FS IV 96 - 1
How a Market Reaches its Equilibrium
Jim Y. Jin
January 1996
ISSN Nr. 0 7 2 2 -6 7 4 8
Forschungsschwerpunkt Marktprozeß und Unter nehmensentwicklung Research Unit
Market Processes and Corporate Development
Zitierweise/Citation:
Jim Y. Jin, H ow a Market Reaches its Equilibrium, Discussion Paper FS IV 96 - 1, Wissenschaftszentrum Berlin, 1996.
Wissenschaftszentrum Berlin fur Sozialforschung gGmbH, Reichpietschufer 50, 10785 Berlin, Tel. (030) 2 54 91 - 0
The paper shows that a market generally converges to its unique equilibrium without managers' super intelligence or natural selection. Information is so imperfect that firms can not effectively learn the market and rivals' strategies.
Given very limited information and bounded rationality all firms follow a simple strategy without solving simultaneous equations. The market structure makes firms' naive decisions eventually "optimal". A Nash equilibrium is reached if every firm estimates the slope o f its demand curve correctly. Monopoly, oligopoly, monopolistic competition and perfect competition outcomes are obtained as special cases. The process works only in price competition, not in quantity competition.
ZUSAM M ENFASSUNG
W ie ein M arkt das Gleichgewicht erreicht
In diesem Beitrag wird gezeigt, wie ein M arkt allgemein zu einem Gleichgewicht kon
vergiert, ohne daß eine "Superintelligenz" der Marktteilnehmer vorliegt bzw. eine natür
liche Auslese stattfindet. Die Information ist so unvollkommen, daß sich Unternehmen nur wenig effektiv Marktkenntnisse erwerben und sich über ihre Konkurrenten informie
ren. Geht man von dieser sehr begrenzten Information und der begrenzten Rationalität aller Unternehmen aus, dann gelangt man zu dem Ergebnis, daß die Unternehmen ihre Strategieentscheidungen ohne Rechenverfahren bestimmen, wie sie üblicherweise die ökonomische Theorie voraussetzt (beispielsweise durch Lösung simultaner Gleichungs
systeme). Die M arktstruktur, die in diesem Beitrag postuliert wird, kann die "naiven"
Entscheidungen der Unternehmen durchaus als optimal ausweisen. In der Analyse wird gezeigt, daß ein Nash-Gleichgewicht dann erreicht ist, wenn jedes Unternehmen den Anstieg seiner Nachfragekurve genau schätzt. M it dieser Annahme kann gezeigt werden, daß die Gleichgewichtsergebnisse für das Monopol, das Oligopol, die monopolistische Konkurrenz und die vollständige Konkurrenz als Sonderfälle erzielt werden können. D er hier beschriebene M arktprozeß funktioniert allerdings nur, wenn es sich um Preiswett
bewerb und nicht um einen Mengenwettbewerb handelt.
I thank H. Albach, R. Amir, F. M. Fisher, W. Novshek, K. Okuguchi, L.-H. Roller and Z. Zhang for the helpful comments. The responsibility for the remaining errors is solely mine.
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I. Introduction
Most markets seem to have orders. Prices tend to be stationary if the market conditions do not change. In a free market economy firms will adjust their actions if they think they can get higher profits. Therefore stationary prices imply that firms view their actions "optimal". When a market reaches this state, one can call it an equilibrium. The belief that a market tends to reach an equilibrium goes back far before Nash. The classic school such as Adam Smith and Karl Marx claimed this based on free entry and exit. Cournot first presented the equilibrium as an outcome o f firms' optimal decisions rather than the magic of an "invisible hand".
There are 4 basic market equilibrium models, perfect competition, monopolistic competition, monopoly and oligopoly. The perfect competition model assumes firms choose outputs taking the market price as given. It fails to explain how the equilibrium price comes out. Monopolistic competition yields an equilibrium only if there exists free entry. Before the "invisible hand" makes every firm break
even, we do not know where the market ends up. A monopoly market reaches its equilibrium when the monopoly firm chooses the optimal price or output to maximize its profit. All these three only represent very special markets. The oligopoly model is more general and covers the other three as special cases when the number o f firms changes from 1 to infinity. An oligopoly market reaches its equilibrium when all firms' actions are optimal against each other.
