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proportions production function as follows: production function

where the price of input is 1 and z2 is supposed to be a fixed factor of production. I've been having trouble finding the cost function because if z2 isn't included in the production function, I'm not sure how I'd approach finding the cost function.

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$z_2$ is included in the production function. Please see the production function $f:\mathbb{R}^2_+\rightarrow\mathbb{R}$: \begin{eqnarray*} f(z_1,z_2) = \left(\min(\lfloor z_2\rfloor, 1)\right)z_1^\alpha = \begin{cases} z_1^\alpha & \text{if } z_2 \geq 1 \\ 0 & \text{if } z_2 < 1\end{cases} \end{eqnarray*} where $\lfloor z_2\rfloor$ is greatest integer less than or equal to $z_2$, and $\alpha > 0$ is given.

Cost minimization problem of a competitive firm is defined as follows: \begin{eqnarray*} \min_{z_1\geq 0,z_2\geq 0} & \ w_1z_1 + w_2z_2 \\ \text{s.t. } & f(z_1,z_2)\geq q\end{eqnarray*} where $q\geq 0$, $w_1>0$ and $w_2>0$. Solving the problem, we get the conditional input demand as follows: \begin{eqnarray*} (z_1,z_2)(w_1,w_2,q) = \begin{cases} (0,0) & \text{if } q = 0 \\ \left(q^{\frac{1}{\alpha}},1\right) & \text{if } q > 0\end{cases} \end{eqnarray*} Therefore, the cost function is \begin{eqnarray*} C(w_1,w_2,q) = \begin{cases} 0 & \text{if } q = 0 \\ w_2+w_1q^{\frac{1}{\alpha}} & \text{if } q > 0\end{cases} \end{eqnarray*}

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