Markup Pricing versus Marginalism.pdf

Markup Pricing versus Marginalism: A Controversy RevisitedAuthor(s): Catherine LangloisSource: Journal of Post Keynesian Economics, Vol. 12, No. 1 (Autumn, 1989), pp. 127-151Published by: M.E. Sharpe, Inc.Stable URL: http://www.jstor.org/stable/4538176 .

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CATHERINE LANGLOIS

Markup pricing versus marginalism: a controversy revisited

Introduction

As economists gathered evidence on firm price-making practices, a controversy developed over the compatibility of these rules with the marginalist principle of profit maximization, and indeed, the relevance of the neoclassical approach altogether. In this paper, the marginalist and cost controversies are re-examined in the light of available cost and demand elasticity estimates for the automobile industry. This confrontation suggests an alternative to the neoclassical vision of the firm that can account for markup-over-average-cost-of-goods-sold pricing and integrate the empirical evidence on cost and price elasticity of demand. The firm described in this paper views inventory holding as the strategic quantity variable and maximizes average profit per unit time over the time it takes to sell output inventory. In the standard approach, by contrast, the relevant strategic quantity variable is production, and profit is maximized within a predetermined time interval. Thus production costs and demand must have the same temporal dimension (costs associated to weekly production are matched to weekly demand). But this constraint can be bypassed if the firm is thought of as maximizing profit made on the sale of a level of inventory, since the choice of a price changes sales time, and thus the temporal dimension of the demand

The author is at the Graduate School of Business Administration at the University of California, Berkeley. She thanks Dennis Quinn and an anonymous referee for extensive feedback.

Journal of Post Keynesian Economics / Fall 1989, Vol. 12, No. 1 127



128 JOURNAL OF POST KEYNESIAN ECONOMICS

relationship the firm chooses to consider. It turns out that per-unit-time profit maximization yields a formula for optimal markup over average costs (where average costs are here defined as the production cost of goods in inventory divided by inventory), which has been shown to be compatible with the inventory and pricing behavior of General Motors, Ford, and Chrysler (Langlois, 1989). In this paper, compatibility of the model with automobile industry estimates of price elasticity of demand is established. These estimates are shown to be compatible with average per-unit-time profit maximizing behavior if average costs are downward sloping over the relevant range of outputs. The automobile industry is one where scale economies and large production runs are characteristic of the production process. This provides strong evidence for a cost structure where marginal cost lies below average cost over the full range of outputs.

In section I, we draw upon Lee's (1984) extensive survey of the marginalist controversy to summarize the debate and introduce the nature of this contribution. Section II introduces the cost controversy and discusses temporal commensurability of costs and demand. Interpreta- tion of the econometric estimates of price elasticity of demand in the light of the cost controversy leads to the modified specification of firm behavior presented in section III. Section IV provides econometric evidence to substantiate the behavioral hypothesis of section III.

I. The marginalist controversy revisited

In the decade that followed World War II, the economics profession engaged in a spirited controversy over the relevance to real world practice of marginalist pricing principles. Marginalist theory prescribes choice of a price to ensure profit maximization. The facts as presented by the authors, such as Hall and Hitch (1939) or Andrews (1949), pointed to pricing rules that emphasized average costs and the conventional nature of the allowance made for profit. Thus, firms were described as applying some "standard" markup over average total costs, a fact that raised the issue of compatibility between business practice and the predictions of marginalist price theory.

In defense of marginalist principles was the argument that real world practice did not imply rejection of the profit maximizing model if the facts were correctly interpreted. Thus Robinson (1950) argues that, in order to decide on the best use of its resources, a producer can make a



MARKUP PRICING VERSUS MARGINALISM 129

"rough and ready test," which "is to calculate the direct costs, to add the costing margin at which the firm's resources will be reasonably fully employed, and reject those uses in which the 'costs' so calculated exceed the probable selling price" (Robinson, 1950, p. 776). The facts, according to Robinson, reflect the procedure used by firms to choose the most profitable strategy through elimination of the less profitable alternative. Machlup (1946), among others, goes one step further and argues that the facts can be interpreted directly in marginalist terms: if marginal and average costs coincide, and markup (chosen with some consideration for demand conditions) is interpreted as a proxy for the inverse of the price elasticity of demand, then average cost-oriented pricing procedures are fully compatible with marginalist principles.

While authors such as Machlup (1946) offered strong marginalist rationalization of average cost-oriented pricing rules, others, such as Gordon (1948), suggested generalization of marginalist principles to account for uncertainty, imperfect information, and alternative objective functions to integrate the observed pricing rules. Interestingly, the basic vision of the manufacturing firm as simultaneously producing and selling a continuous flow of output was not challenged by either party in the debate. Yet, firms hold inventory, a fact that to this day is recognized by the marginalist school as difficult to rationalize (see, for example, Blinder, 1986).

The model developed below transforms the neoclassical vision of the firm to integrate the holding of inventory as a strategic response of the manufacturing firm. This approach relies on the assumption that consum- ers respond to the availability of commodities, and firms hold inventory to endow their output with this characteristic. Thus production and pricing decisions will take explicit account of a firm's desired or target level of inventory. The model developed in this paper is descriptive of the particular level that a firm could choose for target inventory, but it does not make explicit the dynamic process of adjustment of production and price that must take place to maintain inventory at its target level. The hypothesis presented and empirically tested in what follows is that the level of target inventory is chosen together with its selling price so that profit is maximized over the time it will take to sell the inventory. The firm achieves this goal by maximizing profit per unit of time, optimization which generates a formula for optimal markup over the average cost of goods in inventory, where average cost is the average production cost of the goods themselves. Thus, by transforming the




neoclassical vision of the firm to integrate inventory, direct cost pricing is predicted using marginalist terms.

