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Static games - cont
I. Cournot model of duopoly
Let q1 and q2 denote quantities of a homogeneous product produced by firms 1 and
2 respectively.
Let P (Q) = a − Q be the
inverse linear-demand function, where Q =
q1 + q2. Total cost to firm i for producing qi
is Ci(qi) = cqi; there are no fixed costs. We
assume c < a. Suppose both firms choose
their quantities simultaneously.
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We have a normal-form game with I = {1, 2},
Si = [0, ∞) (Since P (Q) = 0 for Q ≥ a, neither firm will produce more than a). Assume
each firm’s payoff is simply its profit:
πi(qi, qj ) = qi[P (qi + qj ) − c] =
qi[a − (qi + qj ) − c]
In a 2-player normal-form game, the strategy pair (s∗1, s∗2) is a Nash equilibrium if for
every player i, ui(s∗i , s∗j ) ≥ ui(si, s∗j ) for every
feasible strategy siSi.
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In the Cournot duopoly model, the equivalent problem becomes:
max πi(qi, qj∗) = max0≤qi<∞
qi[a − (qi + qj∗) − c]
Assuming qj∗ < a − c, the FOC for firm i0s
optimisation problem is:
∗
qi = 1
2 [a − qj − c]
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If (q1∗ , q2∗ ) is to be a Nash equilibrium, then
firms’ quantity choices must satisfy:
1 [a − q ∗ − c];
q1∗ = BR1(q2) = 2
2
∗
q2∗ = BR2(q1) = 1
2 [a − q1 − c]
Solving these equations, we get
q1∗ = q2∗ = (a−c)
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Question: why is each firm producing half
the monopoly quantity, Q2m , where Qm =
(a−c)
2 , not an equilibrium of this game?
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II. Bertrand model of duopoly
Consider a duopoly market structure for a
homogeneous good, where firms choose what
price to charge. Let the demand function
facing firm i be:



 a − pi , p i < p j
0, pi > pj
qi =


 (a−pi) , p = p
i
j
2
Assume no fixed costs and constant marginal
costs c, with c < a.
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We have a normal-form game with I = {1, 2},
Si = [0, ∞), with a typical strategy si being
a price choice pi ≥ 0.
As for the Cournot model, assume each firm’s
payoff is simply its profit:
πi(p
i, pj ) = qi(pi, pj )[pi − c] =


