Microeconomics 2nd Edition by Douglas Bernheim

Exercise 4

Suppose TM Inc. is the unique firm with access to a technology for producing time machines (so it is a monopolist in the time machine market). It uses gidgets as its only input, whose price is R per unit. TM's monthly production function is
where X is the quantity of gidgets. The inverse demand function for time machines is given by P ( Q ) = 10 Q , where Q is the number of time machines it sells per month.
a. Taking the price R of gidgets as given, find TM Inc.'s profit-maximizing price of time machines and the quantity sold each month (your answer should depend on R).
b. What is the demand function of TM Inc. for gidgets as a function of R
c. Suppose now that Gcorp is the monopoly supplier of gidgets and that TM Inc. is the only firm that demands gidgets. Gcorp's cost of producing Q gidgets is given by the cost function C ( Q ) = AQ , where A 0 is a constant. Suppose that sales of gidgets work as follows: Gcorp sets its price, and then TM Inc. takes that price as given and chooses how many gidgets to buy. What is Gcorp's profit-maximizing gidget price Given this gidget price, how many time machines are sold each month and at what price
d. What is the consumer surplus in the market for time machines How does it change as A changes
e. If TM and Gcorp merge and act as a single (monopoly) firm, what price will it set for time machines What happens to consumer surplus, producer surplus, and aggregate surplus