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Oligopoly and Luce’s Choice Axiom Simon P. Anderson and André de Palma∗ October 3, 2011 Abstract We apply Luce’s choice axiomatic framework to oligopoly pricing of quality diﬀer- entiated goods. The demand system is a probabilistic comparison of surpluses across products. Zero demands arise naturally, in contrast to the related CES and Mixed Logit models. With asymmetric products, high mark-ups and high demands are driven by high quality-costs. The oligopoly price equilibrium delivers a simple surplus-split prop- erty. We reconcile the model with standard consumer theory by introducing income, and hence generate a Representative Consumer formulation, which has a quadratic form in a central case. We further introduce a preference representation based on the Gabszewicz-Thisse vertical quality formulation. Keywords: Choice Axiom, Probabilistic choice models, Oligopoly, Pricing, Repre- sentative consumer. JEL Codes: D11, D43, L13. ∗ University of Virginia, sa9w@virginia.edu; and Ecole Normale Supérieure de Cachan and Centre d’économie de la Sorbonne, andre.depalma@ens-cachan.fr. The ﬁrst author thanks the NSF for support. We thank Jacques for his perpetual inspiration. 1 1 Introduction Luce (1959) postulated a choice probability system which has endured in psychology, eco- nomics, marketing, and regional science.1 It stems from Luce’s Choice Axiom, and oﬀers a simple formulation of choices in terms of scale values attributable to alternatives. These scale values are treated by Luce as exogenous, though their values are typically determined by the economic decisions of other actors, most notably ﬁrms. Our intention in this paper is to model the endogenous determination of these values by specifying and solving the pricing game between ﬁrms. Indeed, the simplest economic formulation of Luce’s choice probabili- ties is to use surpluses as scale values, with the “surplus” here measured as a “reservation price” minus price paid. This is the formulation that we investigate in this paper as a basic demand system in the context of ﬁrms selling diﬀerentiated products. Luce’s model is based on the Choice Axiom, and highlights the IIA property (ﬁrst pointed out by Debreu, 1960, in the famous Red-Bus/Blue-Bus paradox). Both the Logit and CES models (see Anderson, de Palma, and Thisse, 1988a) embody the IIA property. Several properties of the formulation we present are shared with those models. However, there are also substantial diﬀerences. For example, Logit and CES models have been criticized because positive amounts are always consumed regardless of prices (a feature shared with mixed logit models), and this aspect is not always apparent in data (on, say, import consumption by sector across countries: see Baldwin and Harrigan, 2007). The current speciﬁcation implies choice probabilities are zero when prices get above “reservation values.” We apply the demand model that is generated by the Luce approach to the problem of competing oligopolists selling products that are diﬀerentiated by qualities, as manifested in the “reservation values.” Equilibrium existence is readily demonstrated because strategy sets are compact, prices being naturally comprised between marginal cost and reservation value, and proﬁt functions are strictly quasi-concave. The latter property is easily shown using the monotonicity of a core function within the proﬁt derivative. The oligopoly prices generate a simple surplus-split property between ﬁrms and con- sumers. Under asymmetric ﬁrm qualities and marginal costs of production, there is a simple ranking between mark-ups, equilibrium outputs, and ﬁrm qualities and costs. The quali- tative nature of this last relation is shared with the Logit oligopoly model, and this is one instance where those similar roots are manifested here. Speciﬁcally, ﬁrms with high mark- ups are associated to high demands and high proﬁts. What drives these relations is a high quality-cost margin. These relationships sustain once we introduce non-purchase options into the basic Luce framework. Such “outside” options also enable us to soften some of the stronger properties of the model. Our next contribution is to reconcile the proposed Lucian model with standard consumer theory. This we do by introducing income into the choice probability system. This enables us to generate a Representative Consumer formulation for the choice probability system, which has a simple functional form and is quadratic in a central case. 1 See also the perspective 20 years later (Luce, 1977). 