COMPARING FINANCIAL

SYSTEMS

Lesson 24

The Limitations of Markets 2

What you will learn in this lesson

Some features of equilibrium in GEI.

Some efficiency problems of GEI.

Models with spanned securities.

The shareholder unanimity hypothesis.

Relevance of Corporate Finance and security design.

The role of asymmetric information, adverse selection and moral hazard.

Definition of equilibrium in GEI An equilibrium consists of:

a) an m-tuple of consumption and portfolio plans (𝒙𝑖0, 𝒕𝑖);

b) an mn-tuple of production plans (𝒚𝑗𝑖 ),

c) a vector of share prices q such that, for every i:

1. (𝒙𝑖0, 𝒕𝑖) maximizes 𝒖𝑗

∗ 𝑥𝑖0, 𝒕𝑖; 𝒚𝑖 subject to the budget constraint:

(where the value of the firm is net of the cost of investment, reflected in the last term in

the budget constraints of the initial shareholders).

2. 𝑦𝑗𝑖 maximizes MV (𝒚𝑗

𝑖1) + (𝒚𝑗𝑖0) subject to 𝒚𝑗 ∈ 𝑌𝑗 for every j;

3. 𝒒𝑗 ≤ MV (𝒚𝑗𝑖1) for all i and j, with

where MV (y1) = maxi=1, . . . ,m {μi(xi) · 𝒚1}

4.

Some features of equilibrium in GEI

Despite the simplicity of the model, this definition of equilibrium appears rather

complicated, even though it simply requires that consumers maximize utility, that

producers maximize value, and that markets clear.

Part of the complexity arises from the fact that we have to allow for the valuation

of new securities that are not actually observed in equilibrium.

To do so we have to build into the definition of equilibrium some analysis of the

consumer's portfolio behavior

the first-order conditions for an optimal portfolio.

In this kind of equilibrium with incomplete markets, many of the standard results

of finance continue to hold.

Efficiency

However, a GEI allocation is not Pareto efficient in the usual sense.

In fact, Pareto efficiency requires that marginal rates of substitution be equalized

across individuals

μi(xi) = μj (xj) ∀ 𝑖, 𝑗.

With a limited number of linearly independent securities to trade,

there is no reason for this criterion will be satisfied.

Constrained Pareto efficiency

It is often argued that the Pareto efficiency criterion is inappropriate for an

economy with incomplete markets.

The incompleteness of markets is the result of transaction costs or other frictions

to which a central planner (as a possible alternative to markets) might also be

subject.

The central planner should be constrained to make use of the transaction technology

when attempting to improve on the equilibrium allocation.

Constrained Pareto efficiency.

Constrained Pareto efficiency

An equilibrium allocation is said to be constrained Pareto efficient if there does

not exist an attainable allocation consisting of first-period consumption,

production plans, and portfolios that Pareto dominates the equilibrium allocation.

Constrained Pareto efficiency characterizes the equilibrium in the sense that every

equilibrium allocation is constrained Pareto efficient and every constrained

Pareto-efficient allocation can be decentralized as an equilibrium with lump-sum

transfers.

Gale (1982) demonstrated that the equilibrium allocation in the GEI model is

constrained Pareto efficient.

Non robustness of costrained Pareto efficiency

However, this kind of effciency is not robust to the introduction of more than two

periods or more than a single good at each date.

For example, an economy with two goods, two periods, and no uncertainty, in

which goods can be traded only on spot markets, can have multiple Pareto-ranked

equilibria (Hart,1975).

A much deeper result is that in a two-period, pure exchange model with a fixed set

of linear securities denominated in terms of goods, equilibrium is generically

inefficient in a very strong sense (Geanakoplos and Polemarchakis, 1986).

More precisely, it is shown that if markets are effectively incomplete and there are

two or more goods at each date, then it is generically possible to make all agents

better off by redistributing goods and securities at the first date.

Non robustness of costrained Pareto efficiency

In fact, with two or more goods, changes in the allocation of goods and securities

at the first date have an effect on prices at the second date.

These endogenous price changes, which are not taken into account by agents

making decisions in the first period, create pecuniary externalities.

With complete markets, pecuniary externalities do not "matter" because marginal

rates of substitution are all equalized, but with incomplete markets they do.

This is what makes it possible for all the agents to be made better off.

Only if there is only one good at each date (a very unrealistic hypothesis),

there are no relative prices at the second date.

There is a unique relationship between first-period decisions and second-period

allocations, and no possibility of pecuniary externalities.

Decentralization

One of the most important lessons of the ADM model is the sufficiency of prices

(or the information contained in prices) as a guide for efficient decision making.

