13. Game Theoretical Models of Economic Competition
First, we consider a model of a monopoly.
It is assumed that the monopoly chooses the quantity of a product
q to produce in a season.
The costs of production are cT (q).
The price resulting from the demand curve is p = g (q) = d −1 (q).
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Model of a Monopoly
The monopoly chooses the level of production, in order to
maximize profits.
The profit function is given by z(q) = pq − cT (q).
The optimal level of production can thus be found by
differentiation.
Note that in the case of a monopoly which knows the demand
curve, choosing the price to sell at is equivalent to choosing the
quantity to sell.
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The Symmetric Cournot Game
In the case of competition, choosing the price at which to sell is
not equivalent to choosing the amount to produce, since given the
amount produced by one firm, based on classical theory, the price
still depends on the amount produced by other firms.
Assume that two firms produce an identical good. Firm i produces
qi units per season.
The price of the good is determined by total supply and all
production is sold at this ”clearing price”. It is assumed that
p = A − B[q1 + q2 ], (A, B > 0).
The costs of production are assumed to be cT (q) = kq for both
firms, i.e. constant marginal costs of production.
The payoff of a firm is taken to be the profit obtained (i.e. profit
per unit times the number of units sold).
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The Symmetric Cournot Game
The payoff obtained by Firm 1 is given by
R1 (q1 , q2 ) = q1 [A − Bq1 − Bq2 − k] = q1 (A − k) − Bq12 − Bq1 q2 .
By symmetry, the payoff obtained by Player 2 is
R2 (q1 , q2 ) = q2 [A − Bq1 − Bq2 − k] = q2 (A − k) − Bq22 − Bq1 q2
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Best Response Functions
In this case, the optimal level of production of a firm depends on
the amount that the other firm produces.
Given the amount produced by the other firm, we can calculate the
optimal output of a firm using calculus.
It can be shown that any feasible extremum of these profit
functions (i.e. where both the quantity produced and the price are
positive) are maxima.
Let B1 (q2 ) denote the best response of Player 1 to q2 .
Let B2 (q1 ) denote the best response of Player 2 to q1 .
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Nash Equilibrium of the Cournot Game
At a Nash equilibrium (q1∗ , q2∗ ), we have
q1∗ = B1 (q2∗ ); q2∗ = B2 (q1∗ ).
Thus, at a Nash equilibrium Player 1 plays her best response to
Player 2’s strategy and vice versa.
The Nash equilibrium is a solution of the following system of two
linear equations.
∂R1
= 0;
∂q1
∂R2
=0
∂q2
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Example 13.1
Suppose that the demand function for a particular good is given by
p =3−
Q
,
1000
where Q is total production and the unit cost of production is 1.
i) Find the optimal level of production, price and profits of a
monopoly.
ii) Find the equilibrium levels of production, price and profits of
two identical firms producing the same good.
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Example 13.1
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Example 13.1
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Example 13.1
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Example 13.1
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The Stackelberg Model - Moves Made in Sequence
This is identical to the Cournot model, except that it is assumed
that one of the firms is a market leader and chooses its production
level before the second firm chooses.
The second firm observes the production level of the first.
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The Stackelberg Model
Suppose Player 2 moves after Player 1 and observes the action
taken by Player 1. The equilibrium is derived by recursion.
Player 2 should choose the optimum action given the action of
Player 1.
Hence, we first need to solve
∂R2 (q1 , q2 )
= 0.
∂q2
This gives the optimal response of Player 2 as a function of the
strategy of Player 1, q2 = B2 (q1 ).
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The Stackelberg Model
We now calculate the optimal strategy of the first player to move.
If Player 1 plays q1 , Player 2 responds by playing B2 (q1 ).
Hence, we can express the payoff of Player 1 as a function simply
of q1 , i.e. R1 (q1 , B2 (q1 )).
In order to find the optimal action of Player 1, we differentiate this
function with respect to q1 .
Having calculated the optimal value of q1 , we can then derive q2 .
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Example 13.2
Derive the equilibrium price, profits and production levels in the
Stackelberg version of the previous example.
