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General Equilibrium with Free Entry: A Synthetic Approach to the Theory of Perfect Author(s): William Novshek and Hugo Sonnenschein Source: Journal of Economic Literature, Vol. 25, No. 3 (Sep., 1987), pp. 1281-1306 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/2726028 Accessed: 21-09-2017 21:11 UTC

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This content downloaded from 131.215.23.153 on Thu, 21 Sep 2017 21:11:36 UTC All use subject to http://about.jstor.org/terms Journal of Economic Literature Vol. XXV (September 1987), pp. 1281-1306

General Equilibrium with Free Entry: A Synthetic Approach to the Theory of *

By WILLIAM NOVSHEK Purdue University

and

HUGO SONNENSCHEIN

Princeton University

The essay is dedicated to the memory of Tjalling Koopmans (1910- 1985). Our purpose has been to communicate to the nonspecialist an aspect of that has been developed since Koopman's masterful exposition of general equilibrium theory (1957). It is very difficult to do the job as well as Professor Koopmans, but he urged us all to trty.

I. Introduction shallian analysis unquestionably domi- nated the theory of . Although this THE TWO DISTINCT THEORIES of perfect probably remains true, a growing num- competition, the Marshallian and the ber of papers in the applied areas adopt Arrow-Debreu-McKenzie (ADM), as- a variant of the ADM framework. Only sume -taking behavior as a funda- in this context can they discuss the im- mental. Filling blackboards with partial portance of interactions among markets equilibrium diagrams, professors empha- and the of wealth. size the Marshallian theory at the under- The Marshallian and ADM theories graduate level. As students progress, have many striking and essential differ- their teachers introduce them to the ences, for example: Arrow-Debreu-McKenzie theory, Gerard Debreu's Theory of Value (1959), the 1. The ADM theory specifies a fixed framework in which "highbrow theorists" finite number of firms. The Mar- explore the relation between perfect shallian theory postulates a pool of competition and economic efficiency. firms, any number of which may be During the first half of this century, Mar- active in the . 2. The ADM theory postulates that the technology of each firm is con- * We gratefully acknowledge the support of the National Science Foundation and our universities. vex, which rules out increasing re- We also thank Susan Elms for editorial assistance. turns to scale. The Marshallian the-

1281

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ory postulates that the ence price. The synthetic theory allows curve of each firm is U-shaped, both for marginal firms and for the firm based on the assumption of fixed to recognize its influence on price; it pro- costs of or regions of in- vides a precise general equilibrium creasing . framework for positive analysis and a 3. The ADM theory assumes price- framework in which to demonstrate the taking behavior regardless of the classical theorems of . number of firms. The Marshallian In contrast to both the ADM and Mar- theory assumes it only if the effi- shallian theories, we integrate into our cient scale is small relative to de- analysis the leading classical explanation mand. for price-taking behavior rather than as- 4. ADM theory, a general equilibrium sume it. We use the term perfect compe- theory, relates perfect competition tition to describe a situation in which and economic efficiency. Most of firms are arbitrarily small relative to their Marshallian analysis, partial equi- markets. Here firms perceive and take librium, ignores intermarket ef- account of the price effect of their mar- fects. keted quantities. As firms become small 5. Finally, the ADM theory is a static relative to the market, we observe in ac- one, to which the adjoinment of dy- cordance with Cournot that their influ- namics via a tatonnement is not very ence on price disappears and it is the satisfactory. The Marshallian analy- limit of this that we call perfect competi- sis of equilibria is a dynamic one tion. Because price-taking behavior is ex- in the sense that entry and exit of plained along the lines of the Cournot firms cease in equilibria. theory, what we offer might be better termed a Cournot-Marshallian-ADM This essay explains a new theory of per- synthesis. ' fect competition, a synthesis of the ADM The Marshallian perspective enriches and Marshallian theories, and summa- general equilibrium theory. A new con- rizes the recent work of many researchers dition effects a close correspondence be- (for example, Philippe Artzner, Carl Si- tween the general equilibrium model mon, and Hugo Sonnenschein 1986; Oli- and standard intuition. Loosely speaking, ver Hart 1979; Andreu Mas-Colell 1974, it requires that provide the proper 1983, 1986; William Novshek 1980; entry signals for firms and is a conse- Novshek and Sonnenschein 1978, 1980, quence of the dynamic aspect of Marshal- 1983, 1986a, 1986b; Kevin Roberts 1980; lian analysis. With each firm associated and Sonnenschein 1982). (See also the with the use of an unpriced and nondi- Symposium Issue, 1980 and the refer- visible resource, sometimes referred to as ences therein.) entrepreneurship, in equilibrium the re- The new theory enables those econo- turns to that factor must fall with entry mists who work in the world of U-shaped and rise with exit. Such a condition might average cost curves and free entry to study intermarket effects and the decen- 1 It is also important to acknowledge that small tralization and efficiency of perfect com- efficient scale and free entry, while sufficient to guar- antee price-taking behavior in the limit, may not be petition. We believe the traditional necessary (see for example Michael Spence 1983). ADM theory dismays many Marshallians The extent to which free entry alone is sufficient because it contains no role for marginal (for the price-taking conclusion) is a much debated issue and is not discussed here. We believe that with- firms and insists on competitive behavior out the assumption of small efficient scale the case independent of the firm's ability to influ- for price-taking behavior becomes less compelling.

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seem axiomatic for standard Marshallian specification. Next, analysis where we cannot imagine its vio- we will use these models to motivate our lation, but it is here that a general equi- definition of partial equilibrium markets librium perspective is important. In gen- (and general equilibrium economies) in eral equilibrium an increase in a which firms are small. This notion lies commodity price, for example, a , at the heart of our definition of perfect has wealth effects via the changing value competition: Perfectly competitive mar- of the endowment that may increase the kets (perfectly competitive economies) amount demanded of the commodity, for are markets (economies) in which firms example, leisure, whereas in the typical are arbitrarily small relative to the mar- partial equilibrium analysis of the firm ket (economy). such wealth effects are ignored. In gen- A. A Stylized Arrow-Debreu-McKenzie eral equilibrium the condition that prices Model provide the proper entry signals elimi- nates certain ADM equilibria. Hence, a Our development follows Tjalling combination of the Marshallian perspec- Koopmans' (1957) classic exposition of a tive and the general equilibrium per- Robinson Crusoe economy. There are spective leads to better economics. two commodities, leisure and food. Rob- Before turning to the theory, a final inson uses his leisure as an input to pro- word on the mathematical aspects of this duce food according to constant returns paper. The papers on which we have to scale technology. Employing a con- based this essay are rather technical. vention of the Arrow-Debreu-McKenzie Also, in its general form, the model we theory, we denote labor input as a nega- have in mind is more complicated than tive quantity and food output as a positive the model of equilibrium found in De- one. Summarizing the technology with breu's classic Theory of Value (1959). In an appropriate choice of units, Figure 1 order to explain the results to the indicates that one unit of labor input nonspecialist, we must forfeit either gen- yields one unit of food output. Robinson erality or precision. We will sacrifice gen- has 24 hours of leisure, his entire initial erality and frame many of our arguments endowment of resources, to offer as labor in a well-developed example. Although input. Adding his leisure endowment to we loosely state our propositions, appli- the technology set, we obtain the set T cation to our example yields precision. of possible aggregate supplies. Each Occasionally the included proofs hint at point in T, a bundle (1, f), contains 1 units the arguments necessary for the general of leisure and f units of food and satisfies case. The interested reader can find f?-' 24 - 1 (to be feasible, 1 units of leisure proper statements and proofs in the ref- implies no more than 24 - 1 units of la- erences (in particular, Novshek and Son- bor, and food is produced 1: 1 from la- nenschein 1986b). bor). Indifference curves, connecting equal- II. The ADM and Marshallian Models ly preferred combinations of leisure and food, represent Robinson's prefer- To begin we present stylized versions ences. In the economy of Figure 1, a of the ADM and Marshallian models in unique "best attainable point" exists, order to accentuate the relation between which we have denoted by x. the ADM and Marshallian formulations. A price system, a nonzero non-nega- In particular, we will demonstrate that tive vector (pl, pf) of commodity prices, the ADM framework incorporates the defines the dollar value of each bundle

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Food

(1 -24, F)

Technology Set (1 , 1)

Labor 24 Leisuire Figure 1

(1, f) as (pl, pf) * (1, f) = pil + pff. Given of labor and producing 24 - 1 units of the price system (1, 1), the vector x = food. It pays 24 - 1 for the labor and (1, f) maximizes the dollar value of sells the food for 24 - 1, which earns supply at 24. Observe that (1, f) is not zero profit. From his ownership of the a unique maximizer of the dollar value firm, Robinson receives no dividends. of supply because each point of the north- However, as a holder of labor resource east boundary of T has the same value. he receives offers from the firm (24 - 1 If the value of the supply action is dis- units) and from himself viewed as a con- tributed to Robinson, then at prices (1, 1) sumer (I units) for all 24 units and thus he will be able to afford bundles with 1 earns an income of 24. As a consumer, andf non-negative that satisfy the budget Robinson uses his 24 units of income to inequality pll + pff - 24. To maximize purchase 1 units of leisure and f = 24 his Robinson demands (1, f), and - 1 units of food. Both markets clear, so at prices (1, 1) the profit-maximizing and all accounts balance. supply and the utility-maximizing de- The interaction between markets mand coincide. This is the unique equi- above is explicit. For example, the wage librium for the ADM model. Interpret- determines the supply of labor and the ing the technology set as that available demand for food through the relative to a competitive firm, and the economy price of leisure and food and the value as a private ownership one in which Rob- of Robinson's initial endowment of labor. inson owns both the firm and the single Developed with the classical theorems scarce input labor, we can describe the of welfare economics in mind, the ADM equilibrium as such: At prices (1, 1) tak- model of general equilibrium supplied ing these prices as given the firm maxi- precise conditions under which (a) equi- mizes profit by purchasing 24 - 1 units librium is efficient in the sense of Pareto

