Question

Two firms, A and B, sell widgets to a market of 100 buyers. The firms' widgets are undifferentiated, and the firms know each others' costs and capacities. Furthermore the firms are playing a one-time pricing game; widgets are obsolete after this one selling opportunity. Each buyer is interested in purchasing a single widget, and has an RP of $10.

Firm A has lower costs than Firm B, but also has lower capacity. Specifically, their (constant) marginal costs and capacities are as follows.

	Marginal Cost	Capacity
Firm A	5	30
Firm B	7.1	100

   Finally, assume that each firm can only post prices in whole dollar amounts. (This question is motivated by firms selling through coin-operated vending machines. It is too costly to stock such machines with pennies, so sellers must set prices in fixed increments of larger denomination coins.) Throughout this entire problem, firms may only choose prices of $1.00, $2.00, $3.00, ..., up to $10.00. Firms may NOT use prices such as $1.50, $2.99, $3.83, etc.

(2a.) Find equilibrium prices for this one-time pricing game. (As usual: all buyers go to the firm with the lowest price. However if Firm A's price is no higher than Firm B's price, Firm A serves only 30 buyers, and the rest go to Firm B. Your answer should be a pair of prices, one for each firm.)

(2b.) Suppose that before the pricing game starts, Firm A can build a production plant that would have full capacity of 100 units. (Marginal costs would remain the same, and Firm B would see Firm A's new capacity. Then firms would simultaneously set some equilibrium prices.)

Ignoring the cost of building, how much profit would Firm A earn if it expanded?  Firm A would now earn profits of ______________. (You must show your calculations for credit. The same assumptions apply as the previous part, except now the firms split the market 50/50 if they price equally.)

Answer 1

2a. For the equilibrium price in the first case, firm A will price it at 10$ and sell its maximum capacity of 30. Firm A will know Firm B's MC and capacity, if firm B prices it at 8$,it will have be able to sell to all 100 buyers,and get a profit of 0.9*100=90$,however if Firm B will experience highest profit when it also prices it at 10$ and gets 70 buyers for a profit of 2.9*70=203$. Note that if Firm A prices it at 10$, Firm B will be able to price it at 9$ (or 8$) and get a total profit of 1.9*100=190 (or 90$ with 8$) but it is less than 203 and so Firm A will know that firm B will also price it at 10$ maximizing both of their profits. =

This the equilibrium set of prices in this case is

10$ for Firm A and 10$ for Firm B

2b If firm A builds the production plant it would have a capacity of 100 and both firms know each other's MC and capacities. Now firm B cannot charge less than 7.1 as it will be making a loss, so if Firm A charges 7$, it will get a profit of 2*100=200$

Now if they both sell at 10$, firm A will have a profit of 5*50=250$,however if it tries to undercut firm B and sells at 9$, it will have the entire market and a profit of 4*100=400$, so firm B will also price it at 9$

now firm A can price it at 8$ for maximum profit(3*100>4*50) and so firm B would have to price it at 8$, and in this case firm A's profit will be 3*50=150$ (If B also prices it at 8)

Again firm A would have a higher profit if it priced it at 7$ for a profit of 2*100=200$ and sell to all buyers.

In this situation firm A will have the highest profit if it prices it at 10$ and sells to 50 buyers for a total profit of 250$, thus it will price it at 10$ and have a profit of 250$

Hope it helps. Do ask for any clarifications required.