Lecture 6Market design

This session begins by examining your working knowledge of strategic behavior in auctions and monopoly markets. You will participate as a subject in several experiments that require you to place bids and asks, make market orders, and in one instance set some capacity constraints before trading.

Then we will continue our analysis of competitive equilibrium which we started yesterday, focusing on institutional mechanisms that sometimes succeed, and sometimes fail, in inducing the competitive equilibrium allocation of resources.

Applying competitive equilibrium

When there are large numbers of traders on both sides of an exchange market, and there is no uncertainty about product quality, experimental outcomes from limit order markets are quite close to the competitive equilibrium allocations.

Alternatively if there are many traders on one side of the market, but only a small number on the other side, the few may have scope to capture rents through strategic play.

But what happens when there are only a small number of players on both sides of the market?

From bargaining to markets

When there are large numbers of traders on both sides of an exchange market, and there is no uncertainty about product quality, experimental outcomes from limit order markets are quite close to the competitive equilibrium allocations.

Alternatively if there are many traders on one side of the market, but only a small number on the other side, the few may have scope to capture rents through strategic play.

But what happens when there are only a small number of players on both sides of the market?

A framework

There are twice as many subjects as there are “markets”.

Each person is endowed with one unit of the “good” from one and only one “market”.

The economy is endowed with exactly two units of each “good”.

Each person makes offers to buy a good from another market and/or sell his own good.

Valuations

We consider four ways of calculating the value of the match to the buyer. It is:

1. the product of the indexed values of the market.

2. the squared difference of the indexed value of the two markets.

3. the inverse of the squared difference of the indexed value of the two markets.

4. The sum of the two markets

Competitive equilibrium and information

Competitive equilibrium economizes on the amount of information traders need to optimize their portfolios. Indeed a peculiar feature of competitive equilibrium is that in some situations it fully reveals private information to those who are less informed about market conditions.

Fully revealing prices

Consider an economy in which Condition A might or might not be true, and suppose only a fraction of the investing population have this inside information.

For example, suppose Condition A is: “The firm has won a major sales contracts, which will increase its dividend flow quite substantially.”

A competitive equilibrium reveals to everybody whether the condition is true or not, so that those without the inside information act as if they are fully informed when they trade.

Why does competitive equilibriumreveal private information? Suppose the uninformed cannot infer the

state of the world from the competitive equilibrium price.

Then the quantities they wish to trade cannot depend whether whether Condition A holds or not.

In this case only the demands and supply of the informed are affected by whether the condition holds.

If they are collectively large enough to exert an influence on the competitive equilibrium price, then it will depend on whether the Condition holds or not.

This contradicts the premise that the uninformed cannot deduce the state of the world from the competitive equilibrium price.

Therefore the only equilibrium can be those in which everyone knows whether the condition is true or not.

What does the experimental evidence show?

As we have seen this is not the case in a sealed bid auction with differential information, and it seems implausible in a double auction too.

We play several games in which the competitive equilibrium fully reveals private information.

Some examples

Private information about the limit order market book

Private information about the distribution of private valuations Information about preferences Information about idiosyncratic quality

Differential information about the value of the common shock Information about aggregate quality

Price discovery process

Consider the following example. There are two groups of players, respectively called the suppliers and demanders of an asset.

All demanders have the same valuation of the asset, either

Suppliers are endowed with a random amount of the asset, but the proportion with each is random too, so total supply is unknown.

Adding dimensions to uncertainty

The reason why the uninformed segment of the population can infer the true state of affairs in the example above is because there is a mapping between the shock and the resulting competitive equilibrium price.

In our last examples on this topic we introduce a second shock.

Those people who know one of the shocks can infer the other from the competitive equilibrium price. Those who know neither can only form estimates of what both shocks are from the competitive equilibrium price.

The value of insider trading

We have just shown that there is no value from having private information about the prospects of a firm in competitive equilibrium if there are no other sources of uncertainty in the market.

This argument implies insiders will not use up resources acquiring information about a firm’s prospects if there is only one source of uncertainty.

Only by recognizing multiple sources of uncertainty is there value from investing resources in market intelligence to gain inside information about the common value of an asset.

Financial MarketsThis final lecture turn our attention to the solution to limit order market games describing the financial sector. First we study arbitrage, showing the relationships between asset prices it implies. Then we analyze the additional predictions that a competitive equilibrium would impose on the predicted outcomes. To make further headway we discuss motivations for trade in financial markets. Portfolio diversification provides one reason for trade if traders are risk averse, and we explore the trade and pricing implications from risk sharing within competitive equilibrium. The last part of the lecture discusses other motivations for trade, and this leads us into an analysis of the tradeoffs between market and limit order submissions.

Financial Markets

In the final lecture for this course we turn to examine electronic limit order markets, which are amongst the fastest growing markets within the financial sector.

