# Constant returns to scale and cost function: $C(p,ty) = tC(p,y)$

Economics Asked by user270396 on January 2, 2021

How can I prove that for a production function $$F:mathbb X rightarrow mathbb R$$ with constant returns to scale

$$forall xin mathbb X, forall t > 0: F(tx) = t F(x)$$

and with the cost function

$$C(p,y):=min_{x in mathbb X} {p^top xlvert F(x) geq y}$$

it must be the case that $$C(p,ty) = tC(p,y).$$

My first strategy has been to make a constructive proof along the following lines

$$C(p,ty) = min_{x in mathbb X} {p^top xlvert F(x) geq ty} =min_{x in mathbb X} {p^top xlvert F(x/t) geq y},$$
with the second identity using the constant returns to scale assumption. I then define $$z=x/t$$ and rewrite the variable over which to minimize
$$= min_{x/t} {t p^top(x/t) lvert F(x/t) geq y} = t min_z{p^top z lvert F(z) geq y} = t C(p,y).$$

However I am not entirely sure that this proof is solid so I guess my question boils down to whether this proof really is solid or if there is another way to proove it?