This lecture analyzes how well competitive equilibrium predicts industry outcomes as a function the of the production technology, the number of firms and the aggregation of information about supply and demand.

Lecture 4Evaluating Competitive Equilibrium

What happens as we increase the number of firms in the industry?

Up until now we have concentrated on situations where there is just on supplier (or auctioneer) and many demanders (or bidders), or where there is just one demander and many suppliers.

Suppose there are just two firms in the industry (and many demanders). We shall see that their market value depends on whether they compete on price, or on quantity.

Demand and TechnologyConsumer demand for a product is a linear function of price, and that market pre-testing has established:

We also suppose that the industry has constant scale returns, and we set the average cost of producing a unit at 1.

pq 13

Price competitionWhen firms compete on price, the firm which charges the lowest price captures all the market.

When both firms charge the same price, they share the market equally.

These sharp predictions would be weakened if there were capacity constraints, or if there was some product differentiation (such as location rents or market niches).

Profit to the first firmAs a function of (p1,p2), the net profit to the first firm is:

Net profit to the second firm is calculated in a similar way.

2111211 )1)(13(),( ppifpppp

2111 )1)(13(21 ppifpp

12 0 ppif

Market games with price competition

We could try to solve the problem algebraically.An alternative is to see how human subjects attack this problem within an experiment.We have substituted some price pairs and their corresponding profits into the depicted matrix.

Solving the price setting gameSetting price equals 7 is dominated by a mixture of setting price to 5 or 2, with most of the probability on 5.Eliminating price equals 7 for both firms we are left with a 3 by 3 matrix.Now setting price equals 5 is dominated by a mixture of setting price to 3 or 2.In the resulting 2 by 2 matrix a dominant strategy of charging 2 emerges for both players.

The algebraic solution to the price setting game

Notice that neither firm will offer the product at strictly less than cost, other wise they will make a loss.

Suppose one firm offers the product at a price p1 greater than cost. The best response of the other firm is to charge a price between cost and . This proves there is no Nash equilibrium in which either firm charges a price strictly more than cost.

Finally suppose one firm charges at price equals cost. A best response of the second firm is to do the same thing. This is the unique Nash equilibrium to this pricing game.

Quantity competitionWhen firms compete on quantity, demanders set a market price that clears inventories and fills every customer order.

If firms have the same constant costs of production, and hence the same markup, then their profits are proportional to their market share.

This predictions might be violated if the price setting mechanism was not efficient, or if the assumptions about costs were invalid.

Calculating profits when there is quantity competition

Letting q1 and q2 denote the quantities chosen by the firms, the industry price is derived from the demand curve as:

p = 13 - q1 – q2

When the second firm produces q2, as a function of its choice q1, the profits to the first firm are q1(13 - q1 – q2) - q1 = q1[12 - q1 – q2]

The profits of the second firm are calculated the same way.

Market games with quantity competition

As in the price setting game, we could try to solve the game algebraically, or set the model up as an experiment.If we can compute profits as a function of the quantity choices, using the second approach, we can easily vary the underlying assumptions to investigate the outcomes of alternative formulations.

The first order for a firm into the quantity setting game

When the second firm produces q2, as a function of its choice q1, the profits to the first firm are q1(13 – q1 – q2) - q1 = q1[12 – q1 – q2]

Maximizing with respect to q1, we obtain the first order condition [12 – 2q1 – q2] = 0

or [12 – q2] = 2q1

Solving the quantity setting game

In a symmetric equilibrium with both firms producing the same quantity this yields a solution with each firm producing 3q1 = 12 or q1 = 4.

More generally suppose there are N firms in the industry and they each produce the same amount.

Let q1 denote the quantity produced by any given firm and q2 denote the quantity produced by the remaining N – 1 firms.

The first order condition is still0 = 12 – 2q1 – q2 = 12 – 2q1 – (N - 1)q1

= 12 – (N + 1)q1

The competitive limitSolving we have that

0 = 12 – (N + 1)q1or q1 = 12/ (N + 1)

Since there are N firms in the industry this implies the amount supplied by the industry is Nq1 = 12N/ (N + 1)

Compare this with the competitive equilibrium quantity, 12, obtained be setting price equal to marginal cost.

Marginal costsConsider the following two examples in which:

1. There are declining costs for a single good

2. There are scale complementarities in a production mix.

Aggregating information

In the trading games described above, all traders had the same information, and trade occurs because of differences in endowments and preferences.

How is trading affected when there are differences between traders in their information?

With respect to competitive equilibrium:1. How much does competitive equilibrium

reveal of the knowledge demanders and suppliers have?

