggregability. For demand applications, one of the late Terence Gorman's great legacies is an attractive generalization of previous work on representative consumers (Gorman 1953, 1961; Muellbauer 1975, 1976; Deaton and Muellbauer 1980a, 1980b) called a Gorman system of Engel curves or a Gorman system in short (Gorman 1981; Lewbel 1987, 1989). A Gorman system readily satisfies exact aggregation. For both consumer and producer applications, this legacy is deeply embedded in agricultural economics, as evidenced by the many applications that take the Gorman class of functions (additive in functions of the aggregated variable) as the starting point for empirical analysis (Shumway 1995; Shumway and Lim 1993; Green and Alston 1990; Chambers and Pope 1991; LaFrance et al. 2002).

Gorman class functional forms are ubiquitous because they are easily interpretable, flexible, and have the added advantage of possessing convenient aggregation properties. However, many applications to agricultural producer behavior focus on a single commodity (Adams and Behrman 1976; Aradhyula and Holt 1989; Askari and Cummings 1977; Azzam and Yanagida 1987; Braulke 1982; Brester 1996; Burt and Worthington 1988; Debertin and Pagoulatos 1992; Eckstein 1985; Griliches 1960; Hallam 1990; Holt and Moschini 1992; Kaiser, Streeter, and Liu 1988; LaFrance and Burt 1983; LaFrance and de Gorter 1985; Levins 1982; Lohr and Park 1992; Marsh 1994, 1999; Milligan 1978; Nerlove 1958, 1959, 1979; Nerlove and Addison 1958; Rucker, Burt, and LaFrance 1984; and Wickens and Greenfield 1973). In these cases, one can either employ a popular Gorman specification (without the system restrictions) or specify any appropriate representation of interest. In the former case, one might use a functional form that is unnecessarily inflexible because choosing from a system "off of the shelf" may have imbedded restrictions that come from the economic theory of systems, for example, symmetry and adding up of demand, expenditure or cost, or symmetry from profit maximization. That is, in a single equation, a researcher could not impose any of the cross-equation restrictions such as symmetry or adding up but would typically impose homogeneity (and usually expect but not impose nonnegativity and monotonicity in the own price). If one chooses an arbitrary form, the imposition of homogeneity may seem trivial by merely deflating by some price or index of prices. However, this can easily destroy aggregability (Lewbel 1989). Further, this may not allow a nested approach for testing homogeneity, which is notoriously rejected in empirical work (Deaton and Muellbauer 1980a, 1980b; Shumway 1995).

The approach that we take is to isolate the homogeneity condition within the general Gorman class of supply functions. Language and notation focus on producer supply behavior, but our results apply to a variety of other settings, for example, a factor demand equation or a normalized profit function. (1) To maximize the empirical relevance of the analysis and reduce the notational clutter, a single supply equation is considered. In doing so, we are not advocating the single equation approach, which sacrifices attractive opportunities to model and test supply and demand symmetries, concavity, and adding up. However, because homogeneity is an equation-by-equation concept, the results transfer immediately to any number of equations with the Gorman structure. This approach has the advantage of seeing exactly what restrictions homogeneity alone places on a Gorman system. (2)

Our results are novel and, to us, they are striking. We began this inquiry conjecturing that homogeneity might not substantially restrict or guide the choice of functional form. This turns out to be far from true. We find that supply functions must include explicit sums and products of power, logarithmic, and trigonometric functions and must do so in rather explicit ways. We find that the set of possible homogeneous functional forms depends crucially on the number of functions of the aggregated variable being utilized. That said, there is a great deal of flexibility remaining to measure supply response and a great deal that can be learned from considering homogeneity in isolation from other properties of a supply function. We present a class of functional forms that are homogeneous and can be imposed directly in applied work, or that can be perturbed to test for homogeneity or aggregability. (3)

Gorman Class Functions

The rationale and approach for specifying an aggregable supply system follows Chambers and Pope (1991, 1994). By focusing on supply, q, and given heterogeneity in output price due to temporal and spatial reasons, it is reasonable to specify supply as

