I have an economics question, which i want to solve mathematically. Two countries produce 2 identical goods. They both can allocate a different percentages of overall production towards each good a and b (eg. 90% a and 10% b).
Country S can produce 10a or 6b if they allocate 100% production towards one.
Similarly, country T can produce 5a or 2b.
I am trying to calculate the greatest overall output between these two countries for a and b for both goods. Overall production of a should exceed 7.5 while overall production of b should exceed 4.
This has led to the following equations, where x and y are the percentage allocation of production allocated to each good:
x10a +y5a > 7.5
(1-x)6b + (1-y)2b > 4
I have attempted to solve this problem, but so far have found nothing.
Any assistance would be greatly appreciated. Thank you in advance.
Let $x,y$ be the percentages of overall production towards good $A$ practiced in countries $S$ and $T$, respectively. The overall output is given by $f(x,y)=10 x + 6(1-x) + 5y + 2(1-y) = 4x + 3y +8$, and the production restrictions are $10 x+5y \ge 7.5, \quad 6(1-x) + 2 (1-y) \ge 4$. So, you want to solve the problem \begin{align*} \max & f(x,y)=4x+3y+8\\ \textrm{s.t.}\,\, & 10 x+ 5y \ge 7.5\\ & 6x + 2y \leq 4\\ & 0 \leq x,y \leq 1 \end{align*}
Can you solve it now? The final solution is $x=\frac 13, y = 1$, i.e. country S should allocate $1/3$ of resources to the production of $A$ and $2/3$ to the production of $B$, while country $T$ should allocate all resources to the production of $A$.
[edited: the second restriction was wrong]