Question

Consider a firm that has some transformation function F(y) for L commodities, and the optimal netput y(p) resulting from profit maximization. Claim: y(p) is homogeneous of degree 0 in prices. Prove this claim

Answer 1

Markets have two sides: consumers and producers. The basic unit of activity on the production side of the market is the firm. The task of the firm is take commodities and turn them into other commodities. The objective of the firm is to maximize profits. That is, the firm chooses the production plan from among all feasible plans that maximizes the profit earned on that plan.

In the neoclassical (competitive) production model, the firm is assumed to be one firm among many others. Because of this (as in the consumer model), prices are exogenous in the neoclassical production model. Firms are unable to affect the prices of either their inputs or their outputs. Situations where the firm is able to affect the price of its output will be studied later under the headings of monopoly and oligopoly.

Consider an economy with L commodities. The task of the firm is to change inputs into outputs. For example, if there are three commodities, and the firm uses 2 units of commodity one and 3 units of commodity two to produce 7 units of commodity three, we can write this production plan as y = (−2, −3, 7), where, by convention, negative components mean that that commodity is an input and positive components mean that that commodity is an output. If the prices of the three commodities are p = (1, 2, 2), then a firm that chooses this production plan earns profit of π = p · y = (1, 2, 2) · (−2, −3, 7) = 6.

Usually, we will let y = (y1, ..., yL) stand for a single production plan, and Y ⊂ RL stand for the set of all feasible production plans. The shape of Y is going to be driven by the way in which different inputs can be substituted for each other in the production process.

The transformation function is such that:

F (y)=0 if y is on the frontier < 0 if y is in the interior of Y > 0 if y is outside of Y . Thus the transformation function implicitly defines the frontier of Y . Thus if F (y) < 0, y represents some sort of waste, although F () tells us neither the form of the waste nor the magnitude. The transformation function can be used to investigate how various inputs can be substituted for each other in the production process.