Nonetheless, the information and rationality required for an oligopoly equilibrium are rather unrealistic. Demand and cost functions are assumed common knowledge. To make optimal decisions firms need to solve simultaneous equations. This is reasonable in the original Cournot and Bertrand models where duopolists face the same demand curve and have identical costs. However, to use this paradigm to explain a real market with n asymmetric firms, the common knowledge assumption becomes too strong. There is numerous studies on
knowledge except for a few parameters. Managers are supposed to act more rationally than in the perfect information case. One can hardly claim these models describe firms' behavior more realistically. Although many businessmen might be smarter than we economists, the fact is that they do not solve simultaneous equations while making their decisions. The game we love so much may be just
"the game economists play" Fisher (1989). It does not explain how an equilibrium is reached in a real market.
The evolutionary economics argues an equilibrium as the outcome or market evolution. Firms almost have no information and make decisions by instincts. An equilibrium is reached when only those whose decisions are luckily "right"
survive and the others go bankrupt. This paradigm requires almost infinite bankruptcies and a very long time before an equilibrium is reached. In biology species extinct when its instinct feature could not adjust to the changing environment. One can refer firms' efficiencies as their unchangeable instincts. It is less natural to argue prices or outputs as unfitted features. Hence evolution may tell us why certain firms survive in the long run, but says little why certain prices can be sustained in the short run without exit and entry.
The learning literature shows the Nash equilibrium as the result o f learning instead o f "the fittest surviving". Firms do not act like animals but make sensible decisions. However, as pointed out by Fudenberg and Tirole (1991, Ch. 1): "the reduction in the informational requirements is made possible by the additional hypotheses o f the learning story: Players must have enough experience to learn how their opponents play, and play must converge to a steady state.” Some information commonly assumed in the learning literature still looks demanding.
We take an excellent paper by Kailai and Lehrer (1993) as an example. The paper
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assumes perfect monitor on others' actions. Each player knows his own payoff and has compatible beliefs on others' strategies, i.e. its prior assigns a positive probability to any true strategy chosen by rivals. These are not always guaranteed in the real world, since firms may not know their demand functions and strategy spaces are often infinite. To update beliefs according the Bayesian rule is not trivial for managers. More importantly, a learning process does not converge to a steady state in general.
It would be desirable to further reduce the information required in the learning literature and show the process indeed converges. An early paper by Hosomatsu (1969) explored this road in a homogeneous Cournot model. He argued that to reach an equilibrium (a stationary price) need not to assume firms knowing the demand function and rivals' costs. "It is necessary only that each oligopolist knows the behavior o f the actual market price, . . . makes repeated posterior estimates o f the amount produced by the 'rest o f the industry' by examining the actual market price” (Hosomatsu pp. 117). Under his conditions even if firms do not know own payoffs, do not have perfect monitor and compatible beliefs, do not follow the Bayesian rule, they eventually produce "optimal" outputs given the information they have. However, it is not clear whether his argument can be generalized beyond homogeneous product cases. More crucially, if firms do not overestimate the slope of the demand curve, his condition essentially requires the equilibrium is stable. The stability is only guaranteed in Cournot duopoly, but not in oligopoly.
It was pointed out by Theocharis (1959) that the Cournot equilibrium is unstable with more than 3 firms. His finding intrigued a series research on the stability of the Cournot equilibrium. Fisher (1961) found that adjustment costs and increasing marginal costs are stabilizing factors and may restore the stability. But these factors are not generally granted. Hahn (1963) proved that the equilibrium is
stable with continuous quantity adjustment. Yet firms change their decisions discretely, not continuously. Therefore, for more than three firms with discrete adjustment, zero adjustment cost, constant marginal costs and reasonable slope estimations, we can not claim the market converges to an equilibrium based on Hosomatsu's argument.
On one hand we have special markets requiring reasonable information (perfect competition, monopolistic competition, monopoly and symmetric duopoly). On the other hand we have general markets requiring unrealistic information (asymmetric oligopoly). In between we can compromise the levels o f reality (the imperfect information models and the learning literature). But no model is acceptable on both sides. One can not claim a general market with realistic information tends to reach an equilibrium. How a market reaches its equilibrium has not been explained.
Here we want to fill up the gap. One should not be too discouraged by the instability in quantity competition since most firms choose prices, not quantities.
Quandt (1967) examined the stability problem under continuous price adjustment and found a sufficient stability condition which is guaranteed in duopoly.
Okuguchi (1969) confirmed Quandt’s result with discrete price adjustment. Their results suggest that price competition may be more likely to yield an stable equilibrium than quantity competition. Although they did not prove the stability in a general condition, no one has found a common unstable case under price competition as Theocharis did in quantity competition. This leaves a possibility to explain stationary prices in a real market with realistic information.
