One of the more noteworthy developments in economics over the last twenty years or so is the emergence of equilibrium models of the financial market. Included in this term is the market for financial securities such as stocks, bonds, options and insurance contracts. The chief building block and spur in this evolution has been the economics of uncertainty, which itself is of rather recent origin. The results of this new focus and the activities and synergies it has generated is often broadly referred to as financial economics. It is within this new subfield that various models of the financial market occupy the centre stage.

After a brief summary of models based on analysis in return space in section, “Return Space Analysis”, this essay will focus on the two-period, pure-exchange model of the financial market beginning in section, “The Basic Model”. Conditions under which full efficiency is attained in incomplete markets will be identified in section, “Full Allocational Efficiency in Incomplete Markets”. Finally, section, “Changes in the Financial Market” will trace the welfare and price effects resulting from changes in the financial market.

Return Space Analysis

Much of the earlier work in financial equilibrium focused on pay-off returns rather than total pay-offs or consumption levels. While return space is both a natural and intuitive object of concern, and in fact continues to draw much attention, it faces certain shortcomings in addressing many questions of interest where prices, endowments and consumption pay-offs play a central role. I shall therefore provide only a brief review of the main results in return space before moving on to the consumption-and wealth-oriented models.

The so-called capital asset pricing model (CAPM) was more or less independently developed by Sharpe (1964)), Lintner (1965), and Mossin (1966). It studies a single-period, frictionless, competitive market of financial securities. Assuming that (a) investors’ preferences are a function of only the mean and the variance of the portfolio’s anticipated return (with the mean favoured and the variance disfavoured), (b) investors have homogeneous probability assessments of returns and (c) there is a risk-free asset and that unlimited borrowing is available at the lending rate, three principal results are obtained in equilibrium:

  1. 1.

    The expected return on an optimal portfolio is a positive linear function of its standard deviation of return.

  2. 2.

    The expected return on every security (and portfolio) is a positive, linear function of its (return) covariance with the market portfolio of risky assets (the portfolio which includes x per cent of the outstanding shares of all securities in the market).

  3. 3.

    All optimal portfolios are comprised of the market portfolio of risky assets in conjunction with either risk-free borrowing or lending.

Since the CAPM model is consistent with the von Neumann and Morgenstern (1944) theory of rational choice only under quadratic preferences and/or normally distributed returns, it has left many economists uncomfortable. Nevertheless, it has been the basis of a very large number of empirical studies, which on balance show that the CAPM model provides a rather good first approximation of observed return structures in the financial markets of various countries.

A more recent development is the so-called arbitrage pricing theory (APT) developed by Ross (1976). It posits that security returns are generated by a linear K-factor mode (with K small) in which securities’ residual risks are sufficiently independent across securities for the law of large numbers to apply. APT can therefore be viewed as an extension of the single-index model introduced by Markowitz (1952) and developed and extended by Sharpe (1963; 1967), which in turn, of course, is closely related to the CAPM. Not surprisingly, the APT appears to offer a somewhat better fit than the CAPM or single-index model.

In studying the economics of financial markets, however, the CAPM and the APT frameworks do not offer fertile ground. In the CAPM framework, for example, the capital structures of firms are a matter of indifference. To study these and other questions, we must therefore turn to more comprehensive formulations.

The Basic Model

The earliest models systematically incorporating uncertainty in analysing markets were those of Allais (1953), Arrow (1953), Debreu (1959, Ch. 7) and Borch (1962). They may therefore be viewed as the forerunners of more comprehensive models of the financial market, including the two-period model developed below.

Assumptions

We consider a pure-exchange economy with a single commodity which lasts for two periods under the standard assumptions. That is, at the end of period 1 the economy will be in some state s where s = 1,…, n. There are I consumer-investors indexed by i, whose probability beliefs over the states are given by the vectors πi = (πil,…, πin), where, for simplicity, πis > 0, all i, s.. The preferences of consumer-investor i are represented by the (conditional) functions Uis(ci, wis), where ci is the consumption level in period 1 and wis is the consumption level in period 2 if the economy is in state s at the beginning of that period. These functions are defined for

$$ \left({c}_i,{w}_{is}\right)\geqq 0\quad \mathrm{all}\ i,s $$
(1)

and are assumed to be increasing and strictly concave.