2. Are the predictions of competitive equilibrium theory accurate?

The no-trade theoremSuppose people have differential information about an asset they would all value the same way if they were fully informed.Will any trade take place?Note that if one trader party benefits from the trading then the other party must lose.Since all traders anticipate this, we thus establish that no trade occurs in competitive equilibrium, or for that matter any other solution to a (voluntary) trading mechanism, because of differences in information alone.

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.

Private valuations in an endowment economy

A simple example illustrating how competitive equilibrium aggregates information is in a market where consumer valuations are identically and independently distributed, and aggregate supply is fixed.

Suppose no demander wants to consume more than one unit of the good, and each demander draws an identically and independently distributed random variable that determines their valuation for the first unit.

Demanders are said to have private valuations.

Competitive equilibrium in endowment economies

The competitive equilibrium price does not depend on whether each trader observes the valuations of the others.

Hence every trader acts the same way as he would if he were fully informed about aggregate demand.

We compare two markets, one in which demanders know the private valuations of everyone, and the other in which they don’t.

Fully revealing prices about a common shock

Suppose there are N traders, and the nth trader receives a signal sn about the the state of the economy, where n = 1,2, . . ., N.In general, the competitive equilibrium price vector p depends on all the signals the traders receive.If, however, p is an invertible mapping of all the relevant information s available to traders, then every trader acts the same way as he would if he were fully informed.In this case p(s) has an inverse which we call f(p). Each trader realizes that seeing p is as good as seeing s.In the example above s is aggregate demand, and p is monotone increasing in s.

Differential information about product quality

Now suppose a component of each demander valuation is common, and traders have differential information about that component.

The more favorable the signal to the informed traders, the greater is their demand, and hence the higher is the market clearing price.

As in the previous example, uninformed traders compute their demands, deducing that if the market clearing price is p, then the common component is f(p). Thus informed traders cannot benefit from their superior information in competitive equilibrium.

Implications for trading mechanisms

Some economists have used this theoretical result to argue that markets are good at aggregating the information that traders have about the preferences of demanders and the technologies of suppliers.

Other economists have argued there is limited investment in acquiring new information relevant to suppliers and demanders, because those who use up resources to become better informed cannot recoup the benefit from their private information.

Both arguments implicitly assume that a competitive equilibrium accurately predicts price and resource allocations from trading.

When do competitive equilibrium prices hide information?

There are two scenarios when competitive equilibrium prices are not fully revealing:

1. The mapping from signals to prices p(s) is not invertible. That is, two or more values of a signal, s1 and s2, would yield the same fully revealing competitive equilibrium price if everyone observed the signal’s value, meaning pfr(s1) = pfr(s2).

2. Different units of the product are not identical, although they are traded on the same market, and these differences are observed by some but not all the traders.

A comparative study

We start out with two shocks and explore what happens if:

1. both shocks are observed by everyone

2. only one shock is observed by some of the traders

3. both shocks are observed by somebody but nobody observes both shocks

4. someone observes both shocks and the others observe nothing.

Adding dimensions to uncertainty

The uninformed segment of the population can infer the true state in the previous example because a mapping exists from the competitive equilibrium price to the shock defining the product quality.

In our next example we introduce a second shock.

Those traders who observe one shock can infer the other from the competitive equilibrium price.

Those who observe neither can only form estimates of what both shocks are from the competitive equilibrium price.

Uncertain supply and qualityNow suppose that product quality is only known by some of the demanders, and the aggregate quantity supplied is also a random variable.

In this case an uninformed demander cannot infer product quality from the competitive equilibrium price, because a high price could indicate high demand from informed traders or low supply.

Informed demanders benefit from the fact that demand by uninformed traders is less than it would be if they were fully informed when product quality is high, depressing the price for high quality goods, and vice versa.

Differential information about heterogeneity across units

Suppose the quality of the individual units varies, and that traders are differentially informed it. What would competitive equilibrium theory predict about the price and quantity traded?

Since traders condition their individual demand and supply on their information, more informed traders gain at the expense of the less informed.

The prospect of being exploited by a well informed trader discourages a poorly informed player from trading.

The market for lemonsFor example, consider a used car market.

Suppose there are less cars than commuter traders, and no one demands more than one car.

The valuation of a trader for owning one car is identically and independently distributed across the population.

The quality of each car is independently and identically distributed across the population. Each owner, but no one, else knows the quality of his car.

The amenity value from car travel is the product of the commuter’s valuation and the quality of the car he owns.

Lecture summaryAs the number of firms in a constant cost industry increases, there is convergence to the competitive limit regardless of whether firms compete on price or quantity.

When there are declining unit costs or scope economies, competitive behavior is not sustainable, and price is not equated with marginal cost in equilibrium.

We competitive equilibrium with market outcomes when:

1. private valuations affect a subject’s willingness to trade but not the distribution of demand.

2. differences in individual valuations are solely attributable to the information about the product

3. heterogeneity in product quality is not observed by both sides of the market