In the ADM model, everyone observes a complete vector of contingent-

commodity prices.

These prices are used for the valuation of everything in the economy.

In particular, the firm's production plan is valued like any other commodity

bundle, using the prices for contingent commodities.

In the GEI model, on the other hand, we use the functional MV (y1) to value the

firm, and this evidently contains a lot more information (parameters) than an

ordinary price vector.

Decentralization

However, it is not clear whether we have achieved the same degree of

decentralization in a GEI model as we have in an ADM model.

An interesting question is how much (or how little) information the producers

need in order to choose the right production plans and thus, indirectly, provide the

right securities for the consumers.

In presenting the definition of equilibrium, we treated the producers as if they

understood how much each consumer would pay for a small amount of each

security that could be produced.

This idea can be elaborated by imagining a two-stage game.

The two-stage game

Producers make choices of production plans in the first period and consumers

trade securities in the second.

Corresponding to every choice of production plans in the first period, there is a

distinct subgame in the second.

In each of these second-period subgames, there will be equilibrium prices for each

of the securities produced.

The price functional MV (y1) is just a reduced-form representation of these

subgame equilibrium prices, and the assumption that each producer knows the

prices of every conceivable security is just an assumption of rational expectations

about the subgame that results when a new security is introduced.

Spanned securities

The alternative interpretation is that the auctioneer simply calls out the prices of

all the securities, at which producers can sell as much as they want of each

security.

In practice, however, the assumption that so much information is available for free

seems a little stringent.

For this reason, theorists have looked for simplifying assumptions under which

producers could use the information available about the prices of actually traded

securities in order to infer the equilibrium price of a nontraded security.

One example of this approach applies to securities that are spanned by the

existing set of traded securities.

Spanned securities

The GEI model of equlibrium assumes that consumers can only hold non-negative

amounts of each security:

tij > 0 ∀i, j.

In the traditional account of spanning, on the other hand, it is implicitly assumed

that unlimited short sales are allowed.

In that case, the set of available portfolios must include

(the linear subspace of Rs spanned by the future production plans {𝑦𝑗𝑖}).

The usual no-arbitrage condition requires that the price of a security

must be equal to

Spanned securities

Otherwise it would be possible to construct a riskless arbitrage

a violation of the conditions for equilibrium

any security in the linear subspace Z can be priced by arbitrage.

In spanning, iorder to infer the price functional MV on Z, it is sufficient to know

the prices of the traded securities{𝑦𝑗𝑖}.

As long as the producer wants to choose a production plan in the span of the set of

traded securities, he can calculate the equilibrium price of the new security from

the prices he observes in the market.

However, if the producer wants to venture outside the span of the set of traded

securities, then linearity no longer applies, and the calculation is more

complicated.

Spanned securities

If we assume that every agent has strictly positive consumption 𝑥𝑖 ≫ 0, the first-

order conditions for equilibrium imply that 𝑞𝑗 = 𝜇𝑘 𝑥𝑘 ∙ 𝑦𝑗𝑖1 ∀k, i, j

∀k, i, j

In other words, individual marginal valuations agree on the subspace Z.

For this reason, consumers are interested only in the value of the security, not its

individual characteristics, as long as it belongs to Z.

This also explains why maximizing value and decentralizing decisions are the

"right" thing to do (consumers all agree that this is the right thing to do).

Of course, since all consumers have tailor-made securities, the issue of unanimity

is more or less irrelevant here.

Shareholder Unanimity

In the model of equilibrium with incomplete markets, there is complete unanimity

by definition.

Each producer has identical shareholders who must therefore agree about what the

firm ought to do.

We achieve this outcome by assuming a very large number of producers of each

type and they produce tailor-made securities.

These assumptions are natural in a world of perfect competition, but they are

nonetheless quite strong.

If the number of firms were limited, unanimity might not be assured, and that

would lead to conflicts among shareholders.

Shareholder Unanimity

Grossman and Hart (1979) have developed a theory of the firm designed to deal

with exactly this issue.

Their focus was on developing an account of producer behavior that would

guarantee a limited form of decentralization (constrained efficiency).

When consumers of several different types hold shares in the production plan of a

single firm, the producer must attempt to maximize a weighted sum of their

marginal rates of substitution.

If the representative consumer of type i holds a share tij in the production plan yj,

producer j must choose yj to maximize

This is a somewhat complicated objective function, and from a positive point of

view, it is not clear that the producer will find it easy to use.

Shareholder Unanimity

Even if we can assume that the producer does use this objective function, the

problems do not end there.

In a multi-period context, the identities of a given firm's shareholders will change

from period to period as the shares are bought and sold.