A general market we consider could be monopoly, oligopoly, monopolistic competition, nearly perfect competition or any mixture of them. We try to limit firms' information to what seems guaranteed in most real markets. We will show
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that all firms follow a simple price strategy without solving simultaneous equations and the market converges to a unique stationary price equilibrium.
The next section introduces the model. Section 3 establishes the unique equilibrium. Section 4 proves the market convergence. Section 5 presents the 4 basic models as special cases. Section 6 discusses strategic thinking and the last section suggests some empirical testing.
2. M odel
We make the following assumptions:
(A l) There are n firm s producing n products. Every firm chooses its price to maximize its profit in the current period.
When a firm can not predict any response to its current price decision from its rivals, it is no use to look beyond one period. The possibility o f anticipating rivals' responses will be discussed in Section 6.
Let pit be firm i's price and pt be the n x l price vector in period t. The quantity demanded for firm i's product in period t is xit and the demand vector is xt.
(A2) Firms know their own sales and prices in the previous period. Every firm i thinks o f its unit cost as constant.
The second part o f this assumption can be justified by repeated empirical studies that found "very widespread use o f average cost pricing” in most industries (Hay and Morris pp. 181). "For pricing purpose firm s often think o f their average variable (or average total) cost curves as horizontal" (Hay and Morris pp. 182).
In many cases, firms deliver their products to the market from inventory, not directly from assembly lines. Although products may be produced at different costs, it makes no difference when they are taken from the inventory. Particularly
in big companies, price decisions and production decisions are made by different divisions. Marketing managers normally do not have a clear idea how unit costs are affected by their price decisions.
Let Cj be firm i's cost and c be the n x l cost vector.
(A3) Every firm estimates the slope o f its demand curve as a constant number.
Firms must realize that their demand curves are downward sloping. Otherwise, they would raise prices to infinity. However, it seems unusual that a firm's slope estimation varies as its price changes.
3x,
Let Xj be the absolute value of firm i's slope estimation, i.e. Aj = -Ej(3pit). Let A be a diagonal matrix whose ith diagonal element is A?
(A4) Given p t the representative consumer chooses x t to maximize his surplus u(xt) - p t x r The utility function u(x) is strictly concave and continuously twice differentiable. For any x and y, there exists z in the neighborhood o f x and y such that u'(x) - u'(y) = u"(z)(x-y).
If u(x) is quadratic, u"(z) is constant and the last equation holds everywhere. If u(x) cubic function, the equation holds when z = (x+y)/2. Consumers can hardly evaluate the forth order derivatives of their utilities. The assumption seems holds for most reasonable utility functions.
(A5) The slope estimation is sufficient, i.e. fo r any relevant true slope, dx;, <
°Pit 1 .5 \.
The slope is the easiest thing to know since each firm can estimate it by its price and sales data. So the slope estimation should not be too bad. The assumption seems reasonable if the slope does not vary dramatically. In particular when the utility is approximated by a second Taylor expansion, all slopes are constant.
z
(A6) Prices are strategic complements, > O for all j i.
We want to restrict the information to a realistic level, although real managers could know more than that. We allow firms having more information, but assume that the information is insufficient for a strategy based on solving simultaneous equations. For instance, firms may observe others' prices and realize certain interdependence. Nevertheless, so long as firms can not predict rivals' behavior in a reasonable precision, a price based on imprecise information is very unreliable and may not be taken. Although more information may change firms' strategies, the m inim um information case can serve as a benchmark.
In our model a firm does not know its own payoff function since it does not know its demand function. Nor does it know other firms' payoffs, strategies, previous actions and probably even their existence. It may perceive the impact of rivals' actions as a demand shock due to consumers' preference or a natural shock. The common knowledge is an empty set. Essentially it is not a game at all. However, when every firm acts alone to maximize its profit, the market may gradually reach a stationary position where the price vector repeats itself over time. We call it a stationary price equilibrium. We will show a market converges to a unique stationary equilibrium under reasonable conditions.
3. Stationary prices
Under our assumptions, the first order condition for the consumer surplus maximization is u'(xt) = p t. Since u(xt) is strictly concave, this equation gives the unique optimal consumption xt. We write the demand function for n firms as xt = x(pt) and for firm i as xit = x;(pt).