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S = supply at p* AC F p with eight AC F active firms

MC AC AC D D-demand at p* P* \Q2** . F . ,, F

M SD y

Y y Figure 3a. Figure 2. should be active.2 When each firm is maximizing profit (p = MC) and the num- ber of firms is such that firms have little incentive to enter or exit, we say that optimum, and (b) every Pareto optimum the market is in equilibrium. is an equilibrium after a suitable redistri- Marshall applies his model of perfect bution of ownership. competition when efficient scale (mini- We emphasize that the ADM model mum average cost output) is small rela- does not consider the plausibility of tive to demand. Another strong justifica- price-taking behavior. The model offers tion for the price-taking assumption is descriptions of perfect competition for the fact that the horizontal gaps in the situations in which bilateral supply function at price p* are small rela- (one consumer and one producer) or tive to demand. Here a particular num- single agent maximization (Robinson) ber of active firms who maximize profit should apply. leave insufficient incentive for other firms to enter. The price in the market B. A Stylized Marshallian Model will exceed slightly p*. Alternatively, we We begin with a familiar textbook fig- can consider an approximate equilibrium ure (Figure 2) where we have labeled at price p*, in which a finite number of average cost, , and demand firms maximize profit by offering the effi- AC, MC, and FF respectively. All firms cient scale output to the market, and de- are identical and their number is fixed mand very nearly matches supply. This arbitrarily at 3. The is is illustrated in Figure 3a. zero up to price p*(= minimum AC); at In Figure 3b each firm achieves effi- p* supply is the indicated four point set cient scale at infinitesimal quantity M. (with gaps of length y', the efficient scale, We assume the existence of an un- between the points), and above p* supply bounded mass of available firms. Let D is the horizontal sum of the marginal cost be demand at prices p*. Each firm pro- curves. ducing at efficient scale M achieves an If additional firms can obtain the tech- exact equilibrium at a mass DIM of active nology represented by AC, then the ag- firms. Clearly Figure 3b represents a nat- gregate supply shown is inconsistent with ural limit of markets of the type consid- a situation in which profit-maximizing firms take prices as given. Note that at 2 Clearly, with free entry, an exact price-taking any price above p* any firm in the market equilibrium will exist only in the unlikely event that the value of demand at price p* is an integral multiple may earn a positive profit, and so all firms of the minimum average cost quantity.

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AC p S = supply at p* with a mass DIM of F infinitesimal firms

AC D = demand at p*

F

. ~~~~~~~~.

M r'< S =D y Infinitesimal Scale Figure 3b.

ered in Figure 3a as efficient scale be- systems that lead to a positive output of comes small relative to demand. In the food at a profit maximum take the form equilibrium of Figure 3b a continuum (p, p). (Without loss of generality we will of firms produce. assume p = 1.) If the input price exceeds the output price, active firms operate at C. The Beginnings of the Synthesis a loss; therefore all firms are inactive and Let us first place the Marshallian production is (0, 0), i.e., no input and model in the framework of the ADM the- no output. Alternatively if the output ory to demonstrate that the framework price exceeds the input price, each firm of the ADM theory captures the Marshal- would make a positive profit by using lian ideas. The U-shaped average cost M units of labor input to produce M units corresponds to the firm technology rep- of food output. Hence all firms would resented in the second quadrant of Fig- be active, which requires an unbounded ure 4. The efficient scale is (-M, M), amount of labor. Thus for any price-tak- corresponding to the minimum average ing equilibrium with positive food pro- cost point in partial equilibrium. We ob- duction, input and output prices must tain the set of feasible aggregate produc- be equal. At prices (1, 1) a consumer de- tion possibilities as follows. We start with mands z, and each profit-maximizing firm an unbounded number of potential iden- supplies either (0, 0) or (-M, M). Hence tical firms (analogous to free entry), each aggregate supply (including the endow- with the given production technology. ment) must take on one of the values Suppose n firms are active at each level [(24, 0), (24 - M, M), (24 - 2M, 2M) of input. Select the allocation of inputs .]. In particular, the unique "best at- among firms that yields the maximum ag- tainable point" x is not an equilibrium gregate output. Next vary n to obtain the of the system. maximum aggregate output given the By limiting the number of firms to level of input. Repeat this for each input three (as in the Marshallian example), we level, to obtain the aggregate technology. can find a price system (1, 1 + E) at which Combine this with the initial endowment supply equals demand and excluded to construct T, the feasible set of aggre- firms have little incentive to enter. Fur- gate supplies. Given the indifference thermore, the associated allocation ap- curves depicted in the first quadrant of proximates the "best point" x as illus- Figure 4, no price-taking equilibrium trated in Figure 5. Because the output will obtain. Observe that the only price price exceeds the input price, all three

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Foo(l

M

Tecliiiology p for Onie Firm .

Labor -M 24 -2M 24 -M 24 Leisuire Figure 4

firms supply a positive amount. Point z' is an unbounded mass of available firms. in Figure 5 represents the price-taking As in the Marshallian Figure 3b, an ap- aggregate production plan. His income propriate mass of active firms achieves from ownership of the labor resource, an exact equilibrium at prices (1, 1) at combined with his (positive) dividend which each produces at efficient scale. from ownership of all three firms gives Figure 6 represents a natural limit of Robinson a budget line through z', which markets of the type considered in Figure is his optimal choice. So aggregate supply 4 as the firms become small relative to equals . the market. We point out the similarity Alternatively, at prices (1, 1), we could with the Arrow-Debreu-McKenzie econ- consider an approximate price-taking omy of Figure 1 where the equilibrium equilibrium in which a finite number of and the "best point" coincide. firms produce positive quantities and D. A Perfectly Competitive Economy maximize profit by offering the efficient scale output (-M, M). In this situation We will apply the term perfectly demand nearly matches supply. Figure competitive economy to a regime in 4 essentially illustrates this where the gap which firms are arbitrarily small relative between S3 and z is small. Once again de- to their markets. We adopt the classical mand almost equals the "best point" x. position that consumers have no market In Figure 6 the efficient scale of each power. However, we could have pro- firm is an infinitesimal quantity and there vided a similar treatment in which we

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Foo(d

Optimiial PlanI for a Firmi I1(,1+E at Prices (1, 1 + e) I ( 1, I + e)

Labor -M 24 Leisure

Figure 5.

view consumers and firms symmetrically. quence by E(O) in which firms are infini- As in Marshallian theory, we assume that tesimal. With this interpretation the firms can freely enter and experience in- market in Figure 3b represents the limit creasing returns to scale over some range of markets in Figure 3a. Similarly, the of output to formalize the notion of the economy in Figure 6 is the limit of a firm's being small relative to its market. sequence of economies in Figures 4 and In Figures 2 and 4 three firms, each pro- 5. ducing at efficient scale, nearly satisfy the In each economy of the sequence we demand of consumers when price equals will assume that firms correctly perceive minimum average cost. By adding an the (typically non-negligible) effect on identical twin for each original consumer prices of their output, and maximize to replicate demand, it would take six profit accordingly. Although nonconvexi- firms producing at efficient scale to satisfy ties in the firms' production sets lead to demand approximately at the same price. generic nonexistence of price-taking Continuing in this manner each firm can equilibrium (i. e., nonexistence for all but be made arbitrarily small relative to its "knife-edge" cases), Cournot equilibrium market. A perfectly competitive econ- with entry frequently exists for the Mar- omy is a sequence of economies [E(ctk)] shallian model if efficient scale is small. in which firms become arbitrarily small This suggests that the limit of Cournot relative to their markets. In our formula- equilibria of [E(ak)] is a natural definition tion, Otk is a measure of firm size relative of an equilibrium for the perfectly com- to the economy E(ak), and we let the petitive sequence [E(ak)]. In other size of the firm in our economy diminish, words, we define perfectly competitive ack .> 0. We denote the limit of the se- equilibrium as the limit of Cournot equi-

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Food

z Food

T

Teclhinology for Onie FirimI

Labor 24 Leisuire

Iifinitesiimial Scales

Figure 6.

libria with entry as firms become small of the technology and behavior of the in- relative to the market. dividual firm. Lionel McKenzie (1959), Not surprisingly we can characterize however, speaks of a competitive con- the perfectly competitive equilibria of stant returns to scale aggregate technol- the perfectly competitive sequence ogy. McKenzie assumes forthrightly that [E(ctk)] in terms of its limit economy aggregateE(O) technology T is a convex cone, in which firms are infinitesimal. In light which follows from the additivity and di- of this characterization we will show that visibility of basic production processes.3 firms in perfectly competitive equilib- We say a production set has the additiv- rium act "as if" they take prices as given ity property if for any two possible pro- in E(O). On the other hand, we will show duction plans the joint production plan that other price-taking equilibria of E(O) is also feasible. It has the divisibility exist that are not the limit of Cournot property if any production plan can be equilibria with entry and thus not per- proportionately scaled down. Additivity fectly competitive equilibria of the se- is regarded as axiomatic when all factors quence [E(ak)]. that affect production are listed so that Before continuing our exposition a his- there can be no underlying fixed factor. torical note is in order. In their treatment of general equilibrium theory, Kenneth 3The assumption that T is a convex cone with ver- tex at the origin means that for any two production Arrow and Gerard Debreu (1954) con- plans x, y in T, and for all non-negative numbers a, cern themselves with the representation b, the production plan ax + by is also in T.

