I am working with the following economic model of labour and consumption decisions:
- I have a population whose mass is normalized to one of consumers.
- They derive utility from consumption $c$, supplying labour $l$ and a public good $y$.
- This is summarized by a differentiable utility function $u(c,l,y)$.
- 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 $f(n)$.
- $c_n$, $z_n=nl_n$ and $u_n$ are used to denote the consumption, earnings, and utility level of an individual with skill level $n$.
- 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".
- 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 clarification of integral aggregation would be required. Thanks!
It is not true that $\frac {\partial}{z_n}\sum_{s=0}^{\infty} T(z_s)=1$. It equals $\frac {\partial T(z_n)}{\partial z_n}$, which is essentially person $n$s marginal tax rate.
I don't see why $\frac{\partial y}{\partial z_n}=0$ If person $n$ works harder, their taxes go up and $y$ goes up. This is analagous to the sum at the end.