Problem 1. Incor is currently the only supplier of widgets and earns monopoly rents of $700k. Enterprise is considering entering the market. It would cost Enterprise $100k to set up a factory. However, Incor has threatened to start a price war if Enterprise enters. Your research shows that if Incor follows though on its threat, Incor would earn only $100k from producing widgets, while Enterprise would earn only $25k (before accounting for building costs). However, if Incor does not start the price war, it would make $300k, while Enterprise would earn $200k.
1. Draw this game in extensive form. Be sure to fully label the game tree.
2. What is the subgame perfect equilibrium outcome?
3. Would you advise Enterprise to build the widget factory?
Problem 2. Suppose there are two emergency hospitals in Townville, State ER and University ER. Overall demand (patients) for ER visits in Townville is P = 120 -2Q per day, where Q = qs + qu. Marginal cost at the State ER is cs = 84 per patient. University has access to better equipment so their marginal cost is cu = 70. Suppose University ER is the Stackelberg leader and is able to choose their quantity before State ER. State ER then responds to the choice by University.
1. Write down the profit function for State ER as a function of their quantity of patients (qs) and the choice of University's quantity of patients (qu).
2. Solve for State's optimal strategy in the Stackelberg game.
3. Write down the profit function for University ER as a function of their quantity of patients (qu) only.
4. Solve for university's optimal strategy.
5. Solve for both hospitals quantity of patients served, the equilibrium price, and profit for both firms when the SPNE is played.
6. What is total surplus (Consumer surplus+Profit)?
7. Suppose after State makes their choice of quantity of patients, University is able to react and easily expand or contract their service. What is the outcome of this model and why is it different from above? (No need to solve the model, just intuitively explain what will happen and why).
Problem 3. Two independent ice cream vendors own stands at either end of a 2 mile long beach. Everyday there are 200 beach-goers who come to the beach and distribute themselves uniformly along the water. Every beach-goer one wants exactly one ice cream during the day, and values the ice cream from both stands at $5. All of the beach-goers would rather be sunbathing or in the water, so they have a disutility to walking on the beach of $1 per mile. Early's Ice Cream, the firm at location 0, is an early riser and always posts his price first. Cali Creamery, at location 2, is more laid back and posts her price just before the beach opens (the beach requires all prices
be posted by the time the beach opens). Both firms have a marginal cost of zero.
1. Each individual is also referenced by a location x on the beach between 0 and
2. What are the utilities of purchasing from Early's and Cali for the person at location .75, given that Early's names price pe and Cali names price pc? What are the utilities for each individual as a function of their location on the beach,x?
2. What is the demand for Early's Ice Cream and Cali Creamery given the firms name prices pe and pc?
3. What is Cali Creamery's best response function when Early's posts a price of pe?
4. What is the Stackleberg equilibrium outcome for this market? Report prices, quantities, and profits for each firm.
5. Early's owner feels that his hard work is not paying off, he hires you as a business consultant. He's annoyed that Cali is always undercutting his price and is considering waiting to post so that Cali will not learn his price before naming her own. He wants you to predict how waiting to post his price will affect his profits. What will Early's profits be under this new regime? What advice do you give him?