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The Review of Economic Studies, Ltd.

Competitive Stock Markets Author(s): Louis Makowski Source: The Review of Economic Studies, Vol. 50, No. 2 (Apr., 1983), pp. 305-330 Published by: Oxford University Press Stable URL: http://www.jstor.org/stable/2297418 . Accessed: 28/06/2014 08:13 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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0034-6527/83/00200305$00.50

Review of Economic Studies (1983) L, 305-330

?

1983 The Society for Economic Analysis Limited

Competitive Stock

Markets

LOUIS MAKOWSKI Cambridge University In a perfectly competitive general equilibrium model with many periods, incomplete markets, and trading through time, we show: (a) current net market value maximization is unanimously favoured by shareholders as the objective of the firm (b) this corresponds to maximizing a relatively simple present discounted value formula (c) the formula is used to derive an Arrow-Lind-type result on the absence of a risk premium in the discount factors for valuing investments whose risk is uncorrelated with social risk (d) competitive stock markets are constrained Pareto optimal in the sense of Diamond.

Suppose that a perfectly competitive firm wishes to determine its intertemporal production-and-investment plan in accordance with its shareholders' interests. Will it be able to satisfy this desideratum? And if so, what course should it pursue? In a general equilibrium model with many periods, uncertainty, incomplete markets, and trading through time, we show that 1. All initial shareholders of a perfectly competitive firm will wish that firm to choose a production-and-investment plan that maximizes its current net market value (Theorem 1 below). 2. If shareholders in the firm alter through time because of transactions in the stock market, all ex-post shareholders will unanimously concur with ex-ante shareholders' wishes. So, there is no potential conflict between ex-ante and ex-post shareholders (Theorem 5). 3. As in the case of complete markets, the production and investment decisions of competitive firms do not depend on shareholders' judgements about the probabilities of the various possible events or upon their preferences. Firms choose their production and investment plans to maximize a relatively simple present discounted value (PDV) formula, the discount factors on future (contingent) earnings representing the market's valuation of the firm's future earnings (which, of course, implicitly includes the market's evaluation of the probabilities attached to the firm's contingent earning stream) (Theorem 2). Conclusions 1-3 contrast sharply with much of the popular literature. Contrary to conclusion 3, in Dreze equilibria (see Dreze (1974)) firms seek to maximize a weighted sum of ex-post shareholders' preferences for contingent income. (Since there are only two periods in the Dreze model, ex-post shareholders are well defined.) And in Grossman-Hart (1979) equilibria, firms seek to maximize a weighted sum of initial shareholders' preferences. Contrary to conclusion 1, neither the Dreze or Grossman-Hart weighted sum is generally, unanimously preferred by shareholders since each shareholder would prefer his own valuation to receive all the weight in the firm's decision. And contrary to conclusion 2, in general these two weighted sums conflict; e.g. in Grossman-Hart there exists the problem of ex-post shareholders wanting to change initial shareholders' production decisions. We also show that 4. Competitive stock market equilibria are generally constrained Pareto optimal (CPO) in the sense of Diamond (Theorem 8). Furthermore, the only allocations which 305

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can Pareto dominate a competitive stock market equilibrium are ones that involve movements in directions that cannot be financed because of the incompleteness of markets (e.g. because of the absence of insurance markets) (Theorem 7). Again by contrast, neither Dreze or Grossman-Hart equilibria are generally CPO unless there is "spanning" (e.g. see Example 1 below). The Dreze objective was designed to satisfy the necessary first order conditions for a CPO but these are not sufficient because of a non-convexity ("bilinear form") in the feasible set. Similarly, the Grossman-Hart objective fails to yield a CPO because of this non-convexity. The problem basically is that in both the Dreze and Grossman-Hart models, firms act in a Nash-like fashion and so cannot simultaneously alter their shareholders and production plans. By co^ltrast,in our competitive stock market model such coordination problems due to incomplete markets simply do not arise. Indeed 5. Analogous to the complete market case, a competitive firm chooses its objective according to the following appealing "as if" story: there are markets initially open for all possible objectives the firm could pursue, and the firm just chooses the objective with the highest current price; i.e. it chooses the objective that maximizes its profits (Theorem 6). Of course in reality there are not so many markets, but we show that firms with correct competitive conjectures act as if this characterization held. The fundamental difference between the Dreze or Grossman-Hart model and the current model is that in the former two models firms are not truly competitive. Dreze makes no real attempt to characterize competitive firms. And while Grossman-Hart do, their model involves wh'atKreps (1979) aptly styles "competition without competition": they assume agents have competitive conjectures, but agents' conjectures are generally not correct in the model. By contrast, we assume firms have correct competitive conjectures; i.e. they face perfectly elastic demands for their goods and for any possible security they may wish to issue. The consequence is that both the Dreze and Grossman-Hart (weighted sum) objectives for firms appear as pure inventions: they are simply irrelevant to competitive firms who wish to act in their shareholders' interests! It is widely believed that unless a firm's security is just like many other firms' securities, the firm must have monopoly power (i.e. face a downward sloping demand curve) in marketing its shares. The reader may thus wonder if, to ensure perfect competition, we must not assume "spanning" (i.e. that markets are essentially complete). The answer is no. Indeed, as observed in Hart (1979a, 1979b, 1980) 6. Spanning is fundamentally irrelevant for perfect competition: a firm may be the sole supplier of a security, yet face a perfectly elastic demand for that security (see Examples 1-3 below; some other simple examples appear in Makowski (1982)). As the above would suggest, the analysis in the current paper is in harmony with that of Hart (1979a, 1979b, 1980). These papers by Hart are a later development to Grossman-Hart. Indeed, our conclusions 1-5 extend some of Hart's results to a multiperiod, multi-good setting. But the approach we take here contrasts to Hart's in that we shall be working in a finite model and assuming firms are perfect competitors-using an exact definition-rather than trying to both (a) ensure that firms are perfect competitors by replication, and (b) exposit the principles operating if firms are perfect competitors.' Our analysis of competitive stock markets is based on Joe Ostroy's "no surplus" characterization of perfect competition (e.g. see Ostroy (1980) and (1981)). He has shown that the same basic principle-the no surplus property-characterizes perfectly competitive economies, whether they are finite or large. The importance of this is that one can study the principles of perfect competition using finite models and finite examples, which generally are a lot more transparent-less mathematically obfuscating-than

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large economy models and examples. In the current context, the no surplus property will be the key underlying our definition of a perfectly competitive firm. That remaining in a finite model and concentrating on the principles of perfect competition indeed helps make things clearer has at least one empirical verification. In the current paper, as in Hart (1979a, 1979b, 1980), we assume there are no short sales of firms' shares. In Hart's work this was interpreted exclusively as an assumption to help ensure firms are perfect competitors after replicating. In concentrating on the principles however, it turns out that this assumption really did play a more basic role: even if firms are perfect competitors, no short selling is required for unanimity when there is not spanning (Example 2, below).2 But while the same principles characterize finite and large competitive stock market economies, the no surplus property and hence perfect competition is much more likely to occur-perhaps it's generic-in large stock market economies. The reason for this is that for perfect competition all buyers and sellers must be small relative to their market.3 And in large (i.e. non-atomic continuum) economies such smallness is usually automatically built in via the non-atomicity of agents. Some intuition about the types of environments consistent with perfect competition-e.g. the importance of local linearity in finite examples and of differentiability in large economy examples involving perfect competition-should develop in the reader via the examples in the paper. Comparing finite and large economy cases is interesting because while finite examples of perfect competition must be carefully cooked up-they are certainly not "preference free" (the cost of simplicity!)-they become preference free in the limit-i.e. in the large numbers casemodulo a differentiability assumption. The intuition underlying this claim is mainly developed in Example 1. (The importance of differentiability for perfect competition in large economies is generally not appreciated in the literature. It is the large economy analogue of local linearity since infinitesimal agents "see" a line when looking around them on a differentiable-hence smooth-surface (see Remark 1 in Example 1; also Ostroy (1980), for a more elegant statement).) Using the general PDV formula mentioned in conclusion 3 above, we shall also show that the Arrow-Lind Theorem (A-L) holds under perfect competition: 7. Under perfect competition, the riskless rate of interest is the discount factor the market will use in valuing investments whose risk is uncorrelated to social risk (Theorem 3). Indeed, although it is not widely recognized (perhaps not even by its progenitors), A-L is really a result about perfect competition. In particular, one can find in Arrow-Lind an interesting precursor to Hart's assertion, conclusion 6: "This result (i.e. A-L) is obtained not because the government is able to pool investments (via its spanning possibilities) but because the government distributes the risk associated with any investment among a large number of people (hence each of its investments is perfectly competitive). It is the risk-spreading (i.e. perfectly competitive) aspect of government investment that is essential to this result (not any spanning aspect)." (Arrow and Lind (1971), p. 244, bracketed words added for interpretation.) Analogous to Hart's approach, the proof of A-L in Arrow and Lind (1971) involves a large numbers argument coupled with a differentiability assumption on preferences. The close, abstract connection between perfect competition, differentiability, and large numbers is thus (unconsciously?) exploited by the authors. By contrast, using our approach no large numbers argument is required for the proof of A-L, the assumption of perfect competition is sufficient. Our analysis of competitive stock markets also fits in nicely with the modern treatment of the competitive allocation of "crowded" or "local" collective goods under free mobility. As noted by Dreze, shares in firms may be viewed as local collective goods since all shareholders "share" the firm's earning stream. This led Dreze to invent his weighted sum objective for the firm, a generalization of Samuelson's well-known

