Question

In a market for dry cleaning, the inverse market demand function is given byP= 100 -Q, and the (private)marginal cost of production for the aggregation of all dry-cleaning firms is given by MC = 10 +Q. Finally, the pollution generated by the dry-cleaning process creates external damages given by the marginal external cost curve MEC =Q.

a. Calculate the output and price of dry cleaning if it is produced under competitive conditions without regulation.

b. Determine the socially efficient price and output of dry cleaning.

c. Determine the tax that would result in a competitive market producing the socially efficient output.

d. Calculate the output and price of dry cleaning if itis produced under monopolistic conditions without regulation.

e. Determine the tax that would result in a monopolistic market producing the socially efficient output.

f. Assuming that no attempt is made to monitor or regulate the pollution, which market structure yields higher social welfare? Discuss.

Short answer

1. The output and price of the dry-cleaning firm under competitive conditions would be 45 and $55, respectively.

2. The socially efficient price and output will be $70 and 30, respectively.

3. The tax that would result in a competitive market producing efficient social output would be $1

4. The output and price of a dry-cleaning firm producing under monopolistic conditions would be 30 and $70, respectively.

5. The tax of producing the socially efficient output under monopolistic conditions would be zero.

6. The monopolistic firm will yield higher social welfare.

Step-by-step solution

Price and output under competitive conditions

To find the competitive equilibrium price and output, set demand equal to supply. Since supply is not given, use marginal cost (which is the industry supply function):

$$\begin{gathered} \text{100 - Q = 10 + Q} \\ \text{Q = 45} \\ \text{P = 100 - 45} \\ \text{P = 55} \end{gathered}$$

Therefore, the competitive equilibrium price and output will be $55 and 45, respectively.

Socially efficient price and output

The socially efficient price is output is determined where MSC=MSB. To find a socially efficient price and output, calculate the marginal social cost (MSC), which is equal to the marginal external cost plus the private marginal cost. Secondly, set MSCequal to the market demand function, which is MSB; therefore:

$$\begin{gathered} \text{MSC = MC + MEC} \\ \text{=}\left(\text{10 + Q}\right)\text{+ Q = 10 + 2Q} \\ \text{MEC = MSB} \\ \text{= 10 + 2Q = 100 - Q} \\ \text{Q = 30} \\ \text{P = 100 - 30} \\ \text{P = 70} \end{gathered}$$

Hence, the socially efficient price and output would be $70 and 30, respectively.

Tax under competitive market

If a tax is levied, then the marginal private cost will increase. Hence new MC will be,

$\text{MC =}\left(\text{10 + Q}\right)\text{+ t}$

Now, set this new marginal cost function equal to the efficient price, that is, $70, and substitutes 30 for 10 as quantity. Thus, solve for t:

$$\begin{gathered} \text{10 + Q + tQ = 70} \\ \text{30}\left(\text{1 + t}\right)\text{= 60} \\ \text{1 + t = 2} \\ \text{t = 1} \end{gathered}$$

Hence the tax would be $1.

Price and output under monopolistic market

The equilibrium price and output of a firm will where the marginal revenue is equal to marginal cost. The slope of the marginal revenue curve under monopolistic competition is twice the slope of the demand curve; therefore,

$$\begin{gathered} \text{MR = MC} \\ \text{= 100 - 2Q = 10 + Q} \\ \text{Q = 30} \\ \text{P = 100 - 30} \\ \text{P = 70} \end{gathered}$$

Hence, the monopolistic firm produces a socially efficient output.

Tax under monopolistic market

In the above step, the monopolistic firm produces the socially efficient level of output at the socially efficient level of price. Therefore, there would be no tax levied on the firm under monopolistic competition.

Higher social welfare

The dry-cleaning firm under the monopolistic competition would yield higher social welfare since they produce the quantity that is socially efficient at a socially efficient price. While the dry-cleaning firm does not produce at a socially efficient level, thus they will not yield maximum social welfare.