As every firm i chooses pit in each period t to maximize the expected profit, E J xit(pit-Cj)}, the first order condition is
E i(x ii) - X/pa-Cj) = 0 (1) The second order condition is guaranteed since — = -2X, < 0. Due to firm i's
^Pit2
limited information, it can not predict how other firms' prices change from the last period to this period. It can only estimate its sales taking into account o f its own price change. So its "rational" sales estimation is
Ei(xit) = xiM - XiCPu-Piu) (2)
Substituting (2) into (1), firm i's optimal price is
P it = ° - 5 (P it-i + c i + x it-
A ) (3)
The price strategy is linear in its cost, previous price and sales. It does not contain other firms' demand and cost parameters. Managers need not to solve simultaneous equations because they need not to worry whether their strategies consistent with others'. This simple strategy seems manageable for real businessmen.
The price strategies for all n firms can be written as
pt = O.SCp,.! + c + A-iXj.j) (4)
The prices are stationary if p, = p,.r Denote the stationary price vector by p and the demand vector by x(p). The stationary price vector satisfies
p = c + A-’x(p) (5)
When (5) holds, the output vector is also stationary. (2) implies Ej(xit) = xit for all i. Firms' demand expectations are fulfilled and their prices are "optimal" given their slope estimations. Since no firm want to change its price and all outputs stay constant, we say the market reaches its equilibrium. Then we first prove
Proposition 7: Given (A l) - (A4), there is a unique market equilibrium.
oiy
(i) Prove the existence: Define a mapping f(p): R" —> R" R" in the following way.
For all i, let p 4 be an (n -l)x l vector without p? Define the ith item o f f(p) as a function y; = ^(p.,): Rn+ —> R+ such that
ys - C; - x /y p p J/X j = 0 (6)
The left hand of (6) continuously increases in y;. It is negative when y; is close to Cj and positive when it is sufficiently large. Thus for any p , a unique y} exists satisfying (5). Further, since xi(yi,p.i) is continuous in p.,, f(p) must be continuous in p. Thus a fixed point exists such that f(p) = p. This point satisfies (5) and hence an equilibrium.
(ii) Prove the uniqueness: Suppose p' and y = x(p') also satisfy the stationary condition (6), i.e. p' = c + A_1y. Subtracting them from each other we get
p - p' = A_1(x-y) (7)
Consumer surplus maximization implies u'(x) = p and u'(y) = p'. Under our assumption we can write p - p' = u'(x) - u’(y) = u"(z)(x-y) where u"(z) is a negative definite matrix since the utility is strictly concave. To simplify the presentation, write u"(z) as u". Then we can write x - y = (u")_1(p-p'). Substituting this into (7) we have {A-(u")’1}(p-p') = 0. As A and -(u")-* are both positive definite, so is A - (u")_1. Therefore p - p' = {A-Cu")’1 }‘'O = 0, i.e. p' = p. The
equilibrium is unique given A. ||
Under our assumptions neither the existence nor the uniqueness o f a Nash equilibrium is guaranteed. It is the constant slope estimations that establishes a unique equilibrium. As few managers can evaluate the impacts o f prices on the slopes, a market should have a unique equilibrium normally. The next step is to show a market actually converging to such an equilibrium.
4. Convergence
If a market converges to the equilibrium, the difference between any price vector p t and the equilibrium price p should shrink over time. Since A is positive definite, (pt-p)'A (pt-p) is non-negative for any p t - p and is zero if and only if p t = p. We can use dt = (pt-p)'A(pt-p) to measure the deviation from the equilibrium.
A market converges if dt is strictly decreasing in t for any p, p.
Using (4) and (5), we have
p l - p = 0,5{pt.1 - P + A-^xCpjJ-xCp)]} (8) Similar to the previous case, we can write p t_x - p = u'Cx,.^ - u'(x) = u"(xt.r x).
Thus xt.j - x = (u'^C Pt.j-p). Let I be an identity matrix. (8) can be written as pt - p = 0.5{I+A-1(u")-1}(pt.1-p ) (9)
Then we can write dt can be written as a function of p t_j - p:
dt = 0.25(pt.j-p)'{ A+2(u")-1+(u")-1A-1(u")-1 }(p,.i-p)
= dM - 0.25(pt.1-p)'(3A-2(u")-i-(u")-iA-i(u")-1)(pt.1-p) (10) Obviously dt strictly decreases if the matrix M = 3A - 2(u")_1 - (u'^-’A '^ u ")’1 is positive definite. W e can prove (see Appendix A) that M is positive definite if 3A + (u")"1 is so. Therefore we get
Proposition 2\ Given (A l) - (A4), a market converges to its unique equilibrium if
3A + (u")'1 is positive definite.