Markup pricing is, of course, well explained in non-neoclassical terms by authors such as Eichner (1973) or Sylos-Labini (1962), who empha- size the dynamic aspects of competition to explain firm pricing behavior. Crucial to the post Keynesian approach is the idea that pricing strategies are embedded in historical time, since they are designed to accommodate capital accumulation by drawing out sufficient funds to cover the costs of growth. In such a framework, costs are the crucial determinant of price, and demand constrains profitability, and hence growth, instead of determining price in response to a non-zero per-unit-time excess demand as the neoclassical model requires. Insofar as the model presented in this paper makes use of neoclassical ideas (profit maximization, demand function), it can be viewed as opposing the non-neoclassical theories of markup. However, this work deviates from the neoclassical vision in two respects: its treatment of time as a decision variable, and its emphasis on compatibility of the model with the data. The paragraphs that follow elaborate on these differences.

Neoclassical theory confines the representative firm's decision-making problem to a temporal framework chosen, a priori, by the theoretician. It is daily, weekly, or monthly profit that is maximized, depending on the assumption made about the frequency of observable demand variations. In this approach, by contrast, the temporal dimension given to demand and profit is endogenously determined: this firm chooses price, inventory, and the time it will take to sell it, not just a price and a level of output to be sold within a given interval of time. By adding a degree of freedom to the representative firm, this model allows for flexible timing of decisions over historical time in response to observed rates of depletion or accumulation of inventory holdings. In other words, target inventory is a concept defined in continuous time, decisions to adjust continuous production or price to accommodate a changing target inventory and inventory-to-sales ratio, being made at intervals determined by demand and cost conditions. By contrast, the neoclassical framework implies that decisions are made with a periodicity that is commensurate with the time interval within which profit is supposedly maximized, and the past essentially leaves no trace, since, in an ideal situation, production and demand have been matched and will have no bearing on future supply or demand conditions.

Compatibility of the proposed model with the data is also of primary concern to this author, and prompts two remarks that differentiate this




empirical approach from the one generally practiced by neoclassical theorists. First, testing of the implications of neoclassical theory involves addressing the unresolved question of observability of the equilibrium concept defined. Because the equilibrium concept developed here relates to a target level of inventory, it can be argued that adjustment to disequilibrium situations occurs fast enough to ensure that these inventory equilibria are indeed observable (Feldstein and Auerbach, 1976, and Maccini and Rossana, 1981, substantiate this claim). Secondly, it has been argued by the marginalist school that the "neoclassical theory of the firm deserves to survive because of its ability to produce verifiable predictions of the qualitative kind" (Blaug, 1980, p. 176). By contrast, the relevance of this approach is linked to its ability to produce verifiable predictions of the quantitative kind. If firm behavior can be described in the terms outlined here, then markup over the cost of goods sold should coincide with the inverse of the price elasticity of the sales time of inventory. This elasticity has been measured for General Motors, Chrysler, and Ford, and its inverse does indeed coincide with markup over the cost of goods sold as recorded by company accounts (see Langlois, 1989). It is one of the objectives of this paper to provide yet further evidence of the compatibility of this approach with the data.

The marginalist debate illustrates the marginalist school's lack of emphasis on verifiable quantitative prediction. Empirical measures of price elasticity of demand were available in the early 1960s, yet the proponents of the marginalist controversy did not appeal to them to refute or confirm the neoclassical approach. Yet, as will now be argued using automobile industry data, the analysis of these elasticity measures provides interesting insights. The demand for automobiles has been the subject of numerous post-war studies, and various authors have estimated price elasticities for the short and long runs (Chow, 1960, is the standard reference for long-run estimates). As shown in Table 1, short-run price elasticity of demand for automobiles in the U.S. has been estimated to lie in the range (-3.1, -1.32). In Machlup's (1946) interpretation of the facts, the inverse of the price elasticity of demand 1 /IqlI should yield profit maximizing markup over marginal cost.

From Table 1, markup is then estimated to lie between 32 percent and 76 percent, a promising outcome for marginalist theory, given estimates of marginal cost in the automobile industry. White states that marginal costs for manufacturing lie between one-half and two-thirds of wholesale

prices (White, 1971, p. 120). This estimate is corroborated by Evans, who




Table 1

Short-run price elasticity of new automobile purchases: summary of studies of the U.S. market

Study Period Estimate

Evans 1948-1964 -3.1

Westin 1953-1972 1. Stock Adjustment Model -1.32 2. Discretionary Replacement Model -1.65

Weiserbs 1929-1970 1. Dynamic Linear Model -2.13* 2. Dynamic Quadratic Model -2.32*

* Uncompensated demand elasticities. Compensated demand elasticities are also reported in Phlips.

