<p>To obtain $\frac{\partial y}{\partial z_n}=0$, you implicitly assume that each consumer takes its own skill level as <em>given</em>. And in such a case, in reality the consumer optimizes with respect to labor supplied only. Perhaps it would be better to optimaze with respect to $l$ given $n$. </p>

<p>Regarding aggregation the discrete analogous would not be what you write but</p>

<p>$$\sum_{s=0}^{\infty}p(s) T(z_s)$$</p>

<p>where $p(s)$ would be the proportion of the population having income $z_s$. </p>

<p>Note that $y = E[T(z_s)]$, the expected value of individual tax burden w.r.t skill level. The density of $s$ plays in the continuous case the role of "population proportions" in the discrete setting</p>

CC BY-SA 4.0
2018-06-22T13:47:37.947Z
2018-06-22T13:47:37.947Z
Alecos Papadopoulos
<p>I am working with the following Economic model of labour and consumption decisions:</p>

<ol>
<li>I have a population whose mass is normalized to one of consumers.</li>
<li>They derive utility from consumption $c$,supplying labour $l$ and a public good $y$</li>
<li>This is summarized by a differentiable utility function $u(c,l,y)$</li>
<li>Agents derive earnings $z$ from suppliying labour according to:
$$z=nl$$
where $n$ corresponds to a "skill level" such that: $n\in[0,\infty[$. Such skill level is distributed according to distribution of skills $F(n)$ with density $f(n)$.</li>
<li>$c_n$, $z_n=nl_n$ and $u_n$ are used to denote the consumption, earnings, and utility level of an individual with skill level $n$</li>
<li>the individuals maximize their utility function subject to a constraint:
$$c_n=z_n-T(z_n)$$</li>
</ol>

<p>where $T(\cdot)$ is a differentiable "tax function"</p>

<pre><code> 7.The public good is produced according to the sum of this tax function:
</code></pre>

<p>$$y=\int_0^{\infty}f(s)T(z_s)ds$$</p>

<p>We can use the equations above to express the maximization problem of the consumer as:</p>

<p>$$\max_{z_n}u\left(z_n-T(z_n), z_n/n, \int_0^{\infty}f(s)T(z_s)ds\right)$$</p>

<p>The first order conditions to this problem write (ommitting the arguments of the function $u$ for notation compactness):</p>

<p>$$u_{c}(\cdot)(1-T'(z_n))+u_l(\cdot)/n+u_y(\cdot)\cdot  \frac{\partial y}{\partial z_n}=0$$</p>

<p>I am interested in studying the last term $\frac{\partial y}{\partial z_n}$. It seems intuitive that $\frac{\partial y}{\partial z_n}=0$ since:</p>

<p>$$\frac{\partial y}{\partial z_n}=\int_0^{\infty}\frac{\partial f(s)T(z_s)}{\partial z_n}ds=0$$</p>

<p><strong>However, I am not sure about the mathematical requirement necessary for the last equation to make sense</strong>. It seems somewhat contradictory to the intuition yo get when aggregating using sums, for instance a discrete analogous would be given by:</p>

<p>$$\sum_{s=0}^{\infty} T(z_s)$$</p>

<p>and it is clear that the derivative of the last expression with respect to $z_n$ would be equal to one.</p>

<p>Could you help me out here? Maybe some broad claryfication of integral aggregation would be required. Thanks!</p>

CC BY-SA 4.0
2018-06-21T23:27:46.080Z
Why is $\frac{\partial y}{\partial z_n} = 0$? Also, an equilibrium condition on the profile $(z_n)$ is missing.
Michael
<taxation><public-economics><optimal-taxation><public-goods>
Aggregating in a Continuum
Weierstraß Ramirez
# Aggregating in a Continuum

(Asked by: username_0 on 2018-06-21)

I am working with the following Economic model of labour and consumption decisions:

1. I have a population whose mass is normalized to one of consumers.
2. They derive utility from consumption $c$,supplying labour $l$ and a public good $y$
3. This is summarized by a differentiable utility function $u(c,l,y)$
4. Agents derive earnings $z$ from suppliying labour according to:
   $$z=nl$$
   where $n$ corresponds to a "skill level" such that: $n\in[0,\infty[$. Such skill level is distributed according to distribution of skills $F(n)$ with density $f(n)$.
5. $c\_n$, $z\_n=nl\_n$ and $u\_n$ are used to denote the consumption, earnings, and utility level of an individual with skill level $n$
6. the individuals maximize their utility function subject to a constraint:
   $$c\_n=z\_n-T(z\_n)$$

where $T(\cdot)$ is a differentiable "tax function"

```
 7.The public good is produced according to the sum of this tax function:

```

$$y=\int\_0^{\infty}f(s)T(z\_s)ds$$

We can use the equations above to express the maximization problem of the consumer as:

$$\max\_{z\_n}u\left(z\_n-T(z\_n), z\_n/n, \int\_0^{\infty}f(s)T(z\_s)ds\right)$$

The first order conditions to this problem write (ommitting the arguments of the function $u$ for notation compactness):

$$u\_{c}(\cdot)(1-T'(z\_n))+u\_l(\cdot)/n+u\_y(\cdot)\cdot \frac{\partial y}{\partial z\_n}=0$$

I am interested in studying the last term $\frac{\partial y}{\partial z\_n}$. It seems intuitive that $\frac{\partial y}{\partial z\_n}=0$ since:

$$\frac{\partial y}{\partial z\_n}=\int\_0^{\infty}\frac{\partial f(s)T(z\_s)}{\partial z\_n}ds=0$$

**However, I am not sure about the mathematical requirement necessary for the last equation to make sense**. It seems somewhat contradictory to the intuition yo get when aggregating using sums, for instance a discrete analogous would be given by:

$$\sum\_{s=0}^{\infty} T(z\_s)$$

and it is clear that the derivative of the last expression with respect to $z\_n$ would be equal to one.

Could you help me out here? Maybe some broad claryfication of integral aggregation would be required. Thanks!

**Tags:** taxation, public-economics, optimal-taxation, public-goods

> **Comments:**
> **username_1**: Why is $\frac{\partial y}{\partial z_n} = 0$? Also, an equilibrium condition on the profile $(z_n)$ is missing.  

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## 1 answers

### Answer by username_2 on 2018-06-22. Score: 2 | Accepted answer

To obtain $\frac{\partial y}{\partial z\_n}=0$, you implicitly assume that each consumer takes its own skill level as *given*. And in such a case, in reality the consumer optimizes with respect to labor supplied only. Perhaps it would be better to optimaze with respect to $l$ given $n$.

Regarding aggregation the discrete analogous would not be what you write but

$$\sum\_{s=0}^{\infty}p(s) T(z\_s)$$

where $p(s)$ would be the proportion of the population having income $z\_s$.

Note that $y = E[T(z\_s)]$, the expected value of individual tax burden w.r.t skill level. The density of $s$ plays in the continuous case the role of "population proportions" in the discrete setting


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