X is a monopolist of a soda source that costlessly burbles forth as much soda as X cares to bottle. It costs X $2 per gallon to bottle this soda. The inverse demand curve for X’s soda is p(y) = 20 – 0.2y, where p is the price per gallon and y is the number of gallons sold.
a) What is the profit function? Find the profit-maximizing choice of y for X.
Profit = Y(P – AC) = Y(20 – 0.2Y – 2) = Y(18 – 0.2Y) = 18Y – 0.2Y^2
MC = 2
MR = 20 – 0.4Y
2 = 20 – 0.4Y
0.4Y = 20 – 2
Y = 45
b) What price does X get per gallon of soda if he produces the profit-maximizing quantity? How much profit does he make?
P = 20 – 0.2(45) = 11
Profit = 18(45) – 0.2(45)^2 = 405
c) Suppose that X’s neighbour, Z, finds a soda source that produces soda that is just as good as X’s soda, but it costs Z $6 a bottle to get his soda out of the ground and bottle it. Total market demand for soda remains as before. Suppose that X and Z each believe that the other’s quantity decision is independent of his own. What is the Cournot equilibrium output for Z? What is the price in the Cournot equilibrium?
I don’t know how to solve for this question.
a) I agree
b) I agree
c) Cournot models can be tricky beasts. But in this example, I believe the soulution is easy because 1) both firms have flat MC curves and MCx < MCz. I believe firm X, in the end, will drive P down to 6 and Z will produced nothing. (I wish I had more time to work the math, but I dont).
Lotsa luck.
To find the Cournot equilibrium output for Z and the price in the Cournot equilibrium, we need to follow these steps:
1. Determine the reaction function for Z:
The reaction function represents Z's profit-maximizing choice of quantity given the quantity chosen by X. In this case, Z wants to maximize profit, which is given by the equation:
Profit_Z = Z(20 - 0.2Z - 6)
2. Solve for the quantity in the reaction function:
To find the Cournot equilibrium output for Z, we need to find the value of Z that maximizes profit. We do this by taking the derivative of the profit function with respect to Z and setting it equal to zero:
d(Profit_Z) / dZ = 20 - 0.4Z - 6 = 0
Solving this equation, we get:
0.4Z = 20 - 6
0.4Z = 14
Z = 35
So, the Cournot equilibrium output for Z is 35 gallons.
3. Find the price in the Cournot equilibrium:
To find the price in the Cournot equilibrium, we substitute the equilibrium output for Z (35 gallons) into the inverse demand curve equation:
P = 20 - 0.2(35)
P = 20 - 7
P = 13
Therefore, in the Cournot equilibrium, Z will produce 35 gallons of soda and the price of soda will be $13 per gallon.
Note: In the Cournot model, each firm assumes that the other firm's quantity decision is independent of its own, so they make their quantity decisions simultaneously based on their profit-maximizing choices.