Externality and Pigovian Tax Example

Related Knowledge background:

• Pigou Tax
• Externality, Externalities and Public Goods
• Externalities with High Transaction Cost

Question Background

A fishery is an open-access resource that allows anyone who wants to enter it and catch fish. The average number of fish a boat can catch each day depends on the number of boats fishing. Specifically, the number of fish a single boat catches is given by $Q=24-0.5N$, where $Q$ is the quantity of fish per boat, and $N$ is the total number of boats on the water. The opportunity cost of fishing is \10$perdayforeachfisherman.Themarketpriceforafishis$$1$.

Question 1

How many boats will fish each day? How much fish will each boat bring in and how much profit will they earn?

Since if this fishery is profitable, there would be more and more boats come in, until there is no profit. ($\pi =0$)

So,

$$\begin{array}{rl}\pi & =TR-TC\\ 0& =(24-0.5N)\ast 1-10\\ N& =28\end{array}$$

So there would be $28$ boats and the quantity of fish would be $10$.

Question 2

What is the efficient number of boats on the water each day?

Since the efficient means the ${\pi }_{max}$, so we have to take the derivative. Also notice, for private, $\pi =14-0.5N$, for public, $\Pi =14N-0.5N^2$. So,

$$\begin{array}{r}\Pi =14N-0.5N^2\\ \frac{\delta \Pi }{\delta N}=14-N=0\end{array}$$

So $N^{\ast }=14$.

Question 3

What is the Deadweight Loss (loss in social gains) from allowing open access to the fishery?

$${\Pi }_{max}=14\ast 14-0.5\ast 14\ast 14=98$$

So the $DWL=98$ (Because the reality is $\Pi =0$)

Question 4

Suppose boats are charged a fee to fish on the lake each day. What fee would lead to the efficient number of boats on the lake?

Suppose there exists a tax $t$, so,

$\pi =14-0.5N-t$

In order to make $N$ to be $14$, so make when $N=14,\pi =0$

So $t=7$

(Similar method about Pigou Tax)