2 Finally, we propose an alternative formulation for introducing income, based on the vertical diﬀerentiation formulation of Gabszewicz and Thisse (1980). This appealing form takes as surplus function a product of the product quality and income net of purchase price. 2 The model 2.1 Consumer Choice Model Coming from a background in mathematical psychology, Luce (1959) viewed the individual decision process as inherently probabilistic, but in a manner that could be described by mathematical formulae. In common with the economics formulation of utility theory, Luce envisioned the use of scale variables that would reﬂect the attractiveness of alternatives, although not in the economics fashion of always choosing the option with the greatest scale. Instead, Luce postulated axioms, most importantly the Choice Axiom, which implied that the probability an option would be selected was an increasing function of its (positive) scale, , and decreasing in the scales attributed to other alternatives. Luce proved that his axioms imply that the probability, P , that alternative is chosen from a choice set is:2 P = X = 1 (1) ∈ Luce does not specify how the ’s are determined. Indeed, as argued in the Introduction, they are often endogenously determined through the actions of other agents, most notably (for our context) by the ﬁrms choosing the prices of the various competing alternatives. With this application in mind and considering the economic context for the consumer of which alternative to choose, the simplest formulation is to use the following speciﬁcation for the scale values: = max ( − 0) = 1 (2) where is the price of option , and we understand the individual to be interested in buying a single unit of the commodity (a motor car, or a chocolate bar with afternoon tea).3 The 2 Luce’s (1959) axioms are as follows. Let P () be the probability that the individual selects from the subset , with ∈ . The most important axiom is the path independence property which states that P () = P () P () with ∈ ⊆ . The second axiom is: if and are such that P{} () = 0 then P () = P \{} ( \{} ). Therefore, the analysis can be restricted to the case where P{} () ∈ (0 1). Under those assumptions, Luce’s theorem (Luce, 1959, p. 23) states that the choice probability of alternative in subset is given by the ratio between a factor which depends on only, and the sum of all other factors (the normalization factors): () P () = X () ∈ with () a positive real-valued function. 3 Another simple form that could bear further investigation is = . 3 term is naturally interpreted as a reservation price insofar as the individual will not buy if , and, since the individual only buys one of the options available under the choice rule (1), the terms − will be interpreted throughout this paper as Lucian surpluses. However, these are not traditional economic consumer surpluses in the usual sense of the term because the consumer does not necessarily buy with certainty the alternative with the greatest surplus under the choice rule (1). Note that it is implicitly assumed that income is not binding here, which makes most sense if the good under consideration is a small fraction of income (like a chocolate bar, though not, for most of us, a motor car). This case is also more pertinent when income eﬀects are eﬀectively small. We discuss below how income can be introduced into the model, and how income eﬀects aﬀect choice probabilities. The choice probability system, which is also the demand system faced by the ﬁrms, provides a simple relation between the Lucian surpluses. For the analysis that follows, we parameterize the basic Luce model (1) by ascribing a power 0 to the scales, in (2). This means that the larger is , then the more discriminating is the consumer in terms of choosing the option with the greatest Lucian surplus. Thus, in this paper, we suppose that the choice probabilities, P , follow the form:4 ( ) P = P = 1 (3) =1 ( ) with ≥ 0 and with = max ( − 0) ≥ 0. If all the are zero, then we assume that P = 0 for all . Note that choices are equally split among options when all the ’s are equal. We also ﬁnd the equal-split result for unequal ’s when → 0 (and then P → 1). This case corresponds to a lack of discriminatory power of the economic variable − (see (2)). At the other extreme, consider → ∞. If alternative is dominated (i.e., there is some for which ), then lim→∞ P = 0. If = max=1 , then lim P = 1 →∞ where ∈ {1 } is the number of alternatives which have the (common) largest value of . These choice probabilities satisfy the well known IIA property (with the blue-bus/red bus paradox articulated by Debreu, 1960) that the ratio of any two choice probabilities is independent of the availability (or price) of any third alternative. Note that the denominator of the expression in (3) is always positive for all once we introduce an active outside option, considered below as = 0 (and so it is active when 0 0). The choice probability system obeys two key properties, which are readily veriﬁed by diﬀerentiation. Lemma 1 For all 0 and 0: 4 In the standard Luce model above, = 1. 