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6Product Diversity in a Vertical Distribution Channel under Monopolistic Competition *
Igor Bykadorov1, Sergey Kokovin2 and Evgeny Zhelobodko3
1 Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
Acad. Koptyug avenue 4, Novosibirsk, 630090, Russia and Novosibirsk State University, Novosibirsk, Russia E-mail: [email protected]
2 Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
Acad. Koptyug avenue 4, Novosibirsk, 630090, Russia and Novosibirsk State University, Novosibirsk, Russia E-mail: [email protected] 3 Novosibirsk State University Pirogova str. 2, Novosibirsk, 630090, Russia E-mail: [email protected]
Abstract. In Russia the chain-stores gained a considerable market power.
In the paper we combine a Dixit-Stiglitz industry with a monopolistic retailer. The questions addressed are: Does the retailer always deteriorate welfare, prices and variety of goods? Which market structure is worse: Nash or Stackelberg behavior? What should be the public policy in this area?
Keywords: monopolistic competition, Dixit-Stiglitz model, retailer, Nash equilibrium, Stackelberg equilibrium, social welfare, Pigouvian taxation.
1. Introduction
In 2000-s, Russia and other developing markets of FSU have shown dramatic growth of chain-stores and similar retailing firms. Inspired by Wal-Mart and other successful giants abroad, Russian food traders like Perekrestok and Pyaterochka has gained considerable shares of the market and noticeable market power, both in Moscow and in province. This shift in the market organization was suspected by newspapers for negative welfare effects, accompanied by upward pressure on prices and inflation. Public interest to this question is highlighted by anti-chain-stores bill currently passed in Russian Parliament (Duma).
Leaving aside the empirical side of this question, this project focuses on constructing and analyzing the adequate model of vertical market interaction, suitable for Russian retailing markets of food, clothes and durables. We step aside from the traditional models of monopolistic or oligopolistic vertical interaction or franchising (see, e.g. review in (Tirole, 1990)), and stick instead to more modern monopolistic-competition representation of an industry in the Dixit-Stiglitz spirit (Dixit and Stiglitz, 1977), but combined with vertical interaction. This combination is rather new, being pioneered by Chen (Chen, 2004) and Hamilton and Richards (Hamilton and Richards, 2007), as described below in the literature review.
* The work on this paper was supported by an individual grant R08-1071 from the Economics Education and Research Consortium, Inc. (EERC), with funds provided by the Eurasia Foundation (with funding from the US Agency for International Development), The World Bank Institution, the Global Development Network and the Government of Sweden.
Our departure from the former two papers is that it is the retailer who is exercising the monopsony power after market concentration, while the production is organized as monopolistic-competition industry with free entry. This hypothesis seems rather realistic, at least for developing markets. Indeed, there are numerous evidences in economic newspapers (see (Nikitina, 2006), (Sagdiev et al., 2006), (Slovak Republic, 2007), (FAS, 2007) cited in our literature review) that each of several big retailers has much stronger bargaining power than quite numerous manufacturers and importers of consumer goods like sausages, shirts, etc. (even such big international companies as Coca-Cola are not strong enough to enforce their terms of trade to Russian retailers).
To reflect such relations, our stylized model of market concentration considers a monopsonistic and monopolistic retailer (as a proxy of an oligopsonistic/oligopolistic retailer). This monopolistic intermediary deals with a continuum [0,N] of Dixit-Stiglitz manufacturers on one side of two-sided market and a representative consumer on another. Each manufacturer has a fixed cost and a variable cost, he produces a single variety of the “commodity” and sets the price for this variety. The most natural timing of the model is the retailer’s leadership, i.e., the retailer starts with announcing her markup policy correctly anticipating the subsequent manufacturers’ responses, and simultaneously chooses the scope of varieties/firms to buy from. At the next stage the manufacturers come up with their prices and then the market clears the quantities. Both sides take into account the demand profile generated by a consumer’s quasi-linear utility function. Another situation is when the retailer can also impose entrance fee on producers or/and consumers.
One or another monopolistic organization of the industry is compared to the preconcentration situation where multiple shops operate. For simplicity, each variety is assumed to be sold through one shop. There can be three kinds of situations, ordered by increasing market power of the producers.
(1) The myopic (Nash) behavior of both manufacturers and retailer(s) who are unable to predict and influence the market. Which model is closer to reality, we tried to find out from empirical market papers (see literature review below), but did not come to a definite conclusion. So, both pre-concentration versions remain discussed.
(2) Leadership of manufacturers, who choose their wholesale prices correctly anticipating the best-response retailer’s markup added later on.1
(3) Vertical integration from the side of producer, who owns the shop selling its variety or, equivalently, dictates the terms of trade and uses entrance fee or other tools to appropriate whole profit.2
The questions addressed in our paper are: Does the emergence of the monopson-istic retailer enhance or deteriorate welfare, and how much? Which retailer behavior is worse: usual monopoly or price discrimination with entrance fees? What should be the guidelines for public regulation (if any) in this area?
1 Such interaction is somewhat similar to the concept of “common agency” considered in contract theory (see e.g. (Bernheim and Whinston, 1986)), but our leaders are in different position because each manufacturer is dealing with one or more small shops, knowing their costs. Instead of common agent (hypermarket) for all producers which should be assumed in “common agency” approach.
2 This situation seems less realistic than other two, but used to compare with vertical integration from retailer’s side.
In general the question is cumbersome, but definite results of this kind were obtained for the case of quasi-linear quadratic utility like in Ottaviano, Tabuchi and Thisse (Ottaviano et al., 2002), that means linear demands. For this model (in contrast with common wisdoms of politicians) it turns out under rather realistic assumptions that:
(i) Market concentration always enhances social welfare through softening the “double marginalization effect”, this enhancement can work through lower prices and bigger consumption, or through adjusting the inefficiently low or inefficiently high number of varieties.
(ii) Under market concentration, further enforcement of the monopolist’s market power by allowing for price discrimination (entrance fees) further enhances social welfare. Even the first-best (Pareto) optimum is guaranteed if the monopolist is able to use entrance fees on both sides: for producers and consumers. The latter situation turns out equivalent in welfare and (socially optimal) consumptions to the integrated monopoly, which means that one firm owns both production and retailing of the whole industry.
(iii) The governmental regulation of a simple monopolistic intermediary through capping the markup - enhances welfare. However, such regulation is not needed, for price discriminating monopolist (even under entrance fees on manufacturing side only). If the government uses Pigouvian stimulation, then not taxes but subsidies to the monopsonistic retailer enhance welfare.
Additionally, in comparisons of various situations we tell what happens to prices, quantities and diversity under changing market organization.
To explain surprisingly positive influence of certain steps in increasing the retailer’s market power, we can mention two general ideas. First, more market power softens the effects of double marginalization. Second, when the industry involves monopolistic competition, some sort of monopolistic behavior is present anyway. Therefore, the retailer’s market power do not aggravate welfare losses of this kind, but instead internalize the externalities on the supply side. When the main decisionmaker in the industry is powerful enough, she internalize them better, sometimes as good as the social planner would.
In the sections that follow we first describe more extensively the literature and motivation for our approach, then introduce the model and finally describe the results.
2. Literature Review
There is a broad literature on vertical interaction between a producer and a retailer, see reviews in (Perry, 19S9) and (Rey, 2003); they study various economic consequences of such interaction. The early classical paper of Spengler (Spengler, 1950) explored the simplest case of Stackelberg game between two monopolists, one above the other, that entails “double marginalization.” In essence, the second monopolist adds her own markup to the monopolistic price of the first one and further deteriorates social welfare, see (Tirole, 1990). A broad range of papers relaxed the Spengler’s assumptions of homogeneous commodity, single producer and single retailer, which is natural and realistic. The difference among them lies mainly in various models of the oligopolists interaction. First of all, there is a strand of spatial Hotelling-type models (Hotelling, 1929) and another class of “representative consumer” models like Dixit-Stiglitz one (Dixit and Stiglitz, 1977). Among the spatial
models, Salop (Salop, 2006) started with the circular city model with one producer and several retailers distributing themselves around this city to meet the demand of the continuous population. The main result is that under reasonable assumptions all consumers are served and the inefficiency found by Spengler disappears. Therefore there is no welfare motive for integration between the producer and the retailers. In contrast, Dixit (Dixit, 19S3) modified this model to include production activity of “retailers” who used also other production factors. Then there is a welfare reason for integration, because it reducers the inefficiently-big number of retailers-producers and increases welfare. Further these ideas were developed in (Mathewson and Winter, 19S3).
Another approach explores the idea of representative consumer in Dixit-Stiglitz manner. In particular, Perry and Groff (Perry and Groff, 19S9) use constant-elasticity-of-substitution (CES) utility function of the consumer, defined on many discrete varieties, each produced by a single retailer. The main result is that such competition brings both distorted prices and distorted number of retailers, and interestingly, integration of two stages of production turns out welfare-deteriorating because the decrease in the varieties outweighs the decreasing prices. Here, like in Dixit’s circular city, the retailers are the low-level producers also, modifying the commodity, otherwise a consumer buying from all retailers would look strange. Another step in this direction is Chen (Chen, 2004) where a multi-product monopolistic producer at the first stage chooses the number of varieties produced and therefore the number of retailers, because one-to-one correspondence remains. Afterwards he performs a bargaining procedure with each of them, and finally the markets clear taking into account the substitutability among the varieties. The number of differentiated goods turns out smaller than the constrained social optimum; the retailer’s countervailing power lowers consumer prices but exacerbates the distortion in product diversity.