Sources: Evans, M., Macroeconomic Activity. Theory, Forecasting and Con- trol. An Econometric Approach. Harper and Row, 1969. Westin, R., "Empirical Implications of Infrequent Purchase Behavior in a Stock Adjustment Model," American Economic Review, June 1975, p. 393. Weiserbs, estimates reproduced in Phlips, L., Applied Consumption Analysis, pp. 201 and 207.

states marginal costs to be "about two-thirds of the manufacturer's price" (Evans, 1969, p. 172). The econometric estimates of 1 / Irll are generated using retail data, however, and are thus not directly comparable to manufacturer markups over marginal costs. But marginal cost data at the retail level is elusive. Nevertheless, values obtained for 1 /lr\ cannot be held against the marginalist hypothesis, particularly if marginal cost in the automobile industry lies below average cost.

Data on automobile manufacturing costs suggests that marginal costs are indeed below average costs. The importance of scale economies for the manufacturing and retail of automobiles is well documented (White, 1971; Pashigian, 1961; Davisson and Taggart, 1974), and data for markup over direct cost of goods sold is available. For the 1950s and 1960s, cost of goods sold was on the order of 80 percent of the wholesale price (weighted average of Big Three direct cost of goods sold for the period 1953 to 1972 as recorded by Moody's Industrial Manuals is 78.6




percent). Marginal cost, representing only one-third of wholesale price, falls short of average cost. These cost data relate to costs incurred given industry output. The fact that marginal cost is less than average cost at that output level suggests that, over the whole range of outputs, the marginal cost curve lies below a presumably downward sloping average cost curve, at least for manufacturing. Studies of automobile retailing suggest a similar pattern for costs (see Pashigian, 1961; Davisson and Taggart, 1974). Thus, the low values found for ITll are not incompatible with the industry's cost structure. However, in a world of declining average costs over the whole range of outputs, marginal revenue and marginal cost may never intersect to yield an optimal markup equal to 1 / ITl. The cost controversy that developed alongside the marginalist debate was concerned with precisely this point, and it is to a summary of this controversy and an analysis of its contribution that we now turn.

II. The cost controversy and the commensurability of costs and demand

The debate over the shape of the short-run costs faced by firms began with Eiteman's (1947) discussion of costs. The central question was the practical relevance of the concept of marginal cost. As Lee (1984) reports, Eiteman "had never heard any price setter even mention marginal costs." And, according to Eiteman, this was attributable to the difficulties involved in measuring marginal cost as soon as a firm used multiple processes. But even if marginal costs were measurable, the shape of the cost curves faced by a typical firm may deny marginal costs any practical relevance in terms of pricing strategy. Marginalist principles, according to Eiteman, can only be implemented if minimum average cost obtains at outputs well below capacity to guarantee an intersection between marginal revenue and the upward sloping portion of marginal cost.

Eiteman and Guthrie (1952) questioned 366 firms and found 316 to believe average costs to decline with increased output, reaching a minimum at or close to full capacity, with average and marginal cost curves simply stopping at full capacity. The implication is that marginal cost lies below average cost at all levels of operation and "cannot intersect the marginal revenue curve (1) if the average revenue curve is horizontal or (2) if the average revenue curve is high and relatively elastic" (Eiteman and Guthrie, 1952, p. 832). Ritter (1953) pointed out at the time that this did not invalidate marginalism as a doctrine prescribing rational




behavior, since profit maximization does not require equality of marginal revenue and marginal cost if a discontinuity prevents it. More to the point, Eiteman and Guthrie's finding limits the set of quantity levels over which costs are defined and in most instances imposes a comer solution to the neoclassical profit maximization problem. But, if profit maximizing does not occur at the intersection of marginal revenue and marginal cost, the optimal markup over marginal cost is no longer equal to the inverse of the price elasticity of demand. And this same conclusion holds if firms are assumed to make intertemporal tradeoffs to, for example, preserve market share, since profit maximization, thus constrained, no longer implies equality of marginal cost and marginal revenue.

Thus, the failure of real world firm behavior to conform to the marginal-cost-equals-marginal-revenue rule does not necessarily mean that profit is not maximized. The implication is rather that the inverse of the price elasticity of demand need not be aproxyfor markup over marginal cost. The issue is an important one, since numerous authors base empirical work on the assumption that 1 /1,9I does indeed capture markup over marginal cost (see, for example, Sumner, 1981, or Rosse, 1970).

With reference to automobile industry data, the fact that marginal costs lie below average costs could explain the low values found for Irl. But for these values to confirm the hypothesis that firms set marginal revenue equal to marginal cost, it must be argued that the schedules intersect. In what follows, it is shown that, as long as demand for the firm's output is not horizontal, an intersection between marginal cost and marginal revenue always exists if the temporal commensurability of costs and demand is bypassed. Moreover-and more importantly-it turns out that this dissociation suggests a model of firm behavior that, when confronted with the data, can be reconciled with the empirical evidence on 11 for the automobile industry.

Costs defined in the standard manner as production costs have a temporal dimension. It is the cost of daily, weekly, or monthly production that is described. The demand and marginal revenue curves associated to such costs must, of course, have the same temporal dimension. To the cost of daily production is associated daily demand for output. It is this necessary temporal commensurability of costs and demand that pre- cludes any guarantee of existence of a point of contact between marginal cost and marginal revenue in the cost configuration revealed by Eiteman and Guthrie (1952).

