Consider a market with two horizontally differentiated firms, X and Y. Each has
ID: 1151921 • Letter: C
Question
Consider a market with two horizontally differentiated firms, X and Y. Each has a constant marginal cost of $ 20. Characterize attributes of the Bertrand equilibrium in prices in the market.
a) Assume the two products are Coca-Cola and Pepsi. To what extent are the products different and perfect substitutes?
b) How does each firm having the same marginal cost affect the normal Betrand equilibrium analysis for this problem?
c) Without attempting to solve the following two demand equations, Qx = 100 - 2Px + 1Py Qy = 100 - 2Py + 1Px What can you say about the optimal reaction functions for firms X and Y?
d) Where will the only stable Betrand equilibrium price be in relation to the constant marginal cost of $20 if the firms faced capacity constraints?
Explanation / Answer
Each firm has a constant marginal cost of $ 20.
a) Assume the two products are Coca-Cola and Pepsi. Products are substitutes because they have similar taste (soft drinks), serve the same purpose and are offered at similar prices. They are different with respect to design and packaging and slight variation in the taste.
b) Since each firm is having the same marginal cost, both of them need to charge a same price and price cannot vary.
c) From the following two demand equations, Qx = 100 - 2Px + 1Py Qy = 100 - 2Py + 1Px the optimal reaction functions for firms X and Y will be a function of the price of their rival. That is, P1 will be a function of P2 and vice versa
d) The only stable Betrand equilibrium price is found as 46.67
Profit functions are
?x = PxQx – cQx
= (100 - 2Px + Py)Px – 20(100 - 2Px + 1Py)
= 100Px – 2Px^2 + PxPy – 2000 + 40Px – 20Py
= 140Px – 2Px^2 + PxPy – 2000 – 20Py
?y = PyQy – cQy
= (100 - 2Py + 1Px)Py – 20(100 - 2Py + 1Px)
= 100Py – 2Py^2 + PxPy – 2000 + 40Py – 20Px
= 140Py – 2Py^2 + PxPy – 2000 – 20Px
Profit is maximized for prices
?x’(Px) = 0
140 – 4Px + Py = 0 or Px = 35 + 0.25Py
?y’(Py) = 0
140 – 4Py + Px = 0 or Py = 35 + 0.25Px
Solve these BRFs to get
Px = 35 + 0.25*(35 + 0.25Px)
Px = 43.75 + 0.0625Px
Px = 46.67
Py = 35 + 0.25*46.67 = 46.67
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