4 The own-price demand derivative is: P P (1 − P ) = − 0, = 1 (4) The cross-price demand derivative is: P P P = 0, = 1 6= (5) The ﬁrst part shows demands slope down; the second part shows that products are substitutes. We now turn to the ﬁrm side of the analysis. 2.2 Firm pricing There are ﬁrms, and diﬀerentiated products. Firm = 1 sells product at constant marginal cost of production, , and the “quality” of ’s product, , is exogenous. We assume that for all ; otherwise (as argued below) a ﬁrm would be inactive. Firm ’s net proﬁt when it sets price and its rivals set prices − , is ( − ) = ( − ) P = 1 Firms compete in prices, and we seek a Bertrand Nash price equilibrium. Note that the best response price of ﬁrm satisﬁes: . When = , Firm is left with no proﬁt, while when = , the consumer buying good is left with no surplus and so never buys. The ﬁrst-order derivative for pricing, using (4), is: ( − ) P (1 − P ) = P − ( − ) = 1 (6) This is simply the extra proﬁt from the existing consumer base, P , minus the mark-up lost on the consumers who switch out due to the price rise. For 0, we have µ ¶ ( − ) P (1 − P ) = − ( − ) = 1 (7) (1 − P ) The factor before the brackets is positive, and hence the sign of ( − ) depends on the sign of the term in brackets. The ﬁrst term in the bracket ( (1−P ) ) is strictly positive for and zero for ≥ . This term is decreasing in since both and P are decreasing in . The second term in the bracket (minus the mark-up) is also decreasing in ; it is positive if and only if ≥ . There is therefore a unique intersection to the equation (1−P ) = ( − ) corresponding to Firm ’s best-response price: = + . (8) (1 − P ) Because the proﬁt function is strictly increasing for prices below the intersection ((7) is positive),5 and decreasing for prices above it ((7) is negative), the proﬁt function is strictly 5 ( − ) Firm ’s best-reply price is always above its marginal cost since 0. 5 quasi-concave, given 1 [ ] . This indicates that indeed (8) is the best response. The proof of existence follows. Firm ’s strategy is the compact set [ ]. Its proﬁt function is continuous in and strictly quasi-concave in ; hence reaction functions are continuous and map from the compact and convex set (a hypercube) [1 1 ] × × [ ] into itself: Kakutani’s ﬁxed point theorem then ensures there is a ﬁxed point, which is a Nash equilibrium. Proposition 1 There exists a price equilibrium. In equilibrium, all ﬁrms for which are active, and have strictly positive mark-ups. The other ﬁrms are inactive. The equilibrium prices solve the system = ( − ) , = 1 (1 − P ) We prove uniqueness, below, in a slightly more general case. The next Proposition shows how mark-ups, Lucian surpluses, outputs, and proﬁts are ranked. To do so, we ﬁrst label ﬁrms by decreasing quality-cost margins, so that 1 − 1 ≥ ≥ − . Proposition 2 (Quality). The equilibrium rankings of the endogenous economic variables reﬂect the quality-cost rankings and conversely - in any equilibrium: 1. Firms’ mark-up ranking: 1 − 1 ≥ ≥ − ; 2. Lucian surplus ranking: 1 − 1 ≥ ≥ − ; 3. Market size ranking: P1 ≥ ≥ P ; 4. Proﬁt ranking: 1 ≥ ≥ ; where any strict inequality implies a strict inequality, and any equality implies an equality in the above rankings. Proof. Suppose that − ≥ − for some pair { }. By (8), then (1−P ) ≥ (1−P ) . Since (1−P ) is increasing in − (note that P is increasing in − ), then − ≥ − . But then the last statement ( − ≥ − ) together with the ﬁrst ( − ≥ − ) implies that − ≥ − . Clearly, if − ≥ − then we cannot have − − and so − ≥ − implies − ≥ − . The proﬁt rankings follow from the mark-up and market size rankings. Similar arguments show that − − implies strict inequality in the other rankings, while − = − implies equality in the other rankings. Hence a ﬁrm delivering a higher quality-cost earns a higher equilibrium mark-up. This in turn means that it has a higher equilibrium choice probability, and so delivers a higher net (Lucian) “surplus” to consumers, and earns more proﬁt. In a broader context, insofar as a free entry equilibrium (with all ﬁrms facing the same entry cost) tends to select the ﬁrms with highest proﬁts, Proposition 2 suggests that the highest quality-cost ﬁrms would enter the market. For a ﬁrst appraisal of Proposition 2, suppose that all ﬁrms delivered the same quality, , but diﬀered by production cost, . Then, ﬁrms with lower costs would set larger mark-ups, but they would also set lower prices. These lower prices ensure higher equilibrium demands. 