PRUDUCTION: THE REVENUE FUNCTION

As explained at the end of Chapter 1,we divide commodities a priori into goods and primary factors. Let v=(V₁,V₂,…….V_m) be the aggregate vector of net inputs of primary factors, and x=(x₁,x₂,……x_n)the vector of total net outputs of goods. Technological considerations tell us which combinations (x,v) are feasible. We rule out increasing returns to scale and all increasing marginal rates of substitution and transformation. In other words, we assume a convex technology, i.e. a convex set of feasible(x,v). So long as domestic production distortions are absent, this formulation encompasses considerable generality；for example, it allows intermediate goods to exist and be netted out in vertical integration. The general presumption is that goods are tradeable and fators are not, but we later indicate some simple generalizations.

Production decisions will maximize total profit, In particular, for given prices p of tradeables and quantities v of non-tradeables, the problem will be to choose a technologically feasible x to maximize the value of output. This is the inner product p.x or p^Tx.

The optimum x is clearly dependent on the specifield p and v ; write it as a function x=x(p,v).The corresponding maximized value of output also becomes a function of p and v. This is called the revenue function, written

r(p,v)=max{p.x | (x,v) feasible}

=p.x(p,v) (1)

This function embodies all relevant properties of the technology, and is particularly convenient for establishing some properties of output supplies and factor prices that we need. We summarize such properties here, giving only some simple supporting arguments in each case. Formal proofs can be found in the references cited in the nots at the end of the chapter.

Otput supplies

For a while we shall fix v, and consider revenue as a function of p alone. The first point is that it is a convex function. The economic idea can be expressed simply: with the input vector fixed, technological feasibility of an output vector is unaffected by a price change. It is thus always possible to maintain a fixed output vector, and make revenue a linear function of price. If there is any possibility of changing output composition along a transformation frontier, this will be used for the purpose of maximizing revenue, and then revenue will increase faster than linearly as prices change. To put it a bit more precisely, let p’ be any price vector, and suppose the corresponding maximum revenue is attained by choosing an output vector x’. For any other price vector p, since x’ remains feasible, the maximum revenue can be no less than p.x’. So we have

r(p’,v)=p’.x’ and r(p,v) ≥p.x’

This can be rearranged into the form

r(p,v)-r(p’,v) ≥(p-p’).x’ (2)

Figure 2.1 shows the dependence of revenue on any one price, say the jth. In the compared p.x’, the only term involving p_j is p_jx_j’, so the graph of that expression is a straight line with slope x_j’.The revenue function lies everywhere above this line ,and the two coincide at p_j’. If r has a jth partial dervative at this point, therefore, it must equal x_j’. If r has a kink at this point, its slope on the left-hand side of p_j’ must be less than x_j’,and that on the right-hand side greater, Thus the linear function is a tangent in a generalized sense; call it a supporting line, or in mang dimensions, a supporting hyperplane. The Appendix provides a somewhat more detailed treatment of this concept.

Collecting components, we have the result that if r is differentiable at p’, then r_p(p’,v)=x’. Since p’ could be any point, we can omit the prime and say that the optimally chosen supplies of goods are obtained by differentiating the revenue function wite respect to their prices,i.e.

x(p,v)=r_p(p,v) (3)

We will often assume differentiability of r with respect to p, i.e. single-valueed supply functions. However, there are important instances where this assumption is not valid, i.e . r has a kink, the supporting hyperplane is not unique ,and supply choices are not unique either. We shall return to this question when necessary.

Now let p’ and p” be any two price vector, and x’ and x” corresponding revenue-maxmizing choices. We have from (1) on taking p=p” that

r(p’,v)=p’.x’ and r(p”,v) ≥p”.x’

On interchanging the roles of p’ and p”

r(p”,v)=p”.x” and r(p’,v) ≥p’.x”

From these .we see at once that

(p”-p’).(x”-x’) ≥0 (4)

In other words, prices changes are non-negatively correlated with the resulting output supply changes. This is a natural multi-dimensional generalization of the familiar property that supply curves cannot be downward-sloping.

To conclude the study of r as a function of p ,we note that r is homogeneous of degree 1 in p for fixed v; this folloes at once from the definition of r, since a proportionate change in all prices does not change the optimizing quantity choices. Now Euler’s Theorem gives∑_j p_j 2r/2p_j=r, or

p.r_p(p,v)=r(p,v) (5)

Next suppose r is twice differentiable. Since it is a convex function, the matrix r_pp of the second-order partial derivatives 2²r/2p_j 2p_k must be positive semi-definite. Also, each 2r/2p_j is homogeneous of degreee zero in p, therefore applying Euler’s Theorem we have ∑_k(2²r/2p_j 2p_k)p_k=0,or ,in matrix notation

r_pp(p,v)p=0 (6)

Factor prices

Now fis p,and study r as a function of v alone. We begin by showing that r is a concave function of v: this is a natural extension of a concave production function for scalar output, and follows from our assumed convexity of the technology. Consider any two vectors v’,v”, and let x’,x” be the corresponding optimum output choices. Since the set of feasible (x,v) has been assumed convex, the average output (x’+x”)/2 is feasible given the average inputs (v’+v”)/2,and therefore revenue (r(p,v’)+r(p,v”))/2 is clearly attainable. The best choice for (v’+v”)/2, whatever it might be ,can yield no less. The graph of the function in a diagram like that used earlier must therefore appear as shown in Figure 2.2. Its slope at v’ can be interpreted by a simple economic argument.The slope shows the effect on the revenue of making available an extra unit of v_i and arranging production optimally. Since the original v’ was itself deployed optimally, the value of its marginal product in all uses has been equalized. To first order, therefore, the same value is to be had by employing the extra unit in any use, and thus the additional revenue is simply this common value marginal product, which is just the shadow price or demand price of the input. Employing this argument for each component, and writing w for the vector of the shadow prices of the factors, we have

w(p,v)=r_v (p,v) (7)

If a vector v of factor inputs is being used, a profit-maxmizing user will be willing to pay w for further marginal quantities of these factors, in accordance with (7). In other words. (7)gives us the inverse demand functions for factors. To determine their equilibrium prices, we must introduce supply considerations. A particularly simple case is that of perfectly inelastic factor supplies. In equilibrium, the fixed supply vector v must equal the employment or input, and then the equilibrium factor price vector w can be found by using this v in (7) .

If r is not differentiable with respect to v, the left-hand and right-hand derivatives will provide bounds within which a non-unique w must lie. More important is the possibility of flat portions in the graph of r as a function of v, at least over some ranges. In such cases, over them , w will be independent of v. This is an aspect of the question of factor price equalization that has received a great deal of attention, and we will examine it in greater detail shortly.

Since the graph of the concave function r cannot lie above its tangent at v’, we have, using (7)

r(p,v”)≤r(p,v’)+(v”-v’).w’

Similarly, drawing the tangent at v” and interpreting its slope as the shadow price w” at that point, we have

r(p,v’)≤r(p,v”)+(v’-v”).w”

Adding these inequalities and simplifying, we find

(w”-w’).(v”-v’)≤0 (8)

i.e. demand prices of fators are non-positively correlated with their quantities, which is a generalization of the normal idea that the derived demand curves for factors cannot slope upwards.

If there are constant returns to scale,we have two further results. First, r is homogeneous of degree one in v for each fixed p, and Euler’s Theorem yields the familiar result of imputation of output into factor incomes:

v.r_v(p,v)=r(p,v) (9)

Further,each 2r/2v_i is homogeneous of degree zero in v, and defining a matrix of second-order partial derivatives, we write

r_vv(p,v)v=0