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Divisibility might be thought of as analo- order to simplify the analysis we assume gous to the assumption that commodities some special structure for the cost func- are infinitely divisible. McKenzie does tion C(y), namely that total costs are zero not explicitly mention firms, but we if production is zero while, if production might regard them as producing outputs is positive, costs consist of both strictly from inputs according to one of an infinity positive fixed costs CO and variable costs of basic processes. When efficient scale v. We assume variable costs increase is infinitesimal relative to the economy, with output at an increasing rate. these processes can be expanded or con- (C) C(y) = O, if y = O, and C(y)= tracted continuously in the aggregate by CO + v(y) if y > 0, where CO > varying the mass of firms using each pro- 0, and for all y ' 0, v' > 0, cess, and this corresponds to the situation v" 0. We also assume average in E(O). Thus our foundations rest on the cost is minimized uniquely at McKenzie interpretation of general equi- y = 1. librium theory which we elaborate by ex- plicitly modeling firms and by providing An inverse demand function F speci- Cournot-like foundations for competitive fies demand by associating a price [F(y)] behavior. with an amount (y) placed on the market. We assume

III. The Partial Equilibrium Synthesis (F) F is twice continuously differenti- able and F(y) = C(1) implies F'(y) An appropriate introduction to the no- # 0. tions of a perfectly competitive economy and perfectly competitive equilibrium is These are regularity conditions that en- consideration of a partial equilibrium able us to use the calculus in our analy- market. Besides being a natural stepping sis. (C, F) specifies the basic Marshallian stone to our synthesis, it provides a rigor- market. We assume there is no bound ous foundation for the Marshallian the- on the number of firms with access to a ory. Explicitly, we establish the exis- particular cost function. This captures tence of equilibrium for the partial the idea of free entry. Of course in equi- equilibrium market without either the librium, demand will limit the number need for introducing the approximate of firms using a technology. equilibrium notions described above or A perfectly competitive sequence of infinitesimal firms. Furthermore, firms markets [M(ak)] is a sequence of markets correctly perceive their influence on in which firms become small relative to price. the market. Let C be the cost function We first develop the notion of a per- for a firm in market M(1). The corre- fectly competitive sequence of partial sponding average cost is ACI(y) = C(y)/ equilibrium markets. This is a sequence y for y > 0. We take a representative of Marshallian markets for a single homo- sequence of markets by rescaling the av- geneous good where firms decrease in erage cost functions: In market a, the size. As suggested above, we define a output ay has the same average cost as perfectly competitive equilibrium as the output y in market 1. We can accomplish limit of Cournot quantity setting equilib- this by defining an a-size firm corre- ria-with-entry of the markets in the se- sponding to C as a firm with cost function quence. Ca(y) = aC(y/a). An a-size firm has aver- Let us begin by returning to the Mar- age cost ACj,(y) ACI(y/a), and attains shallian specification as in Figure 2. In minimum average cost uniquely at out-

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put a. For each a > 0, C, and F, we profit given the production plans consider a market with a countable infin- of all other active firms. ity of firms with technology Ca, facing (b) entry is not profitable; that is market inverse demand F. We denote n this market by M(ot). As a -> 0, firms F yj +Y)Y+ - Cj(y) ?0 for become small relative to the market, and all y ' 0. the aggregate production possibilities converge to the constant returns to scale Cournot equilibrium with entry is an "ex- case diagrammed in Figures 3b and 6. act" equilibrium of the model where Given the cost function- C and the inverse firms do not take prices as given. Individ- demand function F, we can define a per- ually maximizing profit, firms (noncom- fectly competitive sequence of markets petitively) supply exactly the quantity by any sequence of strictly positive real demanded by consumers at the Cournot numbers less than or equal to one, where equilibrium price F(Y yj). the sequence, representing firm size, Finally, we define the equilibrium out- shrinks to zero. We denote the limit mar- put of the perfectly competitive se- ket by M(O). In particular, if the sequence quence [M(ak)] as the limit of YYi(tk) of firm sizes is 1, 1/2, 1/3, . . . then the where [Y1(atk), * * *, Ynk (0)] is a Cour- perfectly competitve sequence of mar- not equilibrium with entry of the market kets may be thought of as resulting from M(ak). These are called perfectly com- repeated replication of demand followed petitive equilibria. by the representation of output in per- Our definition of perfectly competitive capita (actually per-replication) units. equilibrium formalizes the idea that per- Cournot equilibrium requires that the fect competition represents a limiting quantity actions of firms maximize profit case of regimes in which firms can influ- given the quantity actions of all other ence price. Among economists interested firms. In equilibrium no firm makes neg- in rigorous foundations our definition ative profit, because exit is possible and may not at first be acceptable. They ob- yields zero profit. Similarly, the assump- ject because some or all of the markets tion of an unbounded set of firms with that form a perfectly competitive se- access to the technology, only a finite quence may lack a Cournot equilibrium. number of which can be active (because Hence our definition is meaningless. If of fixed costs), implies that in equilibrium demand is downward sloping, however, no inactive firm can enter and make a then a Cournot equilibrium exists even- positive profit. Stated precisely, a (pure tually in any sequence of markets [M(ak)] strategy) Cournot equilibrium with entry forming a perfectly competitive se- for the market M(ot) is an integer n and quence. By assuming that demand slopes a set of positive outputs (YI, Y2 . .. downward our equilibrium concept will Yn) such that: apply. Theorem 1 identifies the condi- tions "demand price equals minimum (a) (YI, Y2, , Yn) is an n-firm Cour- per unit cost" and "demand slopes down- not equilibrium (without entry); ward" in the limit market M(0) as the that is, for all i = 1, 2 . . ., n, characteristics of perfectly competitive F ( yj + Y)Yi - Co(yi) F equilibria for a sequence [M(ak)]. Free entry and exit determine the mass of ac- ( Yj + YY- Ca(y) for all ytive '0. firms endogenously. Average cost No active firm can choose another curves are U-shaped (and so firm technol- production plan y and earn greater ogy is not convex). The theory does not

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assume price-taking behavior. Equilib- p F rium is exact, production equals demand, and all firms maximize profit. AC1/2(y) AC1(y) Theorem 1. (See Novshek 1980.) Given the cost function C satisfying (C), the inverse demand function F satisfying (F), and the perfectly competitive se- p* [1< '4 \F quence [M(ak)], the following conditions are equivalent: 12 y y (la) y* is a perfectly competitive equi- librium for [M(ak)], and Figure 7. (lb) F(y*) = C(1) and F'(y*) < 0. For the case of partial equilibrium, the gue that A does not constitute an equilib- preceding result describes precisely our rium because an infinitesimal firm in approach to perfect competition. Per- M(O) can enter and make a positive profit. fectly competitive equilibria are the limit First, observe that in any Cournot equi- points of Cournot equilibria of the Mar- librium for the market M(ot) all firms shallian markets [M(ak)]. make non-negative profit so aggregate This theorem establishes that these output must lie in [0, y*] or [A, y**]. perfectly competitive equilibrium quan- Second, inactive firms must not profit by tities in M(0) equate the inverse demand producing a, so aggregate Cournot equi- price of consumers and minimum aver- librium output plus a must lie in [yI, y] age cost and satisfy downward-sloping or [y**, oo)]. This implies that aggregate demand. For the case of globally down- output must lie in either the interval [ y* ward sloping demand, the perfectly com- - a, y*] or [y** - a, y**]. For small a petitive equilibria of the sequence neither interval is near A. In fact this is [M(ak)] coincide with the unique Walra- how one proves that la implies lb. The sian equilibrium in the derived constant hard part of the theorem is to show that returns to scale market M(O). On the for a sufficiently small, if F(y*) = C(l) other hand, if inverse demand is not and F'(y*) < 0, then M(ot) has a Cournot globally downward sloping,4 then there equilibrium with entry with aggregate may be Walrasian equilibria of the de- output in [y* - a, y*]. rived (constant returns to scale in the ag- The above argument highlights the dif- gregate) market M(O) that are not by our ferences of our equilibrium concept and definition competitive equilibria. These the Walrasian equilibrium in the limit equilibria fail because we require that market. In the Walrasian theory firms entry and exit also be at rest. take prices as given, and in perfect com- Figure 7 illustrates this point. In Fig- petition we justify this as an approxima- ure 7, the points y* and y** are equilibria tion. The approximation applies so long of the perfectly competitive sequence as firms are small relative to their market [M(ak)], but A is not, even though the and thereby have little influence in most demand price of consumers and mini- any specification of strategic variables. mum average cost coincide at A. We ar- Here the ability of each firm to affect price diminishes; however, in general 'This becomes a more interesting possibility when each firm will have some influence that demand is a function of both price and wealth (which in turn depends on price as in the derivation of the will affect the entry decision. If the entry supply of labor). of a firm will drive up price and make

