On the third law of demand.

Razzolini, Laura ; Shughart, William F., II ; Tollison, Robert D. 等

I. INTRODUCTION

The Alchian and Allen (1967, 63-64) theorem, sometimes elevated to the status of a third law of demand, as in Bertonazzi et al. (1993), is a clever application of the fundamental principle of economizing behavior. As usually stated, the theorem suggests that by reducing the price of higher quality relative to lower quality versions of the same good, a fixed transportation charge will trigger a predictable shift in the mix of quality grades purchased by consumers in distant markets, as compared to the mix purchased locally. Producers will respond rationally to this difference in behavior. Relatively more high-quality goods will be included in outbound shipments, leaving more low-quality goods to be sold nearer the point of origin. Hence, other things being the same, the theorem predicts that it will be harder to find "good" apples in the State of Washington, a prime apple-growing region, than in, say, New York City, where "bad" apples are comparatively more expensive. The first law of demand still holds, of co urse, in that fewer apples of both types will be consumed in New York than in Seattle, ceteris paribus.

Similarly, given that the cost of a babysitter will be the same no matter where the parents of young children decide to spend the evening Out, they are more likely than an otherwise identical childless couple to choose an upscale restaurant over an inexpensive eatery and to go to the theater rather than to the movies. Moreover, if both couples plan to see a play or a concert, the couple with children will opt for better seats.

The foregoing examples indicate that the Alchian and Allen theorem is rich in empirical implications. In what follows, however, we show that placed in the context of a market model, its range of applications is narrower than has been acknowledged in the literature heretofore. More specifically, our analysis demonstrates that a fixed charge unambiguously reduces the price of a higher-quality good, relative to a lower-quality version of the same good, only when that good is sold by a perfectly competitive, constant-cost industry. Under increasing-cost conditions or imperfectly competitive industry structures, by contrast, it is possible for relative prices either to be unaffected by the addition of a fixed charge or, indeed, for the lower-quality good to become relatively cheaper in distant markets, depending on the elasticity characteristics of relevant market demand and supply functions.

Our analysis is in no way intended as an attack on Alchian and Allen or their fine textbook, which has taught more than one generation of economists to be better price theorists. We do not quarrel in the least with the pedagogical value of their theorem, which has been immense. Our purpose is to identify some plausible scenarios under which the third law of demand may fail to hold.

The present article is thus in the spirit of Yang and Stitt (1995), who show that the Ramsey (1927) rule for optimal commodity taxation is changed in important ways when the assumption of constant production costs is relaxed. It is organized as follows. In the next section, the fundamentals of the Alchian and Allen theorem are described in more detail and its predictions are situated within the existing literature. Section III explores the theorem's operations under alternative cost conditions and industry structures. Factors complicating the analysis of the effects of a fixed charge on the relative prices of high-and low-quality goods are also addressed in this section. Finally, section IV contains some concluding remarks.

II. SHIPPING THE GOOD APPLES OUT

To borrow an example from a classic statement of the Alchian and Allen theorem, (1) suppose that "good" Washington apples sell for 10 cents each in Seattle and "bad" apples each cost a nickel there. Because the price of a good apple is twice that of a bad apple, Seattle's consumers must sacrifice the opportunity of buying two bad apples to obtain a good one. Now suppose further that, regardless of quality, it costs 5 cents to ship an apple to the East Coast and, moreover, that the transportation charges are fully passed on to consumers. Under these assumptions, good apples sell for 15 cents in the East and bad apples sell for 10 cents. The price of a good apple in the East is only one and a half times--not two times--the price of a bad apple. Good apples are cheaper on the East Coast, and consumers there will consequently want to buy more of them. Hence, although fewer apples will be consumed overall on the East Coast than on the West Coast (to reiterate , the first law of demand still applies), shipments to markets in the East will include more generous mixes of good apples.

