Question

Solve the Bertrand problem

3.2 Bertrand
Now suppose that instead of competing on quantities, the two restaurants are competing by setting prices following the Bertrand Oligopoly Model. Assume both restaurants have MC = $5 and total market demand for gyros is 1000.
A. What is the Nash-Bertrand equilibrium in this market?
B. What would the Nash-Bertrand equilibrium in this market be if Sam’s has a marginal cost of $5 and Ali Baba’s has a marginal cost of $4 per gyro, assuming no fixed costs?
C. What are two ways this Nash-Bertrand equilibrium is inconsistent with real oligopoly markets?

Now suppose that the two restaurants use advertising to differentiate their products. The marginal cost for each restaurant stays the same at $5 for both. Suppose Sam’s faces a demand function of qs = 1000−10ps + 10pab. Suppose Ali Baba’s faces a demand function of qab = 1000−5pab + 10ps.
D. What is the Nash-Bertrand equilibrium now?

3.1 Cournot & Stackelberg
Suppose there are two restaurants selling gyros on the same block in Davis, Sam’s and Ali Baba’s, and they are competing with each other following the rules of the Cournot Oligopoly Model. Suppose inverse market demand for gyros is p = 600−Q. Suppose Sam’s has cost function C(qs) = q2 s and Ali Baba’s has cost function C(qab) = 9qab.
A. What is the Nash-Cournot equilibrium in this market (quantities) and what is the price in the market?
B. What are the profits of the two restaurants at the equilibrium (assume zero fixed cost)?
C. Graph the marginal cost, demand, residual demand, and marginal revenue curve for Sam’s. Show the equilibrium price and quantity for that restaurant.
Now suppose Sam’s is the first mover following the Stackelberg Oligopoly Model, but everything else stays the same as in the above problem.
D.What is the Nash-Stackelberg equilibrium in this market (quantities) and what is the price in the market?
E. What are the profits of the two restaurants at the new equilibrium?
F. Graph the marginal cost, demand, residual demand, and marginal revenue curve for Sam’s. Show the equilibrium price and quantity for that restaurant.

Answer 1

(a) In the Bertrand model, P=MC. As MC= $5 for both restaurants the price, P= $5

Now, if P1<P2 => Q1= Q & Q2=0

If P2 < P1 => Q1=0 & Q2=Q

If P1= P2 => Q1 = Q2 = 1/2 * Q

since P1 = P2 then the equilibrium quantity for each restaurant will be 1000/2 = 500.

Equilibrium Price = $5

(b) Now, Sam has marginal cost of $5 and Ali Baba has marginal cost of $4.

Ps= $5 and Pab= $4. Since Pab < Ps, Ali baba has the entire market demand of 1000.

(c) In a Bertrand competition, firms compete for prices, and prices are fixed equal to marginal cost. In a real oligopoly, firms maximize their profit using reaction function.

(d)

Sam faces demand, Qs= 1000-10Ps + 10Pab

Ali Baba faces demand, Qab= 1000-5Pab+10Ps

Since, MCs= MCab = $5 => Ps=Pab= $5

So, Qs=1000-10*5+10*50 => 1000

Qab = 1000 -5 *5+10*5 => 1025

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