This content downloaded from 131.215.23.153 on Thu, 21 Sep 2017 21:11:36 UTC All use subject to http://about.jstor.org/terms Novshek and Sonnenschein: General Equilibrium with Free Entry 1293 entry profitable, then inactive firms will profit using an "inverse demand" func- take account of this effect and enter. In tion F. In the next section we will care- our model small firms correctly perceive fully construct this function. The idea is their influence; therefore equilibrium re- this: To each vector of quantity actions quires that no firm can drive up prices of all firms, y, the "inverse demand" by entering. function associates a price vector, p, such that the competitive consumer sector's IV. The General Equilibrium Model, excess demand, given prices, p, and the Preliminaries income generated by the consumers' div- idend payments (their fraction of the In this section we discuss the general profit or loss for each firm), exactly equilibrium model on which we base our matches the aggregate quantity action of concept of perfectly competitive equilib- the firms. Thus the payoff for each firm rium. In the next section the model will is a well-defined function of its own pro- be presented in the context of a simple duction plan and the production plans example. This facilitates the exposition of other firms. Although some firms may and should illustrate the close relation- be making losses, for any vector of pro- ship between the partial equilibrium and duction plans employed by the firms, general equilibrium models. prices adjust so that all markets clear. A A perfectly competitive sequence of Cournot equilibrium exists if (1) each economies [E(ot)] is analogous to a per- firm takes a feasible action in its produc- fectly competitive sequence of markets. tion set; and (2) each firm maximizes prof- As a converges to 0 firms become arbi- its given F and the actions of other firms. trarily small relative to the economy and The assumption of nonconvex technolo- in the limit economy, E(O), are infinitesi- gies (the general equilibrium analog of mal. Each economy E(ot) has an un- U-shaped average cost) implies that only bounded set of potential firms. This pro- a finite number of firms have nonzero vides our notion of free entry. In any actions in equilibrium. Note that inactive equilibrium for E(ot) only a finite number firms are available but cannot make posi- of firms can be active, so inactive firms tive profits by entry. Hence, the entry always exist and can test the profitability process is at rest in an equilibrium. Thus of entry. No firms enter if they cannot we have a description of Cournot equilib- gain positive profit by doing so. rium with free entry for each economy In each economy E(ot), we use Cour- E(ot). not-Nash equilibrium in quantities as our For each economy E(ot), consider the solution concept. As in Marshall, we treat set of aggregate firm actions correspond- consumers as a competitive, price-taking ing to Cournot equilibria of E(ot) relative sector to focus on the role of firms and to "inverse demand" F. We define the entry. In the Cournot tradition, quantity perfectly competitive equilibria of the se- setting provides a tractable basis for our quence of economies [E(ot)] to be the lim- analysis and, as was the case in the partial its of Cournot equilibria of the E(ot) econ- equilibrium model, it avoids the obvious omies. Explicitly, a price vector p* and problem of nonexistence of equilibrium an aggregate production vector y* form that arises when firms set prices in condi- a perfectly competitive equilibrium of tions of production under increasing re- the sequence [E(ot)] provided that for an turns. When firms consider a quantity "inverse demand" selection F satisfying action (a vector of input and output lev- certain conditions, y* is the limit of a els) they evaluate the corresponding sequence [y (a)] as a converges to zero,

This content downloaded from 131.215.23.153 on Thu, 21 Sep 2017 21:11:36 UTC All use subject to http://about.jstor.org/terms 1294 Journal of Economic Literature, Vol. XXV (September 1987) where y(a) is an aggregate production quence, equilibrium production maxi- corresponding to a Cournot equilibrium mizes profit relative to the equilibrium of E(a) (relative to F). This coincides with prices. We will show that the equilibria the partial equilibrium model of Section of the perfectly competitive sequence III. satisfy all the conditions of an ADM equi- The partial equilibrium results in Sec- librium of the limit economy E(O). Also, tion III depended on a condition of we will show that the classical welfare downward sloping demand. The results theorems still hold in our framework: ev- for general equilibrium will depend on ery perfectly competitive equilibrium of an analogous condition, called DSD. the sequence is Pareto efficient and every Prices determined by the "inverse de- Pareto efficient allocation for the se- mand" selection F must give proper en- quence can be supported as a perfectly try signals. At a point satisfying the ADM competitive equilibrium of the sequence. equilibrium conditions, additional entry must lead to new prices at which the en- V. The General Equilibrium Synthesis trants make losses. Because input and output prices change, the DSD require- We adopted our general equilibrium ment is that the effect of all the price model, with the exception of production changes leads to a loss for the entrant. sets (and our sequence of economies ap- Walras had something similar to DSD proach), from the standard Arrow-De- in mind (Leon Walras [1874-77] 1954, breu-McKenzie model. The economies p. 225): E(a) are composed of consumers and firms. Each consumer receives an initial [U]nder free competition, if the selling price endowment of and has preferences of a product exceeds the cost of the productive over potential bundles of goods. For ex- services for certain firms and a profit results, entrepreneurs will flow towards this branch of ample, the first two diagrams of Figure production or expand their output, so that the 8 show representative indifference quantity of the product [on the market] will curves for two consumers, A and B, in increase, its price will fall, and the difference an economy with two commodities, lei- between price and cost will be reduced; and, sure and food, each of which can be con- if [on the contrary], the cost of the productive services exceeds the selling price for certain sumed only in non-negative amounts. firms, so that a loss results, entrepreneurs will Both consumers prefer to consume the leave this branch of production or curtail their two commodities in fixed proportions, output, so that the quantity of the product [on person A at 1: 1, person B at 1:2. A par- the market] will decrease, its price will rise ticular utility function that assigns utility and the difference between price and cost will again be reduced. equal to the minimum of l and f to a bundle with l units of leisure and f units In the next section we will see how of food represents person A's prefer- our framework allows us to analyze a logi- ences. Similarly, one that assigns utility cally precise general equilibrium model equal to the minimum of 21 and f to the with nonconvex technologies where the bundle (1, f ) represents B's. They receive number of active firms is determined en- identical endowments of (1, 0), contain- dogenously and demand equals supply ing one unit of leisure and no food. exactly. We do not assume price-taking To construct a general equilibrium behavior. Rather, when a converges to analog of U-shaped average cost and free zero the ability of a firm to affect price entry we differ from the standard as- becomes arbitrarily small. At any per- sumptions on the producer sector in two fectly competitive equilibrium of the se- important ways. Each firm's production

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Food Food 21 Fouotput

Leisure Leisure Labor -1 (0, 0) Input

Indifference Curves Indifference Curves Technology for a Single for A for B Firm in E(1), [{(0, 0), (-1, 1)}] Figure 8.

set, the set of possible production plans with production plan (0, 0), or it can use for the firm,5 has two components. The one unit of leisure as labor input (nega- first is the origin (the production plan tive by convention) to produce one unit with no inputs and no outputs) and the of food as output with production plan second component is bounded away from (-1, 1). Observe that the production set the origin, compact and strictly convex. {(0, 0), (-1, 1)} contains all possible in- By compact we mean there is some num- put-output vectors for the firm. The firm ber that is a bound for the magnitude cannot scale the production level up or of any input or output level in any feasi- down to produce (-2, 2) or (-1/2, 1/2): ble production plan (boundedness), and there is an indivisibility in the production if a sequence of feasible plans has a limit, process at the firm level. the limit is also feasible (closedness). By Our model differs from the standard strictly convex we mean that for any two ADM one in a second way. We assume production plans in this component, any that there is free entry with no bound weighted average of the two plans is in on the number of possible firms. Each the interior of this component. See Fig- economy has an infinity of potential ure 9. This assumption provides a simple firms; a countable infinity of firms exist production set analog of U-shaped aver- in each economy E(ot), and a continuum age cost in partial equilibrium. Inclusion of firms in the limit economy E(O). In of the origin in the production set guaran- the Cournot equilibria of E(ot) only a fi- tees free exit. For our example we will nite number of firms operate so that there use the simplest version of this type of will always be additional firms to check production, an on-off technology. The the profitability of entry. third diagram of Figure 8 shows the tech- By rescaling technologies in a manner nology of a typical firm in E(1). The firm analogous to the rescaling of cost in the has only two options. It can be inactive partial equilibrium model of Section III, we generate a sequence of economies

S Each production plan is a vector with a complete [E(a)] converging to a limit economy specification of all inputs and outputs for that plan. E(O). For our example, each E(ot) contains

