<p>In your maximization problem the resident chooses the level of pollution <span class="math-container">$u$</span>. If he can really do this then the maximization problem has no solution, since <span class="math-container">$U$</span> becomes infinite for <span class="math-container">$u\rightarrow 0$</span>.</p>
<p>I guess actually the resident doesn't choose the level of pollution, which rather seems to be an exogenously given parameter here. But then <span class="math-container">$(x^*,k^*)$</span> doesn't depend on <span class="math-container">$u$</span>, since <span class="math-container">$u$</span> only scales the utility function. Then showing that utility decreases in pollution is trivial, and also the additional assumption at the end is redundant. Is this really the original version of the question?</p>

CC BY-SA 4.0
2021-04-20T10:08:54.497Z
Or does the resident own the factory, with both pollution $u$ and his income $W$ increasing in production? But then we need some additional assumptions...
VARulle
CC BY-SA 4.0
2021-04-20T10:47:52.747Z
I had similar concerns, VARulle. It doesn't make much sense to me that the agent would choose the pollution levels, but if we don't incorporate it somehow, then the question becomes trivial. I will add the rest of the question to see if it helps. Thank you!
Pedro Cunha
CC BY-SA 4.0
2021-04-20T09:54:31.057Z
2021-04-20T09:54:31.057Z
VARulle
<p>A factory releases a toxic pollutant which causes two types of damage to a representative area resident whose utility function is <span class="math-container">$U(S,H,x)= a \cdot \log(S) + b \cdot \log(H) + c \cdot \log (x)$</span> where <span class="math-container">$S$</span> = health level; <span class="math-container">$H$</span> = market price of residential property, a measure of quality of housing, such that <span class="math-container">$b \cdot \log(H)$</span> represents the satisfaction of the act of residing in the property. In the first place, the health level is moderately reduced by the greater recurrence of allergy symptoms such as coughing and throat soreness, which is given by the function <span class="math-container">$S = s_0 \cdot \frac{k}{u}$</span>, <span class="math-container">$u$</span> being the amount of the toxic pollutant and <span class="math-container">$k$</span> being the amount of health services. The constant <span class="math-container">$s_0$</span> is positive. The second effect is the deterioration of the painting and external aspect of surrounding properties, which occurs via the function <span class="math-container">$H = h(u), \frac{\operatorname{d}\!h}{\operatorname{d}\!u} &lt; 0$</span>. The agent's budget constraint is <span class="math-container">$p \cdot x + r \cdot k = W$</span>, where <span class="math-container">$p$</span> is the price of the consumption good. All relevant markets are competitive.</p>
<p>Argue that <span class="math-container">$\frac{\partial U(k^*(u), h(u), x^*(u))}{\partial u} = \frac{a}{S(k^*)} \cdot \frac{\partial}{\partial u} \left(\frac{s_0 \cdot k^*(u)}{u}\right) + \frac{b}{H(u)} \cdot \frac{\operatorname{d}}{\operatorname{d}\!u} H(u) + \frac{c}{x^*} \cdot \frac{\partial}{\partial u} x^*(u)&lt; 0$</span>, that is, a large exposition to pollution indeed leads to a loss in welfare assuming that <span class="math-container">$\frac{\partial}{\partial u} \left(s_0 \cdot \frac{k^*(u)}{u} \right) &lt; 0$</span> and using the information provided in the question.</p>
<p>What I tried to do is: first we setup the maximization problem:
<span class="math-container">$$
\underset{k,u, x}{\operatorname{max}} \,\, U(k, h(u), x) = a \cdot \log \left[s_0 \cdot \frac{k}{u}\right] + b \cdot \log[h(u)]+ c \cdot \log (x) \\ s.t. \quad p \cdot x + r \cdot k = W
$$</span>
which gives us the following first order conditions:
<span class="math-container">$$
\begin{cases}
\frac{\partial \mathcal{L}}{\partial k} = 0 \rightarrow a \cdot \frac{\frac{s_0}{u}}{s_0 \cdot \frac{k}{u}} - \lambda \cdot r = 0 \therefore \lambda = \frac{a}{k \cdot r} \\
\frac{\partial \mathcal{L}}{\partial x} = 0 \rightarrow \frac{c}{x} - \lambda \cdot p = 0 \therefore \lambda = \frac{c}{x \cdot p} \\
\frac{\partial \mathcal{L}}{\partial u} = 0 \rightarrow a \cdot \frac{u}{s_0 \cdot k} \cdot \left(-\frac{s_0 \cdot k}{u^2}\right) + b \cdot \frac{1}{H(u)} \cdot H'(u) = 0 \therefore a = b \cdot \frac{H'(u)}{H(u)} \cdot u = b \cdot \epsilon_u \\
\frac{\partial \mathcal{L}}{\partial \lambda} = 0 \rightarrow p \cdot x + r \cdot k = W\end{cases}$$</span>
