4 A PROFIT FUNCTION MODEL OF THE COMPOUND FEED FIRM
4.1.2 Differences Suggesting a Successful Profit Model
Three differences between the cost and profit model, relevant with respect to the failure of the cost function approach, can be distinguished. Two of them follow from the fact that the exogeneous variables of the dual profit function are input and output prices rather than input prices and output quantities.
First, the partial derivatives of the profit function for the output variable yield the condi-tional supply equations rather than marginal cost. This is promising since one – though incon-sistent – way of reaching estimability of the cost function model only fails because of problems in measuring marginal cost,78 whereas supply quantities are available. Derivation for input prices leads to conditional demand equations in both cases, which are however not equiva-lent.79
78 See section 3.4.1 above.
79 See section 4.1.1 above.
Secondly, the dual profit function is linearly homogeneous in all its exogeneous variables, whereas the dual cost function is generally inhomogeneous in the output variables or homoge-neous of degree one in both input prices and output quantities in the case of constant returns to scale, i.e. homogeneous of degree two at total, respectively. This lack of linear homogeneity of the cost function contradicts the possibility to infer from strong separability to an additive structure of the behavioral function, which, as shown elsewhere, exists for the profit function.80 Furthermore, from the outset it foils the plan to find a simple decomposition of the profit func-tion by Euler's theorem in order to make aggregate non-component cost measurable on the other hand.81 Hence, there is enough incentive to work out a profit function model of the com-pound feed firm. This is the more indicated since, apart from these chances with respect to the concrete problems in modeling compound feed behavior, a profit function approach is gener-ally superior to a cost function approach.82
But, as a third major difference to the cost function approach, a new strand of problems arises since the profit function is not defined for constant returns to scale; the profit maximiza-tion problem is unbounded in this case. Thus, since the constant returns to scale hypothesis for the component micro function cannot be avoided, the whole profit function project finally fails.
As in the cost function chapter above, "skipping the wrong way" is not favoured for reasons which are explained in the programmatics section 1.3.1 above.
4.2 Structural Assumptions
After the extensive discussion of the previous chapter it is of course no longer possible to assume an innocent perspective and postulate a desired model specification as was undertaken in section 3.2 above before carefully examining the adequacy of relevant structural assumptions and the functional representation consequences of these. Hence, one proceeds immediately to the list of structural assumptions on the profit function, which in any case look quite similar to their cost function analogies. The assumption of equal non-component cost-total cost propor-tions C KDV QR DQDORJ\ LQ WKH SURILW IXQFWLRQ FDVH DQG WKXV LV RPLWWHG
80 See FEGER/MÜLLER 1999.
81 See section 3.4.1 above.
82 See section 4.1.1 above.
4.2.1 Weak Separability
Generally, it cannot be inferred from a separable cost structure to a separable profit structu-re without further considerations. An example is provided by Feger's and Müller's establish-ment of a CES profit representation of separability in the binary and extended partition, which may not be transferred to the cost function for the different homogeneity properties of the cost and profit function.83 Their proof is utilized in section 3.4.1 above and in section 4.3.1 below.
Hence, the relevant equations have to be examined anew. In analogy with the cost function approach, weak separability of the compound feed firm's profit function requires checking the following six equations for the mutually exclusive and exhaustive sectors w1 (corresponding to index set I1) including all component prices, w2 (corresponding to index set I2) denoting all non-component prices, and output price vector p (corresponding to index set J):84
Π.1 ∂
These assumptions can be justified as follows:
Π.1 In analogy to C.1, profit maximizing output mix depends on component prices because the compound feeds are generally characterized by different component intensities; the
83 See FEGER/MÜLLER 1999.
84 See section 3.3.1 above.
entrepreneur will ceteris paribus always substitute away from the compound feed which becomes more expensive due to a rising component price.
Π.2 In analogy to C.2 and in opposition to Π.1, output mix is independent of changes in non-component prices because processing is the same for all feeds.
