Maximize area of rectangle within another rectangle (with additional constraints)

Question

I'm looking at wage and employment data for a region. I have created a chart to show how total wages grew between two periods, $t_0$ and $t_1$. The axes are Total Employment (E) and Average Wage (W). Total wages for t0 and t1 are the products of $W_0\cdot E_0$ and $W_1\cdot E_1$, respectively.

In my case, the values of areas $W_0$, $E_0$, (W_0\cdot E_0) and ($W_1\cdot E_1$) are fixed. Given those fixed areas, what I want is to find the values of $W_1$ and $E_1$ that yield the greatest area of $C$ in the chart, which is given by $(W_1-W_0)(E_1-E_0)$.

I know this is an optimization problem, but it's been so long since my last calc course that I just don't remember where to begin. Many thanks.

Answer 1

You want to maximize $$F=(W_1-W_0)(E_1-E_0)$$ with $$W_0E_0=A \qquad W_1E_1=B$$ where $A$, $B$ and $W_0$ are known.

So $E_0=\frac A{W_0}$ and $E_1=\frac B{W_1}$. This makes $$F=(W_1-W_0)\left(\frac B {W_1}-\frac A {W_0}\right)$$ Computing the derivative with respect to $W_1$ and making it equal to $0$ gives, after simplifications $$A W_1^2=B W_0^2 \implies W_1=W_0\sqrt{\frac B A}$$

Answer 2

The method of Lagrange multipliers would yield a solution, but this can be done easier:

Let $W_1E_1 = c$. Let $x$ denote $W_1$, then $E_1 = c/x$.

You now want to find some $x$ that maximizes the function $C(x) = (x-W_0)(c/x-E_0)$. The derivative of this function is $cW_0/x^2 - E_0$, and for a maximal $x$ this derivative must be $0$.

Solving $cW_0/x^2 - E_0 = 0$ yields $x = \sqrt{cW_0/E_0}$.

Now $W_1 = x = \sqrt{cW_0/E_0}$, and $E_1 = c/x = \sqrt{cE_0/W_0}$.