Question

A firm has a production function given by f(x1,x2,x3,x4)=min{2x1+x2,x3+2x4}. What is the cost function for this technology?

Answer 1

Production function is $f(x_1,x_2,x_3,x_4)=min(2x_1+x_2,x_3+2x_4)$

$x_1\space and\space x_2$ are perfect substitutes, and $x_3\space and\space x_4$ are also perfect substitutes.

If $w_i$ denotes the price per unit of input $x_i,$ then

$x_1$  is used to produce the output when $\frac{w_1}{2}<w_2$   and $x_2=0$

$x_2$  is used to produce the output when $\frac{w_1}{2}<w_2$   and $x_1=0$

$x_3$  is used to produce the output when $w_3<\frac{w_4}{2}$   and $x_4=0$

$x_4$  is used to produce the output when $w_3>\frac{w_4}{2}$   and $x_3=0$

So if one wants to produce one unit of output when the input prices satisfy $\frac{w_1}{2}<w_2$ and $w_3<\frac{w_4}{2}$ then one must employ $x_1=\frac{1}{2}$ and $x_3=1$ so as to minimize cost, and the cost is given by $\frac{w_1}{2}+w_3$

When the input prices satisfy $\frac{w_1}{2}>w_2$ and $w_3<\frac{w_4}{2}$ then one must employ $x_2=1$ and $x_2=1$ in order to minimize cost, and the cost is given by $w_2+w_3$

For other two combinations.

we can write the cost of producing one unit of output as

$min(\frac{w_1}{2},w_2)+mim(w_3,\frac{w_4}{2})$

More generally, if you want to produce q units, cost function is

$C(w_1,w_2,w_3,w_4,q)=[min(\frac{w_1}{2},w_2)+min(w_3,\frac{w_4}{2})]q$