I need help finding the cost function and input demand functions from the following production function: $$f(x_1,x_2)=\min(2x_1+x_2,x_1+2x_2).$$
The proposed answer I am seeing from my professor is: $$c(w_1,w_2,y)=\min \left(w_1, w_2, w_1+\tfrac{1}{3} w_2\right) \times y.$$He also had different answers for the input demand functions based on the ratio of input prices being equal to $2$ or $\frac 1 2$ for each point on the isoquant. I am not sure how he got these answers. Can anyone help with this problem?
Best Answer
Cost minimisation of the firm is defined as follows:
\begin{eqnarray*} \min_{x_1 \geq 0, x_2 \geq 0} & w_1x_1+w_2x_2 \\ \text{s.t.} & f(x_1,x_2)\geq y\end{eqnarray*} where $w_1 > 0$, $w_2 > 0$ are input prices that are given, and $y>0$ is the given level of output.
For the production function $f(x_1,x_2)=\min(2x_1+x_2,x_1+2x_2)$, solution $(x_1^c,x_2^c)$ to the cost minimisation problem (also known as the conditional input demand) is given by: \begin{eqnarray*} (x_1^c,x_2^c)(w_1,w_2,y) \in \begin{cases} \left\{\left(y,0\right)\right\} & \text{if } \frac{w_1}{w_2}<\frac{1}{2} \\ \left\{\left(x_1,x_2\right)\in\mathbb{R}^2_+|x_1+2x_2=y \ \wedge \ x_1\geq x_2\right\} & \text{if } \frac{w_1}{w_2}=\frac{1}{2} \\\left\{\left(\frac{y}{3},\frac{y}{3}\right)\right\} & \text{if } \frac{1}{2} < \frac{w_1}{w_2}<2 \\ \left\{\left(x_1,x_2\right)\in\mathbb{R}^2_+|2x_1+x_2=y \ \wedge \ x_1\leq x_2\right\} & \text{if } \frac{w_1}{w_2}=2 \\ \left\{\left(0,y\right)\right\} & \text{if } \frac{w_1}{w_2}>2 \end{cases}\end{eqnarray*} Therefore, the optimal value of the objective function (also known as the cost function) is \begin{eqnarray*} C(w_1,w_2,y) = w_1x_1^c+w_2x_2^c=\min\left(w_1,w_2,\frac{w_1+w_2}{3}\right)y\end{eqnarray*}