<p>Finding TCA and TEC are examples of the common requirement in economics to calculate a total quantity when one knows the corresponding marginal quantity. Mathematically, this is equivalent to finding the area below the curve representing the marginal quantity.  Sometimes this requires integral calculus, but when as here the marginal curves are linear it is simpler to use the formulae for the area of a triangle and a rectangle.</p>
<p>The diagram below plots both MCA and MEC against E. Point X, their intersection, corresponds to the efficient pollution level.</p>
<p><a href="https://i.sstatic.net/65AbJk5B.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/65AbJk5B.png" alt="enter image description here" /></a></p>
<p>The change in effluent required to reach the efficient level is from <span class="math-container">$E=8$</span> to <span class="math-container">$E=4$</span>, so we are interested in areas on the right hand side of the diagram.  Thus the change in <span class="math-container">$TCA$</span> is triangle <strong>a</strong> whose area (1/2 x base x height) is <span class="math-container">$\frac{1}{2} \cdot 4 \cdot 4=8$</span>.  For the change in <span class="math-container">$TEC$</span> we need the rectangle consisting of triangles <strong>a</strong> and <strong>b</strong> (base x height) which is <span class="math-container">$4 \cdot 4 = 16$</span> and also triangle <strong>c</strong> whose area is <span class="math-container">$\frac{1}{2} \cdot 4 \cdot 4=8$</span> making a total of <span class="math-container">$16+8=24$</span>.</p>

CC BY-SA 4.0
2024-05-08T22:32:37.820Z
2024-05-09T19:29:16.287Z
2024-05-09T19:29:16.287Z
Adam Bailey
<p>There is a manufacturer and a fishery. Let <span class="math-container">$E$</span> be the effluent of the manufacturer. The marginal cost of abatement borne by the manufacturer is <span class="math-container">$M C A=8-E$</span>. The marginal external cost borne by the fishery is <span class="math-container">$M E C=E$</span>. Assume that the manufacturer has the right to release effluent subject to government control.
(a) What is the efficient pollution level?
(b) How can an effluent fee be used to achieve the efficient pollution level?
(c) How can bargaining between the two parties be used to achieve the efficient pollution level?</p>
<p>(a) The efficient level of pollution satisfies <span class="math-container">$M C A=8-E=E=M E C$</span>, which gives <span class="math-container">$E=4$</span></p>
<p>(b) Suppose that the effluent fee is <span class="math-container">$F$</span> per unit of pollution. The manufacturer would choose emissions to satisfy <span class="math-container">$M C A=8-E=F$</span>, which gives <span class="math-container">$E=8-F$</span>. Hence, an effluent fee of <span class="math-container">$F=4$</span> would induce the efficient pollution level <span class="math-container">$E=4$</span>.</p>
<p>(c) The increase in the total cost of abatement from <span class="math-container">$E=8$</span> to <span class="math-container">$E=4$</span> is <span class="math-container">$T C A=$</span> <span class="math-container">$\frac{1}{2} \cdot 4 \cdot 4=8$</span>. The decrease in the total external cost from <span class="math-container">$E=8$</span> to <span class="math-container">$E=4$</span> is TEC <span class="math-container">$=4 \cdot 4+\frac{1}{2} \cdot 4 \cdot 4=24$</span>. Hence, the fishery could offer the manufacturer a payment between 8 and 24 to reduce effluent from 8 to 4 .</p>
<p>Question: I don't understand the solution of (c). Can anyone help me explain part (c)?</p>

CC BY-SA 4.0
2024-05-08T19:54:39.217Z
What exactly do you not understand about the solution to (c)?  Is it the calculations leading to $TCA=8$ and $TEC=24$?  Or is it the conclusion drawn from these numbers?
Adam Bailey
CC BY-SA 4.0
2024-05-08T19:56:18.280Z
@AdamBailey I don't know how to get TCA and TEC.
PiVoyager
<microeconomics><public-economics>
Public goods on cost of abatement
PiVoyager
# Public goods on cost of abatement

(Asked by: username_0 on 2024-05-08)

There is a manufacturer and a fishery. Let $E$ be the effluent of the manufacturer. The marginal cost of abatement borne by the manufacturer is $M C A=8-E$. The marginal external cost borne by the fishery is $M E C=E$. Assume that the manufacturer has the right to release effluent subject to government control.
(a) What is the efficient pollution level?
(b) How can an effluent fee be used to achieve the efficient pollution level?
(c) How can bargaining between the two parties be used to achieve the efficient pollution level?

(a) The efficient level of pollution satisfies $M C A=8-E=E=M E C$, which gives $E=4$

(b) Suppose that the effluent fee is $F$ per unit of pollution. The manufacturer would choose emissions to satisfy $M C A=8-E=F$, which gives $E=8-F$. Hence, an effluent fee of $F=4$ would induce the efficient pollution level $E=4$.

(c) The increase in the total cost of abatement from $E=8$ to $E=4$ is $T C A=$ $\frac{1}{2} \cdot 4 \cdot 4=8$. The decrease in the total external cost from $E=8$ to $E=4$ is TEC $=4 \cdot 4+\frac{1}{2} \cdot 4 \cdot 4=24$. Hence, the fishery could offer the manufacturer a payment between 8 and 24 to reduce effluent from 8 to 4 .

Question: I don't understand the solution of (c). Can anyone help me explain part (c)?

**Tags:** microeconomics, public-economics
> **Comments:**
> **username_1**: What exactly do you not understand about the solution to (c)?  Is it the calculations leading to $TCA=8$ and $TEC=24$?  Or is it the conclusion drawn from these numbers?  
> **username_0**: @AdamBailey I don't know how to get TCA and TEC.  

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## 1 answers

### Answer by username_1 on 2024-05-08. Score: 0

Finding TCA and TEC are examples of the common requirement in economics to calculate a total quantity when one knows the corresponding marginal quantity. Mathematically, this is equivalent to finding the area below the curve representing the marginal quantity. Sometimes this requires integral calculus, but when as here the marginal curves are linear it is simpler to use the formulae for the area of a triangle and a rectangle.

The diagram below plots both MCA and MEC against E. Point X, their intersection, corresponds to the efficient pollution level.

[![enter image description here](https://i.sstatic.net/65AbJk5B.png)](https://i.sstatic.net/65AbJk5B.png)

The change in effluent required to reach the efficient level is from $E=8$ to $E=4$, so we are interested in areas on the right hand side of the diagram. Thus the change in $TCA$ is triangle **a** whose area (1/2 x base x height) is $\frac{1}{2} \cdot 4 \cdot 4=8$. For the change in $TEC$ we need the rectangle consisting of triangles **a** and **b** (base x height) which is $4 \cdot 4 = 16$ and also triangle **c** whose area is $\frac{1}{2} \cdot 4 \cdot 4=8$ making a total of $16+8=24$.


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