where <span class="math-container">$$\epsilon_u = \frac{H'(u)}{H(u)} \cdot u$$</span> is the elasticity of the price of the property with respect to the pollution level.</p>
<p>Using the first, second and last equations, we arrive in:
<span class="math-container">$$x = \frac{c \cdot W}{p \cdot (a+c)}, \qquad k = \frac{a \cdot W}{r \cdot (a+c)}$$</span></p>
<p>Now substituting the value we've found for <span class="math-container">$a$</span> in these equations, we have:
<span class="math-container">$$x^\star(u) = \frac{c \cdot W}{p} \cdot \left(\frac{1}{b \cdot \epsilon_u + c}\right)$$</span>
and
<span class="math-container">$$k^\star(u) = \frac{b \cdot W}{r} \cdot \left(\frac{\epsilon_u}{b \cdot \epsilon_u + c}\right)$$</span></p>
<p>Now, taking the derivative of <span class="math-container">$x^\star(u)$</span> w.r.t to <span class="math-container">$u$</span> and rearranging the expression, we find that
<span class="math-container">$$\frac{\partial x^\star(u)}{\partial u} = \frac{c \cdot W}{p} \cdot \left(-\frac{b \cdot \epsilon_u'}{(b \cdot \epsilon_u + c)^2}\right) = -\frac{b \cdot c \cdot W}{p \cdot (b \cdot \epsilon_u + c)^2} \cdot \epsilon_u'$$</span>
and, therefore, the sign of this expression depends solely on the sign of <span class="math-container">$\epsilon_u'$</span>.</p>
<p>Considering the information given in the question, we can see that <span class="math-container">$\frac{\partial U(k^*(u), h(u), x^*(u))}{\partial u} &lt; 0$</span> will occur if <span class="math-container">$\frac{\partial}{\partial x} x^*(u) &lt; 0 \iff \epsilon_u' &gt; 0$</span>. With that in mind, let's try to define the signal of that derivative:
<span class="math-container">$$
\begin{aligned}
\epsilon_u’ &amp;= \left( \frac{H’’(u) \cdot H(u) - H’(u) \cdot H’(u)}{H(u)^2}\right) \cdot u + \frac{H’(u)}{H(u)} \\ &amp;= \frac{H''(u)}{H(u)} \cdot u - \left[ \frac{H'(u)}{H(u)} \right]^2 \cdot u + \frac{H'(u)}{H(u)} \\ &amp;= \frac{H''(u)}{H(u)} \cdot u + \frac{H'(u)}{H(u)} \cdot \left[ 1 - \frac{H'(u)}{H(u)} \cdot u \right]
\end{aligned}
$$</span></p>
<p>Now, since <span class="math-container">$H(u)$</span> is always positive and <span class="math-container">$H'(u)$</span> is negative, <span class="math-container">$\frac{H'(u)}{H(u)} &lt; 0$</span> and <span class="math-container">$1 - \frac{H'(u)}{H(u)} \cdot u &gt; 0$</span>, which means that <span class="math-container">$\frac{H'(u)}{H(u)} \cdot \left[ \frac{H'(u)}{H(u)} \cdot u \right] &lt; 0$</span>. If we further suppose that <span class="math-container">$H(u)$</span> exhibits decreasing marginal returns in <span class="math-container">$u$</span>, then <span class="math-container">$H''(u) &lt; 0$</span> and thus, <span class="math-container">$\epsilon_u' &lt; 0$</span>.</p>
<p>Since I found that <span class="math-container">$\epsilon_u' &lt; 0$</span>, that means <span class="math-container">$\frac{\partial}{\partial u} x^\star(u) &gt; 0$</span> and because of that I cannot ascertain whether <span class="math-container">$\frac{\partial U(k^*(u), h(u), x^*(u))}{\partial u} &lt; 0$</span> or not. What did I do wrong?</p>
<p>I appreciate any help!</p>
<p>EDIT: Adding the rest of the questions, to see if it helps.</p>
<p>a) Find the marginal impact of the degree of pollution <span class="math-container">$u$</span> in the acquired quantities of the consumption goods and the health services.</p>
<p>b.1) Show that the marginal impact of the degree of pollution in the &quot;utility level of equilibrium&quot; is <span class="math-container">$\frac{\partial U(k^*(u), h(u), x^*(u))}{\partial u} = \frac{a}{S(k^*)} \cdot \frac{\partial}{\partial u} \left(\frac{s_0 \cdot k^*(u)}{u}\right) + \frac{b}{H(u)} \cdot \frac{\operatorname{d}}{\operatorname{d}\!u} H(u) + \frac{c}{x^*} \cdot \frac{\partial}{\partial u}$</span>.</p>
<p>b.2) Argue that <span class="math-container">$\frac{\partial U(k^*(u), h(u), x^*(u))}{\partial u} = \frac{a}{S(k^*)} \cdot \frac{\partial}{\partial u} \left(\frac{s_0 \cdot k^*(u)}{u}\right) + \frac{b}{H(u)} \cdot \frac{\operatorname{d}}{\operatorname{d}\!u} H(u) + \frac{c}{x^*} \cdot \frac{\partial}{\partial u} x^*(u)&lt; 0$</span>, that is, a large exposition to pollution indeed leads to a loss in welfare assuming that <span class="math-container">$\frac{\partial}{\partial u} \left(s_0 \cdot \frac{k^*(u)}{u} \right) &lt; 0$</span> and using the information provided in the question.</p>