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necessary conditions for efficient production of collective goods. We have already noted that in our approach the Dreze-Samuelson weighted sum is simply irrelevant. This irrelevance, however, is but an instance of a more general phenomenon. To quote from Ellickson's introduction to Ellickson (1979), "A striking feature of existing studies of local public goods ... is the general irrelevance of Samuelson's approach to the results that have been obtained.... (Under perfect competition, sharing groups) must be populated by consumers with identical marginal rates of substitution, a feature that renders the summation of MRS's superfluous" (Ellickson (1979), p. 47; bracketed words added). In harmony with the above, in competitive stock markets initial shareholders sell their shares to individuals having the highest valuation for the firm's earning stream. Thus, ex-post shareholders in each period form a homogeneous group. However, unlike some Tiebout-type models, there need not exist enough firms in the economy so that there is a security tailor-made for each type of individual's financial requirements. Indeed, there may be just one firm but many types of individuals in the economy. For perfect competition it is sufficient that the equilibrium price of shares in the firm is sufficiently high so that all individuals except ex-post shareholders (i.e. all except "highestvalued users") do not want to buy any of the firm's shares (e.g. see the illustrations in Section 3). The sequel is organized as follows. In Section 1 the model is presented. The definition of a perfectly competitive firm, which will underly the entire analysis, appears in part IE. Then in Section 2 our basic unanimity theorem is displayed and then proofs of all theorems are gathered together in an appendix at the end of the paper, so as to preserve the continuity of the exposition. Section 3 presents some examples, to fix the concepts introduced. Then in Section 4 the PDV formula for pricing shares is developed and applied. And in Section 5 the optimality of competitive stock markets is discussed.

1. THE MODEL A. The basic economic environment There are I individuals in the economy, indexed by i, and F firms, indexed by f or g. The agents live in an environment with uncertainty and many periods. In particular, there is a finite set W of possible states of the world, indexed by w, and T dates, indexed by t = 1, . . . , T. There is also an information structure consisting of a non-decreasingly finer sequence of partitions of the states W, {Wt: t = 1, . . ., T}. At date t it is known which cell A e Wt contains the true state. We assume that W1 is trivial, i.e. W1 contains only one member, W; and WT is the total partition of W. So, {Wt: t = 1, . . ., T} has a tree structure. In all there are E events on this "event tree". Events will be indexed by e, where any e--(t, A). B. Commodity markets The economy has trading through time. In particular, there are C commodities traded in every event e, indexed by c = 1, . . ., C; so the commodity space is R CE. Each firm f has a production possibility set Yf RCE. Each individual i has a consumption set CE X= R ?, a commodity endowment x E R CE and a utility function U' from Xl to R. Each individual i also has an initial endowment of ownership shares in firm f, s f, satisfying sof? 0 and Ei sif = 1 for all f. A note on notation: Given any point z RCE, it will be convenient to understand that z -(ze) where each Ze ERC. It will also be convenient to let z1 z(!,w); i.e. to let event (1, W) be written event "1" for short. So, for example, if z X' then Ze is i's consumption in event e and z1 is his consumption in period one.

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We will assume throughout that individuals are locally non-satiated in each event; i.e. for any x' eX' and any e, there exists an x' eX' that is arbitrarily close to x' and satisfies x' = x' for all events e' ? e yet U1(x-) > U1(xi). An allocation (x, y) E Xi Xi Xf Yf is feasible if Eix = Zix i + f yf. An allocation (x, y) Pareto dominates an allocation (x, y) if U'(x') _ U'(x') for all individuals i, with strict inequality for some i. An allocation is Pareto efficient if it is feasible and no other feasible allocation dominates it. Commodity prices will be represented by p (Pe) E R CE, where Pe will represent the actual or expected spot prices of commodities in event e as e = 1 or e $ 1, respectively. C. Security markets In addition to the C commodities, ownership shares in the firms are traded in each event. So, each individual i must decide on an ownership plan si_ (s f) e RjFE where sf represents the number of shares in firm f that i plans to hold in event e. Any point in R+FE is a feasible ownership plan for i, so trading in shares is unrestricted except that individuals cannot "go short". That is, negative holdings of shares are prohibited. Share prices will be represented by q 3 (qe) E RF ,where qf represents the actual or expected spot price of a share in firm f in event e as e = 1 or e ? 1, respectively. D. Stock market equilibrium A pair (xW,s )X xR7+ is a feasible market plan for individual i given production decisions y (yf) and prices (p, q) if in each event e, (x', s') satisfied the event e budget constraint: PeXe +Zfqe

(Se

)e-

Pe

+f

Se-lPeYe

(1)

where e - 1 represents the event immediately preceding event e. Formally, if e = (t, A) and e $ 1 then e - I-(t - 1, A' uA); and if e = 1 then e - I-O, where "0" indicates the endowment "event", as will become clear shortly. Note that since in equilibrium Ei sif = 1 for all f and e (see the definition of an exchange equilibrium below), the right hand side of (1) incorporates our assumption that each firm f pays out its total net earnings in each event e to its shareholders from event e - 1 to event e. So, since e - 1 0 if e = 1 and since sf indicates initial ownership shares, first period dividends go to initial shareholders. Let X'(y, q) represent the set of all feasible market plans for i given production decisions y and prices (p, q). (For simplicity we write X'(y, q) rather than Xi(y, p, q) since p will remain unchanged throughout the subsequent analysis.) And let X* (y, q) represent those plans that maximize U' in X'(y, q). A five-tuple (X,s, Y p, q) e Xi X x (R +E)

Xf yf

xRCE

xRFE

is an exchange equilibrium given production decisions y (written (x, s, y, p, q) G EE) if (a) for each individual i, (x', s') E X* (y, q) and (b) (x, s, y) is market clearing, i.e. ixi= ixi + f yf (commodity markets clear in each event) and Ei se'= 1 for all f and e (stock markets clear in each event). Note that an EE is a rational expectations, Radnerian-type equilibrium of plans, prices, and price expectations. Also note that shareholder dividends and the share prices q will depend on firms' production decisions. But how do firms make these decisions? We suppose that each firm f has a (fixed) conjecture vf (vf) about how the value of its shares will vary as it varies its market plan, where vf is a function from Yf to RE. That is, if firm f decides to produce any yf E Yf then it conjectures that in event e its shares will sell for vf (yf) per unit. We shall incorporate into our model the assumption that firms' conjectures are both correct and competitive. But this will become apparent shortly.

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Firm f's conjectured net market value in event e if its production decision is yf, is given by rfe(yf) Peyfe+vfe(yf). We shall assume that firms seek to maximize their conjectured current net market value, an objective we shall rationalize in Section 2 as being consistent with shareholders' interests. Based on this assumption a five-tuple (x, s, y, p, q) is a stock market equilibrium (written (x, s, y, p, q) E SE) if it is an EE given y and for each firm f: (a) yf maximizes rr over Y and (b) in each event e, vfe(yf) = qef Note that (b) insures each firm's valuation conjectures about yf are correct in the SE: in each event e, vf(yf) clears the market for f's shares. But vf(yf), for all yf ? yf, has as yet not been restricted. Nor are f's valuation conjectures necessarily competitive. We now fill in these two omissions in the concept of a SE. This will allow us to rationalize why firms seek to maximize their current net market value. E. The assumption of perfect competition As is well known, in an arbitraryWalrasian equilibrium price-taking by firms-operating in their shareholders' interests-is usually not rational since firms can influence Walrasian prices by altering their production decisions. Also, profit maximizing may not be rational if a firm has potential monopoly power. Similarly, in an arbitrary SE price-taking and current market value maximizing-assumed above-is usually not rational. We wish to define a class of SE in which it is-the class of SE in which firms are "perfect competitors". For this purpose, the concept of a quasi-stock market equilibrium will be useful.

