(b) Compare the optimal capital input under uncertainty with the risk-neutral benchmark case. Is the optimal capital choice under uncertainty higher or lower than in the deterministic case? Explain your result using both mathematical reasoning and economic intuition. (10%)

5. (15 %) Consider a product market in which a private firm faces an inverse demand function p(Q) = a - b Q and private cost c/2 Q^2. The environmental damage D(Q) = d/2 Q^2, where a, b, c, d > 0. Assume that each unit of output produces one unit of carbon emissions. Firms engage in Environmental Corporate Social Responsibility (ECSR) by internalizing a fraction γ ∈ [0,1] of units of carbon emissions. The government can impose a per-unit carbon tax t on output. Derive the equilibrium price and how an increase in the degree of ECSR γ affects the equilibrium price.

(b) Compare the optimal capital input under uncertainty with the risk-neutral benchmark case. Is the optimal capital choice under uncertainty higher or lower than in the deterministic case? Explain your result using both mathematical reasoning and economic intuition. (10%)

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