CC BY-SA 4.0
2021-04-20T09:19:12.753Z
I guess $k$ and $r$ is not capital and rental rate here, but quantity of health services consumed and price of health services or something like this?
VARulle
CC BY-SA 4.0
2021-04-20T10:49:49.940Z
Yes. I seemed to have skipped this line when transcribing the question, but it's exactly as you described.
Pedro Cunha
<microeconomics><utility><environmental-economics>
Utility maximization: pollution and housing prices
Pedro Cunha
# Utility maximization: pollution and housing prices

(Asked by: username_0 on 2021-04-19)

A factory releases a toxic pollutant which causes two types of damage to a representative area resident whose utility function is $U(S,H,x)= a \cdot \log(S) + b \cdot \log(H) + c \cdot \log (x)$ where $S$ = health level; $H$ = market price of residential property, a measure of quality of housing, such that $b \cdot \log(H)$ represents the satisfaction of the act of residing in the property. In the first place, the health level is moderately reduced by the greater recurrence of allergy symptoms such as coughing and throat soreness, which is given by the function $S = s\_0 \cdot \frac{k}{u}$, $u$ being the amount of the toxic pollutant and $k$ being the amount of health services. The constant $s\_0$ is positive. The second effect is the deterioration of the painting and external aspect of surrounding properties, which occurs via the function $H = h(u), \frac{\operatorname{d}\!h}{\operatorname{d}\!u} < 0$. The agent's budget constraint is $p \cdot x + r \cdot k = W$, where $p$ is the price of the consumption good. All relevant markets are competitive.

