Long-run and short-run cost curves Cost curves form a staple part of the curriculum of undergraduate microeconomics. Their presentation across textbooks is fairly uniform uniform and has not varied much over the years since Marshallian partial equilibrium analysis was first codified in a set of diagrams. The uniformity and stability of the presentation of the curves belies their controversial beginnings in the 1920s and 30s, as economists struggled to contain Marshall’s realistic descriptive insights and biological analogies into a logically coherent static equilibrium model. In In the battle that played out between descriptive realism and neat formalisms, the latter won out for center stage in textbook studies of cost. Cost curves are drawn with the quantity of a specific product along the horizontal axis and money cost on the vertical. For an a n analysis of perfect competition, the assumption is that each firm faces identical input prices and choices of technology. Under this assumption, the curves can be interpreted to correspond either to the industry as a whole, or to a representative firm. They can represent the total cost of the quantity, the average (per unit) cost, or the marginal cost. For the short-run, total and average costs can be broken down into the portion reflecting the amount spent on factors of production whose quantities can be varied, and the portion reflecting the sunk costs of the fixed factors of production. Further, one can consider long-run cost curves drawn under the assumption that the quantities of all factors can be varied. Drawn together on one diagram, cost curves can appear daunting to students of economics, and even their teachers can have difficulty disentangling them in a comprehensible c omprehensible way. Evidence of the difficulty can be seen in three pedagogical articles on cost curves that have appeared in the Journal the Journal of Economic Education in recent years. (Sexton, Graves and Lee (1993), Boyd and Boyd (1994), Silberberg (1999)) The typical textbook presentation starts with the short-run and shows the average cost curves (total, fixed and variable) along with the marginal cost. The curves are presented in Figure in Figure 1. 1.
Figure 1. Short-run unit cost curves: marginal cost (MC), average total cost (ATC), average variable cost (AVC) and average fixed cost (AFC). The short-run cost curves are normally based on a production function with one variable factor of production that displays first increasing and then decreasing marginal productivity. Increasing marginal productivity is associated with the negatively sloped portion of the marginal c ost curve, while decreasing marginal productivity is associated with the positively sloped portion. The average fixed cost (AFC) curve is the cost of the fixed factor of production divided by the quantity of units of the output, while the average variable cost (AVC) curve cost traces out the per unit cost of variable factor of production. The U-shaped average total cost (ATC) curve is derived by adding the average fixed and variable costs. The marginal cost (MC) intersects both the AVC and ATC curves at their minimum points. Declining average total costs are explained as the result of spreading the fixed costs over greater q uantities and, at low quantities, the result of the increasing marginal productivity, in addition. Increasing average costs occur when the effect of declining marginal productivity overwhelms the effect of spreading the fixed costs. The long-run cost curves, usually presented in a separate diagram, are also expressed most commonly in their average, or per unit, form, represented here in Figure 2. The long-run average cost (LRAC) curve is shown to be an envelope of the short-run average cost (SRAC) curves, lying everywhere below or tangent to the short-run curves. The firm is constrained in the shortrun in selecting the optimal mix of factors of production an d so will never be able to find a cheaper mix than can be found in the long-run when there are no constraints. If there are a discrete number of plant sizes available, the LRAC will be the scalloped curve obtained by joining those parts of the SRAC curves that represent the lowest cost of production for a given quantity.
