A monopolist producer faces the inverse demand curve gi- 250-Q, where P is the price charged and O is the quantity den total cost of producing the good is C(Q) = 50+50Q.
a) Find the profit-maximizing price and quantity, P and Q
b) Calculate the deadweight loss of this monopoly.
c) Suppose a per-unit tax of t is imposed on each unit produce the monopoly. Calculate the new price P** that maximizes monopoly's profit as a function of t and the derivative dP- What economic interpretation can you attach to this derivat Explain!
a) To find the profit-maximizing price and quantity, we need to determine the monopolist's marginal revenue (MR) and marginal cost (MC) functions.
The marginal revenue function can be found by taking the derivative of the inverse demand curve:
MR = d(250 - Q)/dQ = 250 - 2Q
The marginal cost function is given as C'(Q) = d(50 + 50Q)/dQ = 50.
To find the profit-maximizing quantity (Q*), we set the marginal revenue equal to the marginal cost:
250 - 2Q* = 50
2Q* = 200
Q* = 100
Plugging the value of Q* into the inverse demand equation, we can find the profit-maximizing price (P*):
P* = 250 - Q* = 250 - 100 = 150
So, the profit-maximizing price and quantity are P* = 150 and Q* = 100.
b) The deadweight loss of a monopoly is the loss of economic efficiency that occurs when the monopolist restricts output and charges a higher price compared to a perfectly competitive market. To calculate the deadweight loss, we need to compare the monopolist's quantity Q* with the quantity that would prevail in a perfectly competitive market (Qc) at the same price.
The equation for the equilibrium quantity in a perfectly competitive market is given by the inverse demand curve:
250 - Qc = P*
Solving for Qc:
Qc = 250 - P*
Substituting the profit-maximizing price (P*) into the equation, we find:
Qc = 250 - 150 = 100
The deadweight loss (DWL) is the difference between the monopolist's quantity (Q*) and the perfectly competitive quantity (Qc):
DWL = Qc - Q* = 100 - 100 = 0
In this case, the deadweight loss is zero since the monopolist is producing the same quantity as in a perfectly competitive market. This implies that the monopoly is producing at the socially optimal level, and there is no efficiency loss due to the monopoly power.
c) When a per-unit tax t is imposed on each unit produced, the monopolist's profit-maximizing price (P**) will be affected. To find the new price, we need to consider the impact of the tax on the monopolist's marginal cost.
The monopolist's marginal cost after the tax will be: MC' = MC + t.
To find the new profit-maximizing price P**, we set the marginal revenue equal to the new marginal cost:
250 - 2Q = MC' = MC + t
Given that the original marginal cost is 50 (as provided), we can rewrite the equation as:
250 - 2Q = 50 + t
Solving for the new quantity Q**:
2Q** = 200 + t
Q** = 100 + 0.5t
Plugging this value of Q** into the inverse demand equation, we can find the new profit-maximizing price (P**):
P** = 250 - Q** = 250 - (100 + 0.5t) = 150 - 0.5t
The economic interpretation of the derivative dP/dt is the change in price with respect to the tax rate. In this case, the derivative is -0.5, indicating that for every unit increase in the tax rate, the monopolist reduces the price by 0.5 units.
This implies that as the tax rate increases, the monopolist lowers the price to offset the impact of the tax on their profit, resulting in a smaller profit margin per unit. The derivative represents the monopolist's responsiveness to changes in the tax rate, and it helps determine the new profit-maximizing price (P**).