(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.

題組內容

3. (24 %)

Consider a duopoly market that consists of two firms, OpenAI and Google. The competition proceeds as follows:

Each firm has the following cost function:

$$Ci(qi, xi, xj) = (70 - (xi + xj))qi + 2xi^2,$$

where $xj$ is the R&D level the other firm chooses. Let the inverse demand function of the market be $P = 100 - \frac{1}{3}Q$, where $Q = qO + q_G$.

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

(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.

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