<p>The first order condition for individual <span class="math-container">$a$</span> when the price of <span class="math-container">$y$</span> equals <span class="math-container">$p_y(1+t)$</span> is given by:
<span class="math-container">$$
\frac{\partial u_a}{\partial y_a} = p_y(1+t) \left(\frac{1}{p_x} \frac{\partial u_a}{\partial x_a} \right)
$$</span></p>
<p>Rewriting the first order condition for the social equilibrium gives:
<span class="math-container">$$
\frac{\partial u_a}{\partial y_a} = p_y\left[1  + \left(\frac{p_x}{p_y} \dfrac{-
\dfrac{\partial u_b}{\partial y_b}}{\dfrac{\partial u_a}{\partial x_a}}\right)\right]\left(\frac{1}{p_x}\frac{\partial u_a}{\partial x_a}\right)
$$</span></p>
<p>So we see that the two are equal when:
<span class="math-container">$$
t = \frac{p_x}{p_y}\frac{-\dfrac{\partial u_b}{y_b}}{\dfrac{\partial u_a}{\partial x_a}}
$$</span></p>
<p>This is the marginal external cost as you have it, but you have to normalize it by the marginal utility of income for person <span class="math-container">$a$</span> which is equal to:
<span class="math-container">$$
\dfrac{\dfrac{\partial u_a}{\partial x_a}}{p_a}
$$</span>
The division by <span class="math-container">$p_y$</span> is due to the fact that the tax is computed as a fraction of the price of <span class="math-container">$y$</span>.</p>

CC BY-SA 4.0
2021-08-14T16:40:51.213Z
If I were to consider allocating property rights to person b. Let $z$ be the rights to impose this external cost on person b, and $p_z$ be the price of such rights. Then person a's problem is
$$\max_{x_a,\ y_a} \ u_a(x_a,y_a)$$
s.t.
\begin{align*}
p_xx_a+p_yy_a+p_zz_a&=e_a,\\
z_a&=y_a.
\end{align*}
In equilibrium, person a's and person b's first-order conditions are equal to $\frac{p_z}{p_x}$. When I rearrange to get a condition similar to that for social maximisation, I find that
$$p_z=p_x\frac{-\partial u_b/\partial y_a}{\partial u_b/\partial x_b}.$$
Is this the same as $t$ above?
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CC BY-SA 4.0
2021-08-12T19:10:50.983Z
2021-08-12T19:10:50.983Z
tdm
<p>Suppose person a's consumption of good <span class="math-container">$y$</span> imposes a negative externality on person b. Person a's utility maximisation problem is
<span class="math-container">$$\max_{x_a,y_a} \ u_a(x_a,y_a),$$</span>
subject to
<span class="math-container">$$p_x x_a+p_y y_a=e_a.$$</span>
The first-order condition is
<span class="math-container">$$\underbrace{\frac{\partial u_a}{\partial{y_a}}}_{\substack{\text{marginal} \\ \text{private} \\ \text{benefit}}}-\underbrace{\frac{p_y}{p_x}\frac{\partial u_a }{\partial x_a}}_{\substack{\text{marginal}\\ \text{private cost}}}=0.$$</span></p>
<p>The social welfare maximisation problem is
<span class="math-container">$$\max_{x_a,\ y_a, \ x_b} \ u_a(x_a,y_a)+u_b(x_b,y_a),$$</span>
subject to
<span class="math-container">$$\begin{align*}
p_xx_a+p_yy_a&amp;=e_a,\\
p_xx_b&amp;=e_b.
\end{align*}$$</span>
The first-order condition is
<span class="math-container">$$\underbrace{\frac{\partial u_a}{\partial y_a}}_{\substack{\text{marginal} \\ \text{private} \\ \text{benefit}}}-\underbrace{ \underbrace{\frac{p_y}{p_x}\frac{\partial u_a }{\partial x_a}}_{\substack{\text{marginal} \\ \text{private cost}}}\ \ + \underbrace{\frac{\partial u_b}{\partial y_a}}_{\substack{\text{marginal} \\ \text{external} \\ \text{cost}}}}_{\text{marginal social cost}}=0,\\\\$$</span>
where <span class="math-container">$\partial u_b/\partial y_a&lt;0$</span>.</p>
<p>An optimal Pigouvian tax should bring the competitive equilibrium to the social optimum. However, for some reason when I rewrite person a's budget constraint as <span class="math-container">$p_x x_a+(p_y+t) y_a=e_a$</span> and set <span class="math-container">$t$</span> equal to the marginal external cost, I do not get the first-order condition for the social optimum when I solve person a's utility maximisation problem again. What am I doing wrong?</p>

