# Find cost function for given production function

I have the following production function

$$f(x_1,x_2,x_3,x_4)=max\{\min\{x_1, x_2), x_3+2x_4\}\}\ge q$$

And I want to find the cost function.

What I think

(1) $$P_1+P_2 and $$P_3/P_4<1/2$$

then $$(x_1^*, x_2^*, x_3^*, x_4^*) =(q,q,0,0)$$

Cost function is $$c(q,w)=q[w_1+w_2]$$

(2) $$P_1+P_2 and $$P_3/P_4>1/2$$

then $$(x_1^*, x_2^*, x_3^*, x_4^*) =(q,q,0,0)$$

Cost function is $$c(q,w)=q[w_1+w_2]$$

(3) $$P_1+P_2 >P_3$$ and $$P_3/P_4<1/2$$

then $$(x_1^*, x_2^*, x_3^*, x_4^*) =(0,0,q,0)$$

Cost function is $$c(q,w)=qw_3$$

(4) $$P_1+P_2 >P_4/2$$ and $$P_3/P_4>1/2$$

then $$(x_1^*, x_2^*, x_3^*, x_4^*) =(0,0,q/2,0)$$

Cost function is $$c(q,w)=qw_4/2$$

I have no idea how to solve such a production function. But I posted what I tried. Please show me the correct solution and how to solve such a function. Thank you.

• Someone has deleted the answer. Let me share it here: One of the ways to solve this problem is almost the same as the one posted here: qr.ae/pvKJuI
– Amit
Commented Sep 9, 2022 at 8:19
• Using this method will give the following cost function for the above production function: $C(w_1, w_2, w_3, w_4, q)=q\min\left\{(w_1+w_2),w_3, \dfrac{w_4}{2}\right\}$ where $w_i$ denotes the price (per unit) of input $x_i$, and $q$ is the quantity of output produced.
– Amit
Commented Sep 9, 2022 at 8:20
• I am glad that the link I provided is helpful in understanding the approach to solve this question.
– Amit
Commented Sep 9, 2022 at 15:12
• Yes for sure @Amit that question with solution is very clear and explanatory :) Commented Sep 9, 2022 at 16:05
• Thanks. Actually moderator of the econ-stackexchange deleted the answer giving the reason that the "link only" answers are against the rules of the econ-stackexchange. So I was just confirming.
– Amit
Commented Sep 9, 2022 at 17:13