Finally, Hamilton and Richards (Hamilton and Richards, 2007) synthesize Hotelling-type and Dixit-Stiglitz-type models, i.e., the spatial-market and the differentiated-goods approaches in modeling the oligopoly competition among multiproduct firms - “supermarkets,” who face a competitive manufacturers. Two kinds of diversities are involved. There is a potentially large number of spatially-diversified supermarkets to choose from, but a consumer enjoys also a product-variety from the product line designed in a (single) supermarket chosen. It is found that increase in product differentiation on the manufacturers’ side need not increase the equilibrium length of the product lines when retailers are specializing. Besides, under both no-entry and free-entry oligopoly conditions for retailers, product variety is undersupplied; several other effects are also found and the impact of excise taxation studied.
In contrast, in our setting, as we have mentioned, the monopolist is the retailer, but these are the producers who are organized as a Dixit-Stiglitz industry. To motivate such unusual approach, we can note that such or similar situation is rather typical for today Russian market. At least, there are many observations that modern supermarkets and hypermarkets have dominant bargaining power in relations with more numerous producers. For example, in (Nikitina, 2006) we read:
“... chain-stores ... capture the substantial part of profit of the small suppliers... These chains force suppliers to participate in various promotional actions, no matter whether such advertising or discounts are useful for suppliers or not... Overwhelming
majority of the retailers require payments from suppliers, not only “payment for the shelf,” but also so-called retro-bonus, i.e. “sale percentage.” According to the agreements between suppliers and retailers, the latter receive 5 % on average ... Another “trap” for a supplier is the obligation to guarantee some sales volume. If the quantity sold turns out lower for some reasons, the supplier has to pay the difference between the planned and the actual sales... As a result, the total markup appropriated by a retailer can be from 30 to 50 % of the cost.”
Similarly, in (Sagdiev et al., 2006) we can read:
“Retailers require from suppliers dozens thousands dollars only to start selling their goods... As the suppliers confess, the price of the “entrance ticket” depends on the producer’s reputation and his spending for advertising. A retailer can make a discount when the manufacturer agrees to spend a lot on promotion of his commodities ... The entrance ticket is not the only payment by a supplier wishing to enter a chain-store network. The total share going to the retailers can be about 35% of the consumer price.” 3
Another important citation can be found, e.g. in the website of the Federal Antimonopoly Service of Russian Federation (FAS, 2007): ’’Transformation of retail trade sector into big trade networks (retailers) allowed the latter, despite their seemingly small market share, dictate the rules of the game and determine the network entry conditions for suppliers and producers.”
Such market power of the retailer in Russia is not surprising. The deficit of trading facilities is artificially aggravated by considerable corruption in trading land and in licensing trade in cities. As mentioned in McKinsey Global Institute survey of Russian retailing (Kaloshkina et al., 2009), during 2002-07 the sales of such big chain-stores as Eldorado, X5, Magnit, Metro Group, Auchan were growing by approximately 50% annually. However, the share of the market in the hands of all chain-stores still remains smaller than 50% in 2010.
This situation is not unique only for Russia. It is also, for instance, in some countries of East Europe, as shows the following citation from the site of Antimonopoly Office of Slovak Republic (Slovak Republic, 2007): “In view of the existing structure of the individual local markets, high barriers to entry (considerable direct and forced investments, sunk costs related to the required advertising and marketing support when entering the market, administrative barriers to entry, time necessary for entry into the market, and so forth), saturation of the individual relevant markets, and the nonexistence of potential competitors, if the concentration were carried out, the undertaking Tesco plc would establish or strengthen its dominant position. Consequently, the undertaking Tesco plc. would not be subject to substantial competition and, given its economic strength, it could act independently with respect to its suppliers, consumers, and competitors.”
Besides, the Slovak case is not unique, see, e.g. (Lira et al., 2008) for the case of Chile.
3 The same article note that “...in international practice, a bonus payed for the entrance into the trading network is known, ... but in less scale... In Germany only small suppliers pay for starting to distribute goods through the supermarkets... In Great Britain the scheme ”money for distribution” is not common, but retailer and supplier can promote the goods together... ”
3. Model
We consider a monopolistic competition model modified to include the two-level interaction “manufacturer - retailer - consumer.”
3.1. Demand
On consumers’ side, there is a representative consumer, endowed with L units of labor supplied to the market inelastically, and labor is the only production factor. There are two types of goods in the economy. The first “commodity” consist of many varieties, for instance, milk of different brands. The second one is the numeraire representing other (perfectly competitive) goods. The income effect is neglected. A general-type quasi-linear utility function of any consumer reflects preferences over two kinds of goods:
U(q, N, A) = V (q,N) + A.
Here N is the length of the product line, reflecting the scope (the interval) of varieties; q(i) > 0 is “quantity” or the consumption of i-th variety chosen by any agent (consumer) and q = (q(i))ie[o,N] is the infinite-dimensional vector or function q(-) : [0,N] ^ R describing the whole profile of varieties, all profiles keeping bold notation hereafter. Variable A > 0 is the consumption of the numeraire good. However, we use this general function V only to formulate the concepts of equilibria. Throughout we study a tractable special case, namely, the quadratic class of utility function formulated below. It was introduced by G.I.P. Ottaviano, T. Tabuchi and J.-F. Thisse (Ottaviano et al., 2002) (see also (Combes et al., 2008)) and became quite popular for modeling monopolistic competition:
U(q,N,A) = aJ^ q(i)di - ^ ^ 7 [q(i)f di - | q(i)di
2
+ A.
Here a, 3 and 7 are some positive parameters, satisfying 3 > 7 > 0, to have U quasiconcave. This condition ensures also that our consumer prefers larger diversity. In contrast, under 3 = 7, only the total quantity of consumption Q = qN but not the diversity per se influences utility. Two quadratic terms here ensure strict concavity in two dimensions, i.e., definite consumer’s choice among varieties and between the two sectors.
The main feature achieved by this three-term construction is that this utility generates the system of linear demands for each variety and linear demand for the whole differentiated sector.
To formulate the budget constraint, p(i) denotes the price of variety i for the consumer (which equals the wholesale price p(i) when there is no retailer), w = 1 is the wage rate in the economy, Pa is the price of the numeraire, also becoming 1 at the equilibrium.
Then the utility-maximization problem of the representative consumer takes the form
U(q, N,A)= V (q, N) + A ^ max
(q,A)
,■ N ,■ N
/ p(i)q(i)di + PaA < wL + n^(i)di + nR,
J 0 J 0
where nM (i) is the profit of i-th manufacturer, while nR is the profit of retailer.
The budget constraint has a natural interpretation. Its right-hand side is Gross Domestic Product (GDP) of the economy with respect to income, while the left-hand side represents expenditures.
As to the solution to the consumer’s problem, in the case when the income is sufficiently large, it does not influence the demand, due to quasi-linear utility, standardly.4
Therefore, for any price-profile p : [0, N] ^ R+ , the individual demand-profile function q* for all varieties is defined as follows (any “profile” is a function w.r.t. varieties’ names i G [0, N] and bold letters p, q denote such profiles, in contrast with certain points p(i), q(i) of the profiles)
Solving FOC of this problem in another direction for each price-profile, in Section 4.1 we derive the inverse demand function p(i, q(i), N, p_i) of any variety i for our quadratic utility. This function p describes how i-th consumer’s price depends upon
i-th quantity q(i) and upon exogenous parameters: number N of competitors and the price profile p_i including all prices but for i. This inverse-demand function is taken into account by producers, to be described now.
3.2. Supply
On the supply side there can be several different situations, modeled differently. Anyway manufacturers sell the goods only indirectly, through some retailer, but either they can interact myopically (Nash equilibrium) or one or another side of vertical relations may be the leader.
In all situations we denote by p(i) wholesale price and by r(i) the retailer’s markup or price margin (we use these terms interchangeably). Therefore the ultimate i-th retailing price amounts to the sum p(i) = p(i) + r(i). Each manufacturer’s cost function takes the following form: C(q) = cq + F. So, i-th manufacturer’s profit-maximization problem takes the form
where p is the wholesale price-profile for all producers and p + r are the retail (consumer) prices; therefore, r is the retail markup profile. In the long run, free entry reduces profits to zero but at this stage profit maximizing determines price and quantity.
Now let us turn to the monopsonistic retailer (emerging after market concentration) or many retailers (existing before market concentration). Their number makes difference only when the retailer is big enough to control the market. In multi-retailer situation, we assume that one retailer may sell one variety or several, but each variety is sold through one retailer. As soon as markup optimization per each variety is similar, it is sufficient to describe the behavior of the “gross retailer”
4 In the opposite case when the income is small, there can be a boundary solution when whole consumer’s income is spent only for the diversified commodity. We, traditionally, ignore this case but for specifying some restrictions on parameters ensuring absence of income effect.
q* (N, p) = argmax
q
(1)
nM(i) = p(i)q(i, p + r) - C(q(i, p + r)) ^ max,
P(i)
representing the whole population of them. Gross retailer’s cost function is similar to that of the manufacturers:
p N p N p N
Cn(q)= p(i)q(i)di + / cnq(i)di + / F^di .
