**Problem : **

Two firms with identical cost structures produce a homogeneous good. Both firms
choose the quantity to produce at the same time, but before then, one firm has
the privilege of announcing its production quantity decision. Explain how the
credibility of this announcement can change the outcome. Do we reach the
Cournot equilibrium or the Stackelberg equilibrium?

The notion of a credible threat is a key notion in game theory. An incredible
threat is an action that is announced but will probably harm the announcer if
s/he takes the action. If the second firm believes the first will actually act
as announced, a Stackelberg equilibrium will occur. Otherwise, a Cournot
equilibrium will occur.

**Problem : **

Two firms have marginal costs of 10. They face a market demand curve of *P* = 100 - 4*Q*. The government imposes a tax of 10 dollars per unit sold. Determine
the Cournot equilibrium quantity.

Assume the tax will be paid by the consumer. The effective demand curve is 90 - 4*Q*.

*R*_{1} = (90 - 4*Q*_{1} -4*Q*_{2})*Q*_{1}

*MR*_{1} = 90 - 8*Q*_{1} -4*Q*_{2}

Setting MR = MC:

*Q*_{1}^{*} = 10 - *Q*_{2}/2

By symmetry:

*Q*_{1}^{*} = *Q*_{2}^{*} = 20/3

**Problem : **

Assume three firms face identical marginal costs of 20 with fixed costs of 10.
They face a market demand curve of *P* = 200 - 2*Q*. Find the Cournot equilibrium
price and quantity.

*R*_{1} = (200 - 2(*Q*_{1} + *Q*_{2} + *Q*_{3}))*Q*_{1}

*MR*_{1} = 200 - 4*Q*_{1} -2*Q*_{2} -2*Q*_{3}

Applying MR = MC:

*Q*_{1}^{*} = 45 - *Q*_{2}/2 - *Q*_{3}/2

By symmetry:

*Q*_{1}^{*} = *Q*_{2}^{*} = *Q*_{3}^{*} = 22.5

**Problem : **

Assume two firms have marginal costs of 20. They face a market demand of *P* = 90 - 3*Q*. Determine the Bertrand equilibrium quantity and price. Now assume
one firm moves ahead of the other. Find the Stackelberg equilibrium and price.

The Bertrand equilibrium is simply the competitive equilibrium of no profits.
Bertrand price is the marginal cost, 20. The Bertrand quantity is 70/3.

The Stackelberg equilibrium is a little more complicated. We calculate Firm 2's
reaction curve in the same way we did for the Cournot Model. Verify that Firm
2's reaction curve is:

*Q*_{2}^{*} = 70/6 - *Q*_{1}/2

To calculate Firm 1's optimal quantity, we look at Firm 1's total revenues.

Firm 1's Total Revenue = *P*·*Q*_{1} = (90 - 3*Q*_{1} -3*Q*_{2})*Q*_{1}

= 90*Q*_{1} -3*Q*_{1}^{2} -3*Q*_{2}*Q*_{1}

However, Firm 1 is not forced to assume Firm 2's quantity is fixed. In fact,
Firm 1 knows that Firm 2 will act along its reaction curve which varies with
*Q*_{1}. Firm 2's quantity very much relies on Firm 1's choice of quantity. Firm
1's Total Revenue can thus be rewritten as a function of *Q*_{1}:

*R*_{1} = 90*Q*_{1} -3*Q*_{1}^{2} -3*Q*_{1}(70/6 - *Q*_{1}/2)

Marginal revenue for firm 1 is thus:

*MR*_{1} = 90 - 6*Q*_{1} -35 + 3*Q*_{1}

= 55 - 3*Q*_{1}

When we impose the profit maximizing condition (*MR* = *MC*), we find:

*Q*_{1}^{*} = 35/3

Solving for *Q*_{2}, we find:
INDEX
*Q*_{2}^{*} = 35/6
/INDENX

**Problem : **

A group of *n* identical firms face a market demand curve of *P* = 2000 - 3*Q*.
*MC* = 100. Show that as *n* approaches ∞, the quantity approaches the
perfectly competitive outcome.

First, identify the marginal revenue by taking the derivative of the revenue for
firm 1.

Total Revenue = *P*·*Q*_{1} = (2000 - 3*Q*)·*Q*_{1}

= (2000 - 3(*Q*_{1} + *Q*_{2} + ... + *Q*_{n}))·*Q*_{1}

= 2000*Q*_{1} -3*Q*_{1}^{2} -3(*Q*_{2} + ... + *Q*_{n})·*Q*_{1}

The marginal revenue is simply the first derivative of the total revenue with
respect to *Q*_{1} (recall that we assume *Q*_{i} for *i* not equal to 1 is fixed).
The marginal revenue for firm 1 is thus:

*MR*1 = 2000 - 6*Q*_{1}1 - 3(*Q*_{2} + ... + *Q*_{n})

Imposing the profit maximizing condition of *MR* = *MC*, we conclude that Firm 1's
reaction curve is:

2000 - 6*Q*_{1}^{*} -3(*Q*_{2} + ... + *Q*_{n}) = 100

=> *Q*_{1}^{*} = 1900/6 - (*Q*2 + ... + *Qn*)/2

We can solve for *Q*_{1}^{*}.

*Q*_{1}^{*} = 1900/6 - (*Q*_{1}^{*})·(*n* - 1)/2

=> *Q*_{1}^{*}((2 + *n* - 1)/2) = 1900/6

=> *Q*_{1}^{*} = 1900/[6(1 + *n*)]

By symmetry, we conclude:

*Q*_{i}^{*} = 1900/[6(1 + *n*)] for all firms i

In our model of perfect competition, we know that the total market output of *Q* = 1900/6 is the zero profit quantity.

*Q* = *n**1900/[6(1 + *n*)]

The limit of *Q* as *n* approaches infinity is 1900/6, as expected.