(g) Find the social welfare under this scenario.

4. (20 %) Consider a firm whose CEO's compensation equals a fraction λ of the firm's operating profit. The CEO is risk-averse and has utility U(W)=ln (W), where W is the compensation. The firm's production function is y=θ√k, where k≥0 is the capital input and θ is a random productivity shock with Probability(θ=2)=1/2, Probability(θ=1)=1/2. (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 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.

6. (15 %) Consider a two-good (X,Y), two-factor (L,K) general equilibrium economy. Production technologies are X = F^X(LX, KX) and Y = F^Y(LY, KY), where both F^X and F^Y exhibit constant returns to scale and are strictly concave. Factor endowments satisfy LX + LY = L̄ and KX + KY = K̄. Consumer preferences are represented by standard Cobb-Douglas functions. (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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