At the beginning of period 1 (time 0), consumer-investors allocate their resources among current consumption ci and a portfolio chosen from a set J of securities indexed by j. Security j pays ajs ≥ 0 per share at the end of period 1 and the total number of outstanding shares is Zj. Let zij denote the number of shares of security j purchased by investor i at time 0; his portfolio Zi = (zil,…, zij) then yields the pay-off

$$ {w}_{is}=\sum \limits_{j\in J}{z}_{ij}{a}_{js}, $$

available for consumption in period 2 if state s occurs at the end of period 1. Investor endowments are denoted (Ci, Zi) and aggregate wealth or consumption in state s is given by

$$ {W}_s=\sum \limits_{j\in J}{Z}_j{a}_{js},\quad \mathrm{all}\;s. $$

The financial markets, as is usual, are assumed to be competitive and perfect; that is, consumer-investors perceive prices as beyond their influence, there are no transaction costs or taxes, securities and commodities are perfectly divisible, and the full proceeds from short sales (negative holdings) can be invested. The number of securities, however, need not be large (although this is not ruled out). Since our focus is on the structure of the financial market, and changes therein, production decisions (and hence the vector of aggregate consumption (C,W)) are viewed as fixed.

If the rank of matrix A = [ajs] is full (equals n), the financial market will be called complete; if not, it will be called incomplete. The significance of a complete market is that any pay-off pattern w ≧ 0 can be obtained via some portfolio z since the system zA = w will always have a solution. (In incomplete markets, in contrast, some pay-offs patterns w ≧ 0 are infeasible). The simplest form of a complete market is that in which A = I (the identity matrix); the financial market is now said to be composed of Arrow-Debreu or primitive securities (as opposed to complex securities.) The main ‘advantage’ of an Arrow–Debreu market is that it never requires the consumer-investor to take short positions, which is generally necessary in a complete market composed of complex securities. Finally, a financial market which contains a risk-free asset, or makes it possible to construct a risk-free portfolio, is called zero-risk compatible.

Under our assumptions, each consumer-investor i maximizes

$$ {u}_i\equiv \sum \limits_s{\pi}_{is}{U}_{is}\left({c}_i,\sum \limits_{j\in J}{z}_{ij}{a}_{js}\right) $$
(2)

with respect to the decision vector (ci, zi), subject to (1) and to the budget constraint

$$ {c}_i{p}_0\sum \limits_{j\in J}{z}_{ij}{p}_j={\overline{c}}_i{p}_0\sum \limits_{j\in J}{\overline{z}}_{ij}{p}_j $$

as a price-taker, where P0 is the price of a unit of period 1 consumption and Pj is the price of security j.

Equilibria and Their Properties

In view of our assumptions, an equilibrium will exist but need not be unique (see e.g. Hart 1974; note also that uniqueness is with reference to the consumption allocation (c, w), not allocation (c, z)). The equilibrium conditions for any market structure A, assuming for simplicity that the non-negativity constrains on consumption are not binding may be written

$$ \sum \limits_s{\pi}_{is}\frac{\partial {U}_{is}\left({c}_i\sum \limits_{j\in J}{z}_{ij}{a}_{js}\right)}{\partial {c}_i}{\lambda}_i\quad \mathrm{all}\;i $$
(3)
$$ \sum \limits_s{\pi}_{is}\frac{\partial {U}_{is}\left({c}_i\sum \limits_{j\in J}{z}_{ij}{a}_{js}\right)\;{a}_{js}}{\partial {w}_{is}}={\lambda}_i{p}_j\quad \mathrm{all}\;i,j $$
(4)
$$ \left({c}_i,{z}_iA\right)\ge 0\quad \mathrm{all}\;i $$
(5)
$$ {c}_i+{z}_ip={\overline{c}}_i+{\overline{z}}_ip\quad \mathrm{all}\;i $$
(6)
$$ \sum \limits_i\left({c}_i,{z}_i\right)=\left(C,Z\right) $$
(7)

where the λi are Lagrange multipliers, (7) represents the market clearing equations, and P0 has been chosen as numeraire, i.e. P0 ≡ 1.

Any allocation (c, z) which constitutes a solution to system (3), (4), (5), (6), and (7) (along with a price vector P and a vector λ) is allocationally efficient with respect to the market structure A – since the marginal rates of substitution for any two securities are the same across individuals. When (c,z) is allocationally efficient with respect to all conceivable allocations, whether achieved outside the existing market or not, (c, z) will be said to be fully allocationally efficient (FAE).

