Lecture 3: Further static oligopoly

7/27/2019 Lecture 3: Further static oligopoly

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Lecture 3: Further static oligopoly

Tom Holden

http://io.tholden.org/
http://io.tholden.org/http://io.tholden.org/


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Game theory refresher 2 Sequential games

Subgame Perfect Nash Equilibria

More static oligopoly: Entry

Welfare

Cournot versus Bertrand

Kreps and Scheinkman (1983)


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L R

K U K U

3:1 1:3 2:1 0:0

Source: Wikimedia Commonshttps://secure.wikimedia

.org/wikipedia/en/wiki/File:SGPNEandPlainNE_explainingexample.svg
https://secure.wikimedia.org/wikipedia/en/wiki/File:SGPNEandPlainNE_explainingexample.svghttps://secure.wikimedia.org/wikipedia/en/wiki/File:SGPNEandPlainNE_explainingexample.svghttps://secure.wikimedia.org/wikipedia/en/wiki/File:SGPNEandPlainNE_explainingexample.svghttps://secure.wikimedia.org/wikipedia/en/wiki/File:SGPNEandPlainNE_explainingexample.svg


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Idea: at any point in time, whatevers happened up tothat point, from that point forward people will playthe Nash equilibrium.

And players know this in advance.

A Nash equilibrium is a sub-game perfect Nashequilibrium (SPNE) if and only if at every node in thegame tree, the actions it specifies at that nodeconstitute a Nash equilibrium of the sub-game. Important: Must hold even for nodes that are never reached

in equilibrium!

Solve for SPNE by backwards induction. Start at the leaves (the last period) and work backwards.


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The Centipede game: Consider two players: Alice and Bob. Alice moves first. At the

start of the game, Alice has two piles of coins in front of her: onepile contains 4 coins and the other pile contains 1 coin. Eachplayer has two moves available: either "take" the larger pile ofcoins and give the smaller pile to the other player or "push" bothpiles across the table to the other player. Each time the piles of

coins pass across the table, the quantity of coins in each piledoubles. For example, assume that Alice chooses to "push" thepiles on her first move, handing the piles of 1 and 4 coins over toBob, doubling them to 2 and 8. Bob could now use his first moveto either "take" the pile of 8 coins and give 2 coins to Alice, or hecan "push" the two piles back across the table again to Alice,again increasing the size of the piles to 4 and 16 coins. The game

continues for afixe num er of roun s or until a player decidesto end the game by pocketing a pile of coins. Source:

https://secure.wikimedia.org/wikipedia/en/wiki/Centipede_game_%28game_theory%29
https://secure.wikimedia.org/wikipedia/en/wiki/Centipede_game_(game_theory)https://secure.wikimedia.org/wikipedia/en/wiki/Centipede_game_(game_theory)https://secure.wikimedia.org/wikipedia/en/wiki/Centipede_game_(game_theory)


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Suppose it is Alices turn and the fixednumber of pushes has expired. Then Alice has no choice but to take the big pile,

which is of size 8 where 2 is the size of the

smaller pile. Now think about what Bob would do the turn

before. At that point, the big pile is of size 4 and the small

pile is of size . So he can either take 4 now, or

wait and get 2 from Alice. Now think what Alice would do the turn

before that, etc.


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Two periods

In period 1, firms simultaneously decide whetheror not to enter the industry.

Those that enter pay a fixed cost of .

In period 2, firms set quantities or prices, andproduce.

We solve period 2 first, then find optimalbehaviour in period 1 given expectations of whatwill happen in period 2.


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Assume that all potential entrants have the samemarginal cost. And assume that monopoly profits( ) are bigger than the entry cost, but less

than infinity.

Suppose one firm decides to pay the entry cost. In the second stage it will choose the monopoly price,

and make an overall profit of .

Suppose more than one firm decides to pay the

entry cost. In the second stage all firms will set price equal to

marginal cost, and thus make an overall profit of (i.e.a loss).


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So only one firm will enter!

We get monopoly precisely because

competition would yield the competitiveoutcome.

When potential entrants have different

marginal costs, which will enter?


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When is welfare improved by the government payingthe entry cost for two firms?

When the DWL due to monopoly is greater than . Recall DWL is the area between the demand and the MC

curves for quantities between the monopoly one and thecompetitive one.

I.e. when

> where:

is marginal cost,

is the monopoly quantity

is the perfectly competitive quantity.

Exercise: simplify this condition in the special case of lineardemand ( = ).


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Assume iso-elastic demand = and thatall firms have constant marginal costs . So under Cournot: =

(shown in last weeks

exercise).

And =

Thus, firm production profits are:

=

1

= 1


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If production profits were less than , at leastone firm would want to deviate to not entering.Thus profits must be greater than .

But it must also be the case that if one extra firmentered, it would make a loss overall.

Thus is the largest integer such that

> .