6 The fact that a ﬁrm has a cost advantage enables it to set a lower price while still setting a larger mark-up; and its proﬁt is higher on both counts. Second, suppose that all ﬁrms had the same costs, but diﬀered over qualities. Then, a higher quality enables a ﬁrm to set a higher price, and so get a higher mark-up, but, by delivering a higher Lucian surplus to consumers, to still get a higher equilibrium demand. This means that high prices are associated to high demands. The reason is that both are related through the ﬁrst-order conditions, and are generated by the economic fundamental higher quality. Qualitatively similar results have been shown by the authors for the Logit model (see Anderson and de Palma, 2001). 2.3 Surplus split The equilibrium is remarkable for the surplus split between consumers and ﬁrms; and we highlight this because it contrasts with the logit and CES models. Let ∆ ≡ ( − ) be the quality-cost associated to Firm . We can rewrite the pricing expression (8) as: ∆ − = ∆ . (9) (1 − P ) + 1 This gives = for the monopoly case (P = 1): a monopolist charges the reservation price of the consumer.6 Clearly (so that 0). An alternative way to write the equilibrium price is in terms of the Lucian surplus: (1 − P ) ∆ − = (10) (1 − P ) + 1 This expression immediately gives rise to the following result (where we recall from Propo- sition 2 that lower sales are made by ﬁrms with lower quality-costs). ∆ Proposition 3 In equilibrium, the surplus, ∆ , is split between a mark-up (1−P )+1 for each (1−P )∆ ﬁrm, and a "Lucian net surplus" for the consumer. Hence the ratio of Lucian to (1−P )+1 ﬁrm surpluses is (1 − P ), which is higher for ﬁrms with lower quality-costs. Equivalently, the share of ﬁrm surplus in quality-cost is higher for ﬁrms with higher quality-cost margins. This result indicates that ﬁrms with higher quality-costs are more successful in appropri- ating surplus. Corollary 1 (i) If → 0, then all prices tend to reservation values ( → for all = 1 ). (ii) If → ∞, then 1 → (1 − 2 ) + 2 while → for all 1. 6 By contrast, a logit or CES formulation for monopoly with no outside good gives an inﬁnite price. 7 Proof. (i) This follows directly from (10). (ii) By Proposition 2, P ( P1 ) is bounded strictly below 1 for 1 so that (1 − P ) → ∞. Hence from (9), → for 1. The same argument applies to Firm 1 if 1 −1 = 2 −2 , so consider henceforth the case where 1 − 1 2 − 2 . From (10), ∆1 1 − 1 = 1 (11) 1+ (1−P1 ) Either (1 − P1 ) goes to zero as → ∞, in which case 1 → 1 : but this contradicts that Firm 1 has the highest consumer surplus (Proposition 2: Firm 2, pricing at marginal cost delivers surplus 2 − 2 , which positive by assumption that 2 is active). Likewise (1 − P1 ) cannot go to inﬁnity because, from (11) then 1 → 1 : but this contradicts that Firm 1 is maximizing proﬁts since it can do better setting a positive mark-up and has the leeway to do so against Firm 2 pricing at its marginal cost. Therefore, the remaining case has (1 − P1 ) reaching a ﬁnite, non-zero limit. It can be shown that P ³ ´ =2 1 X µ ¶ (1 − P1 ) = P ³ ´ ≈ 1 1 + =2 1 =2 ⎛ ⎞ µ ¶ µ ¶ µ ¶ µ ¶ 2 ⎜ ⎜1 + 3 ⎟ ⎟≈ 2 − 2 = + + 1 ⎝ 2 2 ⎠ 1 − 1 | {z } ≈0 This expression is neither zero nor inﬁnite as → ∞. Hence 1 − 1 must tend to 2 − 2 . This case yields the value given in the Corollary. The ﬁrst of these results says that if there is no discriminatory power across products, then all products tend to individual “monopoly” solutions by pricing at reservation values. The second result corresponds to the case of perfect discriminatory power, in the sense that consumers choose the product with the highest net quality (Lucian surplus), − . The result is the classic Bertrand one that all prices tend to marginal cost except the highest quality product, which prices to just beat the keenest competition. This means pricing at that rival’s marginal cost adjusted by the top ﬁrm’s quality advantage. 