A five-tuple(x, s, y, p, 4) is a quasi-equilibriumgiven yf (written(x, s, y, p, 4) E Qf) if it is an EE given y in which (a) for each firm g Of: y maximizes rr' over Y' and (b) for each firm g (including f), in each event e V g(yg) = qg. A Qf may be heuristically thought of as a SE with the exception that one firm, f, is not playing by the "rules of the game". Instead of current market value maximizing, f adopts some "arbitrary" market plan. Firm f should be thought of as testing to see if it can manipulate the market outcome from the SE to some Qf to its shareholders' advantage. We shall show that if f is a perfect competitor then it cannot so manipulate the market outcome; i.e. no Qf dominates the SE for any of f's shareholders. One observation on the definition of a Of: note that (b) requires that the valuation conjectures of firm f about yf are proved correct in the quasi-equilibrium given yf: in each event e vf ()f) must clear the market for f's shares. (Of course, a similar statement also holds for other firms' conjectures in the Qf, but this observation is not of much interest since other firms basically play a passive role in the Qf; e.g. see (2a) below.) Firm f is a perfect competitor in a SE (x, s, y, p, q) if for any y E Yf there exists an (x, s, y,p, 4) E Qf such that for each firm g ? f, in each event e

(2a)

PeYe = PeY and 4e =eq

and for each individual i such that sef > 0 in any event e, there exists an (x*, s* ) EX* (y, 4) satisfying s*fe = 0 in

(2b)

all events e. That is, a firm f is a perfect competitor in a SE if it cannot influence the prices of commodities by altering its market plan (in each Qf p remains the same as in the SE); it cannot influence the monetary dividends paid by other firms or the prices of shares other than its own (condition (2a)); and it cannot influence the price of its own shares in the sense that it always receives the economy's reservation price for its shares (condition (2b)).

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The heart of the definition is condition (2b). In words it says that in any Qf, every individual i who buys some of f's shares in any event is indifferent between buying them and not (there is another point in f's demand correspondence involving no purchase of f's shares). Thus the condition may be thought of as saying that a perfectly competitive firm contributes "no surplus" in marketing its shares; or, alternatively expressed, it "fully appropriates" its surplus. And so it faces a perfectly elastic demand for its shares according to the following basic test: if f raises the price of its shares by any small positive amount in each event e, then no one will be willing to buy any of its shares in any event. The five-tuple (x, s, y, p, q) is a competitive stock market equilibrium (written (x, s, y, p, q) e CSE) if it is a SE and all firms are perfect competitors in the equilibrium. Note that as claimed above, in a CSE each firm f's valuation conjectures vf(yV), for all yf E Y1, are both (a) correct and (b) competitive: for any yf E Yf there is a Qf gven f, (x,,s, p, 4) in which the price of f's shares in any event e, qf, equals ve(3-f) and both (a) clears the market for f's shares and (b) satisfies the "no surplus" or "full appropriation" property (2b). Also note that in a CSE each firm f's conjectures about its influence on other firms' shares or on other prices is competitive: (2a) plus the fact that commodity prices always remain at p. To simplify the exposition, henceforth when we write (x, &,y, p, q) EQf we shall mean not only that (x, s, y, p, q) is a quasi-equilibrium but also that it satisfies (2). 2. RATIONALIZING THE FIRM'S OBJECTIVE We are now in a position to display our basic unanimity result, which rationalizes current market value maximization as being consistent with all initial shareholders' wishes. Theorem 1 (initial shareholder unanimity). If (x, s, y, p, q) e SE and firm f is a perfect competitor in the equilibrium then (a) initial shareholders in the firm unanimously favour currentnet market value maximization as thefirm 's sole objectiveand (b) non-initial shareholders in the firm are indifferent about the firm's market plan. That is, for every

yfE=Yf, x ,y p, q) E=Qf implies (a) for each i such that s f > 0:

Ui(xi){ }U (xi)

as irf (yf){

f}i K(yf)

and (b) for each i such that sg =O, Ui(xi) =Ui(i). The proofs of all theorems are in the Appendix. The intuition underlying Theorem 1 is easy to explain: The perfect competition assumption (2) insures that any change by f from some yf to yf cannot induce a "consumption effect", but only perhaps a "wealth effect" benefiting initial shareholders. Thus part (b) of the theorem is obvious, and so is part (a) since initial shareholders only prefer yf if it involves a positive "wealth effect". (Note increasing rf{ just moves initial shareholders' first period budget constraints out in a parallel fashion-commodity prices always stay at p-so it is obviously better for all initial shareholders.) 3. TWO EXAMPLES OF CSE'S To help fix the concepts introduced, we now present two examples of CSE's. A third example is given in Section 4C. The examples will illustrate the unanimity theorem, and also the possibility of perfect competition without spanning: in all the examples there is but one firm, whose production possibilities are not spanned by existing securities.

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The examples will also provide a convenient platform for short discussions of (a) the nature of technologies consistent with perfect competition (Remark 1 below) and (b) the robustness of the unanimity theorem if the no short sales constraint were relaxed (Example 2). Example 1: Perfect competition without spanning Suppose there is no uncertainty (* W = 1), two periods' (T = 2), and one commodity (C = 1). So there are no futures markets; one can only transfer purchasing power between periods one and two by transacting on the stock market. There is only one firm, with production possibilities Yf = {a (-1, 3): a E [0, 1] c R}. The firm is solely owned by individual k who also has an endowment Xk = (3, 2). All other individuals j,j = 1, . . . , J, are identical, with endowments (9, 2). And all individuals have homothetic preferences; a typical indifference curve for individual k is Uk, for any individual j is Uj in Figure 1. X2

7 Uk

5-

k

xk

I

0

1

2

3

4

5

_N

7\

9

11

X

9iN FIGURE

1

Equilibrium in Example 1

Note that Uj has a flat section of length N/J. This will ensure that the firm is a perfect competitor provided N > 3, which we shall assume. Indeed, the reader will easily verify that (x, s, y, p, q) is a CSE when the firm produces yf = (-1, 3), commodity prices are unity in each period-p = (1, 1)- and share prices are q = 3 in period 1 and q2 = 0 in period 2. (Note that share prices will always be zero in the last period for market clearing: there is no payoff from holding shares bought in period T.) In equilibrium individual k consumes x k = (5, 2) and sells his share. While individuals j each consume and each own equal shares in f, si = jf= 1/J. Firm f correctly = (9-3/J, 2+3/J) conjectures that for any yf-- (yf, Yf) e Yfvf%(yf) = 5f while vf (5f))= 0. Note that x' and xi are on the same indifference curve, so the firm's profits fully appropriate its surplus, as required by (2b) and Theorem 1. More generally, in accordance

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with (2), firm f is a perfect competitor since for any yf E Yf there is a quasi-equilibrium supported by the same commodity prices, (1, 1), and a share price that fully appropriates the firm's surplus and equals its conjectured share price; e.g. the reader will easily verify that if y = (--, 12) then the resulting quasi-equilibrium would satisfy (2) and involve a profit of one rather than profits of two, the gain again going entirely to k. As already observed, the flat section in Uj insures that the firm will always face a perfectly elastic demand for its shares. For example, the demand curve for shares paying dividends of 3 per share in period 2 is sketched in Figure 2 where, since N > 3, the length of the flat section exceeds 1 (i.e. it exceeds the maximum number of shares paying 3 that f can issue). Price in period 1 of a security paying 3 in period 2

3

t

D

Number of securities paying 3 in period 2

N/3 FIGURE 2 Demand for f's shares in Example

1

Remark 1. As noted in the introduction, in this paper we are assuming perfect competition via an exact definition, (2), rather than deriving it from primitive assumptions on the technology. However, a heuristic comment on the nature of technologies (i.e. preferences and production possibilities) consistent with perfect competition may be helpful. In finite economies like that studied in the current paper firms will generally be perfect competitors only if their supplies (including securities) are small relative to aggregate demand and the aggregate net trade set has a "linear section" in the neighbourhood of the equilibrium allocation. In continuum economies, as agents become infinitesimal relative to aggregate markets, differentiability or smoothness replaces local linearity as necessary for perfect competition. This is intuitive since a differentiable function or smooth surface is locally linear. For a derivation of these principles, in a somewhat different context, see Ostroy (1980, 1981), or Makowski (1980c) and its sequel Makowski (1980d). These principles are illustrated in Example 1. First, it should now be clear why Uj has been constructed with a flat section around xj. Second, note that as one increases the number of i's in the economy the length of the flat section in U needed to insure firm f faces a perfectly elastic demand for its securities shrinks (i.e. N/J approaches zero as J approaches infinity). So, in the limit Uj becomes a smooth, strictly convex curve. Remark 2. Both the unique Grossman-Hart Equilibrium and a Dreze Equilibrium in this economy would involve autarky (i.e. everyone consuming their endowment, and

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no one trading on the stock market so that ex-ante and ex-post shareholders coincide) and the firm not producing (i.e. yf = 0). To see this, observe that using the MRS's of the firm's initial owner, k, any production would involve a loss: e.g. the vector (-1, 3) added to xk is to the left of Uk; see arrow in Figure 1 (of course, if added to xi it would be to the right of Uj). So, both the Dreze and Grossman-Hart criteria call for no production. The unique CSE allocation is also among the Dreze equilibrium allocations. But beyond the special case of two period models, this coincidence ends. Indeed, beyond T = 2 it is not clear how to extend the Dreze equilibrium concept since "ex-post" shareholders vary through time. Also note that the CSE in this economy is a constrained Pareto optimum (in the sense of Diamond), while autarky is not. Example 2: The importance of the no short sales assumption This example is the same as Example 1 except Yf = {0, (-2, 1)}. Now the reader will easily verify that in the absence of short sales, autarky and no production is a CSE supported by p = (1, 1) and ql=qf = 0. Firm f correctly conjectures vf (-2, 1) 1, so production would involve a net loss equal to one. X2

