Problem Set #3 - ECN 121A, Spring Quarter 2025
Due: May 13, at 11:59pm. Problem sets are to be turned in through Gradescope
Note: In order to receive credit, you must show your work.
1. Suppose there are two identical firms in a market.
The market demand is given by: Q = 50 − 0.5p
Suppose the marginal cost of each firm is $10 per unit and the firms are competing in a Cournot- type setting (set quantities simultaneously).
a. Graph the reaction functions of both firms on the same graph.
b. What price is the product going to be sold for? What is the equilibrium output and profits for each firm?
c. How much would the profits of each firm decrease ifa third firm was added to the industry and they competed in a Cournot-type setting?
You may use the following formula whenever there are three or more firms:
d. Suppose we are back to two firms in the industry. What would the profits of firm 1 be ifit were to produce one more unit than what you found in part ‘b’ (with firm 2 still producing the same amount you found in part ‘b’)? Why do the profits of firm 1 decrease if it is capturing a larger share of the market? In other words, should there not be a positive relationship between the quantity produced and profits? Briefly explain.
2. We went over a question in class where there were two firms competing in a Cournot-type setting, and the marginal cost of each firm was 20.
The market inverse demand curve was:
P(Q) = 120 - 20Q
The reaction functions of the firms were: q1 = 2.5 − 0.5q2 and q2 = 2.5 − 0.5q1
Suppose the marginal cost of firm 2 increases.
What needs to be the new reaction function of firm 2 for it to produce 0 units in equilibrium? Show your work both graphically and numerically. (Hint: Find where the reaction functions need to intersect for q2 to equal 0.)
3. Suppose the inverse demand of a product produced by two firms is given by: P(Q) = 100 - Q The marginal cost of producing the product is 20 and there are no fixed costs.
The two firms compete simultaneously with prices (Bertrand model).
If the products are identical, what are the prices that each firm will charge? Briefly explain.
4. Suppose there are two firms selling protein bars. Firm one sells ‘AggieBars’ with 10 grams of protein and firm 2 sells ‘DavisBars ’ with 20 grams of protein.
Consumers are distributed uniformly over their preferences for grams of protein between 10 and 20.
Suppose firm 1 sells AggieBars for $2 and firm 2 sells DavisBars for $3
The ‘cost’ to consumers of deviating from their optimal amount of protein is $0.20 per gram.
a. What protein content does the marginal consumer (consumer who is indifferent between
AggieBars and DavisBars) prefer?
The equation for finding the marginal consumer (when the range of product attribute values is 10) is: V - p1 - txm = V - p2 - t(10 - xm )
b. How would the proportion of consumers buying each product change if the cost to deviation
from one’s optimal amount of protein increased (was greater than $0.20 per gram)?
5. Suppose there are two firms in a Cournot type setting (choosing quantity simultaneously). How would an increase in the marginal cost of firm 1 change the equilibrium output of firm 1 and firm 2? Depict your answer graphically with reaction functions and show all of your work.