(b) Following (a), express the profit functions as functions of \( x{O}, x{G} \).

(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.

Now, suppose the two firms agree to collude in Stage 1 but continue to compete in a Cournot manner in Stage 2. That is, the two firms will collude with the R&D input level in Stage 1, and then individually decide the output levels in Stage 2. The two firms agree to coordinate R&D levels to maximize total joint profit. (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.

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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