11\

Uk

U,

5 4

I~~~~~~~~~~~~~X

I

0

1

3

4

5

9-

/

9

11

FIGURE 3

Equilibrium in Example 2 with short selling

But suppose short-sales were permitted. Then if the firm produced (-2, 1), it would create a complete set of markets. Indeed, a quasi-equilibrium (x, s, y, p, q) would result in which yf = (-2, 1) and commodity prices remain p = (1, 1) while the share prices become qf =1 and, as usual, qf =0. Now individual k can consume x = (4, 0) by selling two shares short_1 kf= skf =-2 While individuals j each consume '= (9-3/1J, 2 + 3/J) and each own an equal share of all outstanding shares = 3/J. (including k's short shares), so K= _K

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Note that contrary to Theorem 1, x k is preferred by k to xk, his CSE allocation. While f's production yields a loss, the surplus k gains via the short selling this production allows more than compensates k. Also, note that the example has been constructed so that the extent of short sales are bounded for any value of J (i.e. any size economy). In particular, k's desired short sales when added to f's sales are small relative to aggregate demand so security prices are not affected by k's short selling: in Figure 4, N exceeds 3, the maximum number of shares paying 1 in period 2 that f can issue-i-plus the maximum number of such shares that k can issue-2. Price in period 1 of securities paying 1 in period 2

D

N

Number of securities paying 1 in period 2

FIGURE 4 Demand for f's shares in Example 2 with short selling

Remark 3. The current paper follows Hart's (1979a, 1979b, 1980) in prohibiting short selling of firms' shares. But contrary to a common view (e.g. see Kreps' review of Hart's (1979b), Kreps (1979) pp. 26-28), the example illustrates that this restriction plays a more fundamental role than merely to ensure firms are truly competitive: even if a firm would remain a perfect competitor were the no short sales restriction removed, in the absence of the restriction its initial shareholders would in general want it to follow a course other than net market value maximization. The principle underlying the example is that with short sales the firm may be able to induce an externality by altering its market plan. In the example-as would be the case in general-the increased beneficial short selling possibilities that yf permits are not reflected in the current market value of yf since the short seller k just deals with third-parties and does not affect the firm's share price (in accordance with the perfect competition assumption); the firm only creates an external benefit by opening the market. In terms of no surplus theory, without short sales the assumptions of perfectly elastic demands and no surplus are of equal strength, so a firm's profits measure its surplus (e.g. see Makowski (1980a)). While with short sales, the former assumption is weaker than the latter: a firm may be a perfect competitor yet contribute a surplus in excess of its net market value. So with short sales, a firm's profits need not reflect its surplus and hence may be an unreliable measure for guiding the firm's decisions. Of course, if a firm's security is spanned by other firms' securities-so the firm is not innovating a new type security-then the problem illustrated in the example does not arise. Individuals can always do as well by only selling other firms' shares short (cf. footnote 2).

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316

4. PRICING SECURITIES A. A present discounted value formula for pricing all potential securities when there is perfect competition but there are incomplete financial markets Let us henceforth make the following three, basically technical assumptions: Assumption 1.

For each individual i: Ui is quasi-concave and differentiable.

) implies xz ?0 for X Assumption 2. For each i, y, q and e: (Jxi s)X(yE some commodity c such that Uc (V) = aU1(xi)/a4'c >0 ("each individual consumes some good in each event").

Yf:(,,y, p, 4) E Qf implies tnat Assumption 3. For any firm f and any yfEE in each event e ef>0 for some individual i such that sf = 0 ("initial shareholders are not isolated"). Assumptions 1 and 2 should require no further elaboration. The technical Assumption 3 says that initial shareholders are "not isolated" in that for any firm f, in any quasi-equilibrium Qf the holders of shares in f in each event are not all initial shareholders; i.e. there are some highest valued buyers outside the set of initial shareholders. It will be convenient to let eT index the set of terminal events on the economy's event tree; i.e. the events e (t, A) in which t = T. And if e ? eT, to let e + 1 index the set of possible events immediately proceeding e; i.e. the events e'- (t', A') in which t'= t +1 and A'c A. Using this notion and Assumptions 1-2, it follows readily from the Kuhn-Tucker Theorem (e.g. see Makowski (1979)) that for any y, 4, and (JI, si)c Xi (y q), (x, ) c X* (y, q) iff there exist unique positive constants A such that for all e, c, and f: Uec (x i) _ AePec

If e ?e

where a strict inequality implies J4c= 0.

(3a)

T

Ee+l

+4 +) <e

4e+lli/(Pe+l)e+1

where a strict inequality implies sef =0.

(3b)

If e = eT:0 c qfe where a strict inequality implies sef= 0. Furthermore, for any y- and 4, if X* (y, 4) is a correspondence then all (i, s)

E

X* (y, 4)

have the same lagrange multipliers 4e

(3c)

(see Theorem 5 in Makowski (1979)). Note that the Lagrange multipliers e represent i's implicit prices for income in the various events e, measured in terms of "utils". So 4+l/4 is i's implicit price for income in some event e + 1, measured in terms of event e income. This observation allows for any easy interpretation of the Kuhn-Tucker conditions in (3); e.g. ignoring corner solutions (3b) says that i will buy shares in firm f in each event e ? e T up to the point where their cost, qfe equals the benefit in e + 1 income gained, where the bielniAt is measured by i's implicit prices for incomes in the various e + 1 events. Note that if T, fmust equal zero if anyone is going to hold f's shares since there are no benefits to buying the shares at time T: time has an end after period T! Now (3) can be used to derive a present discounted value (PDV) rule for valuing all potential securities (including each firm's actual security, yf, and all of its possible securities, Yf) in the CSE, (x, s, y, p, q). To proceed, let us call any point d E RcE a security if we are thinking of it as representing a random variable of dividends. And let us define a function V (Ve) from R CE to R E as follows: for any security d c R CE and

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MAKOWSKI

any event e, if e =e

TT

then Ve(d)-0 and if e ve(d)-maxi

Ze+1

317

6e

(4)

4+i(pe+1de+i+ve+i(d))

where Xe+i Ae+1/Ae the A es being the Lagrange multipliers of (3) corresponding to i's CSE market plan (x1,s') EX* (y, q). The function v we shall call the (potential) market value function because it specifies the competitive market value of all potential securities, as we shall shortly discover. It can be analytically characterized by solving (4) backwards from the values at the terminal events e T which we know equal zero. In particular, letting i(d, e) represent any one individual for whom the right hand side of (4) reaches a maximum in event e given d, then (4) may be rewritten: Ve(d) = e+1 AX(!le)(pe+lde+l+ Ve+i(d))

("basic recursive formula").

(5)

And solving (5) backward one finds for each e $4eT and each d ER CE Ve(d) = Ze'>e Aet(d, e)pe'de' i

")

("generalized PDV formula")

(6)

Awhre

(d,e where Ae'(d,e) fHece"<e'e"4+l;e'; and where e'>e means e' is a descendent of e (i.e. e' (t', A') where t'> t and A'c A); e -e" < e' means e" =e or et" is a descendent of e, and it leads to e' (i.e. e"- (t1",A") satisfies t"< t' and A" DA'); finally, e"+ 1; e' represents the event in period t"+ 1 that leads to e'. Now v, as characterized in (6), is the PDV rule we are seeking. For any event e', e' > e, the multiplier Ae'(d, e) in (6) may be interpreted as the competitive market's discounted value in event e of a unit of income in event e', available in an asset paying dividends d. This assertion and interpretation is founded on the following theorem.

Theorem 2.

If (x, s, y, p, q) E CSE then for each firm

f, vf (yf)

= v(yf ) for all

_fE yf.

So, v characterizes all firms' valuation conjectures! And since as we observed earlier these conjectures are correct in a CSE, it characterizes the competitive value of all potential securities. The intuition underlying Theorem 2 can be explained. Unless there is perfect 1s of (3)-which reflect individual i's implicit prices for income in the competition, the Ae various events e in a Qf given yf-will generally change when any firm f changes its production plan and hence moves the economy to a new equilibrium. However given the perfect competition assumption (2)-"no consumption effect"-these implicit prices on event-contingent income will not change when any firm f changes (at least for individuals immune to any "wealth effect" of a change, i.e. the non-initial shareholders in f). Now the Kuhn-Tucker condition for optimal ownership plans, (3b), tells us that the value of any security yf, in any Qf given yf, will be given by a formula like (4)with the proviso that the discount factors may have to be A,+1/Ae rather than Ae+l/Ae But the above discussion indicates that under perfect competition this last proviso is not needed (at least for non-initial shareholders, which is all that matters for the validity of the theorem given Assumption 3). B. Essentially insurable risks and a simplified PDVformula The general PDV formula (6) differs from the formula under complete financial markets in that the discounted factors Ae'(d, e) depend on d. Hence in any event e, Ve is not in general a linear function. To provide a link with the traditional formula, which is a special case, we now show that if d is an "essentially insurable risk" then the discount factors will not depend on d.