J 0 J 0 J 0
Here the first integral shows the expenditures to buy from the manufacturers, while the second and the third item show the retail costs: cr is the number of labor units required from retailer to sell a unit of any differentiated good, Fr is the retailer’s fixed cost (also measured in labor) required to start selling any differentiated good. Then the retailer’s profit-maximization problem is
rN !■ N
max,
NN kr = [r(i) - CR]q(i, p + r)di - / FRdi
00
where r(i) = pR(i) — p(i) is the markup of the i-th goods variety.
We consider the following types of timing, or, rather, types of leader-follower relationship between the manufacturers and the retailer, ordered by decreasing retailer’s bargaining power:
— Leading retailer, i.e., monopolistic competition with strategic behavior of retailer5:
• at first the retailer chooses markups and the scope of product varieties, correctly anticipating the subsequent response of manufacturers;
• then each manufacturer chooses, to enter the market or not and the wholesale price;
— Nash equilibrium, when manufacturers and retailer choose p and r simultaneously and myopically6.
— Leading manufacturer, i.e., monopolistic competition with strategic behavior of manufacturers7:
• at first all manufacturers simultaneously choose, to enter the market or not, and the wholesale prices (the number of firms is determined by the zero-profit condition), correctly anticipating the individually adjusted markup function;
• then the retailer chooses the markup for each commodity (each manufacturer);
Note that, in addition to the retailer presence, another distinction here from the standard monopolistic competition is that the equilibrium number of firms is chosen by the retailer, not by the free entry condition. This assumption is rather plausible, as shown by the following saying (Sokolov, 2006) of an owner of a retail supermarket about their policy concerning the diversity: “... for dried crusts ... we deal with a limited number of suppliers, since increasing the number of brands in such commodity groups does not lead to the increase in total sales... ”
The Nash variant of interaction seems less realistic than two leadership cases of interaction with more far-sighted behavior. But Nash interaction can be interesting as starting point to compare and estimate everybody’s gains and losses from the strategic behavior. Though not as realistic as the previous concept, wise producers still can be a plausible approximation of reality. As we have mentioned in Introduction, this concept is, essentially, the solution concept in Common Agency models (the principals are manufacturers while the agent is retailer). The only distinction is the absence of information
asymmetry among the participants.
— “No-retailer” equilibrium, when manufacturers enforce the terms of trade quite
strongly, or each retailer of a variety just belongs to the manufacturer of related
variety.
We formally express these four concepts in (most natural) symmetric case as follows.
Definitions. 1) Symmetric Nash-equilibrium is a quadruple (pNash, rNash, qNash, NNash) g R+, such that related price profile p = p(i) = pNash solves each manufacturer’s problem under external parameters NNash, p— = pNash, markup rNash maximizes the manufacturer profit under pNash, NNash, while q(i) = qNash solves the consumer’s problem under NNash, p = pNash and NNash satisfies the zero-profit condition.
2) Symmetric RL-equilibrium is a quadruple (pRL,rRL,qRL,NRL) g R^_, such that related price function p(i, r, p-i, NRL) describes the optimal response of each manufacturer to external markup r and parameters NRL, p-i = pRL, markup rRL and NRL maximize the retailer’s profit under this function p, pRL = p(i,rRL,pRL, NRL), while q(i) = qRL solves the consumer’s problem under NRL, p = pRL.
3) Symmetric ML-equilibrium is a quadruple (pML, rML, qML, NML) g R+, such that related markup function /i.(i,p(i), p_j, NML) describes the optimal response of the retailer to i-th price p(i) and parameters NML, p-i = pML, price pML maximize the manufacturer profit under this function n, rML = n(i,pML,pML,NML), while q(i) = qML solves the consumer’s problem under NML, p = pML and NML satisfies the zero-profit condition.
4) NR-equilibrium. Each manufacturer owns the shop, bears joint fixed and variable costs of production and trade, C(q) = (c + cR)q + F + Fr, and maximizes the profit function
nNR(i) = p(i,q(i),N,p-i )q(i) — C(q(i)) ^ max .
To compare the social outcomes of these concepts of equilibria, we formulate the welfare function analogous to the previous one, but for modified retailing cost:
,■ N ,■ N
W = V (q, N) — (c + cR)q(i)di — (F + FR)di.
00
The symmetric solution qMaxW, NMaxW to this optimization program is called socially-optimal quantity and socially-optimal diversity.
Besides, we formulate the consumer surplus as
,■ N
CS = V (q, N) — (p(i) + r(i))q(i)di.
0
Thus we have introduced the model and now start analyzing it.
4. Market Concentration
In this section we approach the question: Does market concentration enhance or deteriorate welfare? We consider two models of pre-concentration market: Nash behavior or strategic behavior of producers, leaving to the reader the choice of more realistic one and discussing other relevant models in the end of section. Both are compared to stylized model of extreme concentration: one retailer owns all shops and behaves as a leader, and to social optimum. Thus, we study:
— “no retailer” equilibrium (NR);
— strategic behavior of manufacturers-leaders (ML);
— Nash equilibrium (Nash);
— strategic behavior of retailer-leader (RL);
— socially-optimal quantity and socially-optimal length of the product line (MaxW).
4.1. Equilibria Characterization
Demand. Under this utility function, the consumer’s problem can be written as following:
fN 3 — Y fN 2 Y fN
Jo ----------------------2 Jq di ~ 2 Jo '
q(i)di
+ A —> max
q,A
NN
/ pn(i)q(i)di + A < L + nM(i)di + nR .
J 0 J 0
Here we analyse only the case when the consumer’s income is sufficiently large and does not influence the consumption of the diversified products.
By standard technique (using the first order conditions, cf. (Ottaviano et al., 2002)) we can express the equilibrium retail price through the parameters as
N
p(i)+ r(i) = a - (3 - Y)q(i) - Y q(j)dj , i € [0,^].
0
(2)
Moreover, solving FOC in the opposite direction, we can obtain the linear demand function for any variety i € [0,N]:
q(i) = a - (b + gN)[p(i) + r(i)] + gP,
where coefficients are
Y
3 + (N - 1)y
3 + (N - 1)y
(3 - Y)[3 + (N - 1)y]
(3)
(4)
and price index
N
P = [p(j) + r(j)]dj
0
expresses the aggregate pricing behavior of all firms, negligibly influenced by firm
i. In symmetric equilibrium p, q, r both above integrals can be simplified as
N
/ q(j)dj = Nq, P = N(p + r).
0
Method of finding (Nash), (ML) equilibria and (MaxW). Using the demands system obtained, we calculate all kinds of market outcomes defined: (Nash), (ML) and socially-optimal (MaxW), and then (RL). All these are found in closed form in Result 2 below, proceeding as follows.
Solution MaxW is found directly from FOC. The method of calculating Nash is also rather usual: system of FOC for all participants is combined with the zero-profit condition, resulting in unique solution. For finding equilibrium ML, we first derive
2
a
1
a
a
the markup function ri(p(i), P, N) which is the best-response of the retailer (a small shop) to any price p(i) (P, N given). Then we optimize prices of the manufacturers having this markup function, and adding the zero-profit condition get the unique equilibrium.
Method of finding (RL) equilibrium. More complicated is calculating equilibrium RL. Here, having in mind demands q(i,r(i),N, P) and optimal price policies p*(i, r(i), N, P) as functions of markup r(i) and diversity N, the retailer jointly chooses profile r = (r(i))ieN and N as the solution to her program in the form
If we, reasonably, assume only symmetric variables p , q , r (bar accent denoting sym-metrization, in particular p* (r ,N) = p* (i,r,N,Np) Vi) this retailer’s problem is simplified as
nR = N[r - CR]q(p*(r,N)+r) - NFr ^ maxfjN,
nM(p*(r,N),r, N) > 0 .
Remark that in symmetric RL case one has the following expressions q and p as functions of r and N:
Strictly speaking, now we have stepped aside from the monopolistic competition model, since N need not be necessarily determined by the free entry or zero-profit condition. More specifically, there can be two types of solution, or regimes that we call artificially restricted market or un-restricted market. The first case means non-negative-profit constraint occurring non-binding, i.e., optimization of the unconstrained function nR resulting in positive manufacturers’ profit nM > 0. In this case the retailer really ignores “free entry” condition imposing instead her own restriction on entry (so, we also ignore free entry within calculations). Otherwise retailer first use “free entry” condition to calculate N(r ) as a function of r, then maximize her profit with respect to r . As we have found by direct calculations, the artificially restricted solution happens if and only if some crucial constant exceeds one:
Fr
F = — > 1.
2 F ~
Thus, postponing its further economic discussion, we can formulate now
Result 1 It is profitable for the retailer-leader to artificially restrict the entry, when her fixed cost exceeds doubled manufacturer’s fixed cost.
nR = /0N[r(i) - CR]q(i, p*(r,N) + r)di - f0N FrcU ^ maxr,N,
^m(p*(i,r(i),N,P),r(i),N) > 0 .
(5)
(6)
This constant F denoting relation of retailing fixed costs over manufacturing fixed cost plays hereafter an important role. To concisely formulate the characterization of all our solutions, we introduce the following auxiliary notations (some of
them being interpreted economically in special section):
a a F -r FK n m— a-c-ctz a - c - cn
With these notations, omitting the intermediate calculations, the formulae characterizing equilibria through exogenous parameters are summarized in the next table. The types of competition (equilibria concepts) are shown in rows, while the equilibrium values of the variables are in columns, the separate column showing the optimal value of welfare function W.
