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

I. 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.

II. 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 input is |N.sup.E~|I.sub.j*M~ = (|Q.sup.E~/|M.sup.E~)|I.sub.j*M~, and the price of the jth input is |W.sub.j~(|N.sup.E~|I.sub.j*M~) = |W.sub.j~|(|Q.sup.E~/|M.sup.E~)|I.sub.j*M~~. Hence the right-hand side of equation (2) can be rewritten as follows.

|Mathematical Expression Omitted~

Combining equations (2) and (3) yields equation (4).

|P.sup.S~(|Q.sup.E~) = ||summation of~ |W.sub.j~|(|Q.sup.E~/|M.sup.E~)|I.sub.j*M~~|I.sub.j*M~ where j=1 to L~/|M.sup.E~ (4)

Overall equilibrium requires equilibrium in the input markets, which is described by equation (5).

|Mathematical Expression Omitted~

The analytic exercise of deriving a long-run industry output supply curve requires that we choose a given value for |Q.sup.E~, and solve for the remaining variables. (Demand is taken as exogenous.) In equation (5), |M.sup.E~ is a direct function of the input prices, and the |I.sub.j*M~ are direct functions of |M.sup.E~ and the input prices. Hence |Q.sup.E~ is the only exogenous variable in equation (5), and all endogenous variables will in equilibrium be a function of |Q.sup.E~. (For ease of notation, these functional dependencies will not be written out explicitly.) The sequence in which the long-run supply curve is derived can be thought of as follows. After a value of |Q.sup.E~ is selected, equation (5) determines the |Mathematical Expression Omitted~, from which |M.sup.E~ is determined. Next, the |I.sub.j*M~ are found from |M.sup.E~ and the |Mathematical Expression Omitted~. With all of the preceding information, equation (4) can be used to determine the value of |P.sup.S~ that corresponds to the value of |Q.sup.E~ initially selected.

III. The Equality of Producer Surplus and Rents

Let |Q.sup.ED~ denote any particular value of |Q.sup.E~ on the industry supply curve. Then producer surplus is calculated by equation (6).

Producer Surplus (at |Q.sup.ED~) = |P.sup.S~(|Q.sup.ED~)|Q.sup.ED~ - |integral of~ |P.sup.S~(|Q.sup.E~)d|Q.sup.E~ between limits of |Q.sup.ED~ and 0 (6)

The term rent will be used to denote the area above the input supply curve and below the input equilibrium price line.(3) Let |Mathematical Expression Omitted~ denote the equilibrium industry quantity of the jth input. If the quantity of output chosen along the industry supply curve for output is |Q.sup.ED~, then rents are calculated as in equation (7).

|Mathematical Expression Omitted~

Profits are zero in equilibrium, so |Mathematical Expression Omitted~. Using this fact, a comparison of equations (6) and (7) reveals that the difference between producer surplus and rents (at |Q.sup.ED~) is given by equation (8).

|Mathematical Expression Omitted~

It will now be demonstrated that the expression in equation (8) equals zero, and hence that producer surplus equals rents.

|Mathematical Expression Omitted~, the equilibrium industry quantity of the jth input, is ultimately a function only of the |Q.sup.E~ selected on the industry supply curve for output. (To see this, first recall that |Mathematical Expression Omitted~, and then recall the comments that follow equation (5).) To reach equation (9), use the fact that |Mathematical Expression Omitted~ is a function of |Q.sup.E~, and employ the change of variables theorem.(4)

|Mathematical Expression Omitted~

To reach equation (10), use the fact that |Mathematical Expression Omitted~.

|Mathematical Expression Omitted~

Equation (11) is obtained by substituting from equation (10) into equation (9), and separating terms.

|Mathematical Expression Omitted~

Next substitute from equation (11) into equation (8). To show that producer surplus equals rents, it needs to be shown that the expression in (12) equals zero.

|Mathematical Expression Omitted~

In the second term of (12), relocate the summation sign inside the integral. Then from equation (4), it follows that the last two terms in (12) are equal. Therefore, in order to show that producer surplus equals rents, it only needs to be shown that the expression in (13) equals zero.

|Mathematical Expression Omitted~

In expression (13), relocate the summation sign inside the integral, and factor |Q.sup.E~/|M.sup.E~ out of the summation. Then it suffices to show that the expression in (14) equals zero.

|Mathematical Expression Omitted~

Recall that |I.sub.j*M~ is a direct function of |M.sup.E~ and the input prices, that |M.sup.E~ is a direct function of the input prices, and that the equilibrium input prices are functions only of |Q.sup.E~. (See the comments that follow equation (5).) Equation (15) follows.

|Mathematical Expression Omitted~

Substitute equation (15) into (14). Then in order to show that producer surplus equals rents, it suffices to show that the expression in (16) equals zero.

|Mathematical Expression Omitted~

If terms in (16) are rearranged slightly, then it suffices to show that the expression in (17) equals zero.

|Mathematical Expression Omitted~

Attention will first be focused on the first term in (17). Reverse the order of the summation signs. Then in order to show that this term equals zero, it suffices to show that equation (18) is satisfied.

|Mathematical Expression Omitted~

Equation (18) is equivalent to equation (19).

|Mathematical Expression Omitted~

In order to show equation (19), it suffices to show (20).

|Mathematical Expression Omitted~

By the envelope theorem, |I.sub.j*M~ is the partial derivative of the total cost function with respect to |W.sub.j~. Using this result and applying Young's theorem leads to equation (21).