More formally, if [p.sub.H] and [p.sub.L] are the unit prices of higher- and lower-quality versions of the same good (with [p.sub.H] > [p.sub.L])' the Alchian and Allen theorem exploits the observation that ([p.sub.H] + t)/([p.sub.L] + t) < [p.sub.H]/[p.sub.L], where t is a constant unit cost of transportation or some other fixed charge that applies equally to both quality grades. (2) The third law of demand also works in both directions: Silberberg (1990, 386) and Bertonazzi et al. (1993) show that the price of the higher-quality good falls relative to the lower-quality good whether the good is shipped to the consumer or the consumer travels to the point of purchase.

As initially expounded, the Alchian and Allen theorem did not consider interactions with other goods entering the representative consumer's utility function. (3) Applying the composite commodity theorem, however, Borcherding and Silberberg (1978) prove that in a three-good world where consumers optimize their expenditures over two quality grades of one good plus some other, third good, the addition of transportation costs t will indeed trigger an increase in the amount consumed of the higher-quality good relative to that of the lower-quality good. This result holds provided that the two quality grades are substitutes for one another (the compensated cross-price elasticity of demand between them is positive), (4) and that the higher- and lower-quality versions of the one good have the same interactions with the third good. "Same" means that ceteris paribus increases in the prices of the higher quality and lower quality good have identical effects on the quantities consumed of the other good (i.e., their compen sated cross-price elasticities with respect to the composite commodity are equal in magnitude). Stated in reverse, the Alchian and Allen theorem fails to hold only in an asymmetrical case where the third good is a better substitute for the higher-quality good than it is for the lower-quality good, a possibility that seems "empirically insignificant" to Silberberg (1990, 388). (5)

A second complication arises when t represents a charge for a complementary good or service that enters the consumer's utility function directly and that, in principle, can be purchased separately. Consider, for example, the impact of the cost of traveling to New York City on the prices of first-class versus second-class hotel accommodations or of full-sized versus economy-sized rental cars. The analysis of such cases would seem to depend on the extent to which purchases of the higher- and lower-quality versions of the good in question respond to changes in the price of a third good (airline tickets, say). Indeed, Cowen and Tabarrok (1995) argue that the Alchian and Allen theorem may not hold when a product with two quality grades is bundled with some other good. However, Umbeck (1980) and, more recently, Bertonazzi et al. (1993) reach the opposite conclusion, namely, that the theorem helps explain some common forms of commodity bundling. Packaging airline reservations with rental cars, hotel rooms, or cruise ship cabin assignments not only lowers the relative prices of higher quality grades of these goods at the traveler's destination but also ensures that the third law of demand operates by making it less likely that the airfare will be treated as a sunk cost on arrival, thereby nullifying its impact on relative prices. (6)

In sum, comparative-static analyses of responses to the introduction of a fixed charge that raises the prices of two quality grades of the same good by the same absolute amount thus suggest that other things being the same, the utility-maximizing consumer will tend to buy relatively more of the higher-quality grade. This result seems to be robust under the stated conditions, the only proviso being that, when the prices of both quality grades rise, consumers do not substitute some other product in greater proportion for the higher-quality good than they substitute for the lower-quality good.

Although the usual analysis of the Alchian and Allen theorem examines the behavior of the utility-maximizing consumer, it is silent on the behavior of the seller. (7) The literature has failed to ask and answer the following question: Faced with a fixed cost of shipping two quality grades of the same good to a distant location, under what circumstances will the profit-maximizing firm actually raise both prices by the same absolute Amount? Does the Alchian and Allen theorem operate generally when the firm and industry are taken into account, or is its application restricted to particular market conditions? More specifically, under what circumstances can the opposite of Alchian and Allen's conjecture occur? Is it possible for more of the bad apples to be shipped out because [p'.sub.H]/[p'.sub.L] > [p.sub.H]/[p.sub.L], where [p'.sub.H] and [p'.sub.L] are the new prices inclusive of transportation costs? These questions are explored in the following section.

III. THE ALCHIAN AND ALLEN THEOREM IN A MARKET CONTEXT

The analysis of the impact of a fixed charge t on the relative prices of higher- and lower-quality versions of the same good has much in common with textbook treatments of the incidence of an excise (per-unit) tax. In both cases, the critical question is, if marginal production costs rise by t, how much of the increase will be passed on to consumers in the form of higher prices and how much of it will be borne by sellers in the form of lower profits (or inframarginal rents)?