(1) q = [k.summation over (k=l) [[alpha].sub.k](w)[h.sub.k](p)

with K smooth functions of input prices, [[alpha].sub.k]: [R.sup.n.sub.++] [right arrow] R, k = 1 ..., K, times K smooth functions of output price, [h.sub.k]: [R.sub.++] [right arrow] R, k = 1, ..., K. (4) We also could add fixed inputs or technical change variables. However, to avoid notational clutter, for the present such shifters can be subsumed in a and h. The functional form in (1) defines the Gorman-class of functions. Given that supply functions are positive valued and increasing in the output price, we expect that [[summation].sup.K.sub.k=1] [[alpha].sub.k](w)[h.sub.k](p) > 0 and that [[summation].sup.K.sub.k=1] [[alpha].sub.k](w)[h'.sub.k](p) > 0.

Homogeneity (0[degrees]) of q in (w, p) is most basically described by (5)

(2) q(tw, tp) = q(w, p), [for all] t > 0.

Given the smoothness assumption, homogeneity also can be written in terms of the Euler equation (Allen 1938)

(3)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

One could immediately impose homogeneity by dividing by p or a [w.sub.i] in (1). In the first case, deflation destroys aggregability, at least in the conventional sense, by creating terms that are functions of w/p (Lewbel 1989). In the latter case, the variables are p/[w.sub.i] and [w.sub.j]/[w.sub.i]. This generally destroys independence of an aggregate output price index from the elements of w. If testing homogeneity is of interest, most researchers would prefer a nested test, which is what our development provides. This is opposed to an arbitrary form in which deflated and nondeflated models are compared with a nonnested test. Thus, the next section focuses on the restrictions on each of the functions in (1) that lead to homogeneity. This requires solving the differential equation in (3).

The Main Result

In the Appendix to this article, the proof of our main result is derived by solving the Euler equation for homogeneity when the supply equation has the Gorman form. The results depend on the number, K, of functions included in the supply function. The explicit results for K [less than or equal to] 3 are highlighted because this appears to be parsimonious and sufficient for most empirical representations. (6)

PROPOSITION 1. (The Main Result): Let the supply function take the Gorman form, q = [[summation].sup.K.sub.k=1] [[alpha].sub.k](w)[h.sub.k](p), with K smooth, linearly independent, functions of input prices, w, and K smooth, linearly independent, functions of output price, p. If q is 0[degrees] homogeneous in (w, p), then each output price function is either: (i) [p.sup.[epsilon]] with [epsilon] [member of] R; (ii) [p.sup.[epsilon]] [(ln p).sup.j], with [epsilon] [member of] R, j [member of] {1, ..., K}; (ii) [p.sup.[epsilon]] sin ([tau] ln p), [p.sup.[epsilon]] cos ([tau] ln p), with [epsilon] [member of] R, [tau] [member of] [R.sub.+], appearing in pairs with the same {[epsilon], T} for each pair; or (iv) [p.sup.[epsilon]] [(ln p).sup.j] sin ([tau] ln p), [p.sup.[epsilon]] [(ln p).sup.j] cos ([tau] ln p), with [epsilon] [member of] R, j [member of] {1, ..., [1/2K]}, [tau] [member of] [R.sub.+], and K [greater than or equal to] 4, appearing in pairs with the same {[epsilon], j, [tau]} for each pair, where [1/2 K ] is the largest integer no greater than l/2K. If K [member of] {1, 2, 3}, then the supply of q can be written as:

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) K = 2

i. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

iii. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(c) K = 3

i. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

iii. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

iv. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In each case except (c) iii, where [[alpha].sub.2](w) is homogeneous of degree zero, each [[alpha].sub.i](w) is positively linearly homogeneous for i = 1, 2, 3.

Beginning with K = 1, where there is no possibility of complex roots to (3), a simple power function of p emerges. Only a single linearly homogeneous index in w is required. One might consider p/[[alpha].sub.1](w) > 0 to be the "real price," in which case monotonicity of supply in the output price requires [[epsilon].sub.1] > 0. If [[epsilon].sub.1] < 1, then the smaller is this elasticity, the greater is the concave curvature of supply in real price, while if [[epsilon].sub.1] > 1, then the larger is this elasticity, the greater is the convex curvature of supply in real price.