(b) True/False/Uncertain: Is \(\succsim\) convex?

(c) True/False/Uncertain: Is \(\succsim\) homothetic?

Let $p1, p2$ be the prices of the first and the second good, respectively, and let $m$ be the total income that can be spent. (d) Find the demand functions of two goods \( x{1}(p{1},p{2},m),x{2}(p{1},p{2},m) \) under this preference.

(e) Let \( p2 = 1 \). Suppose \( p1 \) changes from \( 1/2 \) to \( 2 \). Find the equivalent variation and the compensating variation.

2. (9 %) Consider two players in a game: Legislator (L) and the Judiciary (J). The Legislator and the Judiciary negotiate a bill $B$ to be executed from a policy space $\mathcal{B} = [0,10]$, where $B0 = 0$ is the status quo. Legislator's favorite policy is $BL^* = 9$, and Judiciary's favorite policy is $B_J^* = 3$. The