Rents, producers' surplus, and the long-run industry supply curve.

• Author: Frasco, Gregg

1. Introduction

   The term producer surplus conventionally has been used to refer to the area above the competitive industry supply curve for output and below the equilibrium price. For the short run, there is general agreement as to the meaning of this area. In the short run, this area equals the payments to the firm in excess of the firm's total variable costs. Such payments are often referred to as quasi-rents. For the long run, there is no general agreement as to the meaning of this area. Some claim that this area has no particular meaning except under very special circumstances. For example, see Mishan |7, 221-22, 225-30~ and Formby |1, 316, 323-24~. Others argue that this area is equal to the rents paid to factors of production. For example, see Shepherd |9~ and Helmberger and Rosine |2~.(1)

   Helmberger and Rosine |2~ were the only ones to present an analytical example to support their position. They demonstrated that producer surplus will in fact equal rents if (i) an aggregate production function is used, (ii) the production function has four inputs, is of the Cobb-Douglas variety, and exhibits constant returns to scale, and (iii) two of the industry input supply functions are log-linear and two are perfectly elastic.

   The current paper brings the analysis down to the level of the individual firm, and significantly extends the result of Helmberger and Rosine |2~. It is demonstrated that producer surplus equals rents whenever the individual firm has a U-shaped long run average cost curve, regardless of (i) the form of the industry input supply functions, and (ii) the number of inputs. Throughout the paper, the analysis is general, that is, no specific functional forms are used.

2. Derivation of the Long-Run (Inverse) Industry Supply Curve

   The long-run industry supply curve for output is a locus of equilibrium points. Hence the industry quantity of output will be denoted by |Q.sup.E~, and the long-run industry supply curve will be denoted by |P.sup.S~(|Q.sup.E~). Assume there are L inputs, and let |W.sub.j~ denote the price of the jth input. The long-run average total cost function will be written as LATC (q, |W.sub.1~, ..., |W.sub.L~), where q denotes the output of the individual firm. Assume that the long-run average cost of the firm is U-shaped, and let |M.sup.E~ denote the firm's minimum efficient scale in equilibrium. Industry equilibrium will occur at a multiple of |M.sup.E~. Because the long-run industry supply curve is an equilibrium horizontal summation of the firms' minimum average cost outputs, it follows that equation (1) must be satisfied in any equilibrium. |N.sup.E~ denotes the number of firms in equilibrium.

   |Mathematical Expression Omitted~

   Equation (1) simply states that in any equilibrium, if industry output is |Q.sup.E~ = |N.sup.E~|M.sup.E~, then the price on the industry supply curve must equal each firm's average cost when each firm produces |M.sup.E~.(2) Equation (1) can be replaced by equation (2).

   |Mathematical Expression Omitted~

   Let |I.sub.j*M~ denote the firm's cost-minimizing quantity of the jth input when |M.sup.E~ units of output are produced. Also let |I.sub.j~ denote the industry quantity of the jth input, and |W.sub.j~ (|I.sub.j~) denote the industry (inverse) supply function for the jth input. In equilibrium, the total industry use of the jth...