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318

Analogous to (4), for any e and d c R CE let us define a function v T VeT(d) O and if e ? e ve(d)

-Ze+l

ei+1(Pe+1de?1+

vei+

(ve) as follows:

(d)).

(7)

The function v represents i's private valuation of security d. It may be characterized analytically as v was above. Namely, by backward induction one can easily verify that for all e ?eT and any dCRCE Ve(d)=

e'>e

A ' (e )Pe'de'

(8)

where Ae(e)--fle

-e"<' Ae+l;e'

Note that v' is a linear function. We shall say that d is an essentially insurable risk in the CSE (x, s, y, p, q) if the potential market value of d, v (d), equals each individual's private valuation of d. So, D * --{d c R CE: v e(d) = Ve (d) for all i and e } represents the set of essentially insurable risks in the CSE. Intuitively, d c D * means that in the CSE no individual would want to further insure any other individual against risks in the fixed proportions of d; or, still speaking intuitively, all individuals' marginal rates of substitution for event-contingent income are equal, when weighted in the fixed proportions of d. T Now, from (8) and the definition of D * it follows trivially that for any e ? e for any d cD*

Ve(d)=e'>eAe'

(e)pe'de'

("simplified PDV formula")

(9)

where i can be any, arbitrarilyselected individual. Formula (9) is of course the traditional PDV formula, the one that one would expect when markets are complete. One would like to know what types of securities can be valued using the simpler PDV formula (9) even when markets are incomplete. I know of two such types: (a) If d were an available security with unrestricted short selling then d CD *. So if one extended the current model to include securities other than shares, e.g. bonds, then these would be in D *. (Incidentally, such an extension is completely straightforward, see Makowski (1981).) (b) Those securities whose return is uncorrelated with social risk are in D* (see the next section). We shall say that markets are essentially complete in the CSE if D* R CE. When markets are essentially complete, (9) tells us that the simplified PDV formula may be used to price all potential securities. Note that markets may be "essentially complete" even when there are not actually a complete set of insurance markets; so, this is a generalization of the usual, complete markets concept. It is sufficient to allow for a Walrasian characterization of CSE's (see Theorem 9 below). Also, in the next section we shall discover an interesting family of examples involving essentially complete, although actually incomplete, CSE's; so the generalization is not purely pedantic. C. The Arrow-Lind Theorem The general PDV formula (6) can also be specialized to prove an Arrow-Lind-type result. Namely, if the pattern of returns on a security d is uncorrelated to the pattern of individuals' risks, then in each event e the potential price of the security, ve(d), just equals the sum of the expected returns on the security in each succeeding period (given e has occurred) discounted by the riskless rates of interest prevailing in each succeeding period (given e has occurred). We will also show that if the return on d is uncorrelated to individual risk then d is an essentially insurable risk, i.e. d E D*.

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For the current purpose (and the current purpose only), we shall need to assume

Assumption4. {bt: t = 2,..., T} cD *, where bt e R CE is a risklessbond in period t, i.e. a security yielding pebe = 1 for all e = (t, A) and Pe'be' 0 for all e' = (t', A'), t' # t ("there are riskless bonds essentially available in each period"). Assumption 5. Foreachindividual i and eachx' EX': U'(x') =YEir, V'(x'), where Vi is quasi-concave and differentiable, and rwrepresents the agreed upon probability in event 1 of state w occurring (so rw 0O and Ewrw= 1) ("individuals are expected utility maximizers"). Note that in the statement of Assumption 5 I have used the following notation: given where each zwt equals Ze for Zwt ... any point z (Ze) E R CE, Z ZwT) ERC, (ZwX ... e satisfying e = (t, A) and w E A. So, given x' E Xl, x4' represents i's consumption sequence in state w. The return on a security d is uncorrelated with social risk in the CSE (x, s, y, p, q) if conditions (lOa)-(lOc) below hold: ,

W = (W' x Wd)

and

,

{Wt}t=1,..,T={WI

(lOa)

X Wt }t.T,

where WI (respectively, Wd) represents the set of states relevant to individuals (respectively, states relevant to d), and (WI} (respectively {Wt }) is an information structure corresponding to WI (respectively, Wd). Letting eI (respectively, ed) index the set of events on the event tree {WI} (respectively, {Wd}), note that (lOa) implies any event e corresponds to the simultaneous occurrence of a pair of "relevant events", (e I, ed). Thus, to let e = (eI, ed) mean e =((A', Ad), t) where e = (A , t) and (Ad, t). Similarly, to let e I' e mean e = ', e d) for some e d; and, to let ed E e mean (e', ed) for some eI. We may now formulate

it will be convenient

ed e=

_

de =ded,

a constant, for all e such that e d E e

Pe =Pe',

a constant, for all e such that e Iee

and for all i there is some x-' eXX (y, q) satisfying Xe =X e ,

a constant, for all e such that eI E e.

Finally, letting r' (w I) (respectively rd(wd )) represent the agreed upon probability in event 1 of state w' E WI (respectively, wd E Wd) occurring, rw = r (w I)r (w)

for all w=(w,w)EWI

(lOc)

x Wd,

where r'(w )O,

and

rd (wd)O,

EwIrI(wI)=

EWd

rd(w)

1

Observe that the expected return of security d in period t' given that event e -(t, A) has occurred (t < t'), may be expressed as exp (dt,; e)

Yw.A

(eP)dw r(e)

(11)

where r(e) -

w.Arw.

Theorem 3 (Arrow-Lind). Suppose (x, s, y, p, q) e CSE satisfying Assumptions 4 and 5. And suppose the yield on some security d e R CE iS uncorrelated with social risk. Then in every event e = (t, A), t # T, Ve(d) = EtJ>t Ve(bt) exp And d eD*, i.e. d is an essentially insurable risk.

(dtc; e).

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(12)

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Note that the formula (12) can be translated into one explicitly involving interest rates. Let e = (t, A) and let t' > t. Then Re (t'), defined by 1 + Re(t') I

= le V()bhtR

is the rate of interest in event e on riskless income in period t'. It is a short/long rate as t' equals/exceeds t + 1. From (5) one sees that as usual 1Re(t) Hee"ce 1 1 1~~+ Re" 1+ReW where Re"`Re"(t be written as

?+1) is the short, riskless rate of interest in event e". Thus, (12) may Ve(d) =

exp (d,; e)

1 ?Re(t')

or as an equivalent expression involving only short rates. The intuition underlying Theorem 3 can be explained. Perfect competition implies locally linear indifference curves in the neighbourhood of the equilibrium allocation (recall Remark 1). And if d is uncorrelated to social risk then d is basically like a sure bond for individuals, at least locally. So in view of individuals' local linearity and given Assumption 4, the riskless rate of interest is the right weight for discounting d's return provided d involves only a small addition to each individual's portfolio relative to his flat section. (In Arrow-Lind's (1971) proof, local linearity is guaranteed by a limiting large numbers argument plus a differentiability-smoothness-assumption on preferences. From Remark 1 above, this should be understandable. For us it's easier: (2) does the trick.) Note the Arrow-Lind Theorem implies that if the returns on all production plans are uncorrelated with social risk, any firm's objective in the CSE is merely to maximize its expected present discounted value using the riskless rates of interest to discount expected future earnings. Such a lack of correlation will occur if there is only productive risk in the CSE. Formally, let us say there is only productive risk (i.e. no individual risk) in the CSE if (a) pw = Pw for all w, w'cE W and (b) for each i there exists an xl E X* (y, q) such that x =x for all w, w'e W. Now, it is easy to verify: Theorem 4 (a sufficient condition for essentially complete markets). If in the CSE (x, s, y, p, q), there is only productive risk (no individual risk) and Assumptions 4 and 5 are satisfied, then any possible security d E R CE is uncorrelated with social risk. So in view of Theorem 3, the potential market value of any possible security is given by (12) and markets are essentially complete-D* = R CE. Note the theorem tells us that if there is only productive risk in a CSE then, not only will the returns on any possible production plan be uncorrelated to social risk, but markets will also be essentially complete. This latter fact implies there exists a simple but interesting family of CSE's. Any CSE satisfying Assumptions 4 and 5 in which (i) there is only one firm and (ii) all individuals have a riskless commodity endowment, will have essentially complete markets no matter what the firm does-even if there is no insurance available against any productive risks! Consequently, in such an economy no insurance against risk is needed to achieve a pure Pareto optimum. (To see the claim, observe that (ii) plus Assumption 4 imply that when the firm isn't producing, a Walrasian pure exchange equilibrium can be achieved. And since the firm is a perfect competitorthink of it as small relative to the economy-when it produces prices don't change; so condition (a) for "only productive risk" is satisfied. And so is condition (b), since (2b)