Result 2 The market outcomes under several market organizations and parameters are characterized as
quantity q price p markup r
Nash Qne C + (?Ar_E ' /3-7 D - /3_7 + qNE ^
ML V2 — • (1NE c + qNE ■ V2 • /3-7 D - /3 ,V2 + <?№£ p
RL, J- > 1 (In e ■ VT C + qNE ■ P-jVF D cn + qNE -y
RL, J- < 1 Qne c + (?Ar_E • /3-7 cn + qNE ^ + /3-7(.?" — 1)j
MaxW qNE\/2 + 4 J7 - -
diversity N consumer surplus CS
Nash hib~3^) (p 3/?_7) ■ ^-D /?_7) ■ 8.7/3_7
ML D - 2V2/3-7 v^7 ~/3-7) (-D V -/3-7j 8-7/3_7
RL, J- > 1 Tr~4i3-' 2i (£» 4^/3-7) • (i> 2V^/3-7) • 8.7^
i?L, J- < 1 D — 2f3-1(Jr + 1) 27 (£ - 2(Jr + l)/3_7) • (£ - 2F/3—,,) • ^
MaxW 7 ( V2+4J7 -
welfare W
Nash (D 3/3_7)-(£> 3+r-/3-7)-8^
ML (£ - 2a/2/3_7) • (£ - ^«2 • /3-7) • ^
RL, J- > 1 (*> 4a/F/3_7) • (_D 43tf-^)-8X
i?L, J- < 1 (£» 2^_7(T-+1)).(£» 4+r-/3-7)-8.7T,
MaxW (-^ ^2 + 4Jr/3_7^ •
retailer’s profit hr manufact. profit 7r_/v(
Nash (_D — 3/3—j,) 0
ML (D 2V2^)(D 2a/2/3_7 • 1+2'F) ■ 4.7^_t 0
RL, J- > 1 Fn — ~F
RL, J- < 1 (D 2/MJ-+1)) -4.1 0
Here in every equilibria cases one has W = CS + N • 7r_/v( + t^tz-Moreover, one has for NR case:
quantity q price p markup r diversity N W
NR INTi. c + CM + fM - (DM - 2) ^ iP2M ~ 3 ' Dm + 2) • Hm.
where
IF + Fn
mn = 4 —^-----------
, fM = \J{F + Fn) ■ (P—y)
n _ a - c - cn TT _ (F + Fn) • (/?_7)
// 7-i-ti-\--!~n--V 7 r> '
y/(F + Fn) • (/?-7) 2 • 7
Remark that although prices and profits are not relevant for MaxW case, we can calculate these values “artificially”. Indeed, as in the case of perfect competition, “assigned consumer price” (p + r) equals total marginal costs (c + cr). Therefore, total “assigned profit” of retailer and all producers is negative (due to fixed costs), i.e. always less than in all another market structures (ML, Nash, RL). Hence, obviously, “assigned consumer surplus” is bigger than welfare (CSMaxW > WMaxW). So this consumer surplus is bigger than consumer surplus in all another market structures.
Total quantity of consumption Q = qN also can be easily derived from q, N.
Before comparing these equilibria we should discuss the interpretation of the parameters.
4.2. Interpretation of Parameters F, qNE and D
Fjz
First, let us interpret the important parameter T = —— , named so far as “preva-
2 F
lence of retailing fixed cost over the manufacturing fixed cost.” It would be helpful if we intuitively located this parameter in some area greater than 1 or smaller, not to study all mathematically feasible cases, but unfortunately, it is not easy.
What could F mean in reality? In the model, both constants Fr, F relate to whole production of any variety. So, their relation can be perceived as relation of shares of fixed cost within the price of a single variety. Up to our sketch data about the retail prices of Russian food items, the retailer’s markup is usually between 20% and 40% with average about 25% of the ultimate price, the rest 75% going to the manufacturer. Thus, if we believe in similar (about 15-20%) profitability in manufacturing and retailing, total cost-share c of manufacturer in the price of a commodity is about 4 times bigger than the retailer’s cost-share cR. Both costs can be divided into a fixed part modeled here as F/q( i) and a variable cost, i.e., c = c + F/q(i), CR = cR + Fr/q(i). Can FR be bigger than F when c/cR « 4?
At the first glance, it is quite improbable, because on the manufacturing side the fixed cost includes essential capital expenses (like renting buildings), which can
Y
be about 1/2 of total cost. However, in the long run capital becomes a variable cost. Then the “fixed cost,” more or less independent of the production volume, amounts mainly to advertising and intellectual capital of a company: analytics and main specialists with their accumulated knowledge. This cost can be below 10% of total cost. It is unclear, is such share similar or much bigger in retailing. Here “fixed cost,” spent on each variety, more or less independently of the retailing volume, amounts to maintaining the shell devoted to this variety, costs of bargaining with the manufacturer, of book-keeping this item, and certain advertising efforts of the shop. Naturally, selling a bigger diversity is more costly, under the same volume of sells. We would say, that if it is more costly almost proportionally to N, then the share of fixed cost FR/q(i) in retailer’s cost cR is close to 1. Elaborating this idea further, we can note that our fixed-plus-linear cost model F + cq is only a linearization of some real cost functions C(q) and CR (q). To make a good approximation, we need not ask a manufacturer about her capital stock and advertising expenses. Instead, we should ask about her current production q and total cost (the point (q, C(q)) of linearization) and about the derivative C (q) = c: “How costly it could be to increase production for 10-15% or how much you can economize from a decrease.” Then our “fixed cost” parameter F could be calculated as F = C(q) — qC (q). The same for retailer. In this view, concavity or convexity of real cost functions can make F, FR bigger or smaller, positive or negative, with small respect to expenses like advertising cost. Summarizing this discussion, we understand what kind of real data would be helpful to calibrate our stylized model and reduce the area of search for realistic effects. But we are unable to calibrate it now even approximately, so, all values of F are analyzed below. It turns out that this relation between manufacturing and retailing fixed cost essentially change the character of behavior in the industry.
Having explained this, another parameter q^E = \/F!ft____________r is explained as the
quantity of individual variety emerging at Nash equilibrium (Nash serves as a reference point for other equilibria). We observe that quantity qNE depends only on the manufacturer’s fixed cost and consumer’s variety-loving, being independent of the retailer’s parameters. Observe that the bigger is the entry barrier of fixed cost, the larger market space is captured by a single producer, that looks reasonable. Besides,
the stronger is variety-loving 3_____Y, the smaller is each manufacturer’s share of the
market at the Nash equilibrium, that also looks reasonable. The table shows that
similar reactions to these parameters F, 3_______Y are demonstrated also by equilibria
types ML and RL, only the individual quantity itself can be bigger or smaller than at Nash.
We observe also that wholesale prices at various regimes of competition include production cost and some additional term proportional to the individual quantity, which depends positively upon the fixed cost. Analogously is expressed the retailer’s markup.
Different is the formula for the diversity N, which positively (and linearly) correlates with parameter D, named so especially to highlight this correlation.
a — c — cr a — c — cr
y/F qNE
In the numerator there is the “interval for feasible prices” in this industry, i.e., the chocking price a (upper limit on the willingness to pay) minus the marginal cost of production and retailing c + cr. If this magnitude is zero, the industry cannot survive. The greater it is, the bigger is the pie of potential wealth that
can be split among the players in the game and deadweight loss. Besides there is
the “preference for diversity” parameter 3______Y. Naturally, D and the diversity both
increase when the numerator increases. The increasing denominator which is our “fixed cost of manufacturing” F brings the opposite natural effect. The smaller is F, the more competitive is production in the industry (entry is cheaper). Thereby, Dc describes the welfare potential of the industry multiplied by some specific measures of manufacturing competitiveness and love for diversity.
On the other hand, looking on expression a~c~CK we see that a market where
’ ° v qNE
individual quantity is bigger would have smaller diversity than another one, that means total quantity Q = Nq being more rigid with respect to changing parameters than individual quantity and diversity.
There can be also interpretation of D in terms of socially-efficient quantity qMaxW _ qNE^J2 + 4J7 obtained already and the minimal-profit quantity qMm that guarantees nonnegative total profit to retailer and any manufacturer, found from non-negative profit condition (a — c — cR)qMin — (F + Fr) > 0.
qMaxW
Combining these, we find that D = /3_____Y\ T + —------Min . Now we see that the
V 2 q
potential of the industry to generate profit also positively affect D and, respectively, the diversity, but instead of price-type expression a — c—cr of this potential we have
qMaxW
the quantity-type expression q Min . Again, preference for diversity /3______Y affects D
q
and the diversity positively.
In addition, the above formula allows to linearly express the socially-optimal
diversity ]\[MaxW through the potential of the industry:-----^ = 2p—NMaxW-у
l3-~/V'F+2 p—і
qMaxW
2 = —Min , that once again highlights tight connection of parameter D with
diversity.
4.3. Market Concentration Impact on Welfare, Quantities, Prices and Diversity
Now we are ready to compare the outcomes of equilibrium types. Here we are mostly interested in the welfare effects caused by market “monopolization” by big chain-stores, after situation modeled as RL emerged.
Which model or equilibrium concept better describes situation in Russia before the concentration that occurred in 2000-s: NR, Nash or ML? Definitely, retailers did exist before chain-stores and each had certain degree of local market power in a city district. So, for our comparison, we reject NR model which assumes monopolistic competitive retailers8. To choose between Nash and ML models, we guess, did manufacturers exercise some market power in bargaining with each retailer, or did they react myopically to given markup r in the market? The first hypothesis seems most plausible, and occupy our main attention, but the second hypothesis is also analyzed, with similar outcomes.