Instead of dealers mediating between buyers and sellers anyone in good standing can submit and sell limit and market orders.

What can we say about the portfolio management when financial assets are traded on a limit order market?

Arbitrage Pricing The optimal exploitation of arbitrage

opportunities puts bounds on the best prices quoted in the limit order book.

We say two bundles of securities are payoff equivalent if the difference between their payoffs at the end of the game is zero with unit probability, that is for (almost) all possible histories of returns.

For example, if one asset

Arbitrage Pricing The optimal exploitation of arbitrage

opportunities puts bounds on the best prices quoted in the limit order book.

We say two bundles of securities are payoff equivalent if the difference between their payoffs at the end of the game is zero with unit probability, that is for (almost) all possible histories of returns.

Arbitrage bundles of securities that offer the same payoffs should be traded.

It should not be possible to sell one bundle of securities and purchase another payoff equivalent bundle and make a net profit.

What can we say about the portfolio management when financial assets are traded on a limit order market?

Competitive equilibrium in limit order markets

A hallmark of competitive equilibrium is that one price (at any given time or in any given state) clears the market.

A second defining feature of competitive equilibrium is that markets clear. In other words all players trade as much as they wish at the competitive equilibrium price.

How are these features evident when a limit order market is in competitive equilibrium?

No limit order transacts at a noncompetitive equilibrium price

If a limit order market is in competitive equilibrium, no limit sell order placed above, and no limit buy order placed below, the competitive equilibrium price transacts. Therefore it is only optimal to place limit orders at the competitive price.

Hence a competitive equilibrium is compatible with books containing only outstanding buy orders, or containing only sell orders, but not containing both buy and sell orders simultaneously.

More generally, the bid ask spread is spuriously created and maintained entirely by nuisance orders that have no probability of execution.

All limit orders transact If a limit order market is in competitive

equilibrium, the only time a limit order placed at the competitive equilibrium price does not transact is when the trader is indifferent between execution and non-execution of his order.

Suppose a player who is indifferent between trading at the competitive equilibrium price versus keeping his current allocation is willing to place a market order at that price but not a limit order. Then all limit orders placed at the competitive price transact.

Recognizing competitive equilibrium

These rules provide a benchmark for recognizing when a competitive equilibrium attains.

If the spread is minimal, transactions at the best quotes execute with very high probability, and the probability of executing when a trader places a bid behind the spread declines steeply, then the market approximates a competitive equilibrium.

In this case the sequence of transaction price should be interpreted as the competitive equilibrium price (not the midpoint of the spread, which might be influenced by nuisance limit orders).

TradingThe arbitrage conditions derived above place restrictions on how assets are priced. But under what conditions will any trade take place?

Suppose players would value stocks the same way if they shared common information. This might happen if after the game ends (and possibly during it) the assets yield dividends plus some terminal value when the asset is disposed of.

Players are allocated different endowments of various stocks at the beginning of the game.

Initially we suppose that each player is an expected value maximizer.

Different information

Can differences in their information lead them to trade?

Note that less informed traders would be reluctant to trade at any price with a person who has more information, recognizing they would be exploited if they traded with a more informed trader.

For this reason we conclude that in such a setting no trade would take place.

This result is called a no-trade theorem.

When will trading occur?

The no trade theorem raises the question about when trading will occur.

There are two ways of relaxing this result:

1. Traders may have idiosyncratic reasons for holding some assets as opposed to others. For example incentives driven by corporate governance, and managerial compensation might lead to specialization in portfolios. Similarly Liquidity demands, perhaps related to

2. Traders may care about other moments of the probability distribution apart from the first.

Private values

Traders may have idiosyncratic reasons for holding some assets as opposed to others.

For example incentives driven by corporate governance, and

managerial compensation might lead to specialization in portfolios.

Similarly liquidity demands, perhaps related to different consumption demands might lead traders to realize their capital gains at different points in time.

Diversification

Traders might also care about other moments of the probability distribution apart from the first.

For example, suppose that traders are risk averse, rather than risk neutral.

In this case they would seek to diversify their portfolio.

Limit orders and market orders

Execution probabilities

Picking off risk.

Terms of trade

Theorem on order placement.

Some monotonicity restrictions

Competitive equilibrium under uncertainty

The second part of this lecture turns to portfolio management and asset pricing. Starting with the basic model of inter-temporal consumption smoothing, we derive the fundamental equation for determining how many and what kinds of financial assets to hold. This leads us into a discussion of the most widely used models in portfolio management, including CAPM and APT, theories that can readily be tested using experimental methods.

Demand for financial assets

Individuals and households hold wealth in financial securities to defer consumption.

For example parents save for the education of their children, individuals save for retirement, and the wealthy bequest future generations with their largesse.

