(c) Following (b), find each firm's best response function of R&D input level in Stage 1.

(d) Following (c), find each firm's equilibrium level of R&D input.

(e) Find the social welfare under this scenario, where the social welfare is defined as the sum of the consumer surplus and the firms' profits.

(f) Find the optimal level of each firm's R&D input under collusion. Compare your answer with (d). Intuitively explain the difference.

(g) Find the social welfare under this scenario.

(a) The output price is \( p \) and the unit cost of capital is \( r > 0 \). Please derive the CEO's optimal choice of capital \( k \). (10%)

(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 \( \frac{c}{2} Q^2 \). The environmental damage \( D(Q) = \frac{d}{2} Q^2 \), 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 \( \gamma \in [0,1] \) 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 \( \gamma \) affects the equilibrium price.

(a) Show that any competitive equilibrium allocation must satisfy production efficiency. (7%)

(b) Now suppose production of good X involves a factor-use externality (e.g., the output of X depends on an aggregate factor use term in addition to the firm's own inputs). Explain how the competitive equilibrium deviates from efficiency. (8%)

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