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Example 13.2
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Example 13.2
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Example 13.2
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Example 13.2
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The Bertrand Model of Price Competition
Under the assumptions of the Bertrand model, two firms decide
the price at which to sell an identical product.
There is assumed to be no friction on the market (i.e. no costs
incurred in finding the cheapest offer).
Under this assumption, when the firms sell at different prices, the
firm selling at the lowest price satisfies all the demand at that
(lower) price.
If the firms sell at the same price, then the demand is split equally
between them.
The unit cost of production is k in both firms.
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The Bertrand Model
Given that firm 1 sells its product at a price of p1 > k (i.e. at a
price that would give a profit), it would always pay firm 2 to sell its
product at a minimally lower price in order to steal all the demand.
This fact leads to a price war, where the price is driven down to
the unit cost of production.
Both firms sell their good at the same price as the production
costs and hence neither firm makes a profit (or loss).
In the case where one firm (assumed to be firm 1) has lower
production costs than the other (firm 2), then firm 1 lowers the
price to below the production costs of firm 2, which then drives
firm 2 out of business.
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Comparison of Bertrand and Cournot Model
The results from the Cournot model seem a lot more reasonable
than the results from the Bertrand model. However, various
assumptions and changes to the Bertrand model can be made to
make the conclusions more reasonable, namely:
1. The conclusion ”neither firm makes a profit” should
be interpreted as both firms make a ”reasonable
profit, but small enough that no-one else wishes to
enter the market”.
2. There is friction on the market, which ensures that
individuals do not always find the cheapest version of
the product.
3. There is some brand loyalty. Hence, even when there
is no market friction, individuals would not always
buy the cheapest version of the product.
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Comparison of Bertrand and Cournot Model
Assumptions 2 and 3 lead to a model at which the equilibrium
prices are above the unit cost of production, hence both firms
make a profit.
In reality, firms choose both price and the level of production.
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Taxation and Duties Revisited
Although the Stackelberg model may seem unnatural when
modelling competition between firms, especially when talking about
firms of similar size, it is fairly clear that value add tax (VAT) and
duties fall naturally into the frame of the Stackelberg approach.
Firstly, the government announces the introduction (or change) of
a tax.
A firm (or firms) then reacts to this.
It follows that the government is a natural Stackelberg leader in
such games.
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Taxation and Duties Revisited
It is natural to assume that a firm (assumed to be a monopoly)
wishes to maximise its own profits.
The goals of the government may be more difficult to model.
In the simplest case, one may assume that the government wishes
to maximize revenue from the introduction of the tax, i.e.
maximize tq, where t is the tax per unit.
However, the introduction of a tax will have knock on effects on
the economy as a whole (and thus the income of the government
from other sources).
More generally, one can assume that the government also takes the
profits of a firm and consumer surplus into account.
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Taxation and Duties Revisited
Since the firm is assumed to be a monopoly, we may assume that
choosing the level of production is equivalent to choosing the price.
Assuming that the demand curve is linear (or the tax is small), the
firm should increase the price of the good by 50% of the value of
the tax (see next example).
Knowing this, the government can then choose the level of tax in
order to choose the optimal level of tax according to the
appropriate criterion.
Note that when the demand curve is linear, the selling price is ps ,
the maximum price (the price required for demand to disappear) is
pmax and the demand at equilibrium is q(ps ), then the consumer
surplus is 0.5q(ps )[pmax − ps ].
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Example 13.3
Suppose that the demand function for a particular good is given by
q = 3000 − 1000p,
when the unit cost of production is 1.
i) Show that imposing a tax of δ per unit, increases the selling
price by 0.5δ.
ii) Assume that the government wishes to maximize the revenue
from a tax on this product. What should the level of this tax per
unit be?
iii) Assume that the government wishes to maximize the sum of
the revenue from the tax and the consumer surplus. What should
the level of this tax per unit be?
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Example 13.3
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Example 13.3
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Example 13.3
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Example 13.3
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Example 13.3
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