The following problem is originally a standard problem in economics. I understand how to solve it, but would like to better understand how or why it works.
We have the following utility function $u_i(c,l) = c - {l^2 \over 2}$ for individual $i$, where $c$ is consumption and $l$ is hours of labor. The individual faces the budget constraint $c = wL(1-\tau)$, where $w$ is the wage and $\tau$ is the tax rate. To solve for the optimal labor supply, as a function of $w$ and $\tau$, we substitute the budget constraint into the utility function. We get
$$u_i = wl(1-\tau) - {L^2 \over 2}$$
Finally, we take the partial derivative w.r.t l and solve for l.
$$l=w(1-\tau)$$
That is, the individual labor supply is a function of the tax rate. Higher tax, less labor supplied, and vice-versa.
What I don't quite understand is how this works. I guess it has something to do with the slope of a tangent, but I'm struggling with the intuition. My goal is to know how to do this intuitively so that I can apply this solution to other problems instead of just learning to do it mechanically.
Hope that my question is clear, let me know if it needs clarification.