Although the Alchian and Allen theorem does not require the prices of higher- and lower-quality goods to both rise by the full amount of the fixed charge (although this is the usual simplifying assumption), there would seem to be two market situations in which this result would in fact obtain. One is when shipping charges are invoiced separately, that is, pricing is f.o.b. The other is when the good in question is supplied by a perfectly competitive, constant-cost industry, so that the prices to consumers in distant markets rise by the full amount of the cost of shipping the product to them. More generally, the higher-quality good becomes unambiguously cheaper relative to the lower-quality good only if the two prices rise by the same absolute amount (which may be more or less than t), or if the price of the lower-quality grade rises proportionately more than that of the higher-quality grade.

Perfect Competition

Consider a perfectly competitive, increasing cost industry in a simple partial equilibrium setting. The analysis is depicted in Figure 1. In absence of transportation costs, equilibrium in the two markets for lower- and higher-quality grades of a product is attained where supply equals demand and the equation D([p.sub.i]) = S([p.sub.i]), i = H, L is satisfied. Any competitive producer faces a perfectly elastic demand d at price [P.sup.*.sub.i] and initially supplies a corresponding quantity [q.sup.*.sub.i].

Now consider the introduction of transportation costs at rate t per unit. Such charges are added to the marginal cost of each supplier; therefore, each will offer a smaller quantity [q'.sub.i]. Graphically, in the overall markets, the introduction of the transportation charge can be depicted as an upward shift in the supply curve facing consumers by the amount of t. The new equilibrium will satisfy the condition D([P'.sub.i] - t), = S([P'.sub.i] - t), i = H, L.

We need to characterize the changes in equilibrium that result from the introduction of the transportation costs. Differentiating the above equation we get d[p.sub.i] = S'/(S' - D'), where S' = dS/d[p.sub.i], and D' = dD/d[p.sub.i]. More conveniently, we can write this condition in terms of the elasticities of supply and of demand for the two types of goods, viz, d[p.sub.H]/dt = [[epsilon].sup.H.sub.S]/([[epsilon].sup.H.sub.S] - [[epsilon].sup.H.sub.D]) and d[p.sub.L]/dt = [[epsilon].sup.L.sub.S]/([[epsilon].sup.L.sub.S] - [[epsilon].sup.L.sub.D]), where [[epsilon].sup.i.sub.S] > 0 and [[epsilon].sup.i.sub.D] < 0 are, respectively, the elasticity of supply and of demand for a good of quality i = H, L. In general, the greater is the elasticity of supply, the larger will be the price increase induced by the introduction of transportation costs; the greater is the elasticity of demand, the smaller the price increase will be.

The Alchian and Allen theorem is not about any arbitrary pair of goods, however. It is about two quality grades of the same good whose supplies are jointly determined. Thus the theorem applies, for example, to a denim factory where, owing to random variations in the quality of cotton and dyes or mistakes in cutting and sewing, some finished items turn out to be substandard. If, consistent with the theorem, we assume that the supply elasticities of the higher- and lower-quality versions of the good are equal ("good" and "bad" apples come from the same tree, say), then the higher-quality good will become relatively cheaper either if the demand elasticities are the same or if the demand for the lower-quality good is relatively less elastic than the demand for the higher quality good. However, in the empirically plausible case where the demand for the lower-quality good is relatively more elastic than the demand for the higher-quality good, then the effects on relative prices are reversed (i.e., the high-quality good becomes dearer). (8)

Monopoly

The analysis of monopoly proceeds analogously. Consider a single producer supplying two quality grades of a good. The monopolist's cost function is C([Q.sub.H] + [Q.sub.L]) = C(Q), with marginal cost C'. The monopolist faces demand curves [p.sub.H]([Q.sub.H]) and [p.sub.L]([Q.sub.L]). (9) Profit maximization requires choosing quantities to produce so that the marginal revenue in the two markets is the same, namely, [p.sub.L][1 + (1/[[epsilon].sup.L.sub.D])] = [p.sub.H] [1 + (1/[[epsilon].sup.H.sub.D])].