2.4 Symmetric equilibrium Suppose now that all “qualities” are equal to , and all costs are equal to , and let ∆ ≡ ( − ) be the symmetric quality-cost. Then (9) becomes: ∆ − = ¡ 1 ¢ ∆ 1− +1 The equilibrium price is decreasing in (since the demand is more elastic as increases) and decreasing in (more competition drive prices down). When → 0 (no discriminatory 8 power), we have → , while when → ∞ (perfect discriminatory power), we have → . The alternative way to write the equilibrium price in terms of the Lucian surplus (from (10)) is: ¡ 1 ¢ 1− ∆ = − ¡ 1 ¢ 1− +1 Comparing this expression with the previous one indicates that the ratio of Lucian to ﬁrm surpluses is (−1) , which is increasing in . This is a classic competition result that con- sumers gain as more ﬁrms compete. Note though that the limit is not perfect competition but rather “monopolistic competition”: as gets large, the split tends to . 3 Outside option Suppose now there is also a non-purchase (outside) option with ﬁxed “scale” level 0 ≥ 0; for example, eating at home rather than eating in the restaurant. The case 0 = 0 will be seen to concur with the results already found. The choice probabilities then become ( ) P = P = 0 1 (0 ) + =1 ( ) The ﬁrst order conditions look the same as before (see (6)), except with 0 as an argument in P . This gives a polynomial system of equations to solve. The proof of equilibrium existence follows exactly the same lines as before. We here show the uniqueness of the equilibrium. Proposition 4 There exists a unique price equilibrium for the Lucian model with an outside option. Proof. Existence follows from the same argument as the case without an outside option; uniqueness is given now by showing that there is a single solution to the equations deﬁning the (interior) ﬁrst order conditions. The ﬁrst-order conditions solve (6), which can be rewritten as (1 − P ) = = 1 ∆ − or P = 1 − = 1 (12) (∆ − ) Summing over all ﬁrms, and the outside option yields X −1= − P0 (13) =1 (∆ − ) 9 Observe that the RHS of this expression is increasing in , = 1 . Now, the existence of an equilibrium guarantees there is at least one solution to the system of equations (12) above. Suppose there were at least two solutions, denoted with superscripts and . W.l.o.g., suppose that . Then, since the RHS of (13) is monotonic, there must exist a such that . The former relation ( ) implies from (12) that P P , ( ) ( ) or P P . Therefore, this last relation, along with (0 ) + =1 ( ) (0 ) + =1 ( ) P ¡ ¢ P ¡ ¢ implies that =1 . Using the same argument, the inequality =1 P ¡ ¢ P ¡ ¢ implies that =1 =1 , which directly contradicts the earlier statement. Hence there can only be a single solution to (12). The proof above shows a contradiction by assuming there were more than one solution. A standard uniqueness technique (although with limited applicability) is to show that the best-replies constitute a contraction mapping. As shown in Appendix 1, this is not the case here, even though that property holds for the related Logit model (see Anderson, de Palma, and Thisse, 1992, p. 224). The earlier ranking results of Proposition 2 also follow from the same argument as before: any option with garners positive sales, otherwise, it has zero sales. In the symmetric case, recalling that = − we have: µ ¶ (∆ − ) 1 − = 0 + We know that there is a unique solution to this equation: by Proposition 2, any equilibrium must be symmetric, and by Proposition 4, there is only one solution.7 It solves: ( + ( − 1)) +1 − ( − 1) ∆ + ( + 1) 0 − ∆0 = 0 (14) As before, the solution from here with 0 = 0 is = ∆. The monopolistic ( (−1) +) competition limit of (14) is: lim = 1+ ∆, which is independent of the value of 0 . Here →∞ 0 disappears because it is eﬀectively dominated by the variety of all the other options. Note too that the mark-up remains strictly positive in equilibrium, just as for the logit and the CES. The same limit attains when the outside option gets inﬁnitely attractive, i.e., lim = 0 →∞ ∆ 1+ . The intuition is that in either case, any individual ﬁrm is competing with an aggregate competition that is inﬁnitely attractive. Indeed, for both the case of monopolistic compe- tition and an inﬁnitely attractive outside option, the choice probability for Firm tends to (− ) +(− ) with very large in both cases. Eﬀectively, the price term in the denominator is negligible, and we have the monopolistic competition version of the ﬁrm’s problem as max( − )( − ) , which gives the surplus split result (Proposition 3) as − = . − 7 The term on the LHS has a power + 1 and so is the “most convex”: for the other terms on the LHS, the term in has a smaller power, so this cuts the other once; the last term is a constant, and the one before is linear. 