Managerial Economics
Game theory and competition
The goal of game theory is to predict the outcome when
multiple players have conflicting interests and goals. This is
done by understanding what factors can give an edge to a
player or affect the outcome. And by identifying
pitfalls(asymmetric info, opportunistic behaviour, …) that
undermine desirable outcomes and how to mitigate them. In
game theory we assume the players are individually rational
and play to maximize their payoff.
We start by looking at the Strategic game. This is played as a one-shot game where players make a
simultaneous choice (we observe the choice at the end together). This is called a static game. And
there is complete information, meaning that all players know the capabilities of all players and know
the payoffs of all players/strategies. But this doesn’t mean
certainty, players know with what chance a situation can occur.
So there is still uncertainty.
A strategic game can be fully described as a set of players, with a
set of actions and their respective payoffs.
Example:
Here we can say that strategy H is strictly dominated by D. Therefore we can
eliminate strategy H as a rational option. Player 2 knows this and will only use
the reduced game. Here strategy X strictly dominates both Y and T. Player 1
knows this and will now chose strategy D which gives an equilibrium in
dominant strategies <D,X>.
An equilibrium in dominant strategies gives a very reliable prediction, but in
most games you won’t find one. In those cases a Nash equilibrium comes in handy. A Nash
equilibrium is a strategy profile such that none of the players can increase their payoff by choosing a
different strategy, taking the other players strategies for given.
Example:
In this example we can find the Nash equilibrium by looking at the best option for
each player given the strategy of the other player. The best options are shown
and it is easy to see that the only Nash equilibrium is <X,A>.
A market demand function shows how many products are bought for a certain price point. The slope
of the curve shows us the change in sold goods for a given price change. This is called the price
dq
∗p
elasticity of demand =
ε ( p ) =(
q
)/( )
∆q ∆ p
p
=
dp
q ( p)
. For normal goods this will always be negative.
For |ε|>1 = demand is elastic and |ε|<1 = demand is inelastic.
Example: calculate price elasticity of demand for demand curve given by q(p) = 10/p
Derivative = -10/p^2 use in formula
, dq
∗q
ε ( p) =
dp
d ( p)
=
( )(
−10
p
2
∗
p
10
)
=−1
p
We will only look at linear market demands (ex. q(p)=a-bp). It is important to remember that ε(p) is
not automatically = b. It will sometimes be convenient to express the inverted of this function
(ex. p(q)=a/b – 1/b * q). A important point on this curve is the maximum willingness to pay (WTP)
for customers. This is represented by the point A.
Perfect competition is represented as a horizontal line with a constant price ͞p. Infinite suppliers will
provide goods at this price and consumers are willing to pay this price or less. The ε = -∞, for any q,
demand is infinitely elastic.
The single-agent problem is when each individual firm is too small to affect demand.
Each firm will produce for the price ͞p given their cost function C(q) to maximize their
profits.
Max{π(q) = ͞p*q – C(q)} This is when C’(q) = ͞p the moment an extra product
costs the same as the selling price. If the price is sufficiently high enough, there is a
possibility for profits. But when the market also has free entry, there will be no
profits. Because p ͞ will drop to the ATC = average total cost.
When a monopolists sets the price, we will use the same formula
Max{π(q) = p*q – C(q)} but the outcome will be different because we don’t use
͞p. The price is above the marginal cost and dependant on the price elasticity. The
mark-up of the monopoly can be expressed as the Lerner index. With the price = monopoly
price = pM. When q(p) and C(p) are both linear equations, the monopoly price can be found
more easily using the following equation.
In a Bertrand duopoly there are 2 firms producing a homogeneous product, both have the
same cost function and choose their price simultaneously. Customers have a demand and
will only buy from the lowest priced firm or split their demand equal between both if they set an
equal price. A final rule is that firms will only produce and have costs when/if units are sold. For
simplicity we will assume that C’(q) = c, q = a-bp and the firms demand
and profit for their price is given by the following equations.
If p1 is the monopoly price, both firms will have half
the profits. But it is in each firms interest to undercut
, the price to land on the green curve. This undercutting can go up to point p=c. When p<c the firm will
lose money.
