Hopefully, the title isn't too confusing. I have two questions. The first is based on economic theory, but it's really a math question.
Q1: Maximize Government tax revenue
The goal is to maximize Government tax revenue. We are given a labour supply function:
$$l = w(1-t)$$
where l is labour supply, w is the wage and t is the tax (a number between 0 and 1). Then we are asked to construct a Government revenue function (GR) and calculate the tax rate that produces the highest income for the government.
The answer is $$GR = t * w(1-t)$$
And if we take the partial derrivate of $GR$ w.r.t $t$, we get $t^* = {1 \over 2}$, where $t^*$ is the optimal tax-rate.
I realize that the answer gives us the value for t that gives us the maximum pint of a parabola. However, my question is: why is the government revenue function not simply $GR = t*w$? So, for example, if the wage is 100 USD and the tax is 20%, the government revenue will be $0.2*100 = 20usd$. If I didn't know the correct answer that would be my guess. What's the initiative way to understand the logic of this equation?
Q2: Tips on reading material
I would really appreciate if someone could recommend me some reading material to help me (and others) develop some intuition for this. Preferably with some theory and problems that let me practice creating functions. I don't have any formal maths training, so nothing to advanced I guess...
The government maximizes total tax revenue which are given by \begin{align} GR=t\ell(t), \end{align} where $\ell(t)$ is a function of the tax. The level of $t$ that maximizes $GR$ is then simply \begin{align} \frac{\partial GR}{\partial t}=0 \end{align} Computing this derivative, we obtain \begin{align} \ell(t)+t\frac{\partial \ell}{\partial t}=0 \end{align} That is a general expression and makes intuitive sense. A higher $t$ leads to higher tax revenues if labour supply is costant, but it also discourages labour supply since workers get less money for their work (and so we expect $\frac{\partial \ell}{\partial t}<0$).
To obtain the solution in your question, simply note that $\frac{\partial \ell}{\partial t}=-w$. Substituting this in and solving yields the desired solution ($t^{\ast}=1/2$).