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plus only one firm imply no individual's budget constraints, (1), are effectively statecontingent. Thus Theorem 4 implies markets are essentially complete.) The intuition underlying this family of CSE's can be explained: perfect competition, as usual, implies locally linear indifference curves in the neighbourhood of the equilibrium allocation. Given (ii) plus Assumption 4, this means in the neighbourhood of the 450 line-riskless consumption-where individuals are locally risk neutral, given Assumption And 5. But local risk-neutrality means insurance isn't needed for small-local-risks. a perfectly competitive firm can only generate such small risks. We close this section with an example in the above family of CSE's. Example 3. Perfect competition and essentially complete markets without spanning Suppose there are three equally probable states of the world (" W = 3), two periods (T = 2), and one commodity (C = 1). Let us index second period events by h = 1 ... 3. There is one unique individual, k; all others are of the same type and are indexed by j = 1 . .. J, where J- 3. Individual k has a von Neumann-Morgenstern utility function Uk (X k)=

V(X1 ) +h

3V(xk),

where V is any concave differentiable function, and an endowment x individuals j have linear von Neumann-Morgenstern utility functions U1(X')

= X 1 + Eh

k

=

(1; 3, 3, 3). All

Xh

and an endowment xi = (1; 1, 1, 1). Finally, there is only one firm, with production possibilities Yf = convex hull {0, (-1; 2,3,4), (-1; 5, 1, 0)}. The firm is initially owned by individual k. The reader will easily verify that (x, s, y, p, q) e CSE when p = (1; 1,1,1), yf= (-1; 2, 3, 4), q f = 3, and f's conjectures are vf (W)= Eh 3W for all yf E Yf. Individual k consumes 3 in each event and sells his share, while each j consumes xi= 1-3

+2 13

14)

and owns 1/J shares in the firm. The reader will also easily verify that in this example (x, y, p) is a Walrasian equilibrium, where p = (1; 3, 3, 3). Thus although only shares are available in the CSE-which of course do not span R3 -markets are essentially complete, as claimed. To satisfy Assumption 4 individual k's commodity endowment has been "cooked" so that he does not need to buy or sell riskless bonds to finance the same consumption in each period (recall footnote 4). Each individual's implicit sure rate of interest is zero in the example. (Note Assumption 4 is in no way a spanning assumption. Even if riskless bonds were available, two securities wouldn't span R3.) Finally we note that (12) holds; e.g. qf just equals f's expected second period return when producting yf-no discounting needed with a zero sure rate. As in the previous examples, note that U'(xi) = U'(xi) while Uk (xk) > Uk (Xk ), in accordance with Theorem 1. Also note that instead of linear preferences, as in the previous examples we could have given individuals j appropriate preferences with only a linear section (in this case, along the 450 line)-whose length approaches zero as J approaches infinity-without effecting the CSE of the example. D. Unanimity through time Using the general PDV formula (6) we can also now answer a question that may have occurred to the reader after thinking about Theorem 1. In particular, suppose that

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(x, s, y, p, q) E CSE so that in period 1 there is an equilibrium of plans, prices, and price expectations. Will this equilibrium persist through time? That is, will later shareholders in each firm f agree with initial shareholders' desired market plan for f? We now show that if the economy is "intertemporally consistent" and firms remain perfect competitors through time, then the answer is yes. Note that whenever the course of events leads from event 1 to any event e > 1, then {e': e''? e} represents the set of events that are still possible on the "truncated event tree" with initial event e. The economy is intertemporally consistent in the CSE (x, s, y, p, q) if for each e > 1, in the "truncated economy with initial event e" each individual i's consumption set is X'(e) {A; c X': X' E X'eunless e' e} and his utility function remains U'; each firm f's production set is Y'(e) {5f E Yf: yf, = yf unless e' e}; and each individual's feasible ownership plans are {s

RE R+se'-

-if

unless e'?>e } if~~~~~i Sf,

and i's "initial" event e endowment of shares in f is sfie. Now it is easy to show that Theorem 5 (unanimity through time). If (x, s, y, p, q) E CSE and the economy is intertemporallyconsistent then (x, s, y, p, q) remains a SE in each truncated economy with initial event e. That is, for all e > 1: for each individual i, (xi, si) maximizes U in X'(e) subject to (1) holding for all e' i e

(13a)

for each firm f, y f maximizes rf in Yf (e).

(13b)

and

Now, by an argument exactly analogous to that in Theorem 1-it just amounts to translating the origin from event 1 to event e-if any firm f is still a perfect competitor in the truncated economy with initial event e then its "initial shareholders"-{i: sf 1 > O}-will still unanimously wish the firm to maximize its "current" market value-sre. Or, in view of (13b), all ex-post shareholders in each period t will unanimously concur with ex-ante shareholders wishes; hence Theorem 5 is styled "unanimity through time". 5. THE OPTIMALITY OF CSE'S A. The lack of coordination problems in CSE's It is typically argued that with incomplete financial markets, firms will make inefficient market decisions because they do not consider the possibility of simultaneously altering their shareholders and their market plans to suit individuals' financial needs; e.g. Dreze (1974) or Grossman-Hart (1979). Such coordination problems do not occur in CSE's; e.g. Dreze equilibria or Grossman-Hart equilibria are not really competitive stock market equilibria. To make this point, we show that a CSE (x, s, y, p, q) may be thought of as a sort of Walrasian equilibrium in which markets for all current goods and all possible securities are simultaneously open for trading-not just markets for securities y are open-with the restriction that individuals cannot go short on securities except those involving essentially insurable risks (i.e. d E D*) and any firm can issue at most one type of security. Equilibrium occurs when markets for all current goods and all possible securities simultaneously clear.

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323

A consequence of this characterization result is that under perfect competition each firm chooses its objective according to the following appealing "as if" story: Since initially there are markets open for all current goods and all possible securities, any firm will know the current value of any objective it might pursue from looking at current prices; i.e. for any firm f the "price" of any objective yf E Yf just equals piyf +v1(yf) (see (15b) below). And the firm simply chooses the objective with the highest value; i.e. it maximizes current "profits". Alternatively expressed, although markets are incomplete, firms with perfectly competitive, correct conjectures function "as if" markets were complete, except they can only sell their future product as a composite good-i.e. as a security-which can be resold in each event. To proceed formally, let N1 be the family of all functions nti from R CE to RE satisfying n'(d)#O for only a finite number of securities deRCE and n1(d)?O if d-D*. A pair (xi, ni) E Xi x N' is feasible for i under N' (written (x i, ni) E Xi (RCE,v)) if in each event e Ve(d)(ne (d) -

(d)) _

+ ERCE (d)pede (14) n1 p,exe where v is the potential market value function, and no (d) -0 if d?yf for some f, Sf. A Walrasian Stock Market Equilibrium (WSE) is a fiveotherwise ni (d) =Ef;yf=d tuple (x , nI, y,p, v)i=l...I such that

Pexe +

ZdERCE

e-1

for each i, (xi, n,i) maximizesUi in Xi (RCE, v)

(15a)

for each f, yf maximizes pi7f +v1(y ) over all yf e Yf.

(15b)

and Zixi = Zix +Zfyf and EiZn(d)=-in ("all markets clear"). Theorem

6

(xi, n , y,pI V)i=l...Ie

(d) for all dERCE

and all e (15c)

(WSE characterization of CSE's). If (x, s, y, p, q) e CSE then sif WSE, where n (d) 0 if d ?yf for some f and n' (d)--f:yf=d

otherwise. A consequence of Theorem 6 is the intuitive result that if a feasible allocation (x, y) dominates (x, y) it must involve some redistributions in "directions" that cannot be financed (e.g. because of the absence of insurance markets): Theorem 7 (an efficiency property of CSE's). If (x, s, y, p, q) E CSE then there does not exist a feasible allocation (xk,y )-(x + Ax, y + Ay) that Pareto dominates (x, y) and satisfies Ax c D* and Ay cD*. Another consequence is that CSE's are constrained Pareto efficient, in the sense of Diamond (1967). Formally, an allocation will be said to be constrained Pareto efficient if it is feasible and no feasible allocation (xk,y) dominates it that satisfies for all individuals i and all e > 1 Xe

X=e+Zfa

iYe,

where each a if E R+ and Ei a if = 1 for allf. This is just Diamond's definition, "generalized" to apply to environments with T periods and C goods. In words, it says that an allocation is constrained Pareto efficient if there is no feasible allocation that dominates it involving (a) an arbitrary first period redistribution of commodities and shareholdings and (b) all individuals consuming, in each event e ? 1, their original commodity endowment for event e plus event e dividends according to the new shareholdings. Note there is no redistribution of endowments or shareholdings allowed after period one. So beyond

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324

Diamond's special case (i.e. T = 2, C = 1), this is a very weak notion of optimality. When T > 2 or C > 1, Theorem 7 above provides a more interesting "constrained optimality" characterization. However, the following result is of interest when T = 2 and C = 1 since it contrasts sharply with inefficiency results obtained using other equilibrium concepts; e.g. as illustrated in Example 1 Dreze equilibria and Grossman-Hart equilibria are generally not constrained Pareto efficient. Theorem 8 (constrained Pareto efficiency of CSE's). (x, y) is constrained Pareto efficient.

