Question

This exercise brings together several parts of the course: it may help to revise the sections on consumer choice, demand, cost minimization and cost functions. Problem: The market for good X consists of 6400 consumers and one producer (a monopolist). The consumers are all identical and have preferences over X and Y (standing for all other goods) given by ?(?, ?) = √? + ? The associated marginal utilities are ??? = 1 2√? and ??? = 1. The price of Y is $1 per unit, and each consumer has an income of $10. Hint: You can assume that this consumer choice problem has an interior solution for any price that is relevant to this market. The monopolist can produce X using the production function ?(?,?) = min{?, 2?}. The cost of labor (L) is w=1 and the cost of capital (K) is r=2. Find the optimal price and quantity sold for the monopolist in the market for good X. Hints: you will need to go through the following steps: 1) find the individual demand for good X by each consumer, 2) find market demand for good X, 3) solve the producers cost minimization problem (with Q (quantity of X) as a variable), 4) use this solution to find the cost function, 5) using the cost and market demand functions you have found, derive the optimal monopoly price and quantity. You may need to take several derivatives along the way. Here are some rules to help: 1. ? ?? ?√? = ? 2√? , where a is a constant. 2. ? ?? ?? = ?, where b is a constant.

Answer 1