If one changes the unit o f good i, the ith row and ith column o f 3A + (u")'1 are multiplied by the same number. This does not affect the positive definiteness of the matrix. So the convergence condition is independent of the units o f the products. This is an advantage dealing with heterogeneous goods.
The matrix 3A + (u")’1 tends to be positive definite if any \ increases. A market is more likely to converge if firms estimate the slopes more conservatively (high
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absolute values). If firms significantly underestimate slopes (A.j very small), 3A + (u " )1 becomes negative definite as (u " )1 dominates. Hence a market does not converge if firms underestimate their slopes too much.
Let us consider the meaning o f (u " )1. Given z, one can solve a price vector p = u'(z). The implicit theorem implies 3x = (u " )1. Hence the elements o f (u")'1 are
the slopes and cross coefficients o f demand functions at some price vector. Since
3x- 3x-
the utility is concave and prices are strategic complements, < 0 and > 0 for any j i. According to McKenzie (1960), such a matrix is negative definite if and only if it has a dominant diagonal. As (u")'1 is negative definite, it has a dominant diagonal, i.e. there exists dj and dj > 0 such that dj 1 I > X dj 1 for a lii.
j*i aPj
5x 5x
(A5) implies > - ^ 1, i.e. the absolute value o f each diagonal element of 3A + (u")'1 is no less than that in (u")'1. Its off-diagonal elements have the same absolute values as those in (u")_1. Hence the matrix 3A + (u")_1 must have a dominant diagonal as (u")_1 does. Again by McKenzie (1960), a matrix with positive diagonals is positive definite if it has a dominant diagonal. So 3A + (u")'1 is positive definite, (pt.1-p)M(pt_1-p) > 0 and d, < dt_j for any pt p.
For any price vector different from the equilibrium one, the deviation dt strictly falls in each period. The final step is to show that limp, = p. Since dt decreases in t and is bounded by zero, it has a limit, i.e. lim(dt.j-dt) = 0. Then (10) implies lim (p,_1-p )M (p ,.1-p) = 0. As M is positive definite, we have limp, = p. This gives us Proposition 3: Given (A l) - (A6) a market converges to its unique equilibrium.
The assumptions we used for the market convergence are sufficient, but not necessary. In fact prices need not be strategic complements. Suppose a market has two "industries". Prices within each industry are strategic complements while
those between industries are strategic substitutes. In this case 3A + (u")'1 is a Morishima matrix and is guaranteed positive definite.
As a matter o f fact, prices need not be strategic complements. So long as (u " )1 has a dominant diagonal, (A5) guarantees the positive definiteness o f 3A + (u " )1.
W hen prices are a mixture o f strategic complements and substitutes, even if (u")-1 has no dominant diagonal, 3A + (u " )1 still can be positive definite. Only when all prices are strategic substitutes, is a dominant diagonal necessary as well as sufficient for the market convergence.
Prices are all strategic substitutes if goods are all complements. Even if so, the convergence is still likely. One can prove that 3A + (u")-1 is positive definite if the
, 3u2
matrix 3u" + A 1 is negative definite. Complement goods imply '^‘x ^ x - 0 f° r j
■*- i. Since u" is negative definite, it must have a dominant diagonal, i.e. there exists dj and dj > 0 such that for all i
, du2 du2
di3xj2 + ^ d x ^ x j < ° 1 j*i 1 J
(11)
Also by McKenzie (1960), 3u" + A 1 is negative definite if and only if it has a dominant diagonal which is negative, i.e. for all i
, ,3u2 1 du2
d* dxj2 + 3Ä? + ^ x ^ X j < ° 1 1 j*' 1 J
(12)
(12) is more likely to hold if l/3Xj is small relative to 3u2 (A5) implies 1 /3 ^ < - 3x- 5x
0 . 5 / ^ . If is constant, such as in a linear demand case, it is plausible to assume
3x- 3x
accurate slope estimation \ As is the ith diagonal elem ent o f (u")'1, we ,3x, 3u2 „ 1 3u2
hold as long as (11) is not too close to zero.
can show (Appendix B) that -1 /^ j1 < $ ° 3 ^ < ’9^2^- Hence (12) is likely to
(11) close to zero implies that when every Xj increases by di; the marginal utility o f all goods change very little. Concavity is eliminated by complementarity. It
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means, for example, when cars and gasoline are bundled together, the marginal utility o f the composite good becomes almost constant. This is not realistic. Hence (11) should hold with certain cushion and (12) is likely to hold. In general it seems 3A + (u " )1 is positive definite if the utility function is sufficiently concave and slope estimations are sufficiently strong. Then one could expect most markets to converge.