The best decisions for each firm are laid out here, with ε being a small discount to land on the green
curve. So the price we will reach in an equilibrium will be p=c. This is a Nash equilibrium, which you
can reach by taking
the intersection of the reaction curves.
No firm will have profit because p = c = C’. This means that 2 firms create the same situation as in
perfect competition. But in reality firms do make a profit for a variety of reasons (no perfect
homogenous products, distance, asymmetric info, …). These conditions make that firms still have
market power.
The cross price elasticity of demand is the elasticity between 2 products. When ε ij>0 =substitute
∆ qi
∗p
goods and εij<0 =complement goods. And ∆ pj j.
ε ij=
qi
When we consider firms selling differentiated goods. The linear demand for their product can be
expressed as qi = Ai + bi1pj + bi2pj + … + + biipj + … + biNpj where bii = own price coefficient of product I
(for normal goods always negative) and bij = are the cross-price coefficient of product i for price j.
Example: take 2 firms with the demand for their products represented as
followed.
q1 = A1 + b12p2 + b11p1
q2 = A2 + b21p1 + b22p2
The prices in the Nash equilibrium are found using the formula to
maximize profit and are shown here to the right. When the b’s are equal
to 0, the price is equal to the monopoly price because the goods are so
different that they are nog substitutes or complements.
price 1
One way to measure a firms market power is mark-up = μ= =
marginal cost 1−LI
Paper 1:
Mark-ups where stable in the US until the 1980’s. Since than there has been an incline in mark-ups
and profitability of firms. But this is not for all companies. The median has stayed relatively the same,
but the upper percentile (superstar firms) has seen an dramatic increase. In the article they do
research in fixed and overhead cost. An increase in fixed cost will bring with it an increase in mark-up
to cover these costs. That’s why they also look into the profitability of the firms, which has also
increased for these top firms. The consequences are that a few big company’s hold a lot of power,
labour-shares have decreased and it is harder for small firms to enter and stay alive in those markets.
This trend has been partially in the decrease of marginal cost and an increase in fixed cost (Spotify vs
cd’s). There needs to be a balance between what the tech advances need and what a low entry
barrier needs.
Product differentiation
Game theory and competition
The goal of game theory is to predict the outcome when
multiple players have conflicting interests and goals. This is
done by understanding what factors can give an edge to a
player or affect the outcome. And by identifying
pitfalls(asymmetric info, opportunistic behaviour, …) that
undermine desirable outcomes and how to mitigate them. In
game theory we assume the players are individually rational
and play to maximize their payoff.
We start by looking at the Strategic game. This is played as a one-shot game where players make a
simultaneous choice (we observe the choice at the end together). This is called a static game. And
there is complete information, meaning that all players know the capabilities of all players and know
the payoffs of all players/strategies. But this doesn’t mean
certainty, players know with what chance a situation can occur.
So there is still uncertainty.
A strategic game can be fully described as a set of players, with a
set of actions and their respective payoffs.
Example:
Here we can say that strategy H is strictly dominated by D. Therefore we can
eliminate strategy H as a rational option. Player 2 knows this and will only use
the reduced game. Here strategy X strictly dominates both Y and T. Player 1
knows this and will now chose strategy D which gives an equilibrium in
dominant strategies <D,X>.
An equilibrium in dominant strategies gives a very reliable prediction, but in
most games you won’t find one. In those cases a Nash equilibrium comes in handy. A Nash
equilibrium is a strategy profile such that none of the players can increase their payoff by choosing a
different strategy, taking the other players strategies for given.
Example:
In this example we can find the Nash equilibrium by looking at the best option for
each player given the strategy of the other player. The best options are shown
and it is easy to see that the only Nash equilibrium is <X,A>.
A market demand function shows how many products are bought for a certain price point. The slope
of the curve shows us the change in sold goods for a given price change. This is called the price
dq
∗p
elasticity of demand =
ε ( p ) =(
q
)/( )
∆q ∆ p
p
=
dp
q ( p)
. For normal goods this will always be negative.
For |ε|>1 = demand is elastic and |ε|<1 = demand is inelastic.