For large markets, this means

.


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What happens to the number of firms in an industryas the size of the market increases?

Here measures the size of the market.

doubles,

demand doubles (at any price).

From the entry condition:

1

Thus, the number of firms grows more slowly that thesize of the market.


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Intuition: if price did not adjust as more firmsentered, then the entry condition would

always keep

constant.

But price is falling as the market gets larger,so firms are less keen to enter.

Thus firms are larger in larger markets. (Ageneral result for Cournot.)


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Campbell and Hopenhayn (2005) Regress average firm size in an industry on number of

firms and assorted controls. Find firms are larger in larger industries.

Bresnahan and Reiss (1991) Call the market size required to support exactly firms. We should have

>+

(I.e. to grow by one firm, market size needs to grow by a

larger amount than the number of firms.) They find this holds for small , but for 4,

+

.

Suggestive of attaining perfect competition.
http://onlinelibrary.wiley.com/doi/10.1111/j.0022-1821.2005.00243.x/abstracthttp://onlinelibrary.wiley.com/doi/10.1111/j.0022-1821.2005.00243.x/abstracthttp://onlinelibrary.wiley.com/doi/10.1111/j.0022-1821.2005.00243.x/abstracthttp://www.jstor.org/stable/2937655http://www.jstor.org/stable/2937655http://www.jstor.org/stable/2937655http://onlinelibrary.wiley.com/doi/10.1111/j.0022-1821.2005.00243.x/abstract


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Total social surplus (consumer + producer)

is:

where is the total

produced with firms.

We want to know if welfare would beincreased by adding more firms. So we differentiate welfare with respect to , which

gives:


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But at the equilibrium number of firms ():

, thus, the derivative of welfare w.r.t.

at is the value of the following expression,evaluated at = :

1 =

1

=

< 0

(as quantity produced by each firm is decreasing in thenumber of firms).

Thus welfare would be increased by decreasingthenumber of firms.


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Intuition: an entrant does not internalise thedamage it does to the desired quantityproduced by other firms. Business stealing effect.

Ceases to hold if new entrants are producingslightly different products.


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Bertrand: firms set prices, quantities adjust toclear the market. The software industry?

Cournot: firms set quantities, prices adjust toclear the market. Without a process of price-adjustment requires a

mysterious auctioneer to set prices.

Large manufacturing, e.g. cars, airplanes, etc?

Characterised by production quantity decisionsbeing performed in advance.


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Period 1: Firms invest in capacity.

Period 2: Firms compete in price subject tothe constraint that production is less or equalto capacity.

Result: like Cournot.

Kreps and Scheinkman (1983)
http://www.jstor.org/stable/3003636http://www.jstor.org/stable/3003636http://www.jstor.org/stable/3003636http://www.jstor.org/stable/3003636


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Firms 1,2 : Each with zero marginal cost for simplicity.

Firm cannot produce more than (exogenous). Firms choose prices .

Market demand .

Suppose firm 1 sets a price < at which

> . Then demand exceeds supply, so some rationingmust occur.


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Suppose consumers all want at most one unitof the good, but they have differentreservation prices.

Also suppose they leave the house to goshopping at a random time throughout theday.

If they leave late they will arrive at firm 1 tofind it has run out of stock.


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consumers want to buy the good at price. But only of them will be able to.

So, the probability of being rationed (and having

to buy from firm 2) is

.

Probability of a rationed consumer being

prepared to buy at is

.

So, residual demand facing firm 2 is:

=


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Residual demandfacing firm 2


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Is this efficient? Suppose Alice values the good at less than but

more than . If Alice is lucky, shell be able to buy the good at .

Suppose Bob values the good at more than . If Bob is unlucky though, he wont be able to buy it

at any price.

So, if Alice met Bob they would want to trade.


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Suppose instead that the consumers with thehighest valuations rush out to the shops first.

So the

consumers with the highestvaluations get to buy from firm 1.

Then the residual demand curve facing firm 2

is just (as is the number ofconsumers with valuations above , whichincludes the already served).


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Residual demandfacing firm 2


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Consider Alice and Bob again. Recall: Alice values the good between and . Bob values the good at more than .

Alice will not be able to buy the good. By the time she arrives the cheap store has sold

out.

Bob will be able to buy the good. Possibly even at .

Alice and Bob will not want to trade.


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Efficiency: Under efficient rationing the marginal consumer values

the good at the price it costs them. (All other consumerspay less than their valuation.)

Under proportional rationing some consumers pay lessthan their valuation.

Consumer surplus: Maximised by efficient rationing. (Consequence of

efficiency.)

Producer surplus: Firm 1 faces the same demand curve in either case. Firm 2 prefers proportional rationing.


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Assume efficient rationing. And linear demand = 1 .

Claim: both firms charging

= 1 is an equilibrium, providing

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Lecture 3: Further static oligopoly