1. MWG 5.C. $$ 9^{ ext {A }} $$ Derive the profit function $$ \pi(p) $$ and supply function (or correspon- dence) $$ y(p) $$ for the single output technologies whose production functions $$ f(z) $$ are given by: (a) $$ f(z)=\sqrt{z_{1}+z_{2}} $$. (b) $$ f(z)=\min \left\{z_{1}, z_{2} ight\} $$. (c) $$ f(z)=\left(z_{1}^{ ho}+z_{2}^{ ho} ight)^{\frac{1}{ ho}} $$, for $$ ho \leq 1 $$.

Question

1. MWG 5.C. $$ 9^{	ext {A }} $$ Derive the profit function $$ \pi(p) $$ and supply function (or correspon- dence) $$ y(p) $$ for the single output technologies whose production functions $$ f(z) $$ are given by: (a) $$ f(z)=\sqrt{z_{1}+z_{2}} $$. (b) $$ f(z)=\min \left\{z_{1}, z_{2}ight\} $$. (c) $$ f(z)=\left(z_{1}^{ho}+z_{2}^{ho}ight)^{\frac{1}{ho}} $$, for $$ ho \leq 1 $$.

Chen Washington · Accepted Answer

为了推导单一产出技术的利润函数和供给函数，需要考虑生产函数、成本和价格。对于不同的生产函数，推导过程如下： (a) 对于 $$ f(z) = \sqrt{z_1 + z_2} $$，利润函数为 $$ \pi(p) = p \cdot \sqrt{z_1 + z_2} - (w_1 z_1 + w_2 z_2) $$，供给函数通过优化利润得到。 (b) 对于 $$ f(z) = \min \{ z_1, z_2 \} $$，利润函数为 $$ \pi(p) = p \cdot \min \{ z_1, z_2 \} - (w_1 z_1 + w_2 z_2) $$，供给函数通过最大化利润得到。 (c) 对于 $$ f(z) = (z_1^\rho + z_2^\rho)^{\frac{1}{\rho}} $$，利润函数为 $$ \pi(p) = p \cdot (z_1^\rho + z_2^\rho)^{\frac{1}{\rho}} - (w_1 z_1 + w_2 z_2) $$，供给函数通过最大化利润得到。

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