10 4 Introducing income So far, we have written choice probabilities solely in terms of net “surpluses” (“reservation prices” minus actual prices). However, to reconcile the model with standard neoclassical consumer theory, which we should have to do to make any meaningful welfare statements, means that we need to introduce an explicit role for income in the demand functions. The simplest way of doing so, and one which we show below is consistent with neoclassical consumer theory, involves just adding income to the terms, so we write = + − , = 1 Now, redeﬁning ∆ = + − , it is clear that all of the previous analysis up till now goes through without any changes, as long as the income constraint is not binding. To explicitly account for this constraint, we therefore write: = ( + − ) I− ≥0 , = 1 (15) where I− ≥0 is an indicator function for whether or not income is breached. Note that might be interpreted as total income, or it could (more mildly) be a sub-income devoted to a sector.8 The behavior of the demands as a function of income diﬀers as to whether is positive or negative. If 0, demands exhibit a jump down to zero at the price = at which income is breached. This is explained further in the next sub-section. On the other hand, if 0, there is no demand discontinuity, and demand (for product ) tends smoothly to zero when approaches + . One can interpret + as the reservation price in this case, so that more desirable goods still have larger values of . Generally speaking, we are interested in the behavior of the demands away from the prices at which income constraints are breached, although the behavior of demands at the critical values does hold some interest. Indeed, if we are looking at motor-cars, budget constraint violations seem to be a relevant issue, so that is loosely consistent with the values of 0. By contrast, for chewing gum, the relevant might be quite negative, in the simple sense that the reservation price for any brand is signiﬁcantly lower than income. As an example, suppose an individual is deciding which of several possible motor scooters to buy. Her monthly RER pass costs 100Euros, and we take that as the value that the motor scooter has to beat. She retains in her choice set all options costing less that 100Euros, and then evaluates them according Luce’s choice rule with scale parameters + − . In this sense, the model above with the indicator function is similar to that of Tversky (1972a,b) where the ﬁrst decision is whether or not the item is aﬀordable. Then, also like Tversky’s model, the decision of which item to buy is probabilistic, though inﬂuenced by both the “quality” scale values, and also prices. In the formulation (15) above, as income, , rises, prices are less important determinants of product choices. This seems a reasonable property of the demand system, corresponding 8 In the latter case, it might be nested within a broader model. 11 loosely (in a neo-classical sense) to decreasing marginal utility of money. However, the scale values are also less important in determining choices. Insofar as richer people might tend to concentrate their purchases on a narrow band of top quality products, the model does not deliver this intuition. Instead, market shares are monotone in and the individual tends to purchase all goods in roughly the same amount as income gets large. The alternative demand formulation in the next Section delivers instead the property that richer people buy higher qualities with higher probabilities. 4.1 Inferior and normal goods The demand system with = + − allows us to study the eﬀects on demand of changing incomes. Namely, the demand for any given product can be a non-monotonic function of income. To see this, as income rises through the product price, demand jumps from a zero to a positive level. Thenceforth it tends to rise with income, at least until a second, higher-priced, product becomes viable, and the ﬁrst one suﬀers a corresponding jump down. This is interesting for a cross-section of individuals of diﬀerent incomes, and tracking the purchase patterns of high-quality goods as a function of income: the standard Logit and CES models have either no income eﬀects (Logit) or unit income elasticity (CES). Indeed, linear Engel curves are quite standard in consumer models. 