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Output ference between partial and general equilibrium inverse demand functions. In partial equilibrium, dollar incomes re- main constant while a single price varies. Y 0 In contrast, the general equilibrium in- Input verse demand function takes full account Production Set of the induced changes in income, through both changes in the value of en- dowments and in the received dividend Figure 9. payments. The assumption of equality across firms a countable infinity of firms with produc- of each consumer's ownership share im- tion sets {(O, 0), (-ao, a)}. Note that a plies that each consumer's wealth de- firm's technology is small relative to the pends only on prices p and aggregate pro- economy when ao is small. The aggregate duction y, and not on the arrangement production set in E(ot) is {(0, 0), (-ot, of production among firms. If consumer a), (-2ao, 2a), . . .}: Aggregate produc- i owns fraction Oi of each firm and has tion depends only on the number of ac- endowment vector wi, then at prices p tive firms. The aggregate production set and the aggregate production y the con- "converges" to the constant returns to sumer receives dividend payment Oi P - scale technology that converts labor in- y and has total income p * wi + Oi p y. put into food output in the ratio 1:1. This (Because inputs are negative in y, the constant returns to scale technology cor- sum of individual prices times planned responds to the aggregate production set inputs or outputs is just the firm's profit in the limit economy E(O) with a contin- at plan y given prices p.) If his corre- uum of infinitesimal firms. (As a closed, sponding vector of demand is Di(p, p - convex set this limit aggregate produc- wi + Oi p - y) then the excess demand tion set satisfies standard ADM assump- D of the consumer sector (the sum of tions for the production set of a single individual consumer's gross demands, firm.) minus the sum of resources owned as ini- For simplicity the consumer sectors of tial endowments by the consumer sector) each E(ot) and E(O) coincide, and con- is the function D = 2 Di(p, p * wi + Oi sumer i owns a fraction of firm t indepen- dent of the firm (i.e., if Oit is the fraction p y) - wi. An "inverse demand" of t owned by i, then Oit = Oi for all t). selection F(y) is a function from aggregate In our example consumer A owns fraction production vectors y to price vectors OA of each firm while consumer B owns F(y) that clear markets given the action 1 - OA of each firm. Let 0A = 3/4. Thus of firms: E Di[F(y), F(y) - wi + Oi E(a) "converges" to the limit economy E(O) in terms of both consumer and pro- F(y) - y] - wi = y. This is the gen- ducer sectors. eral equilibrium analog of partial equilib- In order to define a Cournot equilib- rium inverse demand. The correspond- rium in E(ot) we need a general equilib- ing partial equilibrium version as used rium analog of a partial equilibrium in- in standard theory would fix the verse demand function so that firms can income of consumer i at Ii rather than evaluate the profits corresponding to dif- recognize that price changes affect in- ferent actions. There is an important dif- come [F(y) - wi + Oi F(y) - y]. Even after

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price normalization several price vectors equilibrium the firms pick production may clear markets given y. We assume plans to maximize profit, taking the pro- F selects one of the price vectors. duction plans of all other firms as fixed. We now illustrate an inverse demand Free exit implies that in equilibrium no function in terms of our example. Every firm operates at a loss. Also, no inactive feasible aggregate production plan as- firm has an incentive to enter. Stated sumes the form (-t, t) where t is non- precisely, a Cournot equilibrium with negative. Normalizing prices to sum to entry for economy E(ot) (relative to "in- one (because only relative prices matter) verse demand" F) is an integer n, the the price vector is of the form (1 - p, number of active firms, and a set of p) where p lies between zero and one. nonzero production plans (yl, Y2, With aggregate production (-t, t) and Yn) such that: price (1 - p, p) consumer A has income (1 - p, p) - (1, 0) + (3)(1 - p, p) (a) (Yl, Y2* , Yn) is an n-firm Cour- (-t, t) = 1 - p + (3/4)t (2p - 1) and not equilibrium (without entry); consumer B has income (1 -p, p) - (1, that is, for all i = 1, 2, . . ., n, 0) + (1/4)(1 - p, p) - (-t, t) = 1 - p + yi is a feasible production plan and (1/4)t (2p - 1). If p is strictly between F ( Yj + Yi) * Yi 2 F( yi + zero and one then consumer A has de- y) * y for all feasible y, and mand vector (IA, IA) and consumer B has demand vector [IB/(l + p), 2lB/(1 + P)] (b) entry is not profitable; that is where Ii is the income of consumer i (re- F(2 yj + y) * y ? 0 for all call Figure 8). feasible y. The "inverse demand" F(-t, t) for the example must yield prices that, together Let F be a (continuous) "inverse de- with the resulting incomes, generate ag- mand" function for the economy E(1). gregate excess demand (-t, t) in the con- We define a perfectly competitive equi- sumer sector, to match exactly the aggre- librium for the sequence of economies gate production plan. Solving for F we [E(ot)] as a price vector p* and an aggre- find gate production plan y* such that (1) F(y*) = p* and (2) y* is the limit of aggre- F(-t, t) = gate production plans y(a) corresponding (0, 1) 0? t < 16/15 to Cournot equilibria with entry for econ- [(15t - 16)/(6t - 4), omies 16/E(ot) (relative S tto F).s Observe 4 that [y12J- 9t)/(6t - )]164? every specification ? of perfectly competi- (1, 0) 4/3< t s 2. tive equilibrium includes an underlying inverse demand function. Observe that the inverse demand func- In our example, the use of the on-off tion depends on preferences and the ini- technology greatly simplifies the analysis tial distribution of wealth in the econ- when looking for an equilibrium of the omy. In particular, when we consider a sequence: Each firm's profit depends sequence of economies [E(oa)], the in- only on whether it operates and the total verse demand function is independent number of active firms. In E(ot), N active of a. firms result in each active firm receiving Using "inverse demand" F, firms de- profit ao if N < 16/150t, ao(14 - 12Not)/(3Not termine an equilibrium for E(ot) just as - 2) if 16/150a C N ? 4/3ao, and -a if 4/3ao in the partial equilibrium case. In general < N. Inactive firms always have profit

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zero. To be a Cournot equilibrium each aggregate endowment and also that ag- active firm must have non-negative gregate production belong to the con- profit, so in equilibrium N ' 7/6ao. On stant returns to scale aggregate produc- the other hand, in Cournot equilibrium tion set of the limit economy E(O). An inactive firms must not have an incentive allocation for the perfectly competitive to be active, so in equilibrium N + 1 2 sequence [E(ot)] Pareto dominates an al- 7/6ao. Each E(ot) has a Cournot equilibrium ternative allocation if it makes at least with N(ot) active firms where N(ot) is the one consumer better off while leaving the integer between (7/6a) - 1 and 7/6ao (when remaining consumers at least as well off. 7/6ao is an integer there are two values of A feasible allocation for the sequence N that work). The aggregate production [E(oa)] is Pareto efficient if no feasible allo- plan in the Cournot equilibrium is cation Pareto dominates it. N(a)(-ao, a) which differs from (-7/6, 7/6) Proposition. Equilibria of the per- by no more than ao units of input and ao fectly competitive sequence [E(oa)] are units of output. Thus in our example, Pareto efficient. p* = (1/2, 1/2) and y* = (-7/6, 7/6) is a The standard argument for this result, perfectly competitive equilibrium for the applied to our example, proceeds as fol- sequence of economies [E(ot)]. At these lows. Recall that the perfectly competi- prices consumer A chooses (1/2, 1/2), con- tive equilibrium for the sequence [E(oa)] sumer B chooses (1/3, 2/3), and supply has corresponding allocation (1/2, 1/2) for equals demand. Note that this agrees A, (1/3, 2/3) for B, and aggregate produc- with the ADM equilibrium of the limit tion plan (-7/6, 7/6) that yields zero profit economy E(O). at prices (1/2, 1/2). This means that A sup- Next we would like to prove the first plies one-half of a unit of leisure (as labor classical theorem of welfare economics: input to the production process) and con- Perfectly competitive equilibria for the sumes one-half of a unit of food, while sequence [E(ot)] are efficient in the sense B supplies two-thirds of a unit of leisure of Pareto. We begin by defining a Pareto and consumes two-thirds of a unit of food. efficient allocation for the perfectly com- In order to prove the proposition we petitive sequence [E(ot)]. First observe must show that any allocation that Pareto the way in which the sequence [E(ot)] dominates the equilibrium allocation is converges to E(O): The demand sectors not feasible. of each E(ot) coincide with the demand The set of bundles that A considers sector in the limit economy E(O), and the at least as good as (1/2, 1/2) is XA = {1, aggregate production sets in E(ot) con- f)ll ? 1/2, f ? 1/2}. Similarly, XB = {1, verge to the constant returns to scale ag- f)ll ? 1/3, f 2 2/3}. To make one of A gregate production set in the limit econ- and B better off while keeping the other omy E(O). As a consequence we define at least as well off would require in excess efficiency for the perfectly competitive of 5/6 units of leisure and in excess of 7/6 sequence [E(ot)] partially in terms of the units of food. But with only two units of limit economy E(O). leisure available in the economy, more A vector listing the consumption of than 7/6 units of food must be produced each consumer and an aggregate produc- from less than 7/6 (= 2 -5/6) units of labor. tion plan is called an allocation; if it is One can see immediately that such a pro- feasible for E(O) then we say it is feasible duction plan is not feasible with the one for the sequence [E(ot)]. Feasibility re- unit of food for each unit of labor constant quires that aggregate consumption differ returns to scale aggregate production set from aggregate production by exactly the of the limit economy E(O).

