A monopolist's demand fQuestion 1.
unction is given as Y=2000-10K, where Y is the quantity of output produced and sold and K is the price per unit in Tshs. If the marginal cost is Tshs 100
Show the inefficiency caused by the above monopolist diagrammatically and mathematically.
To show the inefficiency caused by a monopolist graphically, we need to compare the monopolist's output with the socially optimal output level.
1. Graphical representation:
a) Monopolist's profit-maximizing output: Since the monopolist maximizes its profit, it will produce where marginal cost (MC) equals marginal revenue (MR). In this case, since the monopolist's demand function is Y = 2000 - 10K, we can find the monopolist's marginal revenue by differentiating the demand function: MR = dY/dK = 2000 - 20K.
Set MR = MC to find the monopolist's profit-maximizing output level:
2000 - 20K = 100 (marginal cost is Tshs 100)
20K = 1900
K = 95
So, the monopolist's profit-maximizing output level is K = 95.
b) Socially optimal output level: The socially optimal output level, in a perfectly competitive market, is where marginal cost (MC) equals market price (P). Since marginal cost is given as Tshs 100, the socially optimal output level will be at a price of Tshs 100.
Using the demand function Y = 2000 - 10K, we can solve for the quantity at the socially optimal price:
2000 - 10K = 100
10K = 1900
K = 190
So, the socially optimal output level is K = 190.
Now, we can plot these two points on a graph. The monopolist's profit-maximizing output level (K = 95) will be to the left of the socially optimal output level (K = 190). This difference represents the inefficiency caused by the monopolist.
2. Mathematical representation of inefficiency:
To calculate the inefficiency caused by the monopolist, we can compare the total surplus (consumer surplus + producer surplus) at the monopolist's output level and the socially optimal output level.
a) Total Surplus at the monopolist's output level:
Consumer Surplus: To calculate the consumer surplus, we need to find the area under the demand curve (Y = 2000 - 10K) up to the monopolist's output level of K = 95.
Consumer Surplus = 0.5 * (2000 - 10K) * K (from 0 to 95)
Producer Surplus: To calculate the producer surplus, we need to find the area above the marginal cost curve (MC = 100) and below the monopolist's output level of K = 95.
Producer Surplus = (K - 0.5) * (MC) (from 0.5 to 95)
b) Total Surplus at the socially optimal output level:
Consumer Surplus: To calculate the consumer surplus, we need to find the area under the demand curve (Y = 2000 - 10K) up to the socially optimal output level of K = 190.
Consumer Surplus = 0.5 * (2000 - 10K) * K (from 0 to 190)
Producer Surplus: To calculate the producer surplus, we need to find the area above the marginal cost curve (MC = 100) and below the socially optimal output level of K = 190.
Producer Surplus = (K - 0.5) * (MC) (from 0.5 to 190)
By comparing the total surplus at the monopolist's output level and the socially optimal output level, we can determine the inefficiency caused by the monopolist. The difference represents the deadweight loss, which indicates the inefficiency in resource allocation caused by the monopoly power.