Π.3 Being identical with the justification of C.3 and paralleling Π.2, it is irrelevant for the feed mixer to know about price changes for factors which are not components.
Π.4 In analogy with C.4 and paralleling Π.1, knowledge about an output price change will not alter the least-cost recipe for the livestock category-specific compound feed – com-pound feed cost and profit is nonjoint in inputs –,85 but through the product-price indu-ced shift in revenue-maximal product mix the component demand shares shift, too, since in different compound feeds components are utilized with different intensity.86
Π.5 The argument is the same as for C.5: processing technology treats all components the same so that a component price change is meaningless.
Π.6 Paralleling C.6, an increasing compound feed price yielding a shift in product mix and, by Hotelling's lemma, c. p. an increasing total output causes factor substitution in the non-component sector. As in the cost function case, to claim separability here would imply the denial of economies of scale: the non-component sector is the only place where these can be located, because the component sector is, as assumption Π.12 below states, cha-racterized by constant returns to scale.
Thus, it is neither possible to establish a complete weakly separable cost function nor a weakly separable profit function for the compound feed firm. In both approaches, the output and component sector are mutually inseparable so that they could be united in a single sector with all kinds of interactions, i. e there is no separability in the extended partition. For both approaches, the component and the non-component sector are mutually weakly separable.
Hence, both the cost and the production model can be called "separable in the binary partition in inputs".
85 See section 4.1.4 below.
86 Note that in a single product firm the homothetic compound feed technology would be equivalent with sepa-rability of components from ouptut price.
4.2.2 Strong Separability
Now consider the three additional equations constituting strong separability for the profit function:
These assumptions can be justified as follows:
Π.7 The necessary condition for Π.7 expressing strong separability is satisfied by Π.2 and Π.3. Sufficiency is, as in C.7, again implied by the catalyst character of all non-component activities.
Π.8 The argument is the same as for C.8.
Π.9 The argument is the same as for C.9.
4.2.3 Unity Value of the CES Substitution Parameter
If, relying on the separability assumptions maintained above, it were possible to establish a CES profit representation, a literally additive profit function would require a unity valued CES substitution parameter, too. In this case, the profit function entertains a linear relationship be-tween component and non-component profit micro-functions. As with the cost function, this implies a Leontief technology characterized by right-angled isoquants between aggregate com-ponents and aggregate non-comcom-ponents or no substitution between these aggregates. There is argument in favor of this hypothesis in section 3.3.3 above. Thus,
Π.10 !(y) = 1
can be maintained.
4.2.4 Nonjointness, Homotheticity, and Constant Returns to Scale in Components Since the assumptions of production nonjoint in component quantities, homothetic feed composition and constant returns to scale in component quantities all correspond to technolo-gy rather than dual functions, the argumentation in sections 3.3.5 through 3.3.7 applies without limitation in a profit function context. Hence, constant returns to scale in component combina-tion
Π.12 Y
%
x y1, = ⇒0 Y λ λx1, y%
=0 ∀ >λ 0 , nonjointness in component quantitiesΠ.13 yi =Yi1
%
x1 ∀i ,and, formulated in terms of the profit function, homothetic feed composition in the single pro-duct case, which is already established by Π.13,
Π.14 ∂
While, given the partition established in section 3.3 above, a CES form cost representation would not be established even if all equations C.1 through C.9 could realistically be assumed to vanish, i.e. if strong separability of the cost function in the binary and extended partition could be maintained,88 the opposite is the case in a profit function framework: It can be shown that a profit function which is strongly separable in the binary and extended partition in conjunction, i.e. Π.1 through Π.9 vanish, assumes a CES structure.89 Then, by the adequate assumption of
87 The ordering number 11 is omitted to ease comparison between cost function and profit function structural assumptions. There does not exist a profit function analogy to C.11, and the remaining ordering numbers refer to the respective analogy.
88 See section 3.4.1 above.
89 See FEGER/MÜLLER 1999.