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In general, we must use a less direct would earn strictly positive profit at the argument to show that a Pareto-dominat- Cournot equilibrium despite the price ing allocation a is not feasible. We com- change due to entry by a single firm. bine the observation that, relative to the Hence all firms should be active. Simi- price system (1/2, 1/2), the production plan larly, if p* - y* < 0 then some active that matches aggregate net consumption firms in the Cournot equilibria of E(oa) in a must yield a greater profit than does make a loss, and should exit. If p* * y* (-7/6, 7/6) with the fact that (-7/6, 7/6) is = 0 but p* - y > 0 for some y in the profit maximal in E(O). aggregate production set then again some What are the properties of equilibria firms could not have been maximizing of the perfectly competitive sequence in E(ot).) By an interchangeability lemma6 [E(ot)]? How are these equilibria related (see Koopmans 1957, p. 13) adjusted for to the ADM (price-taking) equilibria of the fact that in the limit there is a contin- the limit economy E(O)? The answers uum of firms, any decomposition of y* agree with the partial equilibrium results into feasible actions for the individual in- of Theorem 1: The equilibria of the per- finitesimal firms in E(O) is such that all fectly competitive sequence are those firms' actions must be profit maximizing ADM equilibria of the limit economy that over their production sets relative to satisfy an additional condition, DSD. prices p*. Thus the actions of firms are This condition is related to downward as if they are price takers in equilibrium, sloping demand in the partial equilib- and an equilibrium of the perfectly com- rium case. petitive sequence is an ADM equilib- Consider an equilibrium of the per- rium of the limit economy E(O). fectly competitive sequence (p*, y*) rela- A slight modification of our example tive to the "inverse demand" function, demonstrates the differences between F. In E(ot), a firm has a production set the requirement that in the limit firms comprised of two components, the no- are unable to influence price, and the production component and the produc- assumption of price-taking behavior. The tion one and the "inverse demand" func- equilibria of the perfectly competitive se- tion is continuous in a neighborhood of quence must satisfy an additional condi- y*. In our example the latter component tion, (weak) DSD. The DSD condition of the production set contained the single is necessary to ensure equilibrium in the production plan, (-ao, a). Though we did entry decision. As a converges to zero, not assume price-taking behavior, as ao in the sequence of economies [E(ot)] firms converges to zero a firm's ability to affect become arbitrarily small relative to the price becomes arbitrarily small. For ao market and thus have arbitrarily small > 0 the entry or exit of firms has an effect impact on prices. Therefore, any produc- on price. In the limit this effect disap- tion plan y in aY with p* - y < 0 would pears and the aggregate production plan not (for a sufficiently small) be able to y* must maximize profit over the aggre- change price enough to make a non-nega- gate production set (a constant returns tive profit. Because max p* - y = p* - to scale cone) relative to prices p*, and must satisfy p* - y* = 0. (Because "in- 6 The interchangeability lemma of Koopmans states verse demand" is continuous near y*, that if one maximizes separately a linear functional and the aggregate productions in the se- on two sets A and B then its maximum value on quence of Cournot equilibria converge A + B is simply the sum of the previous maximums. In less mathematical terms it states that centralized to y*, if p* - y* > 0 then, for small ao, planning yields the same results as decentralized some feasible action for a firm in E(a) planning.

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y* = 0, no actions aoy in aoY with p* fourth - y rather than three-fourths of each > 0 are available. In our example, when firm. For some aggregate production an inactive firm becomes active it more than one price vector would clear changes price by approximately at[dF/ the market, so the "inverse demand" F dy(y)] (-1, 1)' where [dF/dy(y)] is the 2 must be a selection from these prices. x 2 matrix of partial derivatives. Because In particular let us examine the ADM an active firm produces a(-1, 1), the en- equilibrium p* = (1/2, '/2), y* = (-/6, trant's profit minus the incumbent's 7/6) which is unchanged because there are profit before this additional entry is ap- zero profits in that equilibrium. Taking proximately aL2 (-1, 1) [dF/dy(y)] (-1, 1)'. the "inverse demand" with F[(-7/6, 7/6)] For the case of n commodities, the ex- = (1/2, 1/2) and which is continuous near pression at2y' [dF/dy(y*)]y is (approxi- the ADM equilibrium y*, we obtain mately) the profit differential (above the F[(-t, t)] = [(16 - 13t)/(4 - 2t), (llt - profit made by an active firm before this 12)/(4 - 2t)] for 12/i1 ' t ' 16/13. Notice additional entry) available to an inactive that p* = (1/2, 1/2), y* = (-7/6, 7/6) cannot firm by switching to aoy in aoY with p* - be an equilibrium for the perfectly com- y = 0 in the E(oa) equilibrium. Weak petitive sequence [E(ot)]: No Cournot DSD requires that this profit differential equilibrium for any small ao has aggregate be nonpositive. When weak DSD fails, production corresponding to t near 7/6. inactive firms could not be profit maxi- Such a Cournot equilibrium would re- mizing for ao sufficiently small. Thus firms quire the number of active firms, N, to would want to enter in E(oa), and y* could be between 12/loat and 16/A3a. In that re- not be a limit point of any sequence of gion each active firm earns a(12Nao - equilibria of E(oa) relative to F. 14)/(2 - Na), which is negative for N The DSD requirement, a static stabil- < 7/6ao. For N ?-7/6ao, by engaging in pro- ity condition, is similar in spirit to Hick- duction, an inactive firm changes from sian perfect stability ( 1939). zero to strictly positive profit. Cournot Each considers changes in a single "mar- equilibria cannot exist near p* = (1/2, 1/2), ket" at a time. Hicksian perfect stability y* = (-7/6, 7/6) for any small ao because requires that a fall in the price of a single weak DSD fails. The ADM equilibrium commodity make demand exceed supply for E(O) is not an equilibrium for the per- for that commodity (with or without other fectly competitive sequence [E(a)]. prices adjusting to clear other markets). Prices give the wrong entry signals in The DSD condition applies to a change this example because of the general equi- in the number of active firms of a particu- librium income effects in E(ot). The entry lar type, with the corresponding change of an additional firm will result in a lower in aggregate production being a simulta- price for labor relative to food yielding neous change in several input and output profit for the entering firm. The neces- commodities. Prices adjust to clear all sary additional labor is obtained by low- markets given the new aggregate produc- ering the wage. Because A owns only a tion. DSD requires that the profit for small fraction of each firm, the reduction firms declines as more firms enter. Static in the value of his leisure endowment equilibrium of the perfectly competitive (as labor) dominates his extra dividend sequence requires DSD. (all active firms have strictly greater We can demonstrate the consequences profit after the entry of an additional firm) of failure of weak DSD in a slightly modi- and his income falls. Thus he reduces fied version of our example. Consider an his demand for both leisure and food example in which consumer A owns one- equally. On the other hand B owns a

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large fraction of each firm, and his larger allocation of the perfectly competitive se- dividend dominates the reduction in quence [E(ot)], there exists a redistribu- value of his leisure endowment. Con- tion of ownership and initial endowments sumer B receives a larger income and so that this allocation is a perfectly com- demands both more leisure and more petitive equilibrium for the sequence food, but in the ratio 1: 2. The extra labor [E'(ot)] obtained from [E(ot)] by the redis- needed for production by the entrant and tribution. to compensate for reduced labor by B We can demonstrate this result in comes from A. B consumes the extra food terms of our example. The first step is produced and the extra food available be- to find all Pareto efficient allocations. Set cause of the reduced demand by A. a utility level us for consumer A. Maxi- Though each consumer and firm is well mize the utility of B subject to the con- behaved in a partial equilibrium sense, straints that the leisure-food allocation to the general equilibrium income effects A yields utility level at least us, and the lead to an analog of upward sloping de- allocation to A and B is feasible given mand providing wrong entry signals. their aggregate initial endowment and The second welfare theorem is a more the constant returns to scale technology difficult theorem in our context. Accord- of the aggregate production set of E(O). ing to the ADM second welfare theorem, Vary the utility level assigned to con- under suitable assumptions, every Pareto sumer A to trace out all Pareto efficient efficient allocation can be supported as allocations. an ADM equilibrium by redistributing By the nature of consumer A's prefer- wealth; however, not every ADM equi- ences (recall Figure 8), his utility de- librium of the limit economy E(O) is an pends on the minimum of the amounts equilibrium of the perfectly competitive of leisure and food he obtains. If his allo- sequence [E(ot)]. In our context we must cation has different amounts of leisure redistribute initial endowments and own- and food, then the excess amount of one ership shares so that prices give correct above the other is wasted. Because that entry signals (DSD holds). Otherwise, excess added to B's bundle might in- entry and exit might not cease near the Pareto efficient allocation, and therefore crease the utility of consumer B, it should that allocation would not be a limit of be clear that for this special example we Cournot equilibria of E(ot), that is, an can fix consumer A's bundle, with equal equilibrium of the perfectly competitive amounts of leisure and food, rather than sequence [E(ot)]. fixing his utility level. If A receives bun- Despite the extra difficulties imposed dle (1 - d, 1 - d), then a total of 2 - by the requirement that prices give cor- 2d units of leisure are used (including rect entry signals, it remains true that the 1 - d units to produce the food con- under rather general conditions (such as sumed by A) so 2d units of leisure re- no ) a Pareto efficient alloca- main. The problem is to maximize B's tion can be supported as an equilibrium utility given those 2d units of leisure and of a perfectly competitive sequence. To the technology that turns labor into food do so, the redistribution of endowments in a 1: 1 ratio. Any bundle (2d - f, f ) and ownership shares must not only gen- with f between zero and 2d can be as- erate the required wealth level for each signed to B, and the problem resembles consumer but also ensure that prices give a standard partial equilibrium consumer correct entry signals. choice problem (see Figure 10) with solu- Proposition. For every Pareto efficient tion (1, f ) = (2d/3, 4d/3). As d varies be-