It turns out that later shareholders may want to change the plans of earlier

shareholders, thus introducing a form of time inconsistency.

There is a large number of publicly traded companies, and it may be that the

number of different types of consumers is not so great but that they can distribute

themselves among firms in a way that allows for something close to unanimity.

However, this question remains unexplored in financial and economic literature.

The Grossman-Hart theory raises some problems, since they require that the

number of firms (not just firm types) be fewer than the number of consumer types

in order to generate a lack of unanimity.

This emphasis on the finiteness of the number of firms makes the assumption of

perfect competition a little strained.

Relevance of Corporate Finance

One of the properties of the ADM model that does not transfer to incomplete

markets is the irrelevance of corporate finance decisionsthe Modigliani-Miller

theorem.

The fact that different consumers have different marginal rates of substitution

across states ensures that they will value the same security differently.

For this reason, the value of the firm will not be independent of the claims issued

against the production plan.

Suppose that the producer has chosen a production plan yj that will be sold in

equilibrium to consumers of type i.

Then the value of the shares in equilibrium will be 𝑞𝑗 = 𝜇𝑖 𝑥𝑖 ∙ 𝑦𝑗𝑖1.

The equilibrium conditions require that no other consumer type k is willing to pay

more for these shares, so that 𝑞𝑗 ≥ 𝜇𝑘 𝑥𝑘 ∙ 𝑦𝑗𝑖1, ∀k = 1 . . . , m.

Relevance of Corporate Finance

However, there may be a way of marketing tranches of to different clienteles in a

way that increases the total value of the firm.

For example, choose z1 and z2 to solve the problem:

It is always the case that ,

If :

the producer would do strictly better to split the firm-issue two claims, promising to

pay z1 and z2, respectively, and sell them to the consumer types who value them most

highly.

Relevance of Corporate Finance

If there were no costs of issuing new claims,

we would soon end up with effectively complete markets.

This kind of arbitrage is profitable as long as there are two types of consumers,

each of whom values a part of the firm more highly than any other type.

To preserve the incompleteness of markets, Allen and Gale (1988) assumed fixed

costs of introducing securities.

As long as the valuations of different consumers are not too different in

equilibrium, the process of introducing new claims will stop before markets are

complete.

Restriction on short sales

Note that also in this case the restriction on short sales is crucial to supporting this

kind of incentive to introduce securities.

If consumers are allowed to make unlimited short sales,

it will be possible to make an arbitrage profit as long as the value of the sum of the

claims is different from the value of the whole firm.

Suppose that some producers of type j issue two securities and some issue only

one (traditional equity shares).

The two classes of firms have different financial structures, but they cannot have

different values in equilibrium.

If they did, a speculator could buy a one-security firm, short the claims of the two-

security firm, and make a profit.

Relevance of Corporate Finance

Since the issuers of two securities need an increase in value to compensate for the

cost of the additional security, we cannot have an equilibrium in which different

producers choose different financial structures.

If a producer thinks that by issuing only a single security, he can take a free ride on the

others and get the same value,

we cannot have an equilibrium in which everyone chooses a two-security financial

structure.

There is something fundamentally wrong with the notion of a perfectly

competitive market with unlimited short sales when there are fixed costs of

issuing securities.

What lies at the bottom of the problem is the assumption of price-taking behavior.

Perfect competition (price-taking behavior) does not make much sense when

unlimited short sales are allowed.

Security Design

However small a single producer may be, he has a large effect on the market when

he introduces a new security if unlimited short sales are allowed.

Although the producer himself is providing a negligible amount of the security, by

allowing others to trade the security, he is introducing a new market that may have a

non-negligible effect on the equilibrium.

Although the new security is in zero net supply, the open interest may be non-

negligible.

In this case, it is possible to rescue the competitive model by assuming that short

sales are not allowed.

When short sales are not allowed, the open interest in the security is limited to the

amount supplied by the producer, which is a negligible amount if the producer is

negligible.

In this way, the effect of each producer remains negligible in equilibrium.

Security Design

Many markets have zero net supply by definition.

For example, in options markets there must be "short sales" in order to have any

trade at all.

In such cases, it is not clear how to rescue the competitive assumption.

Simply opening such a market is going to have a big effect, and we need a theory

of equilibrium with imperfect competition to account for this.

Modeling Imperfect Markets

Allen and Gale (1992) develop a two-stage model of security design with

imperfect competition.

In the first stage, firms simultaneously choose optimal financial structures.

In the second stage, competitive investors bid for the securities issued in the first

stage.

The equilibrium supply of securities is determined in a Nash equilibrium of the

first-stage game.