O f course, a market always changes over time. The utility function, marginal costs and firms' slope estimations may not be constant for ever. A market may not be able to reach its equilibrium before those parameters change. Nevertheless for a new set o f parameters, the market moves to a new equilibrium. For any set o f parameters there is a unique stationary equilibrium which is the converging point.
The m arket discussed here needs not to satisfy the specific features assumed by the conventional models ranging from perfect competition to monopoly.
Nonetheless the stationary equilibrium gives virtually the same outcome. To fulfill our early promise o f a general model, we will show in the next section that the basic 4 models can be covered as special cases.
5. Special markets
The market presented here allows monopoly, oligopoly, monopolistic competition and perfect competition as special cases.
(i) Monopoly: If good i is independent, all the elements in the ith row and ith column o f u" are zero except the diagonal one. Thus all elements in the ith column and ith row o f (u " )1 are all 0 except the diagonal. Firm i is thus a monopoly. (9) becomes
P it-ft = 0.5(1 + ^ 1/Xi)(pit_1-Pi)3x- (13)
Since < 0, 0.5(1 + | j V \ ) < 1- IPit-Pil < IPit r P i l if °-5( 1 + f j j W > - b or Xj > - 3x;
dp/L/3. Since firm i's demand only depends on its own price, the slope estimation should be accurate given price and sales data. So \ > - ^ 7 3 should hold and the monopoly price should converge. Especially with linear demand the slope is
X1 - X- C^X-
constant, ~a---- = -z- 1 for any T. Thus X, should be very accurate and in fact the PiT - Pil-1 5Pi
true profit maximization price is reached immediately in the second period. The monopolist needs not to know its demand function or even to realize that it is a monopoly.
(ii) Monopolistic competition: If all firms are small, the model fits monopolistic competition except that free entry is not necessary. In the Chamberlin model the equilibrium can not be reached when firms make positive profits. Zero profits may prevail in the long run, but certainly not in the short run. The standard monopolistic competition model does not tell us before the long run equilibrium where prices stay or whether they would stay at all. Our model allows positive profits as well as free entry, hence covers monopolistic competition as a special case. Although it is hard to dispute free entry in the long run, the capital mobility is seldom perfect in reality. W henever free entry has not been fully realized, small firms can enjoy stable and positive profits.
The standard oligopoly and monopolistic competition models rely on an arbitrary distinction between firms' behavior. They are either fully aware of the market interdependence or completely ignorant about it. Real firms are not so polarized.
It is hard to draw such a line between "strategic" and "non-strategic" firms according their sizes. In fact bigger firms need not be more strategic than smaller ones. The awareness o f mutual interdependence is not often fully or none. Our model does not distinguish which firms think strategically and against whom. The difference between oligopoly and monopolistic competition can be superficial.
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The big and small firms follow the same strategy (3). The differences in
"performance" are due to "structure", not "conduct".
(iii) Perfect competition: Cournot showed that two firms competing with identical products can make positive profits. Hence the standard theory assumes infinite number o f firms for perfect competition. Bertrand argued that two are enough for the perfect competition outcome because they undercut each other. Our model resembles Bertrand's since firms set prices not quantities. However firms can not undercut each other due to limited information. Nevertheless, the same result can be obtained as a limit case.
Assume that products 1 and 2 are close substitutes. Then the first two rows and
32u 32u
two columns of u" are almost identical because er—e— and z —r— almost equal for
C/XjOX- OX2OX1
all i. The determinant o f u" is nearly zero. Let Cü be the cofactor o f the ith diagonal element o f u". C n and C22 are generally not too small because they do not have two similar rows or columns. The first two diagonal element o f (u")'1, C n /|u"J and C22/|u"| must be infinitely large. The slopes o f firms 1 and 2 must be very flat. Sufficient slope estimations imply very high Xj and X2. The equilibrium condition (7) implies p; - c; = x/Xj. For a finite output, the price margin approaches zero when Xj approaches infinity. Hence if firm 1 and 2 both produce finite outputs, their prices must be close to costs. Profits fall to zero so long as they realize their demand very elastic. They have to lower prices to marginal costs eventually, even if they do not know rivals' existence. Furthermore, when Xj is very large, the price strategy (3) is insensitive to xit.,. Prices stick to the previous ones and do not respond to the demand change, just like price takers. Hence
"price taking" is a limit case o f "price setting".