Argue that $\frac{\partial U(k^\*(u), h(u), x^\*(u))}{\partial u} = \frac{a}{S(k^\*)} \cdot \frac{\partial}{\partial u} \left(\frac{s\_0 \cdot k^\*(u)}{u}\right) + \frac{b}{H(u)} \cdot \frac{\operatorname{d}}{\operatorname{d}\!u} H(u) + \frac{c}{x^\*} \cdot \frac{\partial}{\partial u} x^\*(u)< 0$, that is, a large exposition to pollution indeed leads to a loss in welfare assuming that $\frac{\partial}{\partial u} \left(s\_0 \cdot \frac{k^\*(u)}{u} \right) < 0$ and using the information provided in the question.

What I tried to do is: first we setup the maximization problem:
$$
\underset{k,u, x}{\operatorname{max}} \,\, U(k, h(u), x) = a \cdot \log \left[s\_0 \cdot \frac{k}{u}\right] + b \cdot \log[h(u)]+ c \cdot \log (x) \\ s.t. \quad p \cdot x + r \cdot k = W
$$
which gives us the following first order conditions:
$$
\begin{cases}
\frac{\partial \mathcal{L}}{\partial k} = 0 \rightarrow a \cdot \frac{\frac{s\_0}{u}}{s\_0 \cdot \frac{k}{u}} - \lambda \cdot r = 0 \therefore \lambda = \frac{a}{k \cdot r} \\
\frac{\partial \mathcal{L}}{\partial x} = 0 \rightarrow \frac{c}{x} - \lambda \cdot p = 0 \therefore \lambda = \frac{c}{x \cdot p} \\
\frac{\partial \mathcal{L}}{\partial u} = 0 \rightarrow a \cdot \frac{u}{s\_0 \cdot k} \cdot \left(-\frac{s\_0 \cdot k}{u^2}\right) + b \cdot \frac{1}{H(u)} \cdot H'(u) = 0 \therefore a = b \cdot \frac{H'(u)}{H(u)} \cdot u = b \cdot \epsilon\_u \\
\frac{\partial \mathcal{L}}{\partial \lambda} = 0 \rightarrow p \cdot x + r \cdot k = W\end{cases}$$
where $$\epsilon\_u = \frac{H'(u)}{H(u)} \cdot u$$ is the elasticity of the price of the property with respect to the pollution level.

Using the first, second and last equations, we arrive in:
$$x = \frac{c \cdot W}{p \cdot (a+c)}, \qquad k = \frac{a \cdot W}{r \cdot (a+c)}$$

Now substituting the value we've found for $a$ in these equations, we have:
$$x^\star(u) = \frac{c \cdot W}{p} \cdot \left(\frac{1}{b \cdot \epsilon\_u + c}\right)$$
and
$$k^\star(u) = \frac{b \cdot W}{r} \cdot \left(\frac{\epsilon\_u}{b \cdot \epsilon\_u + c}\right)$$

Now, taking the derivative of $x^\star(u)$ w.r.t to $u$ and rearranging the expression, we find that
$$\frac{\partial x^\star(u)}{\partial u} = \frac{c \cdot W}{p} \cdot \left(-\frac{b \cdot \epsilon\_u'}{(b \cdot \epsilon\_u + c)^2}\right) = -\frac{b \cdot c \cdot W}{p \cdot (b \cdot \epsilon\_u + c)^2} \cdot \epsilon\_u'$$
and, therefore, the sign of this expression depends solely on the sign of $\epsilon\_u'$.