Temporal commensurability between costs and demand is a constraint




that can be bypassed. Instead of interpreting quantity as an amount produced per unit of time, it can be interpreted as a level of inventory. An inventory level is just dated and can therefore be associated to demand relationships with varying temporal dimensions. A stock of output can be sold within a week or a month, depending on the price charged for it. In other words, an arbitrary price-quantity pair can always be fitted on a demand schedule if the temporal dimension of demand is allowed to vary. If marginal production costs lie below average costs, the pattern can be reproduced in the cost of goods in inventory. Thus marginal cost associated to various levels of inventory may lie below average cost. But the possibility of choosing the temporal dimension of demand guarantees existence of an intersection between marginal cost and marginal revenue.

The possibility of choice raises the question of the significance of the particular time frame chosen. In the model developed below, the firm chooses that time frame for the sale of its inventory that maximizes profit made over the period. Thus the temporal dimension chosen for demand is generated endogenously and is non-arbitrary. To clarify the above ideas and introduce the model (which will be developed in section III below), a formal presentation of demand is necessary.

In the standard monopoly model, demand is approached as a one-to-one relationship linking selling price to quantity demanded. According to Debreu (1959), who provides a precise definition, the time interval within which demand will actually be realized is a "compact elementary interval" whose length "is chosen small enough for all the instants of an elementary interval to be indistinguishable from the point of view of the analysis" (Debreu, 1959, p. 29). Thus, in defining the concept of demand, for example, a subdivision of the elementary interval will not reveal variations in quantity demanded that are meaningful from the point of view of the decision maker's strategy; day-to-day variations in demand for a product might be meaningful, while the fluctuations observed from one minute to the next are surely not. The elementary interval is therefore chosen, in what follows, as the unit within which flow demand is measured. Following Debreu's terminology, the elementary interval will be referred to as a date. Demand at date t is referred to as flow demand q. At price p, flow demand of date t can be written

q= q (p, )

Within any date T, flow demand is a one-to-one relationship linking price to quantity.




The firm described below chooses an inventory level and the time it will take to sell it by choice of a selling price. Thus quantity demanded is explicitly associated to a length of time spanning many sequential elementary time intervals, as well as to a price. For example, the demand schedule associated to the time span running from date 0 to date t, (0, t), is derived from flow demand as follows:

q0(p,t)= q(p@,)dx

To simplify notation, the date at which demand over some time span is defined will be omitted. Thus q (p, t) is to be understood as the quantity demanded at price p if t elementary intervals of time elapse from the date for which the concept is defined. Formally, then, demand at any date is approached as a relationship among three variables: price p, quantity q, and time t. Given values for any two of these variables, the dated demand relationships R (p, q, t) yield the value of the third: to a pricep and a quantity q is associated a time span t and a demand schedule q(p, t) such that q = q(, t).1 Moreover, provided some technical assumptions are satisfied, 2 the implicit function theorem guarantees that q = q (p, t) or, for that matter, p = p (q, t) or t = t(p, q) can always be

1Given a price p and a quantity q, sales time t is not uniquely determined if, as flow demand varies through time, the range of prices over which the sales rate is positive varies. This is a purely technical point that the following example illustrates. Suppose expected flow demand for date T is given by

q(T)=(-2 + s- ) p+18 +sin

The flow fluctuates between a peak flow of q = -1.75p + 19 and a trough flow of q = -2.25p + 17 with periodicity 2n. The range of prices over which demand is positive

varies between (0, 10.86) for peak flow and (0, 7.55) for trough flow. The demand schedule that would prevail over a period of t time intervals is

q(t)=f q(T)d= - -2t - c- s+ p+18t-cost+1

It tums out that the demand schedules for t = 5 (q = -9.8209p + 90.7163) and t = 7 (q = -13.9385p + 126.2461) intersect atp = 8.6288, q = 5.9741. Thus this one price- quantity pair is associated to two values of t.

2 If R(p, q, t) has continuous partial derivatives and a- ( Q, , 7) 0, then there exists a

neighborhood U, , such thatp = p(q, t) in U. Similarly for the other variables. neighborhood U(p, q, i) such thatp = p(q, t) in U. Similarly for the other variables.




locally defined. It is to this concept of demand that the firm we now turn to will refer.

m. A model of target inventory and markup

The model outlined in the paragraphs that follow is descriptive of an equilibrium situation, but does not attempt to describe the process by which that situation is reached. In particular, the question of existence of such an equilibrium as the outcome of dynamic interaction between firms in an imperfectly competitive setting is not addressed at all. Thus, the empirical testing of the hypothesis developed here using data from an oligopolistic industry relies on the assumption that the equilibrium described does indeed exist. The model is formally presented for a monopolist, since this simplifies exposition. Extension of the equilibrium concept to an imperfectly competitive firm is briefly discussed in a final paragraph.

Instead of maximizing profit from the sale of elementary time interval production, given demand at that same date, this firm views as ideal the following situation. Given a date, the level of inventory held at that date and its selling price (which is assumed unique), are such that profit made over the time it will take to sell the inventory is maximum. Thus, the choice (p*, q*, t) is optimal for date t if:

* At p* it takes time t to sell q*. * (p*, q*) maximizes profit given the demand relationship of

date T; q, = q, (p, t*). Thus, given t, marginal revenue equals marginal cost.

