1. **Problem Statement:** We need to find the profit maximization for a steel firm producing $S$ units of steel and $x$ units of pollution. 2. **Given Formula:** Profit function is given by: $$\Pi(S,x) = P_s S - C_s(S,x)$$ where $P_s$ is the price per unit of steel, $S$ is the quantity of steel produced, and $C_s(S,x)$ is the cost function depending on steel and pollution. 3. **Step 1: Find $S$ and $x$ that maximize profit.** To maximize profit, we take partial derivatives of $\Pi(S,x)$ with respect to $S$ and $x$ and set them to zero: $$\frac{\partial \Pi}{\partial S} = P_s - \frac{\partial C_s}{\partial S} = 0$$ $$\frac{\partial \Pi}{\partial x} = - \frac{\partial C_s}{\partial x} = 0$$ 4. **Step 2: Solve for $S$ and $x$.** From the first equation: $$P_s = \frac{\partial C_s}{\partial S}$$ From the second equation: $$\frac{\partial C_s}{\partial x} = 0$$ 5. **Step 3: Substitute $S$ and $x$ back into the profit function:** $$\Pi = P_s S - C_s(S,x)$$ 6. **Step 4: Calculate the profit $\Pi$.** **Note:** Without explicit forms of $P_s$ and $C_s(S,x)$, we cannot compute numerical values. The process involves: - Finding marginal cost with respect to $S$ and $x$. - Equating marginal cost to price for $S$. - Setting marginal cost with respect to $x$ to zero. - Substituting back to find profit. This is the general method to solve the problem.