(|Delta~|I.sub.j*M~/|Delta~|W.sub.k~) = (|Delta~|I.sub.k*M~/|Delta~|W.sub.j~) (21)

Substitute (21) into (20) to get equation (22).

|Mathematical Expression Omitted~

Because each conditional factor demand function is homogeneous of degree zero in the input prices, it follows from Euler's theorem that equation (20) is satisfied. Hence the first term in (17) does equal zero.

To complete the proof that producer surplus equals rents, it needs to be demonstrated that the remainder of (17) equals zero. Hence it needs to be shown that equation (23) is satisfied.

|Mathematical Expression Omitted~

It suffices to show that equation (24) is satisfied.

|Mathematical Expression Omitted~

In what follows, the author greatly benefitted from Silberberg |10, 738-39~. Also found to be helpful were Hughes |3~, Portes |8~, and Silberberg |11, 258-77~.

Let M|C.sup.E~ and A|C.sup.E~ denote, respectively, the marginal cost and average cost of the firm in long-run equilibrium. From inspection, it can be seen that the right-hand side of equation (24) equals A|C.sup.E~. It is well known that M|C.sup.E~ = A|C.sup.E~. To prove equation (24), it therefore suffices to demonstrate that the left-hand side of equation (24) equals M|C.sup.E~.

Consider the first-order conditions for the minimization of total cost when the specified level of output is |M.sup.E~. These are given in equations (25) and (26). Here f denotes the production function, |f.sub.j~ denotes the marginal product of the jth input, and |Lambda~* denotes the Lagrangian multiplier. (It is assumed that the first-order conditions are being evaluated at the cost-minimizing input quantities.)

|W.sub.j~ - |Lambda~*|f.sub.j~ = 0 j = 1, ..., L (25)

f - |M.sup.E~ = 0 (26)

Differentiating equation (26) with respect to |M.sup.E~ yields equation (27).

|summation of~ |f.sub.j~(|Delta~|I.sub.j*M~/|Delta~|M.sup.E~) - 1 = 0 where j=1 to L (27)

Substituting from equation (25) into (27) yields (28).

|summation of~ (|W.sub.j~/|Lambda~*)(|Delta~|I.sub.j*M~/|Delta~|M.sup.E~) - 1 = 0 where j=1 to L (28)

Simplifying (28) yields equation (29).

|summation of~ |W.sub.j~(|Delta~|I.sub.j*M~/|Delta~|M.sup.E~) = |Lambda~* where j=1 to L (29)

By the envelope theorem, |Lambda~* equals marginal cost. From equation (29), it therefore follows that in long-run equilibrium, the left-hand side of equation (24) does equal M|C.sup.E~. Hence the proof that producer surplus equals rents is completed.

IV. Conclusions

The area above the competitive long-run industry supply curve for output and below the equilibrium price line, commonly referred to as producer surplus, has been used to help derive policy implications for a wide variety of applied microeconomic problems. Consequently, it is extremely important to interpret the economic meaning of this area in an appropriate fashion.

The current paper has demonstrated that producer surplus will in fact equal rents whenever the individual firm has a U-shaped long run average cost curve. This result holds regardless of the form of the input supply functions, and regardless of the number of inputs.

1. Part (but not all) of the reason for the disagreement is the lack of consensus as to how rent should be defined. The appropriateness of various definitions of rent is beyond the scope of this paper.

2. In accordance with equation (1), a perfectly competitive industry can exhibit constant returns to scale even if the individual firms in the industry do not experience constant returns to scale. Such would be the case if factor price ratios remain constant. In such a situation, |M.sup.E~ does not change, and the cost-minimizing input quantities used to produce |M.sup.E~ do not change. Changes in industry output would then take place via the entry or exit of equally efficient firms, each producing an unchanged |M.sup.E~, and each using an unchanged combination of input quantities.

3. As Mishan |5~ has pointed out, it would be preferable to use Hicksian input supply curves when calculating rents. However, following Helmberger and Rosine |2~, rents will be calculated using the ordinary Marshalian input supply curves.

4. Note that if |Q.sup.E~ = 0, then |Mathematical Expression Omitted~, so the lower limits of integration are indeed valid. For an explanation of the change of variables theorem, see Marsden |4, 301, 306~.

References

1. Formby, John P., "A Clarification of Rent Theory." Southern Economic Journal, January 1972, 315-24.

2. Helmberger, Peter and John Rosine, "Measuring Producer's Surplus." Southern Economic Journal, April 1980, 1175-79.

3. Hughes, Joseph P., "The Comparative Statics of the Competitive, Increasing-Cost Industry." American Economic Review, June 1980, 518-21.

4. Marsden, Jerrold E. Elementary Classical Analysis. San Francisco: W. H. Freeman and Company, 1974.

5. Mishan, E. J., "Rent as a Measure of Welfare Change." American Economic Review, June 1959, 386-94.

6. -----, "What Is Producer's Surplus?" American Economic Review, December 1968, 1269-82.

7. -----. Introduction To Normative Economics. New York: Oxford University Press, 1981.

8. Portes, Richard D., "Long-Run Scale Adjustments of a Perfectly Competitive Firm and Industry: An Alternative Approach." American Economic Review, June 1971, 430-34.

9. Shepherd, A. Ross, "Economic Rent and the Industry Supply Curve." Southern Economic Journal, October 1970, 209-11.

10. Silberberg, Eugene, "The Theory of the Firm in 'Long-Run' Equilibrium." American Economic Review, September 1974, 734-41.

11. -----. The Structure of Economics: A Mathematical Analysis. New York: McGraw-Hill Publishing Company, 1990.