Consider the simplest monopoly case, where there is linear demand, p(q) = a bq a
ID: 1220166 • Letter: C
Question
Consider the simplest monopoly case, where there is linear demand, p(q) = a bq and constant costs c(y) = c.
(a) (1 point) Write out the monopolist profit maximization problem with both y and p as choice variables.
Use the substitution method to find the optimal p the monopolist should charge.
At this price p, how many output, y, will the monopolist sell?
(b) (4 points)
(c) (2 points)
(d) (2 points)
6. Consider a duopoly of a homogenous good, where firm 1 is a “quantity leader” and firm 2 is a “quantity follower.” (Specifically this means firm 1 chooses y1 and THEN firm 2 chooses y2.) Suppose that firm 1 has c1(y1) = 6y1 + 4 and firm 2 has c2(y2) = 4y2 + 5. The market demand for the good is linear given by p(Y ) = a bY , where Y is the aggregate/market output.
(a) (1 point) What type of competition can we model this duopoly as?
(b) (1 point) Consider firm 2, and write out firm 2’s profit maximization problem. Is this constrained or unconstrained optimization?
How does this optimal solution (p,y) compare to the other optimal solution when the monopolist maximizes profit while choosing output? (The method you used for Uncle Rico.) IF you do not know this answer (or don’t believe your answer), you can quickly derive the other optimal solution by setting MR(y) = MC(y) and solving for y.
Explanation / Answer
p(q) = a bq and constant costs c(y) = c.
TR=p.q= (a-bq)q
Profit=TR-TC
=(a-bq)q-c
Profit'=a-2bq
Let Profit'=0
a-2bq=0
a=2bq
q=a/2b
p*=a-b(1/2b)
=a-0.5