A puzzle to be resolved when interpreting these comparisons, is identifying our single retailer in both models (Nash or ML) with multiple retailers in reality. Whenever we believe in local market power remaining approximately the same af-
8 This case can also be interpreted as “manufacturer’s outlet”, i.e each producer has the own “manufacturer’s outlet” store. Of course, this interpretation is valid under proper
definition of marginal (c + cr) and fixed (F + FR) costs.
ter retailing property concentration, the main change could be in costs. Probably, in Russia the costs of selling did not change too dramatically because of ownership concentration itself (leaving aside other causes). Of course, chain-stores economized somewhat on unified book-keeping, unified marketing specialists and, notably, on unified logistics. Still, we prefer to ignore this change and do model the selling cost function Fr + cqR as remaining the same during concentration, for two reasons. First, we want to analytically separate the market-concentration effect from economies of scale in retailing. Second, since the welfare effect of concentration is found being positive, our conclusion should be only enforced if we take into account these economies of scale also.
Effects of Switching from ML to RL. In addition to these precautions, we also should combine inequalities NML > 0, NRL > 0 with equilibria characterization and
get the following conditions on combinations of parameters F and D/ 3______Y ensuring
non-empty market:
_ D/3-7 > 4 • y/T if T > 1, , .
D/3-7 > max{2 • a/2, 2 • (J7 + 1)} if T < 1.
Now, under these restrictions, we can formulate the main results on the market concentration modeled as switching from ML to RL regimes.
Welfare, Consumer Surplus, Retailer surplus. Algebraic manipulations with the equilibria formulae obtained yield the unambiguous welfare conclusion:9
Restricted market: Jr= ypr > 1 => wML < WRL
Unrestricted market: T= ^ < 1 => wML < WRL
Result 3 Generally there are positive social welfare gains from market concentration,, similar in restricted and non-restricted markets.
To comment on this result, the welfare increase may seem quite counter-intuitive from the first glance. Normally one expects social losses from market concentration, and exactly such anti-trust reasons stay behind recent proposal by Russian Duma of the new law restricting chain-stores. Why any gains may occur instead of losses expected?
To answer, we first give general reasons, and then look carefully on quantity and diversity changes during the concentration. Generally, it is a textbook result in IO that a two-tier monopoly causes higher deadweight loss than a simple monopoly. So, when a two-tier monopoly becomes vertically integrated through ownership concentration, it can be quite beneficial for society. In our setting something similar occurs under horizontal concentration, only the changes in the diversity complicate the picture. Indeed, when a leader-manufacturer monopolistically optimizes her price and has in mind the optimal response of the (locally) monopolistic retailer, this situation is rather similar to the two-tier monopoly. In contrast, when the monopsonistic (and monopolistic also) retailer exercise essential market power over the manufacturers, it is somewhat similar to vertical concentration, bringing the decision-making into essentially one hands. In this general view, the welfare gains are not too surprising.
As to consumer surplus CS, the situation is not so simple. The complete solution is given below.
9 Equality WML = WRL in case F < 1 occurs as a degenerate case, only under specific parameters -jp— = 2\/2 and J7 = \/2 — 1.
Result 4 During switching from ML to RL regimes, welfare is changing, depending on three regions of parameters, as follows:
Parameter regions consumer surplus SC
p ^ tt ^ 3\/2 — 2 U > 4J7+2—3a/2 ’ ^ ~ 4 CSRL > CSML
p „ 4-(^+^—1) 4T+2-3\/2 CSRL < CSML
Fig. 1. Comparison of Consumer Surplus between ML and RL regimes.
Figure 1 illustrates these two cases of market-concentration effects in consumer surplus10. Below these regions either NML or NRL is negative, so the model is inappropriate.
Finally, comparing the retailer’s surplus (i.e. retailer’ profit), we can conclude that the situation is obvious and similar to the one for welfare (see Result 2): if the market exists (i.e. NML and NRL are non-negative) then the retailer’s profit is bigger when retailer is leader then when she is follower: nRL > nML ■
Result 5 Generally there are positive profit gains from market concentration, similar in restricted and non-restricted markets.
Now we should look on market concentration more specifically: are the gains in consumption volume, or in the diversity, or in profits (recall that our representative consumer is the owner of all firms) responsible for benefits to the society?
Quantities, prices and diversity. We again perform algebraic manipulations with equilibria formulae to find regions of parameters for several inequations of interest that could explain the welfare gains: qML <? qRL (increase in the consumption of a single variety), QML <? QRL (increase in the total quantity of consumption), pML + rML >? pRL + rRL (decrease in the retail price), NML <? NRL (increase in the diversity). Using again non-empty market condition (7), we get
Result 6 During switching from ML to RL regimes, prices, quantities and diversity are changing, depending on four regions of parameters, as follows 11:
10 In the figures we use the notation D = D/.
11 Note that the last region of parameters includes both restricted (1 <F) and unrestricted markets.
Parameter regions Quantities Retail prices Diversity
T < mm{y/2 1,1 2+/^} qML ^ qRL qml < qrl pML _|_ r^UlL pRL _|_ r^RL nml < Nrl
I D/I3-, j- < /2 y 1 2+2^2 - V/ 1 qML ^ qRL qml < qrl pML _|_ r^UlL pRL _|_ r^RL nml > nrl
a/2 - 1 < T < ^ qML ^ qRL qrl < qml pML _|_ r^UlL pRL _|_ r^RL nml > nrl
« § ^ O' V V § £ pAdL _|_ rML ^ ^_R_L -|_ r^RL nml > nrl
Moreover, in comparison with MaxW, during switching from ML to RL regimes, diversity is changing, depending on four regions of parameters, as follows:
Parameter regions Diversity
1. 2a/2 < D//3_7 < L>fL J\fML ^ J\fRL ^ j\j~MaxW
2. D^l < ZV/3_7 < DfL J\fRL ^ J\fML ^ j\j~MaxW
3.1. i < T < 1, DfL < D//3_7 < DfL J\fRL ^ j\j~MaxW ^ J\fML
3.2. T> 1, Djfi-1<DfL J\fRL ^ j\j~MaxW ^ J\fML
4.1. i < T < 1, L»//3_7 > L j\j~MaxW ^ J\fRL ^ J\fML
4.2. T > 1, D//3-T > ■L j\j~MaxW ^ J\fRL ^ J\fML
where DfIL= 2 • (1 + V2)(l - n Dfh = V2(l+ _ J ,
D¥L = , DfL = 2y/T{\ + 21) (VI + 2F + V2F) .
Figure 2 and Figure 3 illustrate these cases of market-concentration effects in prices, quantities and diversities described in the both tables of the Result 6.
Fig. 2. Comparison between ML and RL regimes in quantities, prices and diversity:
Region 1: gJWi, < gKi, pUL+rUL <pML+rML j\rML <
Region 2: qML < qHL pUL +rHL < pML +rML
Region 3: qHL < qML ^ pM L _|_ ^1V1 L
Region 4: qHL < qML pML +rML < +r«^
Fig. 3. Comparison between ML and RL regimes and MaxW in diversity:
Region 1: j\^ML ^ j\j-RL ^ jyAdaxW
Region 2: jyKL ^ ^ jyMaxW
Region 3: jyKL ^ jyMaxW ^ jyML
Region 4: jyMaxW ^ jyKL ^
To discuss these results, the first question of interpretation is: Why, depending upon parameters F and D, it turns out more profitable for the retailer to shift the equilibrium in one or another direction when she obtains the monopsonistic power? Before obtaining it, each retailer takes the diversity and wholesale price as given when choosing the markup, now the unique retailer forecasts her influence on these variables. The more is relation F of retailer’s fixed cost to manufacturer’s cost, the more is their foregone joint profit when the free entry determines the excessive (from the profit viewpoint) diversity. This the explanation why fraction NRL/NML becomes smaller and smaller with increasing F in Fig.2. Prices and quantities adjust to this general tendency.
Second question is: How, by which logical ties, the social welfare increases through these changes in quantities, prices and the diversity? Generally there are three variables pleasing the consumer: total quantity Q, diversity N and profit of the retailer, owned by the gross consumer (plus zero profit from manufacturers). To this end, in this quasi-linear setting we can reason in terms of usual Marshallian diagram. After symmetrization of each equilibrium, the maximal possible utility gains can be described by the simplified inverse-demand function:
V(Q,N) = a-(j+£j^Q.
We can see that the chocking price a remains the same, but increasing diversity N can stretch the demand triangle rightwards, the right corner approaching the limiting value y/a. In this respect, increasing N enables to increase the common pie of welfare, divided among the consumer, the deadweight-loss and profit ultimately also belonging to consumer. On the other hand, increasing N increases average cost c = c+cR + (F+Fr)N. Geometrically, it shifts up the effective cost line and thus reduces the common pie of welfare. The socially optimal diversity NMaxW expresses the good balance between these two forces. In contrast, any market equilibrium
brings distortion in two respects: (1) non-optimal diversity N and (2) monopolisti-cally reduced total quantity, which brings a triangle of deadweight loss.
Trying to apply this logic, first we observe in Table 3 that at all equilibria the total quantity consumed is strictly less than the socially-optimal quantity, so social loss is always present, and the found positive impact on welfare can be understood as minimizing this loss. Decrease in the diversity is also always present, which can be beneficial when the diversity is excessive.