Example: calculate price elasticity of demand for demand curve given by q(p) = 10/p
Derivative = -10/p^2 use in formula
, dq
∗q
ε ( p) =
dp
d ( p)
=
( )(
−10
p
2
∗
p
10
)
=−1
p
We will only look at linear market demands (ex. q(p)=a-bp). It is important to remember that ε(p) is
not automatically = b. It will sometimes be convenient to express the inverted of this function
(ex. p(q)=a/b – 1/b * q). A important point on this curve is the maximum willingness to pay (WTP)
for customers. This is represented by the point A.
Perfect competition is represented as a horizontal line with a constant price ͞p. Infinite suppliers will
provide goods at this price and consumers are willing to pay this price or less. The ε = -∞, for any q,
demand is infinitely elastic.
The single-agent problem is when each individual firm is too small to affect demand.
Each firm will produce for the price ͞p given their cost function C(q) to maximize their
profits.
Max{π(q) = ͞p*q – C(q)} This is when C’(q) = ͞p the moment an extra product
costs the same as the selling price. If the price is sufficiently high enough, there is a
possibility for profits. But when the market also has free entry, there will be no
profits. Because p ͞ will drop to the ATC = average total cost.
When a monopolists sets the price, we will use the same formula
Max{π(q) = p*q – C(q)} but the outcome will be different because we don’t use
͞p. The price is above the marginal cost and dependant on the price elasticity. The
mark-up of the monopoly can be expressed as the Lerner index. With the price = monopoly
price = pM. When q(p) and C(p) are both linear equations, the monopoly price can be found
more easily using the following equation.
In a Bertrand duopoly there are 2 firms producing a homogeneous product, both have the
same cost function and choose their price simultaneously. Customers have a demand and
will only buy from the lowest priced firm or split their demand equal between both if they set an
equal price. A final rule is that firms will only produce and have costs when/if units are sold. For
simplicity we will assume that C’(q) = c, q = a-bp and the firms demand
and profit for their price is given by the following equations.
If p1 is the monopoly price, both firms will have half
the profits. But it is in each firms interest to undercut
, the price to land on the green curve. This undercutting can go up to point p=c. When p<c the firm will
lose money.
The best decisions for each firm are laid out here, with ε being a small discount to land on the green
curve. So the price we will reach in an equilibrium will be p=c. This is a Nash equilibrium, which you
can reach by taking
the intersection of the reaction curves.
No firm will have profit because p = c = C’. This means that 2 firms create the same situation as in
perfect competition. But in reality firms do make a profit for a variety of reasons (no perfect
homogenous products, distance, asymmetric info, …). These conditions make that firms still have
market power.
The cross price elasticity of demand is the elasticity between 2 products. When ε ij>0 =substitute
∆ qi
∗p
goods and εij<0 =complement goods. And ∆ pj j.
ε ij=
qi
When we consider firms selling differentiated goods. The linear demand for their product can be
expressed as qi = Ai + bi1pj + bi2pj + … + + biipj + … + biNpj where bii = own price coefficient of product I
(for normal goods always negative) and bij = are the cross-price coefficient of product i for price j.
Example: take 2 firms with the demand for their products represented as
followed.
q1 = A1 + b12p2 + b11p1
q2 = A2 + b21p1 + b22p2
The prices in the Nash equilibrium are found using the formula to
maximize profit and are shown here to the right. When the b’s are equal
to 0, the price is equal to the monopoly price because the goods are so
different that they are nog substitutes or complements.
price 1
One way to measure a firms market power is mark-up = μ= =
marginal cost 1−LI
Paper 1:
Mark-ups where stable in the US until the 1980’s. Since than there has been an incline in mark-ups
and profitability of firms. But this is not for all companies. The median has stayed relatively the same,
but the upper percentile (superstar firms) has seen an dramatic increase. In the article they do
research in fixed and overhead cost. An increase in fixed cost will bring with it an increase in mark-up
to cover these costs. That’s why they also look into the profitability of the firms, which has also
increased for these top firms. The consequences are that a few big company’s hold a lot of power,
labour-shares have decreased and it is harder for small firms to enter and stay alive in those markets.
This trend has been partially in the decrease of marginal cost and an increase in fixed cost (Spotify vs
cd’s). There needs to be a balance between what the tech advances need and what a low entry
barrier needs.
Product differentiation