Figure 2. The long-run average cost (LRAC) curve is an envelope curve of the short-run average cost (SRAC) curves. Increasing, constant and decreasing returns to scale are exhibited at points a, b and c, respectively. In the case of constant returns to scale, the most common assumption for production functions, the LRAC curve is horizontal. If it is the case that a larger scale of production is obtained through increasing the number of identical plants, then the long-run cost curve approaches a horizontal line as the number of plants increases (Joseph 1933); that is, the production function will exhibit “asymptotic-first-degree-homogeneity” (Samuelson 1967, p. 131). Long-run average cost curves that are everywhere decreasing will arise in industries with a large fixed cost and constant variable costs. Traditionally, decreasing long-run cost curves have been associated with natural monopolies. More generally, the LRAC curve c an be drawn as a U-shape displaying increasing returns at lower levels of output and decreasing returns at h igher levels. The long-run marginal cost curve is usually omitted in introductory treatments. If it is included, it is drawn so that it lies above the short-run marginal cost curve to the left of their intersection point, and below to the right. Understanding the relationship between the short-run and long-run marginal cost curves can be challenging, and may require conjuring up additional diagrams. Sexton, Graves and Lee (1993) couch their explanation using isocosts and isoquants; whereas Boyd and Boyd (1994) show how the relationship can be explained using the total cost curves. Keppler and Lallement (2006) trace the cost cu rve construction back to Enrico Barone’s 1908 Principi di economia politica. Barone presents a diagram that contains a total cost curve with costs increasing at a decreasing and then increasing rate—equivalent to the textbook U-shaped average cost curve. He represents the marginal cost o f a particular quantity as the slope of a tangent to the cost curve at that quantity, and the average cost of a particular quantity as the slope
of a ray from the origin to the relevant point of the cost curve. Total reven ue is represented on the diagram as a line with the slope equivalent to the price. The construction allows him to identify the profit maximizing quantity as that for which the price equals marginal cost, and the zero profit long-run equilibrium quantity as that for which price equals both marginal cost and average total cost . The first U-shaped cost curve, in which cost is expressed in per unit terms, uncovered by Keppler and Lallement (2006) appears an article by Edgeworth (1913) on railway rates. Since the discussion was of a monopoly firm the relationship between price, marginal co st and average cost found in Barone’s 1908 work is not brought out. The first per unit cost curve construction of a firm in perfect competition bearing a close resemblance to contemporary textbook treatments, was presented in paper by Piero Sraffa published in 1925 in an Italian languag e journal (reproduced in Keppler and Lallement (2006)). Upon reading the article in Italian, Edgeworth commissioned Sraffa to write a similar piece for the Economic Journal --which appeared in 1926 but did not contain the diagram. A virtually identical diagram, however, appears in a related article by Pigou (1928). It is reasonable to suspect that Pigou derived it from Sraffa’s 1925 paper. Roy Harrod (1931) and Jacob Viner (1931) contemporaneously published articles containing diagrams showing the relationship between the LRAC curve and a set of SRAC curves. In his now legendary 1931 Zeitschrift für Nationalökonomie article, Viner had instructed his draughtsman, the mathematican Y. K. Wong, to draw the long-run average total cost curve so that it would run tangent to the bottoms of the short-run U-shaped curves, be downward sloping, and nowhere lie above the short-run curves. Wong insisted that it was impossible to satisfy all three requirements and produced a diagram with the long-run curve lying, on one portion, above the short-run curve. In a footnote, Viner (1931, p. 36n) admits that the diagram was in error in placing the long-run curve above the short-run, but confesses that he did not understand Wong’s mathematical reason for it being impossible to draw the long-run curve no where above the short-run curves while running along the bottoms of the short-run curves: “I c ould not persuade him to disregard his scruples as a craftsman and to follow my instructions, absurd though they might be.” Harrod (1959, p. 262) recounts the experience, when working on his 1931 paper at Oxford, of wandering outside with his curves on Tom Quad in Christ Church in quest of a scientific colleague who could tell him the proper name