# Monopoly with Diminishing Returns to Scale

Andrea Adrenalin runs a bungee jumping business. Her production function for bungee jumps is:

Q = K0.25 L0.25

where K is the number of bungees she uses and L is the number of employees she hires. The price of a bungee is $100 and the price of an employee is$16. In addition, assume that Andrea controls the only location in the world which can be used for bungee jumping. The demand for bungee jumps is given by P = 4000 - 50*Q.

1. Find Andrea's long-run cost-minimizing factor demands for K and L and calculate her total costs for Q=10, Q=15 and Q=20 jumps. Be sure to show your work!
2. Using calculus (beyond the scope of this course), it is possible to show that Andrea's total costs for any Q are given by the following function:

TC = 80*Q2

Verify that this function produces the same total costs you found in part (1). For the remainder of the problem you can use this function and do not need to calculate her inputs of K and L.
3. If Andrea wants to maximize the number of bungee jumps she produces subject to the requirement that she not run a loss, how many jumps should she produce and what should she charge?
4. If instead she wants to maximize profits, how many jumps should she produce and what should she charge?
5. Finally, suppose that Andrea acted like a competitive market and produced where P = MC (or at least as close as possible for an integer number of jumps). What output would she choose? What price would she charge? Would this be more or less efficient than the outcome under the two monopoly cases? Explain.

## Solution

URL: https://wilcoxen.maxwell.insightworks.com/pages/361.html
Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 08/18/2016