4.2 The Representative Consumer We wish to ﬁnd the representative consumer preferences consistent with the demand model where the are given by (15). Consistency requires that Roy’s Identity holds, and that the indirect utility function satisﬁes the appropriate properties. To show the properties are satisﬁed, we explicitly introduce (for this Section) the numeraire, good 0, with price 0 . Consider then the following indirect utility function: µ ¶+1 1 X − ( ) = + + 0 (16) + 1 =1 0 0 where denotes the price vector, = (0 1 ), and denotes the income of the repre- sentative consumer. Then: ( ) Xµ − ¶ 0 = + + and =1 0 0 µ ¶ ( ) − = − + 0 We can then verify that Roy’s identity holds: ³ ´ () + − 0 ( ) = − () = P ³ ´ =P 0 = P =1 ( ) + − + 0 + 0 0 =1 12 Furthermore, ( ) is clearly homogeneous of degree zero in ( ). It remains to show that ( ) is quasi-convex in its arguments. This task is accomplished by showing that ( ) has the stronger property of convexity. Indeed, because the sum of convex functions ³ ´+1 − is convex, it suﬃces to show that + 0 is convex. This is readily done by showing that the Hessian is positive semi-deﬁnite in ( 0 ) for 0. We thus have: Proposition 5 The Lucian demand functions (choice probability system) can be rationalized by the representative indirect utility function: µ ¶+1 1 X − ( ) = + + 0 + 1 =1 0 0 In the symmetric case, we have: µ ¶+1 − ( ) = + + 0 +1 0 0 This version can be used for surplus analysis of the market outcome.9 Returning to the asymmetric case, Fajgelbaum, Grossman, and Helpman (2011) have recently proposed a nested logit model for analyzing trade in diﬀerentiated products. That model, along with the frequently-deployed CES model, implies that demands are never zero (see Baldwin and Harrigan, 2007, for a strong critique of this property). It would be in- teresting to see whether the model proposed in this paper, extended perhaps to a nested version, could provide better empirical veriﬁcation. 5 A Gabszewicz-Thisse Formulation Here we provide a parallel analysis using a variant of the Lucian surplus inspired from the seminal Gabszewicz and Thisse (1980) formulation of vertical product diﬀerentiation. Speciﬁcally, instead of (2) consider now the following speciﬁcation for the scale values: = max ( ( − ) 0) with the choice probabilities again given by (3), i.e., ( ( − )) P = P = 1 =1 ( ( − )) 9 See also de Palma and Kilani (2011) for ways to compute income eﬀects numerically. 13 where the sum is again over all active ﬁrms. Limit properties are similar to those given previously, but now note that ( ) lim P = P →∞ =1 ( ) so that higher quality products have higher choice probabilities even for arbitrarily large income levels. Corresponding to Lemma (1), the key derivative properties are now: P P (1 − P ) = − 0, = 1 (17) and P P P = 0, = 1 6= (18) Note that the only explicit diﬀerence with the previous case is the introduction of the qual- ities. The ﬁrst-order derivative for pricing is now (for 0): µ ¶ ( − ) P (1 − P ) = − ( − ) = 1 (19) (1 − P ) and hence there is a unique best-response price for : = ( − ) , = 1 (20) (1 − P ) A similar proof to the earlier one shows that there is a single solution to this system of equations, which therefore characterizes the unique Nash equilibrium. We now turn to the surplus split, where now the Lucian surplus associated to Firm is ( − ), = 1 . Let now ∆ ≡ ( − ) be the income-cost associated to Firm . The pricing expression (20) can be rewritten as before as: ∆ − = ∆ , (21) (1 − P ) + 1 which gives = for the monopoly case (P = 1) independent of quality. Otherwise, (so that 0). We can also write the equilibrium Lucian surplus as: ½ ¾ (1 − P ) ∆ = (22) (1 − P ) + 1 ∆ Hence, in equilibrium the income-cost diﬀerential, ∆ , is split between a mark-up (1−P )+1 (1−P )∆ for each ﬁrm, and (1−P)+1 for the consumer, so the share of this diﬀerential going to the consumer is (1 − P ), which is higher for ﬁrms with lower equilibrium outputs. 14 In a symmetric equilibrium we have − − = ¡ 1 ¢ − 1− +1 or ¡ ¢ 1 1− ∆ − = ¡ 1 ¢ 1− +1 which clearly shows the split of income-cost going to ﬁrms decreasing as rises. An outside option with ﬁxed “scale” level 0 ≥ 0 is readily introduced, with the choice probabilities becoming ( ( − )) P = P = 1 (0 ) + =1 ( ( − )) Existence and uniqueness of the equilibrium are shown in Appendix 2. Proposition 6 There exists a unique price equilibrium for the Luce-Gabszewicz-Thisse model with an outside option. In the symmetric case, recalling that = ( − ) we have: µ ¶ (∆ − ) 1 − = 0 + There is a unique solution to this equation: by Proposition 6, there is only one equilibrium, and it solves: ( + ( − 1)) +1 − ( − 1) ∆ + ( + 1) 0 − ∆0 = 0 (23) which is almost the same as (14), and the