Lewis Daniel
10/09/2024 · Primary School

1. MWG 5.C. \( 9^{\text {A }} \) 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) \( f(z)=\min \left\{z_{1}, z_{2}\right\} \). (c) \( f(z)=\left(z_{1}^{\rho}+z_{2}^{\rho}\right)^{\frac{1}{\rho}} \), for \( \rho \leq 1 \).

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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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