Long-term equilibrium of a competitive, capitalist economy.

4.4. LP THEORY OF A FIRM IN A COMPETITIVE ECONOMY 45 r ∗ i n X j=1 a ij x ∗ j , then q ∗ i = 0, i.e., the equilibrium price of an input which is in excess supply must be zero, in other words it must be a free good. Remark 2: Returning to the short-term decision problem 4.33, suppose that λ ∗ 1 , . . . , λ ∗ ℓ , λ ∗ ℓ+1 , . . . , λ ∗ m is an optimum solution of the dual of 4.33. Suppose that the market prices of inputs 1, . . . , ℓ are q 1 , . . . , q ℓ . Let us denote by M ∆ 1 , . . . , ∆ ℓ the optimum value of 4.33 when the amounts of the inputs in fixed supply are r ∗ 1 + ∆ 1 , . . . , r ∗ ℓ + ∆ ℓ . Then if ∂M∂∆ i | ∆=0 exists, we can see from 4.22 that it is always profitable to increase the ith input by buying some additional amount at price q i if λ ∗ i q i , and conversely it is profitable to sell some of the ith input at price q i if λ ∗ i q i . Thus λ ∗ i can be interpreted as the firm’s internal valuation of the ith input or the firm’s imputed or shadow price of the ith input. This interpretation has wide applicability, which we mention briefly. Often engineering design problems can be formulated as LPs of the form 4.10 or 4.19, where some of the coefficients b i are design parameters. The design procedure is to fix these parameters at some nominal value b ∗ i , and carry out the optimization problem. Suppose the resulting optimal dual variables are λ ∗ i . then we see assuming differentiability that it is worth increasing b ∗ i if the unit cost of increasing this parameter is less than λ ∗ i , and it is worth decreasing this parameter if the reduction in total cost per unit decrease is greater than λ ∗ i .

4.4.4 Long-term equilibrium of a competitive, capitalist economy.

The profit-maximizing behavior of the firm presented above is one of the two fundamental building blocks in the equilibrium theory of a competitive, capitalist economy. Unfortunately we cannot present the details here. We shall limit ourselves to a rough sketch. We think of the economy as a feedback process involving firms and consumers. Let us suppose that there are a total of h com- modities in the economy including raw materials, intermediate and capital goods, labor, and finished products. By adding zero rows to the matrices A, B characterizing a firm we can suppose that all the h commodities are possible inputs and all the h commodities are possible outputs. Of course, for an individual firm most of the inputs and most of the outputs will be zero. the sole purpose for making this change is that we no longer need to distinguish between prices of inputs and prices of outputs. We observe the economy starting at time T . At this time there exists within the economy an inventory of the various commodities which we can represent by a vector ω = ω 1 , . . . , ω h ≥ 0. ω is that portion of the outputs produced prior to T which have not been consumed up to T . We are assuming that this is a capitalist economy, which means that the ownership of ω is divided among the various consumers j = 1, . . . , J. More precisely, the jth consumer owns the vector of commodi- ties ωj ≥ 0, and J X j=1 ωj = ω. We are including in ωj the amount of his labor services which consumer j is willing to sell. Now suppose that at time T the prevailing prices of the h commodities are λ = λ 1 , . . . , λ h ′ ≥ 0. Next, suppose that the managers of the various firms assume that the prices λ are not going to change for a long period of time. Then, from our previous analysis we know that the manager of the ith firm will plan to buy input supplies ri ≥ 0, ri ∈ R h , such 46 CHAPTER 4. LINEAR PROGRAMMING that λ, ri is in long term equilibrium, and he will plan to produce an optimum amount, say yi. Here i = 1, 2, . . . , I, where I is the total number of firms. We know that ri and yi depend on λ, so that we explicitly write ri, λ, yi, λ. We also recall that see 4.38 λ ′ ri, λ = λ ′ yi, λ , 1 ≤ i ≤ I . 4.40 Now the ith manager can buy ri from only two sources: outputs from other firms, and the con- sumers who collectively own ω. Similarly, the ith manager can sell his planned output yi either as input supplies to other firms or to the consumers. Thus, the net supply offered for sale to consumers is Sλ, where Sλ = J X j=1 ωj + I X i=1 yi, λ − i X i=1 ri, λ . 4.41 We note two important facts. First of all, from 4.40, 4.41 we immediately conclude that λ ′ Sλ = J X j=1 λ ′ ωj , 4.42 that is the value of the supply offered to consumers is equal to the value of the commodities and labor which they own. The second point is that there is no reason to expect that Sλ ≥ 0. Now we come to the second building block of equilibrium theory. The value of the jth consumer’s possessions is λ ′ ωj. The theory assumes that he will plan to buy a set of commodities dj = d 1 j, . . . , d h j ≥ 0 so as to maximize his satisfaction subject to the constraint λ ′ dj = λ ′ ωj. Here also dj will depend on λ, so we write dj, λ. If we add up the buying plans of all the consumers we obtain the total demand Dλ = J X j=1 dj, λ ≥ 0 , 4.43 which also satisfies λ ′ Dλ = J X j=1 λ ′ ωj . 4.44 The most basic question of equilibrium theory is to determine conditions under which there exists a price vector λ E such that the economy is in equilibrium, i.e., Sλ E = Dλ E , because if such an equilibrium price λ E exists, then at that price the production plans of all the firms and the buying plan of all the consumers can be realized. Unfortunately we must stop at this point since we cannot proceed further without introducing some more convex analysis and the fixed point theorem. For a simple treatment the reader is referred to Dorfman, Samuelson, and Solow [1958], Chapter 13. For a much more general mathematical treatment see Nikaido [1968], Chapter V.

4.5 Miscellaneous Comments