for the long-run curve. His quest was successful and allowed him in the 1930’s, unlike Viner, to properly describe the LRAC curve as an envelope. In Foundations of Economic Analysis, Paul Samuelson provides a proof of what he calls the Wong-Viner envelop theorem showing the proper relationship between constrained and unconstrained functions. Samuelson, who had bee n a student in Viner’s class in 1935 recalls how after admitting his error in class, Viner had told him privately after the bell had rung, that he still thought it possible to draw a downward sloping LRAC curve that would smoothly join the bottoms of a set of SRAC curves. In the same reminiscence, Samuelson (1972, p. 9) reports that Schumpeter, as well, continued to be confused about the matter: “He always insisted somewhere in the third dimension of the parametrized surface of U-shaped cost curves, the envelope did go
through Viner’s bottom points.” Silberberg (1999), noting the opacity o f Samuelson’s proof in the Foundations, provides a clarifying explanation using the total cost curve diagram. The confusion leading to the error may be related to the result from the theory of the competitive firm that firms will produce at the minimum point on the average total cost curve after entry and exit has driven economic profit to zero. If the price of the product (P) is such that the marginal cost (MC) for the representative firm intersects the horizontal price line above the minimum point on the SRAC curve, the firm will be earning economic profits and there will be entry and expansion. As more firms enter, and as existing firms expand production, supply of the product will increase and the price will fall. Long-run equilibrium is reached when P = MC = SRAC. If the price intersects the marginal cost below the minimum point on the SRAC, the opposite process of exit and contraction will reduce supply, increase the price and a new long-run equilibrium will be established with the representative firm producing at the minimum point of the SRAC curve. At first sight, it seems reasonable to suppose that the points on the LRAC curve will coincide with the minimum points on the SRAC curves, since they are the long-run equilibrium points. The confusion is due to the conflation of two separate notions of the long-run. In Marshall’s theory of the firm in perfect competition, the long-run, or long period as he termed it, is the hypothetical state that is achieved after economic profits have been driven to zero. The LRAC curve, on the other hand, is tracing out the unit costs of different levels of output un der the assumption that all factors of production can be optimally adjusted. It do es not refer to the longrun equilibrium of the firm and might be better labeled the unconstrained average total cost curve. It may be a recognition of the conflation of the two notions of the long-run that lead Schumpeter to believe that if an additional dimension was added the LRAC curve wou ld run through the minimum points. In any event, the confusion surrounding the curves seems to bear out well Marshall’s (1920, p. 461) warning about the limitations of the “statical theory of equilibrium” in analyzing the progress of industries through time: “limitations …so constantly overlooked, especially by those who approach it from an abstract point of view, that there is a danger of throwing it into definite form at all.”
References
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Harrod, R. (1931), ‘The Law of Decreasing Cost’, Economic Journal , 41 (164), 566-576. ----------- (1959), ‘Review of The Long and the Short ’, Economica, New Series, 26 (103), 260-262. Joseph, M. F. W. (1933), ‘A Discontinuous Cost Curve and the Tendency to Increasing Returns’, The Economic Journal , 43: 171, 390-398. Keppler, J. H. and J. Lallement (2006), ‘The Origins of the U-Shaped Average Cost Curve: Understanding the Complexities of the Modern Theory’, History of Political Economy, 38 (4), 733-774. Marshall, A. (1920). Principles of Economics, 9th ed., New York: The Macmillan Company. Pigou, A. C. (1928), ‘An Analysis of Supply’, Economic Journal , 38 (150), 238-257. Samuelson, P. A. (1972), ‘Jacob Viner, 1892-1970’, Journal of Political Economy, 80 ( 1), 5-11 -------------------- ( 1976}, ‘The Monopolistic Competition Revolution’, in Monopolistic Competition Theory: Studies in Impact, Essays in Honor of Edward H. Chamberlin, edited by R. E. Kuenne, New York: Wiley. Sexton, R. L., P. E. Graves and D. R. Lee (1993), ‘The Short- and Long-Run Marginal Cost Curve: A Pedagogical Note’, Journal of Economic Education, 24 (1), 34-37. Silberberg, E. (1999), ‘The Viner-Wong Envelope Theorem’, Journal of Economic Education, 30 (1), 75-79. Viner, J. (1931), “Cost Curves and Supply Curves,” Zeitschrift für Nationalökonomie , 3, 23-46.