Besides, generally shift in prices and total quantity during the market concentration have the opposite sign. Only in case a/2 — 1 < T < 1/a/2 it seems unnatural: the quantity is growing, when the price is growing, resembling the Giffen effect, probably this effect could be explained through the equilibrium number of manufactures, which decreases.
Further, individual quantities always increase under market concentration, confirming our general reasoning about concentration impact on a single variety: the strategic behavior of the manufacturers, similar to two-tier monopoly, yields lower total quantity than the strategic behavior of the retailer.
Other equilibrium variables including total quantities behave more complicatedly. Under small F (relation of retailing fixed cost to manufacturing one) total quantity Q grows, alike individual quantities. Here the welfare gains can be explained by growing quantity. But it is not the case under large immoderate F. Here both total quantity and diversity go down during market concentration, especially when the retailer starts restricting the entry. So, here the welfare gains can be either because of reducing too big initial diversity NML (of excessive number of producers) or because of growth of profits, ultimately going to the consumer.The retail prices (p + r) grow during market concentration under large F, but decrease under moderate or small F, but they have only indirect impact utility, through transferring the consumer surplus into profits.
As to the diversity, or the number of manufactures, under large and moderate relation F, it decreases. Excessive diversity can be harmful, because requiring too many producers and excessive total fixed costs. In particular, at ML regime the diversity can be excessive, so, reducing the diversity during market concentration can add to positive quantity effect under small F. In the case of very big F such reduction is intentionally forced by the restrictive policy of the manufacturer and can be stronger.
We conclude from this subsection that changes in equilibrium variables explain some mechanisms of welfare gains during market concentration.
Effects of Switching from Nash to RL. For more complete discussion, we perform now the Nash ^ RL version of modeling the market concentration, to show that the effects are somewhat different.
Taking again the precautions NNash > 0, NRL > 0 we get the conditions on combinations of parameters F and D ensuring non-empty markets:
D //3__j, > 4 • VT if .F> 1,
(8)
D/jS-Y > max{3, 2 • (F +1)} if F < 1.
Now, under these restrictions, we can formulate the main results on concentration modeled as switching from Nash to RL regimes.
Welfare. Algebraic manipulations with the equilibria formulae obtained yield, using again non-empty market condition (8), the unambiguous welfare conclusion:
Result 7 During switching from Nash to RL regimes, welfare is changing, depending on three regions of parameters, as follows:
Parameter regions Welfare
\yMash < yyRL
yyNash, > yyRL
( 2 + V^-3-2 A/3-7< 4 yyNash, < yyRL
For the consumer surplus CS, the situation is more simple12:
Parameter regions consumer surplus SC
1. Small: T < — CSNash < CSRL
2. Large: T > — CSRL < CSNash
Figure 4 illustrates these cases of market-concentration effects in welfare and consumer surplus.
Fig. 4. Comparison between Nash and RL regimes in welfare and consumer surplus.
We see that welfare increases during switching from Nash regime to RL regime only under big or small relative fixed cost F, but decreases under moderate one; while consumer surplus increases during the switching only under small relative fixed cost F. As an interpretation of the Figure 4, remark that, since in the middle region W> WRL, Nash outcome here is more “competitive” than outcome under the leadership of retailer.
Quantities, prices and diversity. We again perform algebraic manipulations with equilibria formulae to find regions of parameters for several inequations of interest: qNash <? qRL (increase in the consumption of a single variety), QNash <? QRL (increase in the total quantity of consumption), pNash + rNash <? pRL + rRL (increase in the retail price), NNash <? NRL (increase in the diversity).
Using again non-empty market condition (8), we get
12 Cf. the case of switching from ML to RL, see the Table in Result 2 and Figure 1.
Result 8 During switching from Nash to RL regimes, prices, quantities and diversity are changing, depending on two regions of parameters, as follows:
Quantities Retail prices Diversity
qNash qNash ^ pNash _|_ ^Nash ^ pRL _|_ ^RL pjNash ^ J\J~RL
qNash ^ qRL QRL ^ qNcisJi pNash _|_ ^Nash ^ pRL _|_ yRL pjNash ^ J\J~RL
Moreover, in comparison with MaxW, during switching from Nash to RL regimes, diversity is changing, depending on four regions of parameters, as follows:
Parameter regions Diversity
1. I <5, D//3-J > 3 j\jNash ^ J\JRL ^ j\j~MaxW
2.T>VTy D/p-7 < Drl JSJ'RL ^ j\j~Nash ^ j\j~MaxW
3.1. i < T < 1, DfL < D/P-y < Drl JSJ'RL ^ j\j~MaxW ^ j\j~Nash
3.2. T > 1, D/P-7 < D§L JSJ'RL ^ j\j~MaxW ^ j\j~Nash
4-1. \ < I < 1, D/P-7 > D^L j\j~MaxW ^ JSJ'RL ^ j\j~Nash
4-2. T > 1, D/P-7 > D^L j\j~MaxWL ^ JSJ'RL ^ j\j~Nash
whm Dr = 1 + = 2J- (l + ■
Drl = 2\JT ■ (1 + 2T) (Vl + 2T + V2iF) .
Figure 5 illustrates these four cases of market-concentration effects on quantities, prices and diversity. Remark that in Figure 3 and Figure 5 the fourth regions are the same.
Fig. 5. Comparison between Nash and RL regimes and MaxW in quantities, prices and diversity:
Region 1: QiVasft < qHL pRL _|_ p-K-k < pMash, _|_ rl\ash j\jl\ash ^ j\j-RL ^ j^j-MaxW
Region 2: qHL < QJVasfe pN ash _|_ ^l\ash ^ p^L -|- j\jRL ^ yyiVas/i ^ j\jMaxW
Region 3: qKJ, < QJV“sft pNash _|_ rl\ash ^ p^^ -\- fKL j\j-RL ^ jyMaxW ^ j\jl\ash
Region 4: qHL < QJVasfe pN ash _|_ ^l\ash ^ p^L -|- j\jMaxW ^ j\jRL ^ j\jl\ash
To interpret these regions, we should say that, like in previous figures, the more is fraction F = Fr/F between the retailer’s and the manufacturers cost, the more is
the need to diminish the excessive variety, determined by the free entry without the respect for the retailer’s losses from the excessive diversity (free entry pays respect only to F). When switching to RL regime, the retailer becomes capable to influence this excess (from her viewpoint) diversity directly or indirectly.
We did not compare No-retailer equilibrium with RL-equilibrium because we do not suppose NR case realistic enough. Never substantial share of the retailing market was covered by the shops belonging to producers. Instead we use this model as theoretical reference point: vertical integration of the industry performed from above, from the producer’s side. It is compared in the next section to vertical integration from below, from the retailer’s side.
Generally, in this section we have seen that a conclusion about welfare benefits of market concentration, switching from one or another previous industry organization may depends upon demand functions and technologies, but welfare gains from market concentration is quite plausible. Therefore, in spite of noticeable public complaints about the retailer’s bargaining power (mentioned in Intro), this power and leadership in relations with manufacturers can occur socially desirable. In this case an economist should not advice a governmental intervention against chain-stores and similar practices, recently suggested by Duma. Section 6 devoted to regulation adds arguments in the same direction.
Possible extensions of the setting. We conclude this section devoted to market concentration effects by the question: Which is the most realistic model of pre-concentration retailing? Instead of two alternative models that we could use several other versions. First, we could study a market with direct selling by manufacturers without any retailing shops, but it seems less realistic (as suggested by discussants in our presentations). Second, the case when manufacturers have bigger market power than retailing sector, seems irrelevant to pre-concentration regime in Russia and similar countries, because several manufacturers jointly controlling common retailer seems unrealistic.
It means that manufacturers behave simultaneously but strategically towards the retailing sector and consumers, percept as their common agent in principal-agent relations, and considering externalities onto each other. It seems not a good model for our question. Indeed, before market concentration, typical manufacturer of consumer goods in Russia sold to several shops shared with several other suppliers. For big manufacturers the markup was the result of individual bargaining with a shop (like in ML equilibrium), for small ones it was a market constant (like in Nash equilibrium). Anyway, a common-agency story seems irrelevant. Instead, a more sophisticated and realistic description of concentration would like to undertake two detailizations of the model.
(1) We can consider big and small manufacturers or/and retailers, describing concentration as appearance of more and more big chain-stores among the competitive fringe of small shops.
(2) In addition, spatial structure of the market should be described somehow to explain why in reality the same varieties are sold for much smaller prices in chain-stores outside the city center than in small shops in the center.
However, both extensions wait for another paper.
5. Entrance Fees
Now we describe an extension of our setting motivated as follows. In Intro we have mentioned that trade relations between the manufacturers and the retailers often involves more complex terms than just a markup. Rather typically, to start and maintain selling anything through a chain-store, a manufacturer is forced to pay an entrance fee to the retailer, annually or monthly. Denoting this per-period fee as FE we can look on it as an addition to the manufacturer’s fixed cost, that becomes now F = F + FE . This amount is subtracted from the retailers fixed cost that becomes Fr = Fr — FE . This fee FE is as a new pricing tool optimized by the retailer simultaneously with optimizing the markup and the diversity. The regime studied is when the retailer is leading the game.