From this profit-maximizing rule, we see that the price tends to be higher for the good with a lower price elasticity of demand. That is, the monopolist will charge a higher price in the market with the less elastic demand curve. Hence, [p.sub.H] > [P.sub.L] implies [[epsilon].sup.H.sub.D] > [[epsilon].sup.L.sub.D].

Consider now the introduction of transportation costs t. The monopolist's cost function becomes C([Q.sub.H] + [Q.sub.L]) + t([Q.sub.H] + [Q.sub.L]) = C(Q) + t(Q). First-order conditions for profit maximization now require that [p.sub.i] + (d[p.sub.i]/d[Q.sub.i])[Q.sub.i] - (C' + t) = 0, i = H, L. Solving these conditions, we get the optimal quantities and prices to charge as a function of the transportation costs: [Q.sup.*.sub.L] = [g.sub.L](t) and [Q.sup.*.sub.H] = [g.sub.H](t); [p.sup.*.sub.L] = [p.sub.L]([Q.sup.*.sub.L]) and [p.sup.*.sub.H] = [p.sub.H]([Q.sup.*.sub.H]). As before, we can investigate the effect of the introduction of the transportation costs by calculating d[Q.sup.*.sub.L]/dt = [g'.sub.L](t) and d[Q.sup.*.sub.H]/dt = [g'.sub.H](t); d[p.sup.*.sub.L]/dt = [p'.sub.L](.)[g'.sub.L](t) and d[p.sup.*.sub.H]/dt = [p'.sub.H](.)[g'.sub.H](t).

The implicit function theorem yields [g'.sub.i](t) = 1/[2(d[p.sub.i]/d[Q.sub.i]) + [Q.sub.i]([d.sup.2][p.sub.i]/[d.sup.2][Q.sub.i]) - C'], i = H, L. The term [g'.sub.i](t) is negative by the second-order conditions for profit maximization. The profit-maximizing monopolist, like the competitive supplier, therefore reduces output when marginal costs increase. Furthermore, d[p.sup.*.sub.L]/dt > 0 and d[p.sup.*.sub.H]/dt > 0. That is, the prices of both the higher-and lower-quality goods will increase after the introduction of the transportation costs. More specifically, we have d[p.sup.*.sub.i]/dt = (d[p.sup.*.sub.i]/d[Q.sup.*.sub.i])/[2(d[p.sub.i]/d[Q.sub.i]) + [Q.sub.i] ([d.sup.2][p.sub.i]/[d.sup.2][Q.sub.i]) - C"], i = H, L.

With a linear cost function (i.e., C" = 0) and linear demand schedules (i.e., [d.sup.2][p.sub.i]/[d.sup.2][Q.sub.i] = 0), the prices of both quality grades rise by exactly one-half t (i.e., d[p.sup.*.sub.L]/dt = d[p.sup.*.sub.H]/dt = 1/2), and the higher quality good becomes relatively cheaper, as the Alchian and Allen theorem predicts. On the other hand, with a linear cost function and constant elasticity demand schedules of the form [p.sub.i] = A[Q.sup.1]/[[epsilon].sub.i].sub.i], where [[epsilon].sub.i] is the constant elasticity of demand for a good of quality i, we obtain d[p.sup.*.sub.i]/dt = [[epsilon].sub.i]/([[epsilon].sub.i] + 1), i = H, L. In this case, the price increases by [[epsilon].sub.i]/([[epsilon].sub.i] + 1) [greater than or equal to] 1 times the transportation costs. Therefore, the more elastic the demand schedule is, the more the price will increase. Given our previous observation that [[epsilon].sub.H] < [[epsilon].sub.L], we conclude that the price will increase relatively more for the lower-quality good, the one with the more elastic demand.

