Chapter 19, Problem 19.1P

A firm in a perfectly competitive industry has patented a newprocess for making widgets. The new process lowers the firm’saverage cost, meaning that this firm alone (although still aprice taker) can earn real economic profits in the long run.
a. If the market price is $20 per widget and the firm’s marginalcost is given by $MC=0.4q$ , where q is the dailywidget production for the firm, how many widgets willthe firm produce?
b. Suppose a government study has found that the firm’snew process is polluting the air and estimates the socialmarginal cost of widget production by this firm to be. If the market price is still $20, what is thesocially optimal level of production for the firm? Whatshould be the rate of a government-imposed excise tax tobring about this optimal level of production?
c. Graph your results.

To determine

To find: The number of widgets produced by the firm.

Explanation of Solution

The profit maximizing situation exists at the level where market price is equal to the marginal cost.

Thus, P = MC

  $\begin{array}{c}\text{20 = 0}\text{.4q}\\ \text{q = }\frac{\text{20}}{\text{0}\text{.4}}\\ \text{q = 50 widgets}\end{array}$

To determine

To find: The socially optimal level of production and rate of tax.

Explanation of Solution

The socially optimal level exists at, P = SMC

Given SMC = 0.5q

  $\begin{array}{l}\text{Since, P = SMC}\\ \text{20 = 0}\text{.5q}\\ \text{ q = }\frac{\text{20}}{\text{0}\text{.5}}\\ \text{ q =}\text{40}\end{array}$

The socially optimal level exists at q = 40 widgets.

  $\begin{array}{c}\text{At the social optimal level of Q = 40, }\\ \text{MC = 0}\text{.4 Q }\\ \text{= 0}\text{.4}\times \text{40}\\ \text{=16 }\end{array}$

So the excise tax, T = 20 − 16 = $4

Thus, $4 tax should be applied to bring the optimal level of production.

To determine

To find: The graph for the calculated results.

Explanation of Solution

Below graph shows the price and marginal cost on vertical axis while the quantity is on horizontal axis. Moreover, the gap between MC and SMC shows the tax imposed by the government.