# EPA Air Pollution Policy Analysis ## a. Emission Standard vs. Tax (When $\alpha = \beta$) If $\alpha = \beta$, both firms have identical abatement cost functions: \( C_i(x_i) = \frac{\alpha}{2}(1 - x_i)^ \). Under either an emission standard (equal abatement for both) or an emission tax (same marginal incentives), the cost-effective allocation will be to split abatement equally: $x_1 = x_2 = 5$ tons. Thus, either policy yields the same outcome, and there is no efficiency advantage of tax over standard in this symmetric case. --- ## b. Permit Allocation When $\alpha \neq \beta$ and Permit Price is $.5$ Each firm chooses emissions $x_i$ to minimize cost, given a permit price $p = .5$: - Marginal abatement cost: $C'_i(x_i) = \alpha(1 - x_1)$ for firm 1, $\beta(1 - x_2)$ for firm 2. - Set equal to permit price: - $\alpha(1 - x_1) = .5 \implies x_1 = 1 - \frac{.5}{\alpha}$ - $\beta(1 - x_2) = .5 \implies x_2 = 1 - \frac{.5}{\beta}$ - Total emissions: $x_1 + x_2 = 10$. So, \[ \left(1 - \frac{.5}{\alpha}\right) + \left(1 - \frac{.5}{\beta}\right) = 10 \implies \frac{1}{\alpha} + \frac{1}{\beta} = \frac{2}{9} \] The emission level of each firm: \[ x_1 = 1 - \frac{.5}{\alpha}, \quad x_2 = 1 - \frac{.5}{\beta} \] with $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{2}{9}$ imposed. --- ## c. Cost-Effective Permit Allocation if $\alpha > \beta$ If $\alpha > \beta$, firm 1 has a higher marginal abatement cost than firm 2. From the above, $x_1 = 1 - \frac{.5}{\alpha}$, $x_2 = 1 - \frac{.5}{\beta}$, so $\frac{.5}{\alpha} < \frac{.5}{\beta}$, hence $x_1 > x_2$. Therefore, to achieve cost effectiveness, allocate **more permits to firm 1** (the higher-cost abater) and fewer to firm 2 (the lower-cost abater), so that the marginal abatement costs are equalized at the permit price.

# EPA Air Pollution Policy Analysis Summary ## a. Emission Standard vs. Tax (When $\alpha = \beta$) In the case where both firms have identical abatement costs ($\alpha = \beta$), either an emission standard or a tax would result in equal abatement levels of 5 tons for each firm. Thus, there is no efficiency advantage in choosing one policy over the other in this scenario. --- ## b. Permit Allocation When $\alpha \neq \beta$ and Permit Price is $0.5$ With a permit price of $0.5$, firms will adjust emissions to minimize costs based on their individual abatement costs: - For Firm 1: \( x_1 = 1 - \frac{0.5}{\alpha} \) - For Firm 2: \( x_2 = 1 - \frac{0.5}{\beta} \) The total emissions constraint gives: \[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{2}{9} \] This ensures that both firms balance their emissions according to their respective costs. --- ## c. Cost-Effective Permit Allocation if $\alpha > \beta$ If $\alpha > \beta$, Firm 1 has a higher marginal abatement cost compared to Firm 2. Based on emissions calculated in part b, to achieve cost effectiveness, **more permits should be allocated to Firm 1** (higher-cost abater) and fewer to Firm 2 (lower-cost abater), ensuring that marginal costs are equalized at the permit price.