To be more precise, define the shadow prices\( {R}_{is^{\prime }} \) is by

$$ {R}_{is^{\prime }}\equiv \frac{1}{\lambda}\left({\pi}_{is}\frac{\partial {U}_{is}\left({c}_i,\sum \limits_{j\in J}{z}_{ij}{a}_{js}\right)}{\partial {w}_{is}}\right). $$

It is well known that (3), (4), (5), (6), and (7) plus

$$ {R}_{is^{\prime }}={R}_{1{s}^{\prime }}\quad \mathrm{all}\;i\ge 2,\quad \mathrm{all}\;s $$
(8)

is a necessary and sufficient condition for the market allocation (c, z) to be FAE because (8) insures that the marginal rates of substitution of wealth between any two states are the same for all investors i. (4) may now be written

$$ A{R}_{is^{\prime }}=p,\quad \mathrm{all}\;i. $$
(4′)

Implicit Prices

The equilibrium value of a feasible second-period pay-off vector w will be denoted V(w); thus if w is obtainable via portfolio z, we get w = zA and hence

$$ V(w)=V(zA)= zp= wR= zAR. $$

In the above expression, R = (R1,…, Rn) represents the not necessarily unique set of implicit prices of (second-period) consumption in the various states implied by P since

$$ AR=p. $$
(9)

By Farkas’ Lemma, a positive implicit price vector is always present in the absence of arbitrage and hence in equilibrium. (Arbitrage is the opportunity to obtain either a pay-off w ≥ 0, w ≠ 0, at a cost zP ≤ 0, or a pay-off w = 0 at a cost zP < 0). In view of (4′) and (9), shadow prices are always implicit prices, but a set of implicit prices need not be anyone’s shadow prices.

Full Allocational Efficiency in Incomplete Markets

When the financial market A is complete, systems (4′) and (9) have only one solution, which insures that

$$ {R}_{t^{\prime }}=R,\quad \mathrm{all}\;i. $$

This condition, as noted, is necessary and sufficient to attain FAE. Complete financial markets, while a useful abstraction, are not an everyday occurrence, however. Securities number at most a few thousand, while the relevant set of states is no doubt much larger. This leads us to the question: under what circumstances is FAE attained in incomplete markets? One such case is trivial and will be dismissed quickly: the case when individuals are identical in their preferences, beliefs and (the value of their) endowments. We now turn to three other sets of conditions when this occurs.

Diverse Endowments

Are there any conditions under which individuals with diverse endowments are as well served by a single security in the market as by many? The answer is yes; beliefs must be homogeneous and preferences e.g. of the form

$$ {U}_{is}\left({c}_i,{w}_{is}\right)=\left\{\begin{array}{c}{U}_i^1\left({c}_i\right)+{\rho}_s{U}_i^2\left({w}_{is}\right)\\ {}\mathrm{or}\\ {}{U}_i^1\left({c}_i\right){\rho}_s{U}_i^2\left({w}_{is}\right)\end{array}\right.\quad \mathrm{all}\;i,s $$
(10)

(with ρs > 0), where

$$ {U}_i^2\left({w}_{is}\right)=\left(1/\gamma \right){w}_{is}^{\gamma },\quad \gamma <1,\quad \mathrm{all}\;i $$

That is, preferences for second-period consumption must be separable, isoelastic and homogeneous. Everyone’s optimal portfolio is now of the form

$$ {z}_i={k}_iZ,\quad \mathrm{all}\;i $$

where the ki are fractions. In addition, the equilibrium implicit prices R are now unique and completely independent of the market structure A.

Linear Risk Tolerance

To attain FAE with heterogeneous second-period preferences, we need at least two securities in the market. Two-fund separation occurs in every zero-risk compatible market A under homogeneous beliefs (but arbitrary return structures) when preferences are of the form (10) if and only if

$$ {U}_i^2\left({w}_{is}\right)=\left\{\begin{array}{c}\left(1/\upgamma \right){\left({\phi}_i+{w}_{is}\right)}^{\gamma}\quad \mathrm{all}\;i\\ {}\mathrm{or}\\ {}-{\left({\phi}_{\mathrm{i}}-{w}_{is}\right)}^{\gamma}\, \upgamma >1,\, {\phi}_{\mathrm{i}}\;\mathrm{large},\, \mathrm{all}\;i\\ {}\mathrm{or}\\ {}-\exp \left\{{\phi}_{\mathrm{i}}\;{w}_{is}\right\}\quad {\phi}_i<0,\quad \mathrm{all}\;i\end{array}\right. $$

provided none of the non-negativity constraints on consumption is binding. The optimal policies are now of the form

$$ {z}_i={k}_{i1}{z}^{\prime }+{k}_{i2}{z}^{{\prime\prime} },\quad \mathrm{all}\;i, $$

where the portfolio (fund) z′ is risk-free and portfolio z″ is risky (see e.g. Rubinstein 1974). It is evident that with diverse endowments, preferences must belong to a very narrow family, even when beliefs are homogeneous, in order for FAE to be attained.