(iv) Oligopoly: When there are a few big firms, we have oligopoly. However the market here differs dramatically from conventional oligopoly models in terms of
information and rationality. Nevertheless the equilibrium outcomes are basically the same, as if firms have perfect information except the slopes. W e deprived firm i's information about how its demand is affected by others' prices, about others' demand and costs. This makes little difference in the equilibrium.
We did not specify how firms form their slope estimations. One could assume firms change prices every week and then update slope estimations every year based on price and sales data. After many years slope estimations should be more and more accurate. It seems plausible with linear demand and constant slopes. If slope estimations tend to be correct over time, one can argue that given a sufficient time the Nash equilibrium is the ultimate gravitating point. The equilibrium predicted by non-cooperative game theory is valid even though firms do not play a game at all. As a matter o f fact, even if the slope estimations are not correct, the equilibrium is indistinguishable from a true Nash equilibrium if no one knows the slopes better than firms.
Here we only considered price competition. As discussed in the introduction the earlier studies indicate quantity competition does not yield stationary prices with limited information and bounded rationality. As a common knowledge, most firms set prices, not quantities. It is not too embarrassing to find out that prices tend to be stationary in price competition, but not in quantity competition.
6. Strategic thinking
So far we have not considered firms' mutual awareness as if it does not exist. We argued that the Nash equilibrium in oligopoly does not depend on strategic thinking. Knowing the interdependence is useless if firms only consider one period profits. However, firms indeed often look beyond one period and take into account others' reactions while setting prices. W hen firm i chooses a price, it may
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not anticipate whether and how it will change again in the next period. If it thinks that the price may last more than one period, it may also anticipate other firms' price changes during that time.
If firm i assumes that other firms follow similar strategies as its own, it may realize that its price will affect other firms' prices through affecting their sales.
For those producing substitute goods, firm i's price increase raises their sales.
Then they will raise their prices and increase firm i's demand. Similarly, for complement goods producers, firm i's price rise reduces their sales, pulls their prices down which also boosts firm i's demand. In both cases the impact o f its price increase on its demand is reduced. Realizing this its slope estimation \ becomes smaller. Given limited information firm i would not be able to estimate this strategic effect precisely. It is reasonable to assume that the estimation is also constant. Hence we have a new slope estimation X/ smaller than the original X?
Every firm behaves the same way as before. The same strategy (3) applies and the market converges provided the new slope estimations are still "sufficient"
comparing to the true slopes.
The price margin/output ratio, (p-c^/Xj = 1/X/, will be bigger. Hence the equilibrium outcome become less competitive or more "collusive" than the Nash equilibrium. In this sense "strategic thinking" indeed reduces competition.
Although the new equilibrium looks less competitive, firms are actually not guilty o f collusion. They only need to "rationally" anticipate others' reactions. There is no need for explicit or implicit agreement, coordination, discipline or credible punishment.
This is quite different from quantity competition. W hen firm i increases its output, it anticipates the other firms reduce (raise) outputs if they produce substitutes (complements). Firm i's sales decrease is reduced. This "strategic thinking" should encourage firm i to produce more output and lead to a more
competitive outcome than the Nash equilibrium. To keep prices above the Nash equilibrium level does require coordination, discipline or credible punishment.
This has been discussed extensively in the literature and generated many results such as the "folk theorem". Generally speaking a Cournot-Nash equilibrium can be achieved and collusion can be prevented if the antitrust authority makes coordination and credible punishment impossible. Under price competition, however, a certain "collusive" outcome seems natural and difficult to eliminate.
The key factor o f collusion is to make every firms' slope estimation small. This can be achieved by price responses among firms. If a firm is convinced that its price cut is always matched and its price raise is always followed by rivals, it will choose a higher price without collusive agreement or punishment threat. Stronger responses a firm anticipates, less an incentive it has to lower its price. A price leadership and parallel pricing serve this goal. This simple approach seems more effective and probably closer to the reality than complex punishment schemes in a Cournot super game.
7. Test and conclusion
We tried to present a market and firms' behavior in a realistic way. This opens the opportunity and challenge for empirical testing. The following claims made in this paper might be tested.
(i) Firstly the model gives an uniform price strategy (3) which applies to monopoly, oligopoly, monopolistic competition, perfect competition or any combination o f them. This simple strategy can be tested empirically against the null hypothesis that strategies differ dramatically in different market structures.
One can use short run data to estimate (3). Under the null hypothesis one expects R2 generally low and quite different for firms arranging from monopoly to
19
perfectly competitive ones. Our model predicts R2 generally high and no systematic differences among firms.