About half of the value of the stock market is held by a very small fraction of individuals. Nevertheless more than 50 percent of households hold financial securities of some form or other.

Collectively, these groups, including foreign investors, create the demand for financial securities.

Maximizing present value

Consider the the lifetime utility of a consumer whose labor supply and wage income are determined outside the model:

where: is the period or year

is a subjective discount factor

is consumption in period t

is current utility at that time

1

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Random walk hypothesis

Consider the the lifetime utility of a consumer whose labor supply and wage income are determined outside the model:

where: is the period or year

is a subjective discount factor

is consumption in period t

is current utility at that time

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t tt cu

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Variance Bounds

Consider the the lifetime utility of a consumer whose labor supply and wage income are determined outside the model:

where: is the period or year

is a subjective discount factor

is consumption in period t

is current utility at that time

1

0

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t tt cu

tt

cu

c

Tt

0

10

1,,2,1,0

Financing Expenditures

Consider a model in which people are endowed with private values:

They trade with each other to

The No trade theorem

Consider a model in which people have different information about an asset they all value the same way.

Who will trade?

Diversification and risk sharing in a competitive equilibrium

Traders might also care about other moments of the probability distribution apart from the first.

For example, suppose that traders are risk averse, rather than risk neutral.

In this case they would seek to diversify their portfolio.

Insurance in a competitive equilibrium

First we look at how idiosyncratic risk is spread across traders (and when it breaks down).

Then we focus on pricing in asset markets more generally.

Insurance where there is no private information

First we look at how idiosyncratic risk is spread across traders (and when it breaks down).

Then we focus on pricing in asset markets more generally.

Markets where sellers have private information

Can an entrepreneur insure his new idea,or a person buy health insurance?

Consider a model in which there are many cars of different quality known only to the respective sellers.

Which ones will sell and for how much?

Universal health insurance handles the adverse selection problem but not the moral hazard problem.

Using financial markets to smooth consumption

Consider the the lifetime utility of a consumer whose labor supply and wage income are determined outside the model:

where: is the period or year

is a subjective discount factor

is consumption in period t

is current utility at that time

1

0

T

t tt cu

tt

cu

c

Tt

0

10

1,,2,1,0

Budget constraint

Suppose the person has assets, which can either consumed or invested in J financial securities:

where: is the amount of the jth security bought at the

beginning of period t

is the return on the jth security announced at the end of period t

J

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Maximization problem

Given her choices up until period t, and anticipating her future choices from period t+1 onwards, the consumer chooses consumption ct and her assets qtj to maximize:

subject to her period t budget constraint

where

is her information at time t

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A two period specialization

The essence of this problem is captured by the following 2 period model.

Suppose the investor allocates wt, her wealth in period t, between current consumption and securities which she will sell in period t+1 to support her consumption next period.

The investor chooses her assets qtj to maximize:

where:

is her wealth at time t

J

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First order conditions

The first order conditions for the original problem are:

where:

is the Lagrange multiplier associated with the choice in period t

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The fundamental equation for portfolio choice

Substituting out the Lagrange multipliers, we obtain the fundamental equation of portfolio choice for each asset j:

where:

is the marginal rate of substitution between consumption in period t and period t+1.

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The risk free rate

If a risk free (interest) rate called rt exists, it must satisfy the equation too:

or:

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1

Risk correctionsRecall that form its definition:

Using the formulas

we now obtain:

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1ttjt IrmE

A Beta Representation

Multiplying top and bottom by the the variance of mt and using the formula for the interest rate:

or

Note that tj isthe regression coefficient of the return the jth asset on mt.

We typically interpret as the price of risk and tjas the quantity of risk inherent in the jth asset.

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The mean-variance frontier

Multiplying top and bottom by the the variance of mt and using the formula for the interest rate:

or

We now obtain the inequality

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CAPMNoting that:

where rtw is the return on the market portfolio in period t, the Beta representation described in the previous slide simplifies to the familiar formula

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APTNoting that:

where rtw is the return on the market portfolio in period t, the Beta representation described in the previous slide simplifies to the familiar formula

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Testing the fundamental equation

Since the fundamental equation applies to each investor we can test the basic hypothesis

Notice that the test requires us to

1. take a time series average, not a cross sectional average.

2. look at each individual investor’s consumption, not the average consumption of all investors.

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Testing CAPM

We could also analyze the sample moments of CAPM

The test requires us to

1. take a time series average, not a cross sectional average.

2. calculate the return of each asset for each investor (who typically do not pay the same prices for identical securities in a limit order market)

tttw

ttw

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Ir

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,cov

Testing APT

We could also analyze the sample moments of CAPM

The test requires us to

1. take a time series average, not a cross sectional average.

2. calculate the return of each asset for each investor (who typically do not pay the same prices for identical securities in a limit order market)

tttw

ttw

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Ir

IrrrIrE

var

,cov