Supershares

Two states s and s′ such that Ws = Ws′, i.e., with equal aggregate pay-offs, are said to belong to the same superstate t (Hakansson 1977). If the financial market is complete with respect to the superstate partition T, FAE is attained for arbitrary endowments if and only if

$$ {\pi}_{is}/{\pi}_{it}={\pi}_{1s}/{\pi}_{1t},\quad \mathrm{all}\;s\in t,\quad \mathrm{all}\;i\;\mathrm{and}\;t $$
(11)

and

$$ {U}_{is}={U}_{is\prime },\quad \mathrm{all}\;s\;\mathrm{and}\;{s}^{\prime}\in t,\quad \mathrm{all}\;i\;\mathrm{and}\;t $$
(12)

Note that (10) and (11) require only conditionally homogeneous beliefs and that preferences are insensitive to states within a superstate -beliefs and preferences with respect to superstates are unrestricted.

To complete the market with respect to superstates, three simple alternatives are available (Hakansson 1978). The first is a full set of ‘supershares’, each share paying $1 if and only if a given superstate occurs (superstates are readily denominated in either nominal or real terms). The second and third alternatives are a full set of (European) call options or a full set of (European) put options on the market portfolio αZ or αW, where 0 < α ≤ 1.

It may be noted that a market in puts and calls on a crude approximation to the United States market portfolio, namely the Standard & Poors 100 Index, was opened in 1983. These options are now the most actively traded of all option instruments.

Changes in the Financial Market

Changes in the set of securities available in the financial market are everyday occurrences. Early studies on this subject include those of Borch (1968, Ch. 8), Ross (1976) and Litzenberger and Sosin (1977). To trace fully the effects of such changes involves comparing equilibria, which is a matter of some complexity. However, using the two-period framework of this assay, it is possible to reach some general conclusions on how changes in the market structure from A′ to A″, say, affect welfare, prices and other dimensions of interest in a pure exchange setting.

The Feasible Allocations

One of the critical determinants, not surprisingly, is the change in feasible allocations. Recall that a market structure A is any ‘full’ set of instruments; that is, any set of instruments capable of

$$ F(A)\equiv \left\{w\left|{w}_i\geqq 0,{w}_i={z}_iA,\sum \limits_t{z}_{ij}={Z}_j,\quad \mathrm{all}\, j\right.\right\}. $$

allocating, in some fashion, aggregate wealth W = (W1,…, Wn). The set of feasible second-period consumption allocations w = (w1,…, w1) obtainable via market structure A will be denoted F(A), i.e.

In comparing two market structures A′ and A″ with respect to feasible allocations, there are (since holding the market portfolio αZ is always feasible) three possibilities; either

$$ F\left({A}^{\prime}\right)=F\left({A}^{{\prime\prime}}\right)\left(\mathrm{Type}\ \mathrm{I}\right) $$

or

$$ F\left({A}^{\prime}\right)\subset F\left({A}^{{\prime\prime}}\right)\left(\mathrm{or}\, \mathrm{the}\, \mathrm{converse}\right)\left(\mathrm{Type}\ \mathrm{II}\right) $$

or

$$ {\displaystyle \begin{array}{l}\left\{F\left({A}^{\prime}\right)\cap F\left({A}^{{\prime\prime}}\right)\right\}\subset F\left({A}^{\prime}\right)\\ {}\left\{F\left({A}^{\prime}\right)\cap F\left({A}^{{\prime\prime}}\right)\right\}\subset F\left({A}^{{\prime\prime}}\right).\left(\mathrm{Type}\ \mathrm{III}\right)\end{array}} $$

These three types of changes will be referred to as feasibility preserving, feasibility expanding (or reducing) and feasibility altering.

A sure way to obtain a feasibility expanding change is to make a finer and finer breakdown of existing instruments into an ever larger set of linearly independent (or unique) securities.

Endowment Effects

Since changes in the financial market structure are generally implemented by firms or exchanges and take place when the market is closed, such changes frequently alter investors’ endowments. An example would be a merger, which results in the substitution of new securities for old. It is useful to distinguish between three cases:

  1. 1.

    Strong Endowment Neutrality This occurs if the endowed consumption patterns in the two markets are unaltered, i.e. if

    $$ \left({\overline{c}}_{i^{\prime }},{\overline{w}}_{i^{\prime }}\right)=\left({\overline{c}}_i^{{\prime\prime} },{\overline{w}}_i^{{\prime\prime}}\right),\quad \mathrm{all}\;i. $$
  2. 2.

