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

題組內容

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.

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.

題組內容

(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%)

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