* (p*, q*, t) maximizes, by choice of any two of the three variables, p, q, t, average per-unit-time profit x; where

P q- c(q) 3 t(p, q)

The cost of a level of output inventory q to be held at future date T is

3 The maximization of in guarantees that given optimal t, marginal revenue equals marginal cost. However, what is critical to our argument is the fact that unconstrained maximization of per-unit-time profit yields pricing rules that can be confronted with the data and shown to be comparable with it. It is this compatibility which allows us to conclude that firms can be described as choosing a price to equate marginal revenue to marginal cost, not the existence of this intersection in and of itself. Indeed, if an upper limit qx to inventory of data x is introduced, the firm then solves




represented by c,(q). Thus, time available to adjust inventory to its desired level (the interval spanning the current date to date t), as well as production costs within each elementary interval of that time period, will enter the specification of c,(q). It is in this sense that costs have the three dimensions prescribed by Alchian (1959). But the time interval relevant to the definition of production costs is here independent from the temporal dimension the firm will choose for demand: optimal t,(p, q). Thus production and demand are not constrained to temporal commensurability.

Turning to the resolution of this firm's optimization problem, subscript T is omitted to simplify notation. The firm's objective is to determine the optimal values of three variables: price p, quantity q, and time horizon t. Once values are assigned to two of the three variables, the value of the third follows from the demand relationship R(p, q, t). As a result, average profit per unit of time X can be expressed as a function of any two of the above-mentioned variables. Two formulations are of particular interest:

(i) I ,Pq (p' t) - c (q(p, t))

(ii) X (p, q) = p q- c (q) t (p, q)

In the first case, the firm chooses p and t to solve

Max Xc (p, t) p, t

while the optimization problem associated to the second formulation reads

Max X (p, q) P, q

Expressing first order conditions in each case will yield, among other things, a formula for the optimal price.

p, q - Ct (q) Max t- p, q rt, (p,C) P, q

Sto q< q-c

and, if the constraint is binding, per-unit-time profit is not necessarily maximized where marginal revenue equals marginal cost.



MARKUP PRICING VERSUS MARGINALSM 139

In the first case, the optimal price is obtained by solving the first order condition a n / a p = 0. As t is held constant to solve for p using this condition, the optimization yields the optimal price on the demand schedule defined for time interval t, and the usual formula for the profit maximizing price obtains:

(1) MC 1+ 1/n (p, t)

where MC is marginal cost and t1 the price elasticity of demand. In the second case, a formula for optimal price is obtained by solving

the first order condition a / a p = O. In this case, it is inventory level q that is held constant and the optimal price obtained can be expressed:

(2) AC P l-l/(p, q)

where AC is the average unit cost of goods in inventory, and e(p, q) = (at / ap) x (p/t) is the price elasticity of the sales time of a given level of inventory q.

The standard formula for markup over marginal cost is derived from Equation (1). Equation (2) yields a formula formarkup over average cost of goods sold:

p-AC 1

P e(p, q)

Econometric estimates of 1 / e (p, q) for General Motors, Ford, and Chrysler have been generated (see Langlois, 1989). They are not statistically different from markups over average cost of goods sold as recorded in Moody's Industrial Manuals.

In section IV, further empirical evidence of the compatibility of the above model with the data is presented. The literature provides econometric estimates of r for the automobile industry. Data on markup over average cost of goods sold is recorded in Moody's Industrial Manuals. The theoretical link between elasticities rl and e provides the tool for empirical testing of the model.

Linking T1 and e involves definition of a third concept of elasticity: r = (3 q / t) x (t/q). {, defined holding price constant, measures the percentage change in quantity demanded and sold for a 1 percent change in the time span allowed for demand to be realized. As shown in Appendix




1, elasticities r1, e, and C evaluated at any point (p, q, t) satisfy the relationship

1 1 nr=-Ce or =- -

At the optimal point, therefore, C equals the ratio of markup over average cost to markup over marginal cost. Given estimates of 1/lI and data on markup over average cost, generated econometric estimates of C can provide evidence to reconcile the data with the marginalist principle of equating marginal revenue to marginal cost.

Before turning to the empirical findings of section IV, the limitations of the model presented here must be acknowledged. The above model describes a monopolist. A formal extension of the above equilibrium concept to an imperfectly competitive firm can be achieved by making the demand relationship R depend on all other firms' pricing strategy. Thus, in a Nash equilibrium situation, each firm will choose a markup over average cost of goods sold equal to 1/e, but e will be a function of the prices chosen by all firms in the industry. Therefore, if a Nash equilibrium concept adequately describes real world situations, the formulae derived from the above model are appropriate tools for the empirical testing of the model. Our model does not address the funda- mental issue of dynamics and stability of competitive situations, but this is clearly beyond the scope of this paper.

IV. Empirical findings: a reconciliation of direct cost oriented pricing with the empirical evidence on price elasticity of demand

The model presented in this paper predicts that the ratio of optimal markup over average cost of goods sold to optimal markup over marginal cost is equal to the percentage change in quantity sold for a 1 percent change in sales time given a price:

(3) 1 C Iril

Estimates of 1 /IrlI are provided by a number of authors while markup over average cost of goods sold is recorded in Moody's Industrial Manuals. Our econometric test of the hypothesis that firms behave according to the above model involves estimating C for the automobile industry for post-war samples that cover those used by the authors




Table 2

Per-unit-time sales patterns: three numerical examples

A: declining B: constant C: increasing per-unit-time sales per-unit-time sales per-unit-time sales

t q q t q q t q q

1 10 10 1 10 10 1 10 10 2 9 19 2 10 20 2 11 21 3 8 27 3 10 30 3 12 33

referred to in Table 1. These estimates are then used in conjunction with data on markup and price elasticity of demand to check whether (3) is satisfied. The data used, as well as the copies of the regressions run, can be obtained from the author upon request.