If (x, s, y, p, q) E CSE then

Note that the Dreze bilinear form problem does not get convexified away. There is a non-convexity in the set of feasible allocations in competitive stock market economies. But Theorem 8 tells us that this non-convexity just doesn't matter for the efficiency of CSE's. In particular, although no firm can market a convex combination of securities, in maximizing its net market value it markets the security that is Pareto efficient for it to market. The intuition underlying Theorem 8 is straightforward: Given the no surplus property of perfectly competitive firms, we know that each firm's net market value reflects all the benefits accruing to buyers of its shares. So, since there are no short sales, its net market value must reflect all the benefits accruing to all individuals from its shares. Thus by maximizing net market value each firm maximizes its contribution to aggregate consumer surplus. Or, aggregating over firms, firms' production decisions maximize aggregate consumer surplus; i.e. they lead to a constrained Pareto optimum. B. The relationship of CSE's to ordinary Walrasian equilibria A CSE may be thought of as a Walrasian equilibrium with the extra constraint that individuals' "insurable risks" are restricted to D*. Hence if D* R CE, a CSE should be fully Walrasian. This intuition we now verify. A triple (x, y, pi) is a Walrasian equilibrium under the distribution of wealth (a') E RI (written (x, y, F) E W(a1)) if for each i, x' maximizes U' over all x' e Xi satisfying px i_1

for eachf, yf maximizesfy f over all yf E Yf Zix-

ix+fY

(16a)

(16b) (16c)

and i a,= Eifp-X+ If py*

(16d)

We now show Theorem 9 (Walrasian characterization of CSE's when there are essentially complete markets). If(x, s, y, p, q) E CSEandD* =RCEthen (x, y, F) E W(a'), for some Pand (a'). Of course, if (x, y, p) e W(a1) then (x, y) is Pareto efficient. So, CSE's are Pareto efficient when there are essentially complete markets. (But this was already implied by Theorem 7) APPENDIX Proof of Theorem 1. Since f is a perfect competitor, (x, s, y, p, q) E Qf. So for each i there exists (xx, s*) EX* (y, q) satisfying s fe = 0 for all e. Similarly, since (x, , y, p, q) e Qf there exists (x*, V) eXt (y, 4) satisfying s fe = 0 for all e.

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MAKOWSKI

COMPETITIVE STOCK MARKETS

325

But using (2a), (x*, s*) e X (y, q)

implies (Xi,Si)E=Xi(y,q)

provided sof(pj5fh+qf{): sof(pjyf +qf). So if (a) sgf 0 or (b) f (y) )-

(yf),

then ui (x)

ui (i)-

= ui(xi).

Similarly, one shows that implies (x*,J s*X

(y, q)

provided (a) sof = 0 or (b') r.1(f) ? r1(yf). So, (a) or (b') imply U1(xi)

?

-

U(-i)

Or, combining results, (a) or (b') with equality implies U1(x1)=U=(xf). Furthermore, if (b') holds with strict inequality and sif > 0, then (x* *)EX

(y, q)

and it satisfies i's first period budget constraint with strict inequality. So then U1(x U'(x* ), otherwise (x , s')) EX* (y, q) would be contradicted given i's local nonsatiation i)>

in period 1.

II

Proof of Theorem 2. Let (x, s, p, 4) EOf for any efEYf. And let XI be the lagrange multipliers for (x', s'). Since (, ) EX*(y,' ), A- are also lagrange multipliers for this market plan, by (3c). In particular, using Assumption 2 and (3a), for each e there exists a c, say j, such that -i

Uecj(X*)

Ae=

P ec

But recalling the proof of Theorem 1, (x*,s*) is also in X*(y,q) if sof=0. So if i.e. using Assumption 2 and (3a), s= 0, Atare also lagrange multipliers for (x,); for each e Ae.

Ae = Pec

Since Ai = A- if s8f = 0, the theorem follows immediately from (3b) and (4), given the non-isolation Assumption 3. II Proof of Theorem 3. Given any security d (de) ERcE, let us write de for the monetary return on d in event e; i.e. de =Pede. This will make some of the expressions

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326

REVIEW OF ECONOMIC STUDIES

shorter! Now, to proceed. Observe that (6) may be written Ve(d)

Ze?+ Ae+j

+ -

de+? + Ee?l Ae+1 A

+ Ze +i

i(d,e

(Ze+2 Ae+2 ..

Ai(d,+l)(

(e1(+ee2 AeC

**( *

(A 1)

de+2)

. (zAi(d,e+(T-t-1))A

Ae+(T-t) de++(T

+(T _+t)

-t)

where e =(t,A), e + 1 (t + 1, A') such that A'c A; e + 2-(e + 1) + 1 (i.e. e + 2= (t+2,A") such that A"c A'); and in general e +k -(e +k -1)+1. And observe that if for all i and all e ?

Ze?l Ae+?de+1 = Ve(b)

eT:

exp (dt+?;e)

(A.2)

and for all i and i', and all e = (t, A), t 74TorT-1: AXe+2de+2)= Ve(b

Ee+l Ae+i(eX2

) exp (dt+2; e)

then the theorem follows upon substituting repeated applications of (A.2) and (A.3) into (A.1). So we need only verify (A.2) and (A.3), to which we now proceed. Observe first that given Assumption 5, (3a) implies for all i and all e = (t, A), if Jx eX'(y,q)then

Uec(x ) = ,w sA fw

-A ePec.

Axwtc

And given Assumption 2 there is equality for some c. But letting

e

=(e',

ed)

and

A = (AI, A ), EwEA

So, since

rd(ed)

=

i

w avo8 Xw rfwaVi EwE)A' 8x wtc

EdAd

rd(wd),

r' (w EWdEwdAAd

)rd(w d)

(jXwi

a-Vwtc

for all e A e = r (e )Aer

(A.4)

where Ae

1 -

I

Ew

Pec

rcA,

r (w

'V 1 )

IVa

)-

Xwtc

(the c satisfying Assumption 2). And since Ae+1i

Ae+1 rd(e d+1)e+

rd (ed)A i

=

for all e + 1 (d

A+=

d)

Ae+i

where e + 1 =(e'

+1,

ed +

1)

and Ae +1--Ai

Ae

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(A.5)

MAKOWSKI

COMPETITIVE STOCK MARKETS

327

Now to verify (A.2), observe rd (e d+1)

AA i

Le+? A4+de+i

=

de'd1

Ael+j

= Ee--l

A

rded

Aed?1

Lded+i

exp (dt+i; e)

Ve(bt+1) exp (dt+1; e)

=

since (9) plus (A.5) imply

~ rr(ed+

A

te(b

Ve (b

= Ael eee+A Ee +1 =..e+L

)

1)

A

A'

d (e d) Ae'+i

Eer+l

=Ee'+1

Ae'+l+

Analogously, to verify (A.3) observe that Ee+1

e+1 (Ze+2

Xe+2de+2)

Eerd d

=

Ee+1

Ee+l

-

Ee'1

Ae +1(Ze

Xe+1 (e

d

rd(e d)rl~e2Z? +2

r+2

+2

+2ded+2)

d (d+l)xe r(ed+)e+d+)

AXl+2 exp (dt+2; e))

exp (dt+2;e)

V ve(bt+2)

since using (9) plus (A.5) implies Ae+2) = Ee + 1 Ike+l(Ze+2

) = Ee+1 Xe+1(Ze+2

Ve(b

Ie'+2).

Finally, to show d ED* observe ve (d) equals an expression like (A. 1) with i substituted for each i(d, e), i(d, e 1). + ... , i(d, e + (T - t - 1). So, substituting (A.2) and (A.3) yields ve(d) = Zt'>tVe(bt) exp (dt; e) = Ve(d). II Proof of Theorem 4. Let W' ={w'}, a singleton with r'(w-) 1. And let W= W; {W}={W' x Wt }; and r (wd)-rw if w = (w', w ). Then (10) is satisfied, so Theorem 3 holds. 11 W'x

Proof of Theorem 5.

(13a) follows immediately from (X',S') EX* (y, q).