We want to know: What happens to price and variety because of the two-part tariff practice? Is this practice really harmful for society, as manufacturers and many journalists and politicians suggest in the cited papers (Nikitina, 2006) and (Sagdiev et al., 2006), or not? We are going to show positive gains of the entrance fee. So, the government and legislature should not do what they do now, prohibiting the fee.13
Equilibrium with entrance fee. To support our hypothesis, we modify the retailer’s optimization program and the equilibrium formulae for the case ”RL”. The profit function of i-th manufacturer becomes
nM(i) = [p(i) — c]q(i) — (F + fe ) and the profit of the retailer is now
p N p N
nR = [r(i) — cR]q(i)di — (Fr — Fe )di.
J 0 J 0
Since the retailer is the leader, in symmetric equilibria we have the same expressions as before (see (5)-(6)) for the wholesale price p and quantity q of each variety as functions of markup r and diversity N:
= (r N) = 0b + gN)[a-b(c + r)] a + (b + gN)c-br
q q{, ) 2b + gN ’ P P{, ) 2b + gN '
Therefore, the new profit-maximization problem of the retailer is
nR = N[(r — CR)q(r, N) — Fr + Fe] ^ maxrjN,FB, nM = [p(r, N) — c]q(r, N) — (F + Fe) > 0 .
Since
dnR
dFp,
N > 0, (9)
the objective function should attain a boundary maximum, so restriction on nonnegative profit -km is active in the equilibria, i.e. free entry condition -km = 0 holds.
13 Notably, Russian parliament has just adopted a bill (”Law on trade...”) that prohibits the entrance fees in retailing, because they are supposed harmful for competition and welfare.
Hence
Fe = -F+ \p(r, N) - c]q(r, N) = —F + (10)
b + gN
Substituting this into nR we obtain the unconstrained optimization problem
q2 (r N)
7TR = N[ ’ + (r - cn)q(r, N) - (Fr. + F)] max.
b + gN r,N
Its solution is
= V1 + 2^--------— , rE = cn + qNE - /3_7v/T+2f) .
Surprisingly, for each diversity the optimal entrance fee equals exactly to the fixed selling cost Fe = Fr. The quantity is qE = </at£\/1 + 2F and the wholesale price is pE = c + </jvs/3_7VT^P2F.
Finally, we calculate profit, social welfare and consumer surplus under two-part tariff:
CS- = ^-{D-2VTTW.^)b,
WE =nE + CSE = ■ (D - 2v/TT2F • /3_7) VT+2F ■ /3_7^) .
Comparing all the above formulae with the ones in Result 2 (see page 11), it is easy to derive the following consequences of introducing the two-part tariff.
Result 9 When the leader-retailer introduces the entrance fee,
— pE > pRL, i.e. entrance fee always pushes up the wholesale price, thus compensating the fee to manufacturers,
_ rE < rRL, i.e. entrance fee decreases the markup, and, surprisingly,
— pE + rE < pRL + rRL, i.e. entrance fee always pushes down the retail price!
— qE > qRL, i.e. the entrance fee always increases quantity of each variety,
— Qe > QRL, i.e. the entrance fee increases total quantity of the industry,
— WE > WRL,CSE > CSRL,nE > nRL, i.e. entrance fee always increases social welfare and consumer surplus.
As to the diversity N, under the fee NE can be less or more than NRL, depending on two regions of parameters D and F as follows:
Parameter regions Diversity
-Or >DHL N-E < NRL
dhl > > Drl P — 'Y Ne > Nrl
where bounds of these regions are the following continuous functions of F :
pRL _ f 4%/F, F > 1; jjRL _ f (l + Jy + 2) • VI + 2F • , F > 1;
— I 2(F + 1), F < 1; \ (1 + VTT2F) • VTT2F, F < 1.
Fig. 6. Comparison between RL and Entrance fee regimes in diversity.
Figure 6 explains the comparison in diversity: when NE < NRL and when NE > NRL. One can see that when the retailer’s fixed cost is sufficiently bigger than manufacturer’s one and the slack between the chocking price and costs is small, the entrance fee increases the diversity.
Why the optimal entrance (shelf) fee becomes equal to the retailer’s fixed costs? To understand the result obtained, let us note that an increase in entrance fee (in first approximation) leads to the growth of retailer’s profit (it is easy to see, for instance, from (9)). On the other hand, when entrance fee increases, the fixed costs of producers also increase. These additional costs must be covered by higher
dN
wholesale price or/and by bigger per-firm output. Therefore, generally < 0,
dr e
i.e, the number of varieties (i.e. the number of firms in the industry) decreases14. This tendency hampers the increase in the retailer’s profit. The interaction of these two forces determines the equilibrium entrance fee. For the linear functional form of demand, it turns out that resulting entrance fee equals to the retailer’s fixed costs. Probably (in our opinion), for other functional forms the optimal entrance fee may be bigger or less than the retailer’s fixed costs, depending on the value of these opposite forces.
Another way to understand this effect is the comparison of entrance fee practice with the vertical integration of the industry. We use such comparison below to explain why the entrance fee increases social welfare and even the consumer surplus (in contrast to what Russian legislature thinks about it).
This increase becomes not so surprising, if we recall that generally in many market situations it is not the monopolism per se responsible for the welfare losses, but the inability of the monopolist to use the efficient tools of pricing like perfect price discrimination, etc. In particular, the entrance fee works here exactly like in typical situations where two-part tariff compensates the fixed cost and increases the efficiency of the market through making prices closer to costs. Here the markup works like a price for manufacturers to get the service of retailer. Thus, the welfare increase under entrance fee has the common nature with a two-part tariff.
If we substitute in (10) the expression (see (5)) q = q(r, N) = Kc+r)l ; then
fe+f= (Tr;+v;r7 -s°’^<a
) -0 — .yQ dN
14
More light on increasing welfare is shaded by the following argument. Suppose that starting from situation with entrance fees the next step in retailer’s enforcement is undertaken: vertical integration of the industry. It means that the monopolistic retailer becomes the owner of the whole industry with its manufacturing and retailing. She chooses the retail price and the number of varieties (the product line), bearing all manufacturing and retailing costs. Does this integration change the market outcome and her profit? The surprising answer is “no”.
To show this, we maximize the owner’s profit n = ((pR — c — cR)q(pR, N) — F — Fr )N with respect to the length N of the product line and the retail price pR equal to previous p + r. Here, as previously, the demand for a variety i is qi(pR, N) = a — (b + gN)piR + g PjRdj where a, b, g are dependent on N as in (4). Taking into account that the solution should be symmetric, we replace pi and pj by pR and get
<">
Maximizing profit and simplifying the result we get the market outcome (qMon,pMon, NMon) of such integrated monopoly:
Mon _ lF + Pn_V Mon _ a + C+ Cr _ E , E
Q —\ ---Q--- ~ Q 1 PR — --7>- _ P + T !
i3-
Y
NMon =Pzz(_2+ a-c-c-R \ = ne
27 V y/{F + FR)^j ’
which turns out exactly the same as under retailer leadership with entrance fee. Naturally, profit and welfare also coincide. Thus, we have proved that combination of two pricing tools: entrance fee plus markup is equivalent to integrated monopoly. With their help the retailer fully controls the industry, to the benefit of consumers, neutralizing the shortcomings of two-tier monopoly and resolving the negative externalities among manufacturers.
The next improvement in the same direction, aiming to get first-best Pareto efficiency would be exercising the entrance fee FC for consumers (using two-part tariff instead of linear pricing). To show this, use (5)-(6) under symmetry assumption and maximize such price-discriminating monopolist’s profit
nDMon = ((pDM — c — cR)q(pDM, N) — F — FR)N + FC ^ max
Pdm ,N,Fc
Fc < [a - ~~q{pDM; N) — ^Nq(pDM, N) — pdm]Nq(poM, N).
Here the right-hand side describes the potential consumer surplus from using the diversified industry. This CS is completely captured by the monopolist through the consumer’s entrance fee (nothing remains actually), but it returns to consumer in the form of monopolist’s profit and spent on other goods. It can be observed from the above program that the monopolist just maximizes the consumer’s welfare, and the outcome will be price equal to marginal cost, efficient quantity, see Result 2, and efficient diversity NMaxW :
„DMon „ 1 „ „DMon /0 F -\- FR
PR c + cR, q _J2.-—,
FC
a — c — cr
2y
a — c — cr — J2 • (F + Fr)fi—
N
DMon
ft- 7
27
2
a — c — cr
y/(F + FR)3-
= V2
DMon
= WDMon =(d~ \J2(1 + 2Jz)(j3 - 7))
Comparing with previous results, observe that qDMon = qMaxW > qMon = qE,
N DMon = N MaxW > N Mon = N E W DMon = w MaxW > W Mon. therefore allowing for more powerful and sophisticated monopolist increases both consumption and diversity, thereby increasing welfare (as sophistication typically does in IO).
Our conjecture is that this market outcome can be obtained indirectly, namely a retailer-leader can use both entrance fees (FE for manufacturers and FC for consumers) instead of integrating the industry to arrive at the same first-best outcome as we have just found for integrated monopoly. Then, usage of discount cards payed by consumers in some shops (entrance fees) can be welfare improving.
Finally, let us compare Entrance fee equilibrium with NR equilibrium. It turns out that qNR = qE, Nnr = 2NE, i.e., under NR regime, the quantity equals to the quantity in the situation with Entrance fee, while the diversity is two times more then the diversity in the situation with Entrance fee.
