Question

1. Suppose that identical duopoly firms have constant marginal costs of $16 per unit. Firm 1 faces a demand function of q₁ = 70-2p₁ + p₂, where q1 is Firm 1's output, p1 is Firm 1's price, and p2 is Firm 2's price. Similarly, the demand Firm 2 faces is q2 = 70-2p2 + p1. Solve for the Nash-Bertrand equilibrium.

2. Solve for the Nash-Bertrand equilibrium for the firms( described in above question) if both firms have a marginal cost of $0 per unit.

3. Solve for the Nash-Bertrand equilibrium for the firms ( described in above question 1) if Firm 1's marginal cost is $25 per unit and Firm 2's marginal cost is $15 per unit.

Answer 1

Solution:

a) Solve for the Bertrand equilibrium:

Let c₁ represent the marginal cost for Firm 1, and

Let c₂ represent the marginal cost for Firm 2.

Here c₁=c₂=$16.

A Bertrand competitor maximizes profits by choosing the optimal price, taking into account its competitor`s price. For Firm 1, this means:

Π = max{ p₁q₁(p_1,p₂)-c₁q₁}=max{(p₁-c₁)(70-2p₁+p₂)}.

Profit maximization implies that $\frac{d}{dp_1}$ {(p₁-c₁)(70-2p₁+p₂)}=0, or

(70-2p₁+p₂)-2(p₁-c₁)=0, or 4p₁=70+p₂+2c₁.

By symmetry, we also have 4p₂=70+p₁+2c₂.

Substituting the second expression into the first gives us $p_1=\frac{1}{4}[70+2c_1+\frac{1}{4}(70+2c_2+p_1)],$ or $p_1=\frac{8c_1+350+2c_2}{15}.$

By symmetry, $p_2=\frac{8c_1+350+2c_2}{15}.$

Substituting the marginal costs above gives:

$p_1=p_2=$ $34

$q_1=q_2=$ 66

b) Solve for the Bertrand equilibrium if both firms have a marginal cost of $0 per unit:

Substituting the appropriate marginal costs into the expression above gives  

$p_1=p_2=$ 23.33

$q_1=q_2=$ 46.77

c) Solve for the Nash-Bertrand equilibrium for the firms ( described in above question 1) if Firm 1's marginal cost is $25 per unit and Firm 2's marginal cost is $15 per unit:

Substituting the appropriate marginal costs into the expression above gives :

$p_1=$ $38.66

$p_2=$ $42.67

$q_1=$ $41.33

$q_2=$ $65.33