(ii) Secondly the equilibrium outcome resulted from the uniform price strategy has a common feature. Regardless o f the degree o f competition, from monopoly to perfect competition, a uniform relation exists in the equilibrium: i.e. (pj-c^/x; = 1/A.j. It means that the price elasticity o f demand, Xjp/Xj, is equal to the ratio of revenue to the profit, PjX/Cpj-Cj)^. The standard theory, as the null hypothesis, predicts this relation for oligopoly and monopoly firms, but not necessarily for small or non-strategic ones. One can use long run or average data to see whether big firms fit this relation better than small ones. If there is no significant difference, our model seems empirically supported.
(iii) The Cournot model has dominated the economic literature, especially the industrial organization. The most cited empirical justification is based on the positive profits predicted by the Cournot model, but not by the Bertrand's.
However this assertion is valid only if goods are nearly perfect substitutes.
Physically identical products need not be close substitutes if they are offered in different time, locations, with different services and so on. With differentiated or heterogeneous products, price competition nee not eliminate profits.
To see whether the Cournot or Bertrand equilibrium fits the real market better, one should make quantitative comparison instead o f qualitative one. In a Bertrand equilibrium the price margin/output ratio (pj-cp/X; = - l / ^ 1, while in a Cournot3x-
.... , . . . . d a 9u2 4 , 3u2
equilibrium the same ratio (Pi-cj/X; = As we have shown that > - 1 / |^ . If one can estimate and around the equilibrium, two hypothesis's
can be tested. This may help us get out o f an awkward situation where theorists make policy recommendations based on quantity competition and the results often reverse in price competition.
The main message in this paper is: prices tend to be stationary in most markets, not because o f managers' super intelligence or natural selection. It is the market structure that lead firms' naive decisions equivalent to the most sophisticated ones.
The process works in price competition, not in fictional quantity competition.
Firms' conducts need not differ dramatically with their sizes or market positions.
Performances are directly due to the structures. It might be called a "structure- performance" paradigm.
21
Appendix A, Prove that M is positive definite if 3A + (u " )1 is so.
Pre- and post-multiplying the matrix M by A’172 and denoting -A*172^ ' ) - ^ - 172 by C, we can write A4/2MA-1/2 as
N = 31 + 2C - C2 (Al)
M is positive definite if and only if N is so. Since N is symmetric, it is positive definite if all o f its roots are positive. It is sufficient to show that |N-XI| = 0 holds only if X > 0.
Notice that N - XI = (4-X)I - (C-I)2. Since (C-I)2 is positive definite, |N-XI| = 0 implies 4 - X > 0. Thus we can write
N - XI = -{C-[l+(4-X )1/2]I} {C-[l-(4-X)1/2]I} (A2) So |N-XI| = 0 holds only if |C-[ 1 +(4-X) 1/2]I| = 0 or |C-[ 1 -(4-X) 172]I| = 0. If X < 0, 1 - (4-X)1/2 < -1. Since C is positive definite, its roots are all positive, thus the determinant |C-[l-(4-X)172]I| * 0. When X < 0, 1 + (4-X)172 > 3, |C -[l+(4-X )172]I| = 0 requires that C has at least one root greater than 3. Hence it suffices to show that none o f the roots o f -A-172^ " ) ' ^ ’172 is more than 3, i.e. |A-172(u")',A-172-tI| = 0 implies T < 3.
If 31 + A '1/2(u")'1A-172 is positive definite, so is Tl + A’172^ " ) ' ^ ' 172 with any T > 3.
Then the determinant |A '1/2(u")'1A '172-Tl| < 0 always. Thus the proof will be complete if 31 + A’i72(u")‘1A 1/2 is shown positive definite, which holds true if 3A + (u")_1 is positive definite. Hence M is positive definite if 3A + ( u " ) 1 is so. ||
Appendix B:
Let an (n - l)x l vector bj be the ith column o f ( u " ) 1 w i t h o u t ^ 1, an (n - l)x l vector 3x- ßi be the ith row o f u" without d2u an (n -l)x (n -l) matrix (u");J be the submatrix
of ( u " ) 1 without its ith row and ith column. Then we have
<B1)
^ b , + (u'XiPi = 0 (B2)
M ultiplying (2) by ß ’, we get ^ ^ f r ß ; = -ß;(u")Jßj. Substituting it into (B I) we get32u
dx I ^2U 0 2U 3x- 3 2u
= 1 + ß X u '^ -J ß i/^ . Since (u");} is negative definite and < 0, >
2 —
, J d2U ÖX;
1 and - v ; >
dXj2 dpj li
23
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