The first subsection below discusses the method used to estimate 5, while the second presents the results, and the third confronts them with the available evidence on markup over direct costs and rl.

Methodology

C is an elasticity concept associated to the integral of flow demand. In

other terms, q = t), where q (p, t) = q(p, ) d. Thus, C will at q o be descriptive of the progression of cumulative sales over time, not of the change over time of per-unit-time sales. To fix the ideas, consider the numerical examples in Table 2.

To each date t is associated sales at that date, q, and cumulative sales since date 1, q. The elasticities are computed between dates 2 and 3. If per-unit-time sales are constant (example B), the elasticity of the change, over time, in per-unit-time sales (A q /A t) x t/ q = 0, since this elasticity is, in fact, a measure of the per period growth rate of sales. By contrast

Aq t 10 5/2 At q 1 50/2

If per-unit sales decline, -.2941 growth from date 2 to 3 in example A, r drops below 1 to 0.8696, while for sales growing at + .2174 as in example C, r rises above 1 to 1.1111. Thus barring dramatic fluctuations




Table 3

Estimates of C: Methodology

Sample: 1953.3-1973.5

Multiplicative Model: Dependent Variable LNSALE

Independent Variables 1 C LNTRND LNPRIX LNGPY

Coefficient 12.730330 0.0183887 -2.0451384 0.6263535 (T) (6.9467845) (0.3694518) (-4.6618805) (6.3647833)

Adjusted R2= 0.598306 D.W. = 1.600705

Multiplicative Model: Dependent Variable LNQNT

Independent Variables 2 C LNTRND LNAPRC LNCGPY

Coefficient 15.422906 .= 0.5185682 -2.3902806 0.5353927 (T) (7.5093672) (2.9432297) (-5.2871581) (4.4286640)

A

Adjusted R2 = 0.999878 D.W. = 1.850189 = 0.5185682

Sample 1953.3-1972.4


Independent Variables 3 C LNTRND LNAPRC LNCGPY

A

Coefficient 15.272142 = 0.4899031 -2.3714331 0.5562831 (T) (6.82194796) (2.3536793) (-49388468) (3.8418860)

A

Adjusted R2 = 0.999868 D.W. = 1853300 5 = 0.4899031

in sales, C will oscillate around 1 while the elasticity of the change, over time, in per-unit-time sales will oscillate around zero. Typically, in large markets (and in the automobile industry in particular), percentage fluctuations in sales are modest. Moreover, if short enough periods are considered so that positive as well as negative changes in sales occur from period to period, for example seasonally, the elasticity of the change over time in per-unit-time sales will indeed fluctuate around zero, but with a trend that may not depart significantly from zero. Consequently, if per-unit-time sales are regressed against a time trend to generate this




elasticity, the regression coefficient it corresponds to may turn out to be modestly positive, but the confidence interval surrounding it is likely to contain zero. In regression 1 of Table 3, the logarithm of quarterly automobile sales (LNSALE) is regressed against the logarithm of the time trend (LNTRND), the logarithm of the implicit new auto price deflator (LNPRIX), and the logarithm of personal income (LNGPY). The coefficient on LNTRND is indeed close to zero and it is not statistically significant.

There are two methodological approaches to the estimation of r: 1. Estimate 5 directly using cumulative series to perform the regression. 2. Estimate C indirectly using estimates of the sensitivity of per-unit-

time sales to the time trend. The second method assumes that the functional form chosen to describe

sales at a price permits derivation of C, and if so, that the relevant coefficient is statistically significant. But C is linked to the sensitivity of per-unit-time sales to the time trend, which, as we have argued above, is not likely to be statistically significant. For these reasons, it is the first method that we have chosen to implement.

The first method is not without drawbacks, however. Working with cumulative series introduces serial correlation that must be corrected for. Moreover, a single price must be associated with each level of cumulative sales. If Q, is cumulative sales made over t quarters starting from the beginning of the sample period, p, ideally captures the price at which cumulative sales Q, would have been made had prices remained constant over the period. A simple average of the prices charged over the trelevant quarters is used to approximate p. However, as illustrated by regression 2 of Table 3, this approach provides a direct, statistically significant estimate of C. In regression 2, the logarithm of cumulative sales (LNQNT) is regressed against the logarithm of the time trend LNTRND, average price LNAPRC, and cumulative personal income LNCGPY. By contrast with regression 1 of the same table, all coefficients are significant and with reference to the adjusted R2, 99.98 percent of the variations in LNQNT are captured by the regression. By comparison, only 59.83 percent of the variations in LNSALE are captured by the regression.

