The price elasticity of demand for melocotones is constant and equal to -2. The melocotone market is controlled by two Cournot duopolists who have different cost functions. One of the duopolists has a constant marginal cost of $675 per ton and produces 50% of the total number of melocotones sold. The equilibrium price of a ton of melocotones must be

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Since the price elasticity of demand for melocotones is constant (-2), the demand curve is linear. Each firm will choose its output by assuming the rival's output is constant. The total quantity demanded will be divided by the two duopolists according to their market shares. If one duopolist has a 50% market share, then the other must also have a 50% market share. The total quantity sold will be twice the quantity sold by each duopolist. So the market demand function can be written as:
Q = (P/2) x (-2)
Q = -P
Each duopolist's profit is π = (P - MC) x Q, where P is the price, MC is the marginal cost, and Q is the quantity sold by that duopolist. To maximize profit, each duopolist will choose its output to maximize its own profit given the other's output. The best response function for the duopolist with the constant MC of $675 can be found as follows:
π = (P - 675) x (0.5P)
dπ/dP = 0.5P - 675 - (P - 675)
dπ/dP = -0.5P
-0.5P = 0
P = 0
This is not a valid price, so we must look for another extreme point. Since the best response function is linear, it must be either increasing or decreasing. To determine which, we can check the endpoints of the range of possible prices, which is determined by the lowest MC (belonging to the duopolist with the constant MC) and the highest possible price that the market will bear. This highest possible price is the price at which one duopolist would sell no melocotones, so the other's quantity sold would be the entire market demand of 0.5P. Setting this quantity equal to the other duopolist's output and solving for P:
0.5P = -P
P = 0
This is not a valid price, so we must look for another extreme point:
0.5P = P/2 - 675
1.5P/2 = 675
P = $900
So the best response function for the duopolist with the constant MC of $675 is P = 1800 - Q/2. The best response function for the other duopolist (with some other cost function) can be found in the same way. The equilibrium price and quantity can be found by solving for the intersection of the two best response functions. Alternatively, we can use the fact that each duopolist sells half the total quantity at the equilibrium price, so we can simply add up their profits and set the sum equal to zero. At the equilibrium:
P = $900
Q = 900
Revenue for each duopolist is: TR = P x Q/2 = $450,000
Cost for each duopolist is: TC = 675 x Q/2 = $303,750
Profit for each duopolist is: $146,250
The equilibrium quantity is 900, so the total revenue is $810,000, which is divided equally between the duopolists. Therefore, each duopolist's profit is $146,250.

The Price Elasticity of Demand for Melocotones Is Constant and Equal