Stated in another form, the higher-quality grade becomes relatively cheaper if the demand elasticities for the two quality grades are the same or if the demand for the lower-quality good is relatively less elastic than the demand for the higher-quality good. Neither of these possibilities seems plausible empirically. In fact, the opposite is more likely to be true, namely, that the demand for the higher-quality good will be less elastic than the demand for the lower-quality good. (10) There are numerous reasons for this, including investments in brand-name capital, fewer substitutes for the higher-quality good, and so on. If that is the case, then more "bad" apples will be shipped out because they are relatively cheaper in distant markets.

IV. CONCLUDING REMARKS

Perhaps the best way of thinking about our analysis is that it recasts the Alchian and Allen theorem as a testable proposition. In the same spirit as Cheung's (1973) examination of apple growing and beekeeping and Coase's (1974) inquiry into the history of lighthouses, there is a need to replace casual observation and conjecture with careful empirical studies of whether relatively more good apples are in fact shipped out. (11) Although such studies await the attentions of enterprising empiricists, we note that the proliferation of factory outlet malls along the U.S. interstate highway system suggests that the predictions of the Alchian and Allen theorem are ambiguous. Most of the stores in these malls sell factory "seconds," or dated goods; very few of them (if any) sell "firsts," or offer consumers the range of choices available at traditional shopping malls or department stores. Factory outlets were literally at the factory when these stores first began to appear, as the standard version of the theorem sugg ests they should be. Today, however, it seems that market conditions have changed so that more of the "bad" apples are being shipped out. Although the dominance of seconds in factory outlet malls away from production centers is consistent with our analysis, a systematic study would be useful.

As this article shows, such a study must take market supply and demand conditions fully into account. In particular, the analysis presented herein suggests that the addition of a fixed charge unambiguously reduces the price of a higher-quality good, relative to a lower-quality version of the same good, only when that good is sold by a perfectly competitive, constant-cost industry. Under increasing-cost conditions or imperfectly competitive industry structures, by contrast, it is possible for relative prices either to be unaffected by the introduction of the fixed charge, or, indeed, for the lower-quality good to become relatively cheaper in distant markets. Will a fixed transportation charge cause producers to ship the "good" apples out? The answer is that it depends. It depends on the elasticity characteristics of relevant market demand and supply functions. In the empirically plausible case where the demand for the high-quality good is less elastic than the demand for the low-quality good, more "bad" apples will be shipped out to distant markets because they are relatively cheaper there.

[FIGURE 1 OMITTED]

(1.) Silberberg (1990, 386) staled the theorem in response to a complaint published in the Troubleshooler column of the Seattle Times of Sunday, 19 October 1975, in which an irate consumer demanded to know why the apples available in local markets were so "small and odd looking." "where," the writer asked, "do the big Delicious apples go? Are they shipped to Europe, to the East or can they be bought here in Seattle?" After explaining the impact of shipping charges on the relative prices of good and bad apples on the East Coast, Silberberg remarked that, "It is no conspiracy--just the laws of supply and demand."

(2.) The assumption that transportation costs do not vary with the quality of the good shipped is somewhat unrealistic--higher-quality goods may be packed more securely and handled more carefully. But given that weight and volume are the chief determinants of the costs of transportation, the assumption does not do too much violence to reality. Another possibility arises when transportation charges are subject to economic regulation. In that case, the regulatory agency may promulgate rate schedules wherein shipping charges are not constant but instead vary with the elasticities of demand for different quality grades of a product.

(3.) Gould and Segall (1969), for example, suggested that the theorem might not operate in a three-good world.

(4.) This is also a necessary condition for the Alchian and Allen theorem to hold in the two-good case.

(5.) Silberberg (1990, 389) reasons that the Alchian and Allen theorem would be expected to hold for most goods, assuming that the higher- and lower-quality grades of the same good are "fairly close substitutes" (so that the absolute values of their own- and cross-price elasticities are relatively large) and that these goods are not closely related to the composite commodity (so that the cross-price effects with it are "fairly small, even if not approximately equal").

(6.) The Alchian and Allen theorem, in other words, is about ex ante decision making. It is not about two-part tariffs whereby, as a strategy for replicating the profitability of price discrimination of the first degree, consumers are charged nonrefundable lump-sum fees for the right to purchase one or more products at a discount. The relative prices of higher- and lower-quality grades of a good or service sold under a two-part tariff are unaffected once the fee has been paid.

