Question

A producer has $20$ and $10$ units of labour and capital respectively which he can use to produce two kinds of goods $X$ and $Y$. To produce one unit of $X,2$ units of capital and $1$ unit of labour is required. To produce one unit of $Y,3$ units of labour and $1$ unit of capital is required. Find $X$ and $Y$ ?

Answer 1

Let capital be ${}^{\prime }{a}^{\prime }$ ans Labour be ${=}^{\prime }{b}^{\prime }$

Total capital present $=10a$

Total labour present $=20b$

One unit of $X$ requires $=(2a+b)$

One unit of $Y$ requires $=(a+3b)$

no. of $X$ and $Y$ he can product at atoms

$pX+qY=Totalcapital+Totallabour$

$pX+qY=10a+20b$

$\Rightarrow p(2a+b)+q(a+3b)=10a+20b$

$\Rightarrow (2p+q)a+(p+3q)6=10a+20b$

On comparing : $2p+q=10---\left(1\right)$

$p+3q=20\Rightarrow 2p+6q=40----\left(2\right)$

$\left(2\right)-\left(1\right)$

$5q=30$

$\Rightarrow q=6;2p+6=10$

$\Rightarrow p=2$

$\therefore$ The products on produced $2$ units of $X$ and $6$ units of $Y$ at a given time.