A final methodological issue is that of the sample used to estimate 5. The estimates of C are to be confronted with estimates of short-run rl computed by various authors. But the samples used do not necessarily span full business cycles. Because C captures a trend in the growth of sales over time, it is sensitive to the particular end point chosen for the




sample. A sample running from a peak to a trough will provide a lower estimate of r than would a sample running from peak to peak. To avoid introducing a bias associated with the end point chosen for the sample, C has been estimated over the samples that most closely correspond to those of the authors estimating TI, but span full business cycles. For example, Westin (1975) estimates short-run elasticities for the period 1953 to 1972. C is estimated for the period 1953.3 to 1973.4, the sample that most closely corresponds to Westin's but spans full business cycles (1953.3 and 1973.4 are peak quarters). Regression 3 of Table 3 illustrates the downward bias associated with the choice of sample.

Statistical estimates of C For the reasons discussed above, C is estimated directly using cumulative series. Moreover, a range of values for C is generated to accommodate the multiple estimates of Tr provided by the authors of Table 1. To generate a range of values for C for each sample period, cumulative sales are explained using two different econometric models.

The first explains cumulative sales Q, as a multiplicative function of time elapsed t, average price p,, and cumulative income y,:

Qt=B Pt ,ptly Ps 'u, A A

where u, is white noise. P, is a direct estimate of C. P2 reflects the sensitivity of cumulative sales to a change in price. P2 does not measure a price elasticity of demand, however, since function Q,(t,p,, yt) does not capture a demand function but rather uses points on demand functions defined for various tiIe intervals to describe cumulative sales overtime. For the same reason I3 does not capture the income elasticity of demand but the elasticity of cumulative sales to cumulative income.

The second econometric model explains cumulative sales as a linear function of price, time elapsed, and total income accumulated during the elapsed time period:

Q= A + aco t+ 2p, + cy, + v,

where vt is white noise. Using the second model, C is the ratio of cato the ratio of average Qt to average t. ^

Turning to the econometric estimates of C, Table 4 presents estimates to be confronted to Westin's (1975), Evans' (1969), and Weiserbs' (in Phlips, 1983) estimates of r. Quarterly cumulative sales, in number of




units, QNT, is the dependent variable. A time trend (TRND), the average new auto implicit price deflator series (APRC), and the cumulative current dollar national income (GPY) are the independent variables.4 The multiplicative model is used for regressions, 1, 3, and 5. All coefficients are significant and have the expected sign. An average price increase decreases cumulative sales while an increase in income or in the time allowed for sales to be made increases sales. The additive model is used for regressions, 2, 4, and 6. Again, all coefficients are significant and have the expected sign. All regressions have been corrected for serial correlation using the Cochrane-Orcutt procedure. Adjusted R2 is of the order of 99.9 percent for all regressions.

Test of the behavioral hypothesis If firms choose inventory levels and price so as to maximize per-unit-time profit, then at the optimal point the following relationship holds:

1 1

or 1

m=m+

where m is markup over average direct costs. Thus, given a range of values for , computation of the ratio -C /m will yield an interval within which Tr is expected to lie.

For example, regressions 1 and 2 provide a range for the estimated value of C: .369176 to .8789. An estimate of markup over average cost at the retail level has been constructed using manufacturer markups as a base, and information on dealer costs provided by Bresnahan and Reiss (1985) and Davisson and Taggart (1974). Markup over average costs at the retail level for the period 1954 to 1973 has been estimated at 29.6 percent. Appendix 2 provides a detailed account of the computation. The ratio of -C to markup therefore ranges from -2.97 to -1.25. This is compatible with Westin's estimates of price elasticity of demand as summarized in Table 5.

Regressions 3 to 6 of Table 4 present estimates of C for the sample

4The National Income Accounts are the source for the new auto implicit price deflator and current dollar national income. Wards Automotive Yearbooks are the source for unit sales.




Table 4

Estimates of 5 Sample 1953.3-1973.4


Independent Variables 1 C LNTRND LNAPRC LNCGY

A

Coefficient 14.029798 = 0.3691767 -2.1531173 0.6385347 (T) (6.1594962) (2.0023913) (-4.3748699) (5.1321090)

A

Adjusted R2 = 0.999888 D.W. = 1.897562 5 = 0.3691767

Additive Model: Dependent Variable QNT

Average QNT = 70398.839 Average TRND = 43.5

Independent Variables 2 C TRND APRC LNCGY

Coefficient 44532.420 1422.4177 -667.78975 3.8673892 (T) (1.7941245) (4.0623119) (-4.4233425) (3.0547783)

A

Adjusted R2 = 0.999964 D.W. = 2.066540 , = 0.8789

Sample: 1948.4-1969.4


Independent Variables 3 C LNTRND LNAPRC LNCGY

A

Coefficient 21.570651 = 1.1056435 -3.9007135 0.3005679 (T) (7.133456) (8.2087430) (-5.1475341) (2.9499693)

A

Adjusted R2 = 0.999874 D.W. = 2.115463 r = 1.1056435


Average QNT = 65172.02 Average TRND = 46

Independent Variables 4 C TRND APRC CGY

Coefficient 90346.118 1053.7905 -1203.1536 6.466371 (T) (5.3364858) (6.4495549) (-5.0331825) (6.2957888)

Adjusted R = 0.999964 D.. = 2.081139 0.7438 Adjusted R2 = 0.999964 D.W. = 2.081139 = = 0.7438




Table 4 (continued)