To verify (13b) we need only show that for any e and e + 1 implies

if WfeYf(e +1) (A.6) 7rf+1(yf)_rf +1(f) : 1f since repeated application of (A.6) yields 7rfl(yf) i7rf()f) implies (yf)' iref(yf) for all yf E Yf(e). But using (5) f rfe(yy) wf(3f)

rfe (Y)fPeYfe

+ Z,e+l

_e(r -PeY e + PeYe

+Ve+l(yf))

(e)(+lYfe+l

PeYf + Ze+1 e+l

+ Ee+le

+Ve+1(yW))

(Ye)(pe+lfe+l Ae+

+ 1V+1()

)(Pe+lYef+l

)(Pe+lYfe+l

+

Ve+l(yf))

+A

(Yf e)(e-f

+

f\x

where e is any one event following e and gf E Yf (e), which implies f

PeYe + V((y )

Pete + veo(Y)

||

Proof of Theorem 6. Since (x, s, y, p, q) e CSE, (15c) is obviously satisfied and so is (15b). To verify (15a) assume some i likes (J-', hi)eXX'(RCE, v) better than (x', n'). Let D = {d: n1(d) ? 0 or ni (d) ? 0}. We know that (x', n1) satisfies (3a) and also, given (3b) and (4), that for all e ?eT and d eD. Ee+l Xe+l(Pe+lde+l + Ve+l(d)) _ Ve(d),

with < implying n' (d) = 0.

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REVIEW OF ECONOMIC STUDIES

Since these are just the Kuhn-Tucker conditions for (xi, ni) to be an optimum for i among all (x', E)eX'(RCE, v) where fn' is restricted to be non-zero only in D, by the Kuhn-Tucker Theorem there can be no (Jx',ni') dominating (x', n') in D. II Proof of Theorem 7. Suppose the contrary. And let ni EN1 satisfy n-i= n i except for all e ni(Ax') = n' (x') + 1. Clearly for each i, (x', ni') satisfies (14) for all e ? 1. So since (x', n') is optimal for i, given local non-satiation in period 1 it must be that for all i p1x i + Ed (\fi (d) -n(d)) v1(d) 'plx1.l + Ed no'(d)pid 1, with > for some i. Or, summing over the i and using the fact that (x', n') must satisfy the first period budget constraint with equality, +Ei Vl(AX )>O.

EiPlAxi

But using (9) and the fact that Ei ix' = Ef Ayf, this implies

Pi Ef Ayf +vl(Zf zyf) >O. Or, adding Ef (plyf +v1(yf)) to both sides of the inequality,

Ef plyl +v,Wy))> Ef(plyfl+ vl(yf)) contradicting (15b).

||

Proof of Theorem 8. Suppose the contrary. And, in particular, that (x, y) dominates (x, y). Let fniEN' satisfy fn'(d)=_Eyf:=daf for all e if d = yf for somef; nie(d)=O otherwise. Since (x, n') satisfies (14) for all e ? 1, for some i it must not satisfy (14) for e = 1. Or, using the fact that individuals are locally non-satiated in period 1 and summing first period budget constraints over all i, Ei Pl 1-fi+ Ei Ed (nil (d) - fi'(d))v 1(d) > i p ix- + Ei Ed no(d)pid,.

Or, since Ef yf+

Exi Ef (Pl+Zf +vl())

>f

xi ~

(plyf +vl(yf)).

That is, for some f (15b) must be violated. Proof of Theorem 9. Let be ERCE be a contingent claim in event e; i.e. and bet 0 for all e' ? e. And let Pe vi(b )Pe for each event e; let a =plx1+

EfsO (plyfl +vl(yf))

for each i. We show (x, y, p) E W(a i). First note that EiZa = fjpX +Zf where Ae =Ae(1) if e ? 1 and A1 -1.

(rnf

+v (yf)) = ii +Zf Ze AePeYfe Or since vi(be) = Ae by (9),

sai = Zip.xi+Efp-yf

as required by (16d). Next note that PlY 1 + vl(y )_-P1Yl + vl(y implies Ee AePeYe

-Ee AePeYe

or pyfFYf f. So, (16b) is satisfied.

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Pebe

COMPETITIVE STOCK MARKETS

MAKOWSKI

329

Finally, note (x ', si) E X (y, q) implies S ef-lPeY]fe EeAe[PeXe + Ef Ve(Yf)(Sef- ef 1)]-Ee Ae[Pe4 ie+ Ef or + Ef Ee Ie [Se' - (PeY e + Ve (Yf )SiefVe

Fpx -ft

using (5) and manipulating terms. We need only show that x' eX' and px' 'a'

(Yf ApX

+ Ef SiOfP-Yf =ai

implies U'(x') _ U'(x'). Let

(x, n, y, p, v)i=l .r E WSE and let Fn'= n' except for all e ? 1 fil (be)-nl

(be) =PeX-

Pe

Clearly (x', ni) satisfies (14) for all e ? 1. And using (9) e

(nii (b e)-nfi

(b e))V i(b e) = Ze?

= Eel = Ee?l C-plx

V ((Pe Xie

Ae(PeXe

peX i )be)

PeXe)

( Pee-feXe) i -pix

i

So, it also satisfies (14) for e = 1. But then U'(f') < U'(x') since (x', ni) solves (15a). First version received August 1981; final version accepted July 1982 (Eds.). I have benefited from discussions with Jeremy Edwards, Terence Gorman, Frank Hahn, Oliver Hart, David Kreps, Joe Ostroy, Roy Radner, David Starrett and Joe Wilcox. I have also benefited from the referees' comments. This research was supported by a grant from the (U.K.) Social Science Research Council, which is gratefully acknowledged. NOTES 1. The approach here is similar to that taken in Makowski (1980b), except now we are in a multi-period setting with an explicit stock market. And, the only innovation occurring is that of securities, although one could generalize the current model to include firms innovating products. 2. When there is spanning, short selling doesn't matter for unanimity. It is only in Hart's world of perfect competition without spanning that it does a new, interesting thing. 3. Warning: This does not imply for perfect competition there must be many buyers and sellers of a good. The examples in the text will illustrate that a unique seller having many highest valued buyers will also be small relative to his market-in the relevant sense (i.e. in the sense guaranteeing he faces a perfectly elastic demand for his good). 4. If the current model were extended to include trading of riskless bonds-so riskless bonds were actually available, then Assumption 4 would be satisfied automatically and hence superfluous (recall (a) of Section 4B). The assumption just allows us to forgo the burden of adding bonds. Hopefully, this makes life a little easier for the reader too! REFERENCES ARROW, K. J. and LIND, R. C. (1971), "Uncertainty and the Evaluation of Public Investment Decisions", in Arrow, K. J. (ed.) Essays in the Theory of Risk-Bearing (Chicago: Markhan) 239-266. DIAMOND, P. A. (1967), "The Role of the Stock Market in a General Equilibrium Model with Technological Uncertainty", American Economic Review, 57, 759-776. DREZE, J. H. (1974), "Investment under Private Ownership: Optimality, Equilibrium and Stability", in Dreze, J. H. (ed.) Allocation Under Uncertainty: Equilibriumand Optimality (New York: Wiley) 129-166. ELLICKSON, B. (1979), "Competitive Equilibrium with Local Public Goods", Journal of Economic Theory, 21, 46-61. GROSSMAN, S. J. and HART, 0. D. (1979), "A Theory of Competitive Equilibrium in Stock Market Economies", Econometrica, 47, 293-329. HART, 0. D. (1979a), "Monopolistic Competition in a Large Economy with Differentiated Commodities", Review of Economic Studies, 46, 1-30.

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HART, 0. D. (1979b), "On Shareholder Unanimity in Large Stock Market Economies", Econometrica, 47, 1057-1083. HART, 0. D. (1980), "Perfect Competition and Optimal Product Differentiation", Journal of Economic Theory, 22, 279-312. KREPS, D. M. (1979), "Three Essays on Capital Markets" (mimeo, Institute for Mathematical Studies in the Social Sciences). MAKOWSKI, L. (1979), "A Dual Characterization of Optimal Choices Given Preference and Constraint Sets" (Discussion Paper No. 20, Cambridge Economic Theory). MAKOWSKI, L. (1980a), "A Characterization of Perfectly Competitive Economies with Production", Journal of Economic Theory, 22, 208-221. MAKOWSKI, L. (1980b), "Perfect Competition, the Profit Criterion, and the Organization of Economic Activity", Journal of Economic Theory, 22, 222-242. MAKOWSKI, L. (1980c), "No Surplus in Large Economies" (mimeo). MAKOWSKI, L. (1980d), "Characterizing Perfectly Competitive Sequential Equilibria" (mimeo). MAKOWSKI, L. (1981), "Competitive Stock Markets" (Cambridge University Discussion Paper 52). MAKOWSKI, L. (1982), "Competition and Unanimity Revisited", American Economic Review (forthcoming). OSTROY, J. M. (1980), "The No Surplus Condition as a Characterization of Perfectly Competitive Equilibrium", Journal of Economic Theory, 22, 183-207. OSTROY, J. M. (1981), "Competitive Pricing of Persons" (UCLA Discussion Paper No. 193).

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