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duction (-1 - d/3, 1 + d/3) and prices Food f =21 p* = (1/2, 1/2). Solving for the "inverse demand" selection F which is continuous

2d / near (-1 - d/3, 1 + d/3) and satisfies / F[(-1 - d/3, 1 + d/3)] = (1/2, 1/2) we obtain

4d/3 , F[(-t,t)] = [(3t+Ot-4)/(2d+20t-2), 'I\ 'I\ (2d + Ot - 3t + 2)/(2d + 20t - 2)] for t near 1 + d/3 where 0 is the fraction of each firm owned by consumer A. In E(ac), when aggregate output is near (-1 - d/3, 1 + d/3) the entry of an additional firm changes the profit of each active firm by approximately (-ao, a) aF/at [(-1 - 2d/3 2d Leisure d/3, 1 + d/3)] ot = -90t2/(Od + 30 + 3d Figure 10. - 3) (t changes by ao when a firm enters) which must be negative for DSD to hold. tween zero and one we trace out all Par- To give correct entry signals, we must eto efficient allocations, with d = 0 corre- redistribute ownership shares so that sponding to B's receiving nothing, and consumer A owns fraction 0 > 3(1 - i)! d = 1 corresponding to A's receiving (d + 3) of each firm. This is possible for nothing. Observe that as d varies, the each Pareto efficient allocation corre- aggregate production (-1 - d/3, 1 + sponding to a d value greater than zero. d/3) also varies. That this can be done in general depends With constant returns to scale technol- on a revealed argument.7 ogy, every ADM equilibrium will have It is important to observe that our ver- zero profit and prices p* = (1/2, 1/2). Thus sion of the second welfare theorem re- the Pareto efficient allocations can be quires more flexibility of transfers than supported as ADM equilibrium of E(O) the corresponding ADM result (Debreu only by assigning ownership of 2 - 2d 1959, p. 95). In that theory all of the units of the aggregate leisure endowment required redistribution can be achieved to A and the remaining 2d units to B. by means of a single commodity, which This generates the required wealth levels may be thought of as "government wealth so that each consumer can purchase the transfer." Here more than a single com- appropriate bundle with prices (1/2, 1/2). modity may need to be redistributed be- The ownership share of firms is irrelevant cause the allocation of all of the endow- for the ADM equilibria. ments is important, not just the To support these Pareto efficient allo- purchasing power. Leonid Hurwicz cations as equilibria of a perfectly compe- (1959) refers to the property that every titive sequence [E'(ot)] we must also as- sign ownership of 2 - 2d units of the 7At d = 0 we hit the constraint 0 ' 1. With d = aggregate leisure endowment to A and O and 0 = 1 we have essentially a one-person econ- 2d units to B in order to generate appro- omy; consumer A owns all the aggregate endowment and all of the firms. Because of the kink in the indif- priate wealth levels. However, we must ference curves in our special example, F cannot be also assign the ownership share of the made continuous near (-1, 1) for this case. Thus in firms so that prices give the correct entry this case there is a Pareto efficient allocation that cannot be achieved as a perfectly competitive equilib- signals in E(ot). The equilibrium corre- rium of a sequence [E'(CL)]; however, this is an artifact sponding to d must have aggregate pro- of the fact that preferences are not differentiable.

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optimum is an equilibrium (after redistri- framework, we next consider the quan- bution) as "unbiasedness." Our analysis tity adjustments of a fixed number of ac- suggests a bias of the competitive mecha- tive firms to achieve a short-run equilib- nism beyond that which government rium. In E(ot) we assume a continuous transfers can correct. This bias can be Cournot dynamics. Assuming all other corrected, but the correction in general firms' outputs are fixed, each active firm requires more than the redistribution of changes its output continuously. In the a single commodity. From the viewpoint limit economy E(O), each firm is infinites- presented here, the competitive mecha- imal and this dynamics agrees with that nism has the bias that it can seek out generated by firms viewing price as fixed. only those optima that, with the means In both E(O) and E(ot) the firms do not of transfer at hand, give the necessary account for the adjustments of other firms entry signals. or the changing of price over time. We prohibit exit in the short run, so that an VI. Dynamics active firm cannot produce zero. It must produce from the nonzero component of Marshall and Walras examined a long- its production set (e.g., some inputs may run equilibrium in which all factors vary be fixed in the short run so that even if freely and "flow toward that branch of all outputs are zero some inputs are production" where they can realize prof- used). If F is continuously differentiable its. In the short run the returns to fixed and ot is small, then the firm has little factors do not necessarily equalize. At effect on price. Thus the incentives and each moment, prices clear markets, but behavior of the firm in the short-run dy- these prices reflect only the relative scar- namic for E(ot) "converge to" that in the city of the variable factors. Walras de- short-run dynamic for E(O). scribed the process from which these We build a bridge between the short prices derive by a tatonnement. run and the long run in the following In each E(ox) firms correctly perceive manner. For concreteness, let us refer the prices, F(y), that will prevail given to the fixed factor as entrepreneurship. any aggregate production y. Thus our After each short-run adjustment we as- analysis presupposes an anticipated ad- sume that entrepreneurship flows toward justment of prices to clear markets, a a- higher profits; that is, the number of ac- tonnement for the exchange economy tive firms in each industry changes be- generated by a fixed production vector cause of their profitability. Following y. In each Cournot equilibrium firms an- such a period of factor movement, a new ticipate that the price vector F(y) will re- short-run equilibrium arises with associ- sult from production y. At the equilib- ated prices, which lead to new incentives rium the prevailing price is the right for factors to move, and so on. From price; that is, it equates supply and de- this standpoint the tatonnement occurs mand. Thus, no firm actually changes quickly relative to the short-run quantity output in an equilibrium. We could have adjustment of firms, which in turn takes assumed that firms formed a subjective place faster than the entry and exit ad- F(y), but we required that F(y) actually justment of the entrepreneurial factor. clear markets. For our model of dynamics As in the short-run adjustment, in E(ot), we assume that the adjustment of prices recognizing only their own effect on to clear markets is instantaneous relative price, firms enter or leave, so again, in to the speed at which firms are able to E(O) this entry dynamic agrees with that change production levels. generated by viewing price as fixed. In Following the standard Marshallian both the short and long run, firms behave

This content downloaded from 131.215.23.153 on Thu, 21 Sep 2017 21:11:36 UTC All use subject to http://about.jstor.org/terms 1304 Journal of Economic Literature, Vol. XXV (September 1987) myopically. The entry-exit decision is a reach a short-run equilibrium; and choice of production set oxY or {O} (recall 3. entry and exit at a rate proportional Figure 9). If F is continuously differenti- to the (firm) profit levels in each able and ot is small, then the firm's deci- industry to reach a long-run equilib- sion has little effect on price. Thus the rium. incentives and behavior of the firm in the long-run dynamic for E(ot) "converge The partial equilibrium market M(O) used in Section III adequately illustrates to" that in the long-run dynamic for E(O). The DSD condition suggests a dy- the dynamics that we have in mind. We namic theory of convergence to the long- add to the hypotheses from that section run equilibrium, in which the realign- the condition that F is nonincreasing and ment of the entrepreneurial factor plays there exists a unique y* such that F(y*) a significant role. Because DSD is a nec- = C(1). In this case there is a unique essary condition for equilibrium of the equilibrium. perfectly competitive sequence [E(ot)], Introduction of the dynamics is as fol- lows. For each aggregate output y, F(y) no infinitesimal firm in E(O) can enter with positive profit at an equilibrium. gives market-clearing prices, the adjust- However, we can conceive of a model ment process of which is stable because where out of a long-run equilibrium the F is nonincreasing. In the short run an entry and exit of firms in a sector is pro- active firm must produce a positive quan- portional to the returns to the entrepre- tity whether or not the positive profit- neurial factor available there. This leads maximizing action yields positive profit. us to consider the stability of the equilib- Each active firm increases output when- rium introduced in the previous sections. ever the current price exceeds the mar- It relates to the questions of whether re- ginal cost at the current output level and turns to the homogeneous entrepreneu- decreases output when current marginal rial factors tend to equalize and whether cost exceeds price. This adjustment is myopic profit-seeking behavior moves an also stable because F is nonincreasing economy toward a Pareto optimum. In and all firms have nondecreasing mar- addition, it is relevant to the viability of ginal cost. For each mass of active firms a planning procedure in which central L,u the short-run equilibrium price associ- planners increase production in the most ated with ,u, p(,u), is determined as fol- profitable sectors. lows: Our previous argument that when at 1. Supply, S[p(,u)], is the integral of is small, both the short- and long-run dy- the profit-maximizing actions of the namics in E(ot) will be similar to the dy- ,u active firms at price p(,u), and namics in E(O) assures us that we lose 2. F[S(p(pi)] = p(pi); that is, the inverse no generality by examining E(O) alone. demand of supply at p(,u) is p(,u), Hence, for simplicity we will discuss the or short-run supply equals demand dynamics in terms of the limit economy at p(,u). E(O). Our three stages of dynamics are as follows: Let ur(pi) denote the profit of each ac- tive firm when there are ,u active firms 1. instantaneous adjustment of prices in short-run equilibrium. If DSD holds, to clear markets given any aggregate then a mass of active firms ,u less than production y; (greater than) the mass of active firms 2. output adjustment by a fixed num- in the equilibrium of the perfectly com- ber (mass) of firms, each viewing petitive sequence receives positive (neg- price as fixed at each instant, to ative) profit. Thus in this case the long-

