A monopolist has a cost function, 𝐶(𝑥) = 30𝑥 and a demand curve given by

Question

A monopolist has a cost function, 𝐶(𝑥) = 30𝑥 and a demand curve given by

𝐷(𝑝) = 1000 − 15𝑝.
1.4.1 Derive expressions for marginal cost (MC) and marginal revenue (MR) .
1.4.2 How much will the monopolist charge and how many units of output will it produce?
1.4.3 What is the socially efficient quantity (the quantity that maximises total surplus)?

Answer 1

1.4.1 To derive the expressions for marginal cost (MC) and marginal revenue (MR), we need to understand their definitions and relationships.

Marginal cost (MC) represents the change in total cost when one additional unit of output is produced. In this case, the cost function is given by 𝐶(𝑥) = 30𝑥. To find MC, we need to calculate the derivative of the cost function with respect to 𝑥.

𝑀𝐶 = d𝐶/d𝑥

Differentiating 𝐶(𝑥) = 30𝑥 with respect to 𝑥 gives us:
𝑀𝐶 = 30

So, the expression for marginal cost (MC) is 𝑀𝐶 = 30.

Marginal revenue (MR) represents the change in total revenue when one additional unit of output is sold. The revenue function can be determined from the demand function, as revenue is the product of price (𝑝) and quantity demanded (𝑥). The demand curve is given by 𝐷(𝑝) = 1000 − 15𝑝.

To find the revenue function, we need to express quantity demanded (𝑥) in terms of price (𝑝). By rearranging the demand equation, we have:
𝑥 = (1000 − 𝐷(𝑝))/15

Now, we can express revenue (𝑅) as the product of price and quantity demanded:
𝑅 = 𝑝 * 𝑥 = 𝑝 * (1000 - 𝐷(𝑝))/15

To find marginal revenue (MR), we differentiate the revenue function with respect to 𝑥 and multiply by 𝑥 to account for the change in quantity demanded:
𝑀𝑅 = 𝑥𝑑𝑅/𝑑𝑥

Differentiating 𝑅(𝑝) with respect to 𝑥 gives us:
𝑀𝑅 = (1000 - 𝐷(𝑝))/15

So, the expression for marginal revenue (MR) is 𝑀𝑅 = (1000 - 𝐷(𝑝))/15.

1.4.2 To determine how much the monopolist will charge and how many units of output it will produce, we need to find the profit-maximizing level of output.

To maximize profit, the monopolist will produce the quantity where marginal cost (MC) equals marginal revenue (MR). So we set the two expressions equal to each other:

30 = (1000 - 𝐷(𝑝))/15

To find the monopolist's charging price (𝑝), we can rearrange this equation and solve for 𝐷(𝑝):
𝐷(𝑝) = 1000 - 15𝑀𝐶

Substituting the expression for marginal cost (MC) as 30, we have:
𝐷(𝑝) = 1000 - 15 * 30 = 550

Therefore, the monopolist will charge a price where 𝐷(𝑝) = 550, and the quantity demanded at that price will be produced.

1.4.3 The socially efficient quantity is the quantity that maximizes total surplus. To find this quantity, we need to consider the welfare of both the monopolist and the consumers.

In a competitive market, total surplus is maximized when supply equals demand. Since this is a monopolistic market, the monopolist has market power and charges a price higher than the competitive price. As a result, the monopolist produces less than the socially efficient quantity.

To find the socially efficient quantity, we can set the marginal cost (MC) equal to the demand function and solve for 𝑥:

30 = 1000 - 15𝑝

Simplifying the equation gives us:
15𝑝 = 1000 - 30
15𝑝 = 970
𝑝 = 970/15 ≈ 64.67

Substituting the price (𝑝) back into the demand function, we can find the quantity (𝑥) at the socially efficient price:
𝑥 = (1000 - 𝐷(64.67))/15

Evaluating 𝐷(64.67) gives us:
𝑥 = (1000 - 15 * 64.67)/15

Calculating this expression will give us the socially efficient quantity.