<consumer-theory><public-economics><welfare-economics><externalities><pigouvian-taxes>
Pigouvian tax with general utility function
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# Pigouvian tax with general utility function

(Asked by: username_0 on 2021-08-12)

Suppose person a's consumption of good $y$ imposes a negative externality on person b. Person a's utility maximisation problem is
$$\max\_{x\_a,y\_a} \ u\_a(x\_a,y\_a),$$
subject to
$$p\_x x\_a+p\_y y\_a=e\_a.$$
The first-order condition is
$$\underbrace{\frac{\partial u\_a}{\partial{y\_a}}}\_{\substack{\text{marginal} \\ \text{private} \\ \text{benefit}}}-\underbrace{\frac{p\_y}{p\_x}\frac{\partial u\_a }{\partial x\_a}}\_{\substack{\text{marginal}\\ \text{private cost}}}=0.$$

The social welfare maximisation problem is
$$\max\_{x\_a,\ y\_a, \ x\_b} \ u\_a(x\_a,y\_a)+u\_b(x\_b,y\_a),$$
subject to
$$\begin{align\*}
p\_xx\_a+p\_yy\_a&=e\_a,\\
p\_xx\_b&=e\_b.
\end{align\*}$$
The first-order condition is
$$\underbrace{\frac{\partial u\_a}{\partial y\_a}}\_{\substack{\text{marginal} \\ \text{private} \\ \text{benefit}}}-\underbrace{ \underbrace{\frac{p\_y}{p\_x}\frac{\partial u\_a }{\partial x\_a}}\_{\substack{\text{marginal} \\ \text{private cost}}}\ \ + \underbrace{\frac{\partial u\_b}{\partial y\_a}}\_{\substack{\text{marginal} \\ \text{external} \\ \text{cost}}}}\_{\text{marginal social cost}}=0,\\\\$$
where $\partial u\_b/\partial y\_a<0$.

An optimal Pigouvian tax should bring the competitive equilibrium to the social optimum. However, for some reason when I rewrite person a's budget constraint as $p\_x x\_a+(p\_y+t) y\_a=e\_a$ and set $t$ equal to the marginal external cost, I do not get the first-order condition for the social optimum when I solve person a's utility maximisation problem again. What am I doing wrong?

**Tags:** consumer-theory, public-economics, welfare-economics, externalities, pigouvian-taxes


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## 1 answers

### Answer by username_1 on 2021-08-12. Score: 5 | Accepted answer

The first order condition for individual $a$ when the price of $y$ equals $p\_y(1+t)$ is given by:
$$
\frac{\partial u\_a}{\partial y\_a} = p\_y(1+t) \left(\frac{1}{p\_x} \frac{\partial u\_a}{\partial x\_a} \right)
$$

Rewriting the first order condition for the social equilibrium gives:
$$
\frac{\partial u\_a}{\partial y\_a} = p\_y\left[1 + \left(\frac{p\_x}{p\_y} \dfrac{-
\dfrac{\partial u\_b}{\partial y\_b}}{\dfrac{\partial u\_a}{\partial x\_a}}\right)\right]\left(\frac{1}{p\_x}\frac{\partial u\_a}{\partial x\_a}\right)
$$

So we see that the two are equal when:
$$
t = \frac{p\_x}{p\_y}\frac{-\dfrac{\partial u\_b}{y\_b}}{\dfrac{\partial u\_a}{\partial x\_a}}
$$

This is the marginal external cost as you have it, but you have to normalize it by the marginal utility of income for person $a$ which is equal to:
$$
\dfrac{\dfrac{\partial u\_a}{\partial x\_a}}{p\_a}
$$
The division by $p\_y$ is due to the fact that the tax is computed as a fraction of the price of $y$.

> **Comments:**
> **username_0**: If I were to consider allocating property rights to person b. Let $z$ be the rights to impose this external cost on person b, and $p_z$ be the price of such rights. Then person a's problem is $$\max_{x_a,\ y_a} \ u_a(x_a,y_a)$$ s.t. \begin{align*} p_xx_a+p_yy_a+p_zz_a&=e_a,\\ z_a&=y_a. \end{align*} In equilibrium, person a's and person b's first-order conditions are equal to $\frac{p_z}{p_x}$. When I rearrange to get a condition similar to that for social maximisation, I find that $$p_z=p_x\frac{-\partial u_b/\partial y_a}{\partial u_b/\partial x_b}.$$ Is this the same as $t$ above?  

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