Sample: 1948.4-1973.4


Independent Variables

5 C LNTRND LNAPRC LNCGY

Coefficient 9.4495282 = 0.7657066 -0.9499821 0.3499533 (T) (5.1430586 (4.7955937) (-2.1064163) (3.0954979)

A

Adjusted R2 = 0.999811 D.W. = 2.057361 { = 0.7657064


Average QNT = 79724.366 Average TRND = 54

Independent Variables

6 C TRND APRC CGY

Coefficient 24667.608 1055.7221 -351.69931 5.3452385 (T) (1.9598794) 10.434661) (-2.2546144) (8.5465201)

A

Adjusted R2 = 0.999970 D.W. = 2.178076 { = 0.7151

period 1948.4 to 1969.4, which spans Evans' (1969) sample, and sample period 1948.4 to 1973.4, which spans the post-war sample used by Weiserbs (in Phlips, 1983). A comparison of regressions 1, 2, 3, and 4 reveals that the use of an additive rather than a multiplicative model to explain cumulative sales does not bias the estimate of C in any systematic way. Indeed, for the sample 1948 to 1969, the linear model yields lower estimates of C than the multiplicative model, while the reverse occurs for the sample 1953 to 1974. As shown in Table 5, the range of values generated for 11 includes Evans' (1969) estimate for the sample 1948 to 1969. The 1948 to 1973 sample provides a narrow range for l, which is close to the values found by Weiserbs (in Phlips, 1983). Weiserbs, however, uses pre-war as well as post-war data to generate his demand elasticities. Our sample, therefore, does not overlap his.

The fit, as summarized in Table 5, of estimated price elasticities of demand to the range of values for iT generated independently using estimates of C and values for m, provides evidence that the model presented in this paper is not incompatible with the data.




Table 5

Summary of results

Regression Results Relevant Comparison Study

Sample 1953.3-1973.4 Range for {: [.369, .879] Author: Westin (1975) Markup m: .296 Sample: 1953-1972 Range for iT: [-2.97, -1.25] Price elasticity estimates: -1.65, -1.32

Sample 1948.4-1969.4 Range for r: [.744, 1.106] Author: Evans (1969) Markup m; .302 Sample: 1948-1964 Range for il: [-3.66, -2.46] Price elasticity estimate: -3.1

Sample 1948.4-1973.4 Range for {: [.715, .766] Author: Weiserbs (in Phlips, 1983) Markup m: .296 Sample: 1920-1970

Range for rT: [-2.59, -2.42] Price elasticity estimates: -2.32, -2.13

Conclusion

This paper reviewed the marginalist and cost controversies in light of the post-war statistical estimates of short-run price elasticity of demand for automobiles. These range from -1.32 to -3.1. Data on automobile industry costs suggest that marginal costs lie below average costs. Since markup over average cost at the retail level is on the order of 30 percent, this explains the low values found for ir. But the issue is the validity of the interpretation of 1 / Irll as markup over marginal cost. This would imply that firms indeed choose price so as to equate marginal revenue to marginal cost, but in the cost configuration typical of the auto industry, these schedules may not even intersect. Thus, in the absence of precise data on marginal costs at the retail level, confrontation of the marginal- cost-equals-marginal-revenue rule to the data on 11 would have to remain inconclusive. This paper suggests a modification of the standard firm profit maximization model, which provides additional tools to analyze the data. If firms choose inventory and price so as to maximize per unit time profit, formulae for optimal markups over average as well as marginal costs are generated, and their ratio is predicted to match (, the percentage change in cumulative sales that results from a 1 percent




change in the time allowed for those sales to be realized. In an empirical final section, estimates of ( are generated and are used to show that the values found for rl are compatible with per-unit-time profit maximizing behavior and their inverse are, consequently, proxies for markup over marginal cost.

Appendix 1: Relationship between elasticities rl, e, and 5

Proposition: Elasticities il, e, and C evaluated at any point (p, q, t (p, q)) satisfy the relationship rl = - ( x e.

Proof: Let

d=d.P= [. E= q+ . dt 1. P dp q a3p at dp q

and

A dt . t= at d+at 1 dp t ap aq dp _ t

By definition of l, , and e

(1) =rA +A A

and A A

Replacing e by its value in (1) we obtain the relationship between rl, e, and C:

n =- -

Appendix 2: Computation of automobile industry markup over direct cost at the retail level

The following information and data are used to compute markup:

1. Dealer direct costs represent on average 35 percent of gross margin over manufacturer price (see Bresnahan and Reiss, 1985, and Davisson and Taggart, 1974).




2. The ratio of dealer to manufacturer margin is constant and estimated at .71 (see Bresnahan and Reiss, 1985).

3. Weighted average Big Three markup over direct costs as recorded in Moody's Industrial Manuals is used as a proxy for markup over direct cost. Weights used are firm shares of total production.

Computationfor sample period 1954-1973

Let 100 = manufacturer direct cost Manufacturer markup over direct cost for 1954-1973 is 21.4 percent. Manufacturer price = 127.2

Dealermargin = (.71) x (27.2) = 19.3 = 65 percent of gross dealer margin Dealer costs = 10.4 = 35 percent of gross dealer margin Gross dealer margin = 10.4 + 19.3 = 29.7

Retail price = 127.2 + 29.7 = 156.9 Average costs = 100 + 10.4 = 110.4 Margin over average costs = 46.5

Markup over direct costs = 29.6 percent

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