corresponding properties are the same, mutatis mutandis. When there is no outside good, the equilibrium price is a convex combination of cost, , and income, : ³ ´ (−1) =−³ ´ ( − ) = ³ ´ + ³ ´ (−1) + (−1) + (−1) + | {z } | {z } 1 1 Inspection yields the limit properties lim→∞ = ; lim→0 = ; lim→1 = , and lim→∞ = 1+ + 1+ . The representative consumer preferences consistent with this demand model are (cf. (16)): 1 X ( ) = ( − )+1 + 0 (24) + 1 =1 15 which is again a simple sum of quadratics for the central case of = 1. Roy’s identity is readily veriﬁed: () ( − ) ( ) = − () = =P = P P ( ) + 0 ( − ) + 0 =1 =1 In the symmetric case the representative consumer’s preferences boil down to: ( ) = ( + − )+1 + 0 +1 This simple form enables welfare analysis is a straightforward manner. 6 Conclusions This paper has considered variants on the Independence of Irrelevant Alternatives (IIA) property of the Luce model, of which the CES and Logit are the mainstream examples (see also Laurent, 2007a, 2007b). Further work can develop the theme of this paper by considering other Lucian speciﬁcations of the form P = P ( ) ( ) , with () an increasing function. =1 The paper brings several of Jacques Thisse’s many contributions to economic theory, namely his work on the probabilistic discrete choice model of Luce, oligopoly pricing of diﬀerentiated products, formulating representative consumer counterparts to discrete choice models, and vertical product diﬀerentiation. Useful extensions of the current paper could follow some of Jacques’ other research themes, namely introducing localized competition elements and allowing for spatial diﬀerentiation of consumers. 7 Appendix 1: Contraction Mapping Approach The reaction function is given in implicit form by (8) as (1−P ) − ( − ) = 0, which can rewrite as (1−P ) + − ∆ = 0. Recalling Lemma 4 (here rewritten in terms of ’s), the own-price demand derivative is (for all 0 and 0): P P (1 − P ) = 0, = 1 while the cross-price demand derivative is: P P P = − 0, = 1 6= The slope of the best-reply function for Firm ’s choice of with respect to is given by the Implicit Function Theorem as ³ ´ ¯ P P − ¯ ¯ (1−P )2 ¯ =− 1 ³ ´ + 1 + (1−P )2 P (1−P ) (1−P ) 16 which simpliﬁes to ¯ P P ¯ ¯ (1−P )2 = 0 ¯ 1 (1−P ) +1 + P (1−P ) P P (1−P ) = 1 +1 P ¯ ¯ Hence the desired contraction property, 6= ¯ 1, requires proving that P X P 1 + 1 (1 − P ) 6= Note that: ( ) P X P P X P P = − (1 − P ) 6= (1 − P ) P X P P2 = − (1 − P ) (1 − P ) Note that if = 1, this reads P X P P2 P2 = − (1 − P ) 6= (1 − P ) (1 − P ) P2 = ( − 1) (1 − P ) P2 which needs to be less than 2. However, P be anywhere between 0 and 1, so (1−P ) can be anywhere between 0 and inﬁnity ; so therefore the best reply is not a contraction mapping. 8 Appendix 2: Proof of Proposition 6 Existence follows from similar arguments as used for the other case. Uniqueness is given now by showing that there is a single solution to the equations deﬁning the (interior) ﬁrst-order conditions. Recall that he ﬁrst-order conditions solve (22), which can be rewritten as (1 − P ) = , = 1 ∆ − or P = 1 − = 1 (25) ( ∆ − ) 17 Note that P is decreasing in on the reaction function (25). Summing over all ﬁrms, and the outside option yields X −1= − P0 (26) =1 ( ∆ − ) Observe that the RHS of this expression is increasing in , = 1 . Now, the existence of an equilibrium guarantees there is at least one solution to the system of equations (25) above. Suppose there were at least two solutions, denoted with superscripts and . W.l.o.g., suppose that for a given . Then, since the RHS of (26) is monotonic, there must exist a such that . The former relation ( ) implies from (25) ( ) ( ) that P P , or ( ) +P P . Therefore, this last relation, along 0 =1 ( ) (0 ) + =1 ( ) P ¡ ¢ P ¡ ¢ with implies that =1 =1 . Using the same argument, the P P ¡ ¢ inequality implies that =1 ( ) =1 , which directly contradict the earlier statement. Hence there can only be a single solution to (25). References [1] Anderson, Simon P. and André de Palma, 2001. Product Diversity in Asymmetric Oligopoly: Is the Quality of Consumer Goods Too Low? Journal of Industrial Eco- nomics, 49, 113-135. [2] Anderson, Simon P., André de Palma and Jacques-François Thisse, 1988a. The CES and the Logit : Two Related Models of Heterogeneity. Regional Science and Urban Economics, 18, 155-164. [3] Anderson, Simon P., André de Palma, and Jacques-François Thisse, 1988b. A Repre- sentative Consumer Theory of the Logit Model. International Economic Review, 29, 461-466. 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