    Weak Endowment Neutrality This occurs if the values of the endowments (provided there is a common implicit equilibrium price structure R) are identical in the two markets, i.e. if

    $$ {\overline{c}}_{i^{\prime }}+{\overline{z}}_{i^{\prime }}{p}^{\prime }={\overline{c}}_{i^{\prime }}+{\overline{w}}_{i^{\prime }}R+{\overline{c}}_{i^{{\prime\prime} }}+{\overline{w}}_{i^{{\prime\prime} }}R={\overline{c}}_i^{{\prime\prime} }+{\overline{z}}_i^{{\prime\prime} }{p}^{{\prime\prime} },\quad \mathrm{all}\;i $$

    where R > 0 satisfies A′R = P′ and A″R = P″.

  3. 3.

    Non-Neutral Endowment Changes While the first two cases are rather rare, strong endowment neutrality typically accompanies non-synergistic (pro rata) corporate spin-offs when applicable bonds remain risk-free, as well as the opening of option markets, for example.

The Welfare Dimension

As noted, in comparing different market structures, the comparison which is ultimately relevant is that which compares allocations actually attained; that is, equilibriumn allocations. Using (2), we denote investor i’s equilibrium expected utility in market all i structure A″ by ui″ and his equilbrium expected utility in market structure A′ by \( {u}_{i^{\prime }} \). A comparison of any given equilibrium in market A″ with some equilibrium in some other market A′ must then yield one of four cases:

$$ {u}_i^{{\prime\prime}}\ge {u}_{i^{\prime }},\mathrm{all}\;i,{u}_i^{{\prime\prime} }>{u}_{i^{\prime }},\mathrm{some}\;i\;\left(\mathrm{Pareto}\ \mathrm{dominance}\right) $$
(i)

or

$$ {u}_i^{{\prime\prime} }={u}_{i^{\prime }},\mathrm{all}\;i\quad \left(\mathrm{Pareto}\ \mathrm{dominance}\right) $$
(ii)

or

$$ {u}_i^{{\prime\prime} }>{u}_{i^{\prime }},\mathrm{some}\;i,{u}_i^{{\prime\prime} }>{u}_{i^{\prime }},\mathrm{some}\;i\;\left(\mathrm{Pareto}\ \mathrm{redistribution}\right) $$
(iii)

or

$$ {u}_i^{{\prime\prime}}\le {u}_{i^{\prime }},\mathrm{all}\;i,{u}_i^{{\prime\prime} }>{u}_{i^{\prime }},\mathrm{some}\;i\;\left(\mathrm{Pareto}\ \mathrm{inferiority}\right) $$
(iv)

The task at hand, then, is to identify the conditions under which each of these cases, as well as combinations of these cases, will occur. All comparisons are contemporaneous in the sense that they compare welfare under market structure A″ to what it would be if Á′ were in use instead.

Principal Results

The principal results (Hakansson 1982) may be summarized as follows:

  1. 1.

    Feasibility preserving market structure changes yield either Pareto equivalence or redistributions. To preclude Pareto redistributions we must either have efficient endowments in the first market and strong endowment neutrality, or weak endowment neutrality coupled with unique equilibria. Pareto equivalence is always accompanied by value conservation.

  2. 2.

    Feasibility expanding market structure changes imply either Pareto dominance, Pareto equivalence or Pareto redistributions. To preclude redistributions we must have efficient endowments in the first market and strong endowment neutrality, or weak endowment neutrality coupled with unique equilibria. Value conservation is highly unlikely.

  3. 3.

    Feasibility altering changes in the market structure have unpredictable value and welfare effects.

  4. 4.

    Value and welfare effects are relatively independent.

As noted by Hart (1975), the introduction of multiple commodities or more than two periods is a non-trivial step which may bring about additional complications, such as Pareto-dominated equilibria when feasibility is expanded.

Within the limits of the single-good, two-period model under pure exchange, certain tentative general conclusions concerning common market structure changes can be stated. Even under mild heterogeneity of preferences and/or beliefs, 100 per cent non-synergistic mergers tend to be welfare reducing while (non-synergistic) spin-offs and the opening of option markets tend to be beneficial. The use of risky bonds and preferred stock tends to be virtuous as well, at least apart from bankruptcy costs. Finally, value conservation is a much rarer phenomenon than suggested by Modigliani and Miller (1958) and Nielsen (1978) among others.

See Also