A Short Course in Intermediate Microeconomics with Calculus | 2nd Edition
Chapter 13, Problem 5E
Problem
Laban and Jacob are sheep farmers in a differentiated goods market. The market demand for Laban’s sheep wool is
while the market demand for Jacob’s sheep wool is
Laban’s cost function is C1(y1) = 24y1, while Jacob’s cost function is C2(y2) = 20y2. They compete with each other through their choices of price.
(a) Calculate the equilibrium in which Laban is the price leader in this market, and Jacob is the price follower. Indicate prices, output levels, and individual profits.
(b) How do prices, output levels, and individual profits change if Jacob is the price leader in this market, and Laban is the price follower?
step1/3
Market demand of sheep wool of Laban is
Market demand of sheep wool of Jacob is
Cost function of Laban is
Cost function of Jacob is
step2/3
a)
When Laban is price leader and Jacob is price follower:
Profit of Jacob is
Differentiate with respect to
and equate to zero,
Since Laban is price leader, substitute it in demand function of Laban,
Profit of Laban is
Maximize profit by differentiating with respect to
and equating to zero,
Using
Using
Using
Profit of Laban is
Profit of Jacob is
Therefore,
Price, output levels and profit of Laban is
respectively.
Price, output levels and profit of Jacob is
respectively.
step3/3
b)
When Jacob is price leader and Laban is price follower:
Profit of Laban is
Differentiate with respect to
and equate to zero,
Since Jacob is price leader, substitute it in demand function of Jacob,
Profit of Jacob is
Maximize profit by differentiating with respect to
and equating to zero,
Using
Using
Using
Profit of Laban is
Profit of Jacob is
Therefore,
Price, output levels and profit of Laban is
respectively.
Price, output levels and profit of Jacob is
respectively.