(7.) An exception is Barzel (1976), who explores incentives on the part of the seller to adjust product quality in response to the imposition of a per-unit tax. The burden of a tax levied on loaves of bread, for example, can be reduced by increasing the size of the loaf. Sobel and Garrett (1997) find empirical support for this proposition in the market for cigarettes. An alternative model in which changes in relative prices produce changes in quality is contained in Leffler (1982), who assumes that a product's quality attributes are not fully reflected in its price and hence requires price and quality to be determined jointly. Such price-induced changes in product quality obviously complicate the analysis considerably, and the corresponding shifts in demand can easily confound the Alchian and Allen theorem. However, our discussion stays within the theorem's original framework, which treats quality differences as fully reflected in price differences and holds demand curves in place.

(8.) A price increase of less than t is consistent with equilibrium in a competitive market when producers encounter increasing costs. In that case, demand in the away market is filled by inframarginal firms whose supply prices (inclusive of transportation) are [p.sup.'.sub.i] or less (see Figure 1). The impact of t on the relative price of high-versus low-quality goods then hinges on their respective elasticities of demand. With constant costs, by contrast, the prices of both quality grades must rise fully by t and, hence, the high-quality good necessarily becomes relatively cheaper.

(9.) We are assuming that the two types of good are independent, in the sense that the price charged in one market does not influence the demand in the other market. Assuming alternatively that high- and low-quality grades are substitutes for one another strengthens our results. If, for example, the introduction of transportation charges causes a decline in the relative price of the high-quality good, then consumers would shift their purchases even more so away from the low-quality good in favor of the high-quality good than they would if the two goods are independent in demand.

(10.) Barron et al. (2000, 550), for example, report that "within the Los Angeles Basin area ... the average dealer margin for self-service premium unleaded gasoline was 58.7% higher than the average margin for self-service regular gasoline for the 1992-1995 period." Higher retail margins imply that the demand for premium unleaded gasoline is less elastic than the demand for regular unleaded gasoline, ceteris paribus.

(11.) Bertonazzi et al. (1993) and Sobel and Garrett (1997) have gotten this literature off to a good start.

REFERENCES

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Sobel, R. S., and T A. Garrett. "Taxation and Product Quality: New Evidence from Generic Cigarettes." Journal of Political Economy, 105(4), 1997, 880-87.

Umbeck, J. R. "Shipping the Good Apples Out: Some Ambiguities in the Interpretation of 'Fixed Charge.' "Journal of Political Economy, 88(1), 1980, 199-208.

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LAURA RAZZOLINI, WILLIAM F. SHUGHART II, and ROBERT D. TOLLISON *

* We benefited from the comments of Donald Boudreaux, Dennis Coates, Fred McChesney, Tim Sass, Russell Sobel, John Sophocles, and two anonymous referees. Michael Reksulak supplied valuable research assistance. As is customary, however, we take full responsibility for any remaining errors. We acknowledge financial support from the Robert M. Hearin Support Foundation. Laura Razzolini also acknowledges financial support from the National Science Foundation, grant SBR-9973731. The views expressed in this article are the authors' and do not necessarily represent those of the National Science Foundation.

Razzolini: Associate Professor, Department of Economics, University of Mississippi, P.O. Box 1848, University, MS 38677-1848, and Program Director for Economics, National Science Foundation, 4201 Wilson Boulevard, Room 995, Arlington, VA 22230. Phone 1-703-292-7267, Fax 1-703-292-9068, E-mail lrazzoli@nsf.gov

Shughart: Professor, Department of Economics, University of Mississippi, P.O. Box 1848, University, MS 38677-1848. Phone 1-662-915-7579, Fax 1-662-915-6943, E-mail shughart@olemiss.edu

Tollison: Professor, Department of Economics, University of Mississippi, P.O. Box 1848, University, MS 38677-1848. Phone 1-662-915-5041, Fax 1-662-915-6943, E-mail rdtollis@olemiss.edu