_{1}

^{*}

When inputs in the firm’s production function are pair-wise complements, I show that all variable factors of the firm are output elastic. Via Silberberg’s analysis, this implies that for given output of a competitive firm that marginal cost will rise more than average cost for a factor price increase. Accounting for changes in output through profit maximization and industry equilibrium change in output price, I show that cost pass-through can be larger than one in a competitive industry when inputs are complementary. Because input complementarity seems likely with commodity aggregates like materials, labor, energy, and capital, this could provide an alternative explanation for over cost shifting in commodity-oriented industries like the oil industry and food industries. This approach also allows researchers to abandon the highly restrictive assumption of constant elasticity of demand function facing the firm that is required under imperfect competition with constant marginal costs.

The concept of output elasticity (percentage change in input usage for one percent change in output, holding factor prices constant) plays a prominent role in the theory of the firm. Among other things, the concept is useful in determining whether a firm will increase or decrease its output in response to a change in factor price [

where is the firm’s cost function and subscripts denote partial derivatives. In the last step of Equation (1) use has been made of Shephard’s lemma. Converting Equation (1) to elasticitieswe obtain

where is the elasticity of marginal cost with respect to a change in the i-th factor price, is the output elasticity of the i-th factor, and is the share of total cost of the i-th factor in output valued at marginal cost. When the firm is a price taker, s_{i} is cost share of the i-th factor in total revenue; when the firm is imperfectly competitive in the output market, , where is the total cost of the i-th factor as a share of total cost, is average cost, and MC is marginal cost.

Equation (2) shows there is a one-to-one relationship between the elasticity of marginal cost with respect to factor price and its output elasticity. The elasticity of marginal cost with respect to w_{i} will be larger (smaller) than s_{i} according as is larger (smaller) than 1. In general, we cannot say with certainty what the relationship will be. If the firm is in long-run equilibrium, where marginal cost equals average cost, then and [^{1}.

Linear homogeneity of the long-run production function is a reasonable assumption in light of the replication argument [

In this form, the production function can be used to determine the output elasticity of any variable factor.

In the short run with x_{n} fixed, assume the firm is a price taker in both output and factor markets. Assume also that the firm takes x_{1} as fixed in determining its profit-maximizing input levels^{2}. The first-order conditions for profit maximization conditional on x_{1} are:

Solving these n – 2 equations for the conditional input demand functions yields:

Substituting these functions into the production function yields the conditional supply function

Differentiating Equation (6) with respect to x_{1} yields the expression

This expression implies that

This has implications for the output elasticity of x_{1}, which can be determined through substituting the optimal output-constant demand function for x_{1}, , into Equation (6) to obtain the identity

where now y is assumed to be the profit-maximized value for output. Differentiating the identity with respect to y:

Thus,. This means that if the sign of Equation (7) is negative, then and the output elasticity.

TheoremWhen inputs are complementary in production, all output elasticities of the firm will be larger than one.

Proof. Let

be the Hessian of the production function with respect to from differentiating the firstorder conditions in Equation (4). The comparative statics of the n – 2 conditional input demand functions can be characterized as follows:

where

and .

The solutions in Equation (8) can be written more explicitly as follows:

where is the co-factor of in F. The term is negative because F is a negative definite matrix for the firm to maximize profit. Because the matrix F is a Metzler matrix (off-diagonal elements nonnegative and diagonal elements nonpositive), each of the terms will be nonpositive [

From Equation (2) we also see that an output elastic factor demand implies that the elasticity of marginal cost from an increase in factor price will be greater than the cost of the input as a share of total revenue. This result, while important in its own right, is only valid if output remains constant. To calculate the price effect we must account for the effect the change in factor price has on output as well.

In the identical firm case, the comparative static expression for cost pass-through for a competitive market in the short run can be shown to equal^{3}

where is the elasticity of output price with respect to the i-th factor price, ε is the supply elasticity, and is the demand elasticity of the output. It is useful for our purpose to normalize Equation (10) by redefining the elasticity of price transmission as

This expression now shows how much output price changes for each unit output change in the i-th factor price. For example, if the input is crude oil and output is gasoline, the expression in Equation (11) now shows how much the price of gasoline changes for a change in the price of crude oil per gallon of gasoline.

Equation (11) indicates that cost pass-through can now be larger than 1. This will occur when . This will more likely be the case the larger the supply elasticity relative to the absolute value of the demand elasticity.

To see how plausible cost pass-through larger than 1 can be, consider the cost function derived from the CobbDouglas short-run production function:

where the parameters and represent the cost shares of the three variable inputs with returns to scale equal to. The values chosen are typical of many manufacturing industries for materials, labor, and energy^{4}. The output elasticity for the material input in this case (which equals that for labor and energy because production function is homothetic in this case) is 1/0.8 = 1.25. The value for The elasticity of marginal cost with respect to output is 1/0.8 – 1 = 0.25. Thus, the supply elasticity is ε = 1/0.25 = 4. If the demand elasticity is, then the output price change from a one unit change in materials price per unit output is

.

So even with a very elastic supply curve, about 11% more of the increase in materials price would be passed on to output price.

In the general case of input complementarity in production, I have shown that output elasticities of all variable factors will be elastic. A direct implication of this finding is that with input complementarity, marginal cost will increase more than average cost for an increase in factor price. For somewhat aggregate inputs like labor, materials, energy, capital, we would expect input complementarity to be the rule rather than the exception. It is also noteworthy that the result does not depend on any other restrictions on the production function other than the long-run production function exhibiting constant returns to scale. Another condition leading to this result is homotheticity of the short-run production function [

where recall that k_{i} is the total cost of the i-th factor as a share of total costs.

The main implication for cost pass-through is that we have an explanation based on the cost structure for how there can be more than full pass-through of costs. The usual explanation assumes marginal costs are constant so that cost pass-through depends on the shape of the demand curve. However, it is well-known that only very restrictive forms of the demand function will give rise to more than complete cost pass-through [

Research supported in part by the North Carolina Agricultural Research Service, Raleigh, North Carolina, 27695.