(a − pi)[pi − c], pi < pj
0, pi > pj


 (a−pi) [p − c], p = p
i
i
j
2
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There are four possible equilibrium configurations:
1. p1 > p2 > c. This is not an equilibrium.
Firm 1’s sales and profits are both zero.
Firm 1 could profitably deviate by setting
p1 = p2 − τ , where τ is very small. Then,
π1 = D(p2 − τ )(p2 − τ − c) > 0 for small τ .
2. p1 > p2 = c. This is not an equilibrium.
Firm 2 captures the entire market, but its
profits are zero. Firm 2 could profitably deviate by setting p2 = p1 − τ , where τ is very
small. Then, π2 = D(p1 − τ )(p1 − τ − c) > 0.
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3.
p1 = p2 > c.
This is not an equilib-
rium since either firm (say, firm 1) could
profitably deviate by setting p1 = p2 − τ .
Then, firm 1would capture the entire market, with sales of D(p1 − τ ) and profits of
π1 = D(p1 − τ )(p1 − τ − c). For small τ this
almost doubles firm 1’s sales and profits.
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4. p1 = p2 = c. These are the (unique)
Nash equilibrium strategies.
Neither firm
can profitably deviate and earn greater profits even though in equilibrium, profits are
zero. If a firm raises its price, its sales fall to
zero and its profits remain at zero. Charging a lower price increases sales and ensures
a market share of 100%, but it also reduces
profits since price falls below unit cost. Discontinuities in the profit function make price
competition very sharp.
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The Nash equilibrium to this simple Bertrand
game has two significant features:
1. Two firms are enough to eliminate market power.
2. Competition between two firms results
in complete dissipation of profits.
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These features are the foundation of the
Bertrand paradox: two firms are sufficient
for the competitive outcome. However, marginal
cost pricing as a Nash equilibrium is not robust to variations in the Bertrand game.
Two major variants are product differentiation and capacity limitations.
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II. Bertrand model - differentiated
products
Here, firms 1 and 2 choose prices p1 and p2
simultaneously, and firm i0s demand is given
by:
qi(pi, pj ) = a − pi + bpj ,
where b > 0 reflects the extent to which firm
i0s product is a substitute for firm j 0s product. Assume no fixed costs and constant
marginal costs c, with c < a.
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We have a normal-form game with I = {1, 2},
Si = [0, ∞), with a typical strategy si being
a price choice pi ≥ 0. Again, each firm’s
payoff is simply its profit:
πi(pi, pj ) = qi(pi, pj )[pi − c] =
[a − pi + bpj ][pi − c]
Solution to the firm’s maximisation problem
is:
∗ + c)
p∗i = 1
(a
+
bp
j
2
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Using symmetry, we get the Nash equilibrium p∗1, p∗2 as:
p∗1 = p∗2 = a+c
2−b
Location/Address models
Address models of product differentiation
assume that consumers have preferences defined over the characteristics or attributes of
products that are measurable.
Each attribute defines a dimension and the
measure of all relevant attributes defines a
product.
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The set of all possible products is called the
product space and the total number of attributes defines the dimension of the product space.
In general, let θi be the address of brand i. If
the number of attributes or characteristics
is either one (a line) or two (a plane), then
an address model of product differentiation
is identical to a model of firm location in
physical or geographic space.
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Consumer preferences are distributed in the
same product space.
The address of a consumer represents their
most preferred product.
Tastes are heterogeneous since different consumers have different addresses.
Consumers have completely inelastic demands
in that they will purchase a single unit of
only one brand.
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The utility of a consumer located at address
θ∗ who purchases brand i is
U (θ∗, θi) = V − T (D) − pi,
where D = |θ∗ − θi| is the distance between
the address of the consumer and the address
of brand i, pi is the price of brand i, and V is
the consumption benefit of the consumer’s
ideal product.
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If the consumer’s ideal product is available
and chosen, then D = 0.
If not the consumer must incur either transportation costs (if the model is one of firm
location) or so-called mismatch costs - their
consumption benefits are reduced by the extent to which the product actually consumed
differs from their most preferred product.
The (dis)utility cost or transportation cost
is given by the function T (D).
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Mismatch and transportation costs are assumed to be strictly increasing in distance.
Common specifications: (a) linear in distance, T (D) = kD where k is the transport
cost per unit of distance; or (b) quadratic
in distance, T (D) = kD2.
Brands that are closer to the consumer’s
ideal are preferred. However, a more distant
less preferredbrand might be purchased if it
has a lower price.
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IV. Hotelling competition
Suppose initially there is a fixed number of
firms, equal to N.
Assume that marginal production costs are
c for all products and less than the price, p.
Since we have fixed prices, competition between firms will be over locations or product
specification.
Our objective is to identify the equilibrium
set of locations or products.
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A game-theoretic interpretation is that firms
simultaneously select their location and our
task is to find the Nash equilibrium in locations.
Alternatively, we could assume that firms
enter sequentially and they can costlessly
relocate. We then look for the set of locations such that firms do not have an incentive to relocate given the locations of their
rivals.
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Sets of locations that meet this criterion
will be the same as the Nash equilibria to
the simultaneous location game.
The profits of firm i are simply πi = (p −
c)M li,
where li is the market length, or interval,
served by firm i.
The market interval of i is the length of
the line segment defined by consumers who
purchase from firm i.
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Since M is the density of consumers, M li is
total sales of firm i.
Given that prices are fixed and greater than
the marginal cost of production, firms maximise their profits by maximising sales. Maximising sales requires that firms maximize
their market lengths.
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Suppose that firm i is located at θi. Its two
nearest competitors are located at θi−1 and
θi+1.
Firm i is an interior firm since its market
on either side is limited by the presence of
competitors.
Let x be the location of the consumer between firm i and i − 1 who is just indifferent
between the two firms.
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All consumers in the interval between x and
θi i to that of i − 1. The address of the
marginal consumer x is defined by:
x − θi−1 = θi − x, or x =
θi +θi−1
2
x is half of the distance between firm i and
i − 1.
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Similarly, if y defines the location of the consumer between firm i and i + 1 who is just
indifferent between the products of i and
i + 1, then
li = x + y, or
li =
θi+1 +θi−1
2
Firm i0s interval = (1/2) the distance between nearest competitors on either side. #
consumers gained to the right from moving
toward θi+1 = # lost to the firm at θi−1 on
the left.
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What about peripheral firms? The boundary of their market segment to one side is
determined not by competition with another
firm, but by the end of the market.
The market length of firm i − 1 to its left
equals θi−1, while its interval on the right is
of length (θi − θi−1)/2 - halfway between it
and its right-hand-side competitor i.
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The entire market length of firm i − 1 is the
sum of its LHS and RHS intervals,
li−1 = θi−1 +
θi +θi−1
θi −θi−1
=
2
2
which is increasing in θi−1.
The market share of a peripheral firm increases as it moves closer to its interior
competitor.
It gains sales in the interior,
without losing sales on the periphery.
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Define a firm’s half-market length by its
market length on one side. Then if locations are chosen simultaneously, two conditions must be satisfied for a Nash equilibrium in location:
1. No firm’s market length is less than any
other firms half-market length.
2. The two peripheral firms must be paired.
The peripheral firms must be adjacent to an
interior firm.
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The result that two firms in either product
or geographic space will locate in the middle is often referred to as the ‘principle of
minimum differentiation’ (Boulding).
The principle does not strictly hold when
there are more than two firms.
However,
even when there are more than two firms,
the equilibrium market configuration is characterized by “bunching”.
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