Considering the information given in the question, we can see that $\frac{\partial U(k^\*(u), h(u), x^\*(u))}{\partial u} < 0$ will occur if $\frac{\partial}{\partial x} x^\*(u) < 0 \iff \epsilon\_u' > 0$. With that in mind, let's try to define the signal of that derivative:
$$
\begin{aligned}
\epsilon\_u’ &= \left( \frac{H’’(u) \cdot H(u) - H’(u) \cdot H’(u)}{H(u)^2}\right) \cdot u + \frac{H’(u)}{H(u)} \\ &= \frac{H''(u)}{H(u)} \cdot u - \left[ \frac{H'(u)}{H(u)} \right]^2 \cdot u + \frac{H'(u)}{H(u)} \\ &= \frac{H''(u)}{H(u)} \cdot u + \frac{H'(u)}{H(u)} \cdot \left[ 1 - \frac{H'(u)}{H(u)} \cdot u \right]
\end{aligned}
$$

Now, since $H(u)$ is always positive and $H'(u)$ is negative, $\frac{H'(u)}{H(u)} < 0$ and $1 - \frac{H'(u)}{H(u)} \cdot u > 0$, which means that $\frac{H'(u)}{H(u)} \cdot \left[ \frac{H'(u)}{H(u)} \cdot u \right] < 0$. If we further suppose that $H(u)$ exhibits decreasing marginal returns in $u$, then $H''(u) < 0$ and thus, $\epsilon\_u' < 0$.

Since I found that $\epsilon\_u' < 0$, that means $\frac{\partial}{\partial u} x^\star(u) > 0$ and because of that I cannot ascertain whether $\frac{\partial U(k^\*(u), h(u), x^\*(u))}{\partial u} < 0$ or not. What did I do wrong?

I appreciate any help!

EDIT: Adding the rest of the questions, to see if it helps.

a) Find the marginal impact of the degree of pollution $u$ in the acquired quantities of the consumption goods and the health services.

b.1) Show that the marginal impact of the degree of pollution in the "utility level of equilibrium" is $\frac{\partial U(k^\*(u), h(u), x^\*(u))}{\partial u} = \frac{a}{S(k^\*)} \cdot \frac{\partial}{\partial u} \left(\frac{s\_0 \cdot k^\*(u)}{u}\right) + \frac{b}{H(u)} \cdot \frac{\operatorname{d}}{\operatorname{d}\!u} H(u) + \frac{c}{x^\*} \cdot \frac{\partial}{\partial u}$.

b.2) Argue that $\frac{\partial U(k^\*(u), h(u), x^\*(u))}{\partial u} = \frac{a}{S(k^\*)} \cdot \frac{\partial}{\partial u} \left(\frac{s\_0 \cdot k^\*(u)}{u}\right) + \frac{b}{H(u)} \cdot \frac{\operatorname{d}}{\operatorname{d}\!u} H(u) + \frac{c}{x^\*} \cdot \frac{\partial}{\partial u} x^\*(u)< 0$, that is, a large exposition to pollution indeed leads to a loss in welfare assuming that $\frac{\partial}{\partial u} \left(s\_0 \cdot \frac{k^\*(u)}{u} \right) < 0$ and using the information provided in the question.

**Tags:** microeconomics, utility, environmental-economics
> **Comments:**
> **username_1**: I guess $k$ and $r$ is not capital and rental rate here, but quantity of health services consumed and price of health services or something like this?  
> **username_0**: Yes. I seemed to have skipped this line when transcribing the question, but it's exactly as you described.  

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## 1 answers

### Answer by username_1 on 2021-04-20. Score: 2

In your maximization problem the resident chooses the level of pollution $u$. If he can really do this then the maximization problem has no solution, since $U$ becomes infinite for $u\rightarrow 0$.

I guess actually the resident doesn't choose the level of pollution, which rather seems to be an exogenously given parameter here. But then $(x^\*,k^\*)$ doesn't depend on $u$, since $u$ only scales the utility function. Then showing that utility decreases in pollution is trivial, and also the additional assumption at the end is redundant. Is this really the original version of the question?

> **Comments:**
> **username_1**: Or does the resident own the factory, with both pollution $u$ and his income $W$ increasing in production? But then we need some additional assumptions...  
> **username_0**: I had similar concerns, VARulle. It doesn't make much sense to me that the agent would choose the pollution levels, but if we don't incorporate it somehow, then the question becomes trivial. I will add the rest of the question to see if it helps. Thank you!  

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