So, we find the correspondence between NR and Entrance fee regimes. Therefore, it seems interesting to compare consumer surplus and welfare for these regimes. One has
CSE
F
(d - 2a/1 + 2T ■ /3_7) D,
WE =
3 F
8^.
CSNR = WNR
Hence
CSNR - CSE =
3 F
8^,
■ (jj — 2a/1 + 2J- ■ /3—— — • a/1 + 2J- ■ /3—y^j ,
• i^D — 2a/ 1 + 2J- ■ /3—y^j ■ ■ a/1 + 2J- ■ /3—y^j
■ (3 - 2a/1 + 2J7 ■ /3_7) • (d - | • a/1 + 2J- • /3_7^) > 0
WNR - WE =
F
8l!3-
(d - 2a/1 + 21 ■ /3_7) • D >
0,
i.e., under NR regime, welfare and consumer surplus are bigger then the correspondence ones under Entrance fee regime.
For the government, the ideas of this section suggest refraining from any regulation, when profits are transferred to consumers and redistribution needs does not force the regulation. The next section expands these ideas.
R
Y
6. Governmental Regulation of Retailing
Based on the above analysis, this section considers the outcomes of several possible measures that the state may wish to use, under one or another market situation. Namely, when the retailer’s leadership after market concentration arise, it may tempt the state (the government or legislature) to restrict the market size of each firm, or/and the markup or/and restrict entrance fee when it is used (exactly like the Russian legislature did).
Section 4.3 about concentration showed that restriction of concentration by the state can be detrimental, at least under our assumptions.
As we have seen in the previous section, prohibition of entrance fee for manufacturers is detrimental for welfare. Now we study alternative measures.
6.1. Government Intervention through Capping Markup
Consider the specific situation when the government starts regulating the markup of the retailer-leader who does not use any entrance fees. So, regulated markup r maximizes welfare, being chosen by the government who correctly anticipates the subsequent choices of retailer, producers and consumers. The retailer-leader chooses only variety N within the constraint of non-negative producers’ profits, which is typically binding.
To solve the equilibrium by backward induction, we have seen in (5)-(6) that the output and wholesale price of a firm in the long-run is always
obtaining her best response N = N(r) to the markup regulated by the government. Anticipating this N(r), the government solves the welfare maximization problem
We restrict our study only to the important case when the parameters of the model make “the non-negative manufacturer’s profit” condition nM > 0 active, i.e. the retailer becomes a dummy in the game whereas free entry determines N(r) 15. Then, it is easy to see that under direct intervention of the government into pricing, quantity qd, wholesale price pd, markup rd, diversity Nd and social welfare Wd become
, p = p(r, N)
a + (b + gN)c — br
2b + gN
Having this in mind, the retailer solves the problem
nR = N[(r — cR)q(r, N) — Fr] ^ max,
(12)
nM = [p(r, N) — c]q(r, N) — F > 0
W = W(r) ^ max .
r
15 The class of parameters generating the opposite type of equilibrium (restricted entry) is not so easy to analyse. Besides, the regulating measures hardly can be confined to capping markup in this case, so a richer model is needed.
D ~ 2/3--------------;
Nd = ^^--------- , 74= (D-2/3-t(1 + T)
CSd = (D — 23—r • (1 + F)) • (D — /3-y • (1 + F))
= (d - 2/3_7 • (1 + F)) ■ (d - • /3_7 • (1 + F)^j
2 • (1+ F)2 y3—y 1 + 4J7 ^ 1 + 2J7 F
2-(l+-^)2 '7^ '
Comparing this outcome with the previous results we get the following comparison of the direct intervention of the government with the “RL” (free entry!) case 16.
Result 10 One has
— qd = qRL, i.e direct intervention of the government does not change the quantity of each variety
— pd = pRL, i.e direct intervention of the government does not change the wholesale price
— rd < rRL, i.e direct intervention of the government d,ecrea,ses the markup
— Nd > NRL, i.e direct intervention of the government increases the diversity
— Wd > WRL, i.e direct intervention of the government increases the social welfare
Now, using the above formulae, we can compare diversities Nd NRL with the social optimal diversity NMaxW. A natural conjecture is that socially efficient diversity is the largest among NRL, Nd, NMaxW. Surprisingly, it is not true in general. The table below shows that all three cases NRL < Nd < NMaxW, NRL < NMaxW < Nd, NMaxW < NRL < Nd are possible:
Parameter regions Diversity
1.2(1 + F) < -P- < DMaxW, F < 1 P — ’y J\J~RL ^ j\jd ^ pjMaxW
2.1. DMaxW < F < 0.5 P — ’y J\J~RL ^ pjMaxW ^ j\jd
2.2. DMaxW < -fp-, F > 0.5, -p- < DMaxW J\J~RL ^ pjMaxW ^ j\jd
3. DMaxW < -fi-, 0.5 < F < 1 P — ’y j\jMaxW ^ J\J~RL ^ j\jd
where
DMaxW = _4 _ V2 + AF + 14F + 6 + (4F + 2)3/2
-rprMaxW ^ /—----
D = 3 + V2 + 4F +
—F2 + 2F + 1 3 + yfAF + 2
2F — 1
Moreover (see Figure 7), region 1 where NRL < Nd < NMaxW is rather small, while the other two zones are large.
Thus, we conclude that direct intervention of the government may lead to “over-production of diversity” and in the case when the diversity is already too large, the intervention aggravates “over-production of diversity”.
Further, we can ask: Does the government enhance social welfare when it prohibits entrance fee (previously existing), and simultaneously restricts markup from above? Some ideas for the answer we can find in the table below
Let us recall that we consider here only free entry case, hence F < 1.
16
Fig. 7. Comparison between RL, Capping markup and MaxW regimes in diversity:
Region 1: j\j-RL ^ j\j-d ^ jyAdaxW
Region 2: j\j-RL ^ j\^MaxW ^ j\jd
Region 3: jyMaxW ^ j\j-RL ^ j\j-d
Parameter regions Welfare
D > jjEd P — -Y WE < Wd
2(1 + F) < <DEd, T < 1 P — -Y WE > Wd
”4irrr r>Ed — cl — 5Vl+2^ I 5 0^0 -1 — ^
where D ~ d1+^di+d2 ’ dl - + i+^ - 8 < d2 - -
0 VFe [0; 1].
Comparing such intervention with entrance fee situation, we see on Figure 8 that
capping markup is better for welfare than entrance fee where D = (o. — c — cr)aJ
is big and T = small, but the opposite is true in another region. So, when retailer’s fixed cost are sufficiently bigger than manufacturer’s one and the slack between the chocking price and costs is small, the government should not intervene in this fashion.
Fig. 8. Comparison between Entrance fee and Capping markup regimes in welfare.
6.2. Pigouvian Regulation: to Tax or to Subside the Retailer?
Suppose that the government stimulates the market in Pigouvian way. The retailer pays the tax t per every unit of goods sold. Then the retailer’s profit takes the form
,■ n ,■ N
/ [r(i) — (cr + t )]q(i)di — Fr di.
Jo Jo
The collected taxes are redistributed among the consumers in a lump-sum manner. In the case of negative t it means a Pigouvian subsidy, funded from a lump-sum taxes on consumers in amount NF = r q(i)di.
The welfare function WRL(q(T), N(t)) has the previous form, only equilibrium variables </(t), N(t) become the functions of tax t. They, and welfare, can be found from the previous formulae. Knowing these equilibrium expressions, the government can maximize the welfare function with respect to t, and find the taxation policy that maximizes the social welfare. We did not solve this problem here (though it is possible), finding instead only the sign: Should the government start taxing or subsidizing the retailer for a small amount? In other words, the question is: When it is beneficial for the government to tax the retailer (t > 0), or to subside her (t < 0)?
Result 11 Under the strategic behavior of the retailer, it is beneficial for the government to subside her, i.e. t* < 0.
This fact seems unnatural in anti-trust logics: the state should subsidize a monopolist/monopsonist. However, such effects are rather typical in IO literature and the proofs are simple. We just have ensured that our more complicated situation with varieties keeps the same effect.
7. Conclusion
We study the model where the monopolistic-competition industry is selling many varieties through the monopsonistic retailer. The emergence of such retailer with a two-sided market power is compared to initial situation, when the manufacturers were selling through many retailers having a local selling market power but no buying power. Thereby we model the welfare and marketing effects of the market concentration after the emergence of big chain-stores in Russia and other CIS countries.
In the special case of quadratic valuations (linear demands) it turns out that generally the concentration enhances social welfare. This effect, unexpected for noneconomists, actually is not too surprising, because it reminds the switch from a two-tier monopoly to a simple monopoly. The welfare increase here, however, can go not only through the growth of the total quantity, but in some cases through growing total profits and, notably, through decreasing excessive diversity. The retailer is shown to be generally interested in restricting the diversity, but it is not harmful itself.
In the case of the governmental regulation of the monopsonistic retailer through Pigouvian taxes, it turns out that subsidies instead of taxes are needed, that also reminds similar effect known for a simple monopoly/monopsony.
The anti-chain-stores bill recently suggested by Russian Parliament (Duma) shows the public interest in this type of questions.
Acknowlegments. We are grateful to EERC experts, especially to Prof. Richard Ericson, Prof. Russell Pittman, Prof Shlomo Weber, Prof. Michael Alexeev, Prof. Oleksandr Shepotylo and Prof. Gary Krueger for the attention to the research and useful suggestions.
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