Then, taking the supply of securities as given and allowing unlimited short sales,

prices adjust to clear the competitive securities markets in the second stage.

Letting the number of firms increase without bound does not bring us back to

perfectly competitive pricing, however.

Imperfect competition is essential to allow firms to recover the cost of issuing

complex securities.

Modeling Imperfect Markets

The fact that the model of equlibrium with incomplete information is badly

behaved when we allow unlimited short sales and financial innovation

the fundamental incompatibility of the assumptions of perfect competition, in the sense

of ADM, and the factors that explain the incompleteness of markets in the first place.

Merely assuming that some markets are missing does not lead to a credible theory

of incomplete markets.

If we think about the other reasons that markets are incomplete, this observation

becomes quite obvious.

The category of transaction costs includes not just the cost of recording and

verifying the transaction, but the much more important costs of learning to make

optimal decisions and thinking about the optimal transaction in any particular

context.

Modeling Imperfect Markets

Given the uncertainty that attends any commitment to trade in the distant future

and the difficulty of collecting and analyzing information that might be relevant to

such transactions,

it is not surprising that the number of transactions that are undertaken for the distant

future is small. In fact, they appear to be limited to long-term debt and certain types of

supply contracts.

Similarly, the costs of making transactions contingent on unlikely and very

minutely defined states of nature are likely to outweigh the benefits.

These costs, of course, are present in markets that do exist and may be important

determinants of the kind and volume of trades that go on there.

Whether the markets exist or not, transaction costs are something that should be

taken into account in modeling them.

Adverse selection

It is well known that adverse selection may limit trade or even prevent a market

from operating at all.

Insurance markets and credit markets are prime examples.

What is true for these markets is clearly true for financial markets in general.

Corporate finance has taken on board the adverse selection problem, but it has not

had much impact on the theory of asset markets.

The important point is that markets with adverse selection behave very differently

from markets without adverse selection.

The GEI model, which assumes that some markets are missing but otherwise

maintains the assumptions of the ADM model, is not consistent with the presence

of adverse selection.

Asymmetric information and moral hazard

Even the rational expectations model, which at least introduces asymmetric

information, deals only with uncertainty about states of nature, not the

characteristics of individual goods or traders.

To account for any of these things requires a market with a different structure.

Many risks cannot be insured because of the attendant moral hazard problems.

It is hardly surprising that there are no private markets in which one can insure

against future unemployment or business failure, since these risks depend so

much on decisions made by the insured.

The same kinds of moral hazard affect the financial transactions that we do

observe, for example:

1. moral hazard on the part of the CEO of a company in which an investor has just

bought stock;

2. moral hazard on the part of the investment banker who is advising his purchase.

Markets that are affected by moral hazard will behave very differently from the

Walrasian markets of the ADM and GEI models.

Modeling market imperfections

Careful modeling of these problems is required to understand the uses and

limitations of financial markets.

Many contracts are introduced in futures and options exchanges only to disappear

after a few months because of lack of interest.

It takes a certain minimum volume of transactions to provide adequate liquidity

and cover the fixed costs of running a market.

These factors, which are important in explaining the absence of many markets,

also affect the operation of markets that do survive.

They need to be explicitly modeled if we are to understand the efficiency

properties of these markets.

Conclusions

The rationale for unregulated competitive markets is based on an ideal case,

formalized in the ADM model.

Once imperfections such as transaction costs and asymmetric information are

introduced,

the efficiency of the market mechanism is no longer guaranteed.

Regulation and other forms of government intervention are one answer to market

failures, but there are alternatives.

The development of fifnancial institutions such as intermediaries can take the

place of "missing markets," as can different ways of organizing the financial

functions of firms.

Conclusions

The limitations of the ADM model as a description of reality go beyond the mere

absence of markets, however.

The markets that do exist are characterized by many of the frictions listed.

The structure of these markets has been adapted to cope with transaction costs,

adverse selection, moral hazard, illiquidity, and fixed costs of participation.

Instead of trading a homogeneous commodity at a single market-clearing price,

these markets allow agents to negotiate contracts, often through intermediaries,

taking into account asymmetric information about the nature of the transaction.

In these "markets," information is exchanged, relationships are established,

bargaining and renegotiation occur, search takes place (i.e. - individuals are

matched), innovation and security design are undertaken, and institutional forms

are established.

References

Allen F., Gale D., Comparing Financial Systems, MIT Press,

2001, ch. 5.

Examination questions

What are the main assumptions in the ADM model and why

are they unrealistic?

What are the main results of the ADM model?

Describe the main features of the GEI model.

What are the main market imperfections we shoud introduce

in financial markets models?