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run equilibrium is globally stable under ciding on "who was to deliver what to the long-run adjustment process: dlidt whom and on what date." In this section = 'rr(pu). Let us pause to interpret this. the analysis is dynamic and temporary Suppose that we start with the initial equilibrium in spirit. Markets open for mass of active firms ,u(O). The prices, today's exchange. The prices determined p[,u(O)], provide signals for entry. Over in those markets determine profits, time the mass of active firms changes, which induce factor movements and a dif- inducing a continuum of short-run equi- ferent distribution of firms tomorrow. libria. The mass of active firms adjusts The new distribution of immoble factors toward the final equilibrium at which makes tomorrow's prices different from firms earn zero profits and produce at today's and so profits differ, etc. In our efficient scale. temporary equilibrium analysis we as- To examine these ideas in more detail sume that firms are myopic; they do not we return to our original general equilib- consider how prices will change over rium example. Recall consumer A's own- time. Even with such nearsightedness, ership share of firms is 0. For 0 > 3/7 for our example, when prices provide the there is a unique equilibrium for [E(oa)], correct entry signals and so equilibrium the unique ADM equilibrium of E(O), exists, equilibrium is globally stable. This and DSD is satisfied. For any initial mass of course provides a strong link between of active firms that leaves consumers in the existence and stability theorem, and their consumption sets when each active we consider this to be very much in the firm produces at the efficient point (-1, spirit of both Walrasian and Marshallian 1), the dynamics defined by dpldt = ir(VL) analysis. converge to the unique equilibrium * = 7/6. On the other hand, if O.K 3/7, VII. Conclusion then the unique ADM equilibrium re- mains ,u* = 7/6, but this allocation is not As we have shown, with the right even locally stable. It is not locally stable model it is not so difficult to travel be- for either the first-stage tatonnement or tween the world of the general equilib- for the third-stage entry dynamic using rium theorist and the Marshallian, partial the "inverse demand" given in Section equilibrium analyst. Furthermore, there V. See Lars Svensson (1984). are substantial gains from ! A rigor- The example at hand lends itself to a ous general equilibrium framework in- planning interpretation. Suppose that corporates the Marshallian model, with the central authority provides licenses for marginal firms and U-shaped average opening or closing facilities, but aside costs. Perfect competition pertains to from this allows prices to be flexible. In markets where firms are relatively small. the example we start with a certain mass In this case their ability to influence price of licensed facilities. If 0 > 3/7, then the disappears and price-taking behavior re- procedure that has facilities open when sults from our theory. This is because profits are positive and closed when there in perfectly competitive environments are losses converges to the unique Pareto the production sector mimics the behav- optimal allocation. ior of a manager who is unable to influ- Let us summarize the dynamics. Prior ence price. Notice the enhancement of to this section we were concerned with the classical welfare theorems: Competi- static analysis. We treated time in the tion leads to the number of active firms spirt of Debreu's Theory of Value, by consistent with ; it does dating all commodities and opening all not require the assumption of convex markets only once for the purpose of de- technology for the second welfare theo-

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rem (this is in part because Cournot equi- HURWICZ, LEONID. "Optimality and Informational Ef- librium does not require convexity); it ficiency in Resource Allocation Processes," in Mathematical methods in the social sciences. Eds.: highlights the effects of the distribution KENNEThI ARROW, SAMUEL KARLIN, AND PATRICK of wealth on the entry signals conveyed SUPPES. Stanford: Stanford U. Press, 1959, pp. 27- by prices and so on. Finally, the assump- 46. KooPMANS, TJALLING. Three essays on the state of tion that profits induce entry (and losses economic science. NY: McGraw-Hill, 1957. induce exit) plays a major role in the anal- MAS-COLELL, ANDREU. "Algunas Observaciones so- ysis and suggests a classical but much bre la Teoria del Tatonnement de Walrasen Econo- mias Productivas," Anal. Econ., Enero-Junio 1974, neglected dynamics that fits well with the 21-22, pp. 191-224. Marshallian vision. . "Walrasian Equilibria as Limits of Noncoop- Our analysis suggests that we need not erative Equilibria. Part I: Mixed Strategies," J. Econ. Theory, June 1983, 30(1), pp. 153-70. be content with a general equilibrium . "Notes on Price and Quantity Tatonnement theory that excludes marginal firms and Dynamics," in Models of economic dynamics. Ed.: free entry and hypothesizes price-taking HUGO SONNENSCIIEIN. Berlin: Springer-Verlag, 1986, pp. 49-68. behavior for firms without regard to their McKENZIE, LIONEL. "On the Existence of General strategic opportunities. At the same time Equilibrium for a Competitive Market," Econo- we need not settle for a theoryNof value metrica, Jan. 1959, 27(1), pp. 5471. NovsIIEK, WILLIAM. "Cournot Equilibrium with that does not account for intermarket ef- Free Entry," Rev. Econ. Stud., Apr. 1980, 47(3), fects and relies on indirect measures, pp. 473-86. such as consumer and producer surplus, NovsIIEK, WILLIAM AND SONNENSCIIEIN, HUGO. "Cournot and Walras Equilibrium," J. Econ. The- for its welfare economics. We should be ory, Dec. 1978, 19(2), pp. 223-66. all the more suspicious of a theory that . "Small Efficient Scale as a Foundation for loosely specifies behavior and equilib- Walrasian Equilibrium," J. Econ. Theory, Apr. 1980, 22(2), pp. 243-55. rium. The theory exposited here does . "Walrasian Equilibria as Limits of Noncoop- have its special features, but it is precise, erative Equilibria. Part II: Pure Strategies," J. and it allows us to encompass both the Econ. Theory, June 1983, 30(1), pp. 171-87. . "Quantity Adjustment in an Arrow-Debreu- Marshallian framework and the classical McKenzie Type Model," in Models of economic welfare theorems. With this theory less dynamics. Ed.: HUGO SONNENSCHEIN. Berlin: compromise between descriptive rele- Springer-Verlag, 1986a, pp. 148-56. . "Non-cooperative Marshallian-like Founda- vance and mathematical precision is tions for General Equilibrium Theory," in Contri- required, and as a result there should butions to mathematical economics in honor of be better communication between the Gerard Debreu. Eds.: WERNER HILDENBRAND AND ANDREU MAS-COLELL. Amsterdam: North-Hol- "highbrow" and the "bread and butter" land, 1986b. theorists. RoBERTS, KEVIN. "The Limit Points of ," J. Econ. Theory, Apr. 1980, 22(2), REFERENCES pp. 256-78. SONNENSCIIEIN, HUGO. "Price Dynamics Based on ARROW, KENNETH AND DEBREu, GERARD. Existence the Adjustment of Firms," Amer. Econ. Rev., Dec. of an Equilibrium for a Competitive Economy," 1982, 72(5), pp. 1088-96. Econometrica, July 1954, 22, pp. 265-90. SPENCE, MICIIAEL. "Contestable Markets and the ARTZNER, PIIILIPPE; SIMON, CARL P. AND SONNEN- Theory of Industry Structure: A Review Article," SCIIEIN, HUGO. "Convergence of Myopic Firms to J. Econ. Lit., Sept. 1983, 21, pp. 981-90. Long-Run Equilibrium via the Method of Charac- SVENSSON, LARS. "Walrasian and Marshallian Stabil- teristics," in Models of economic dynamics. Ed.: ity,>" J. Econ. Theory, Dec. 1984, 34(2), pp. 371- HUGO SONNENSCIIEIN. Berlin: Springer-Verlag, 79. 1986, pp. 157-83. SYMPOSIUM ISSUE. "Noncooperative Approaches to DEBREU, GERARD. Theory of value. NY: Wiley, 1959. the Theory of Perfect Competition," J. Econ. The- HART, OLIVER. "Monopolistic Competition in a Large ory, Apr. 1980, 22. Economy with Differentiated Commodities," Rev. WALRAS, LEON. Elements of pure economics. Trans.: Econ. Stud., Jan. 1979, 46(1), pp. 1-30. WILLIAM JAFFE. London: Allen & Unwin, [1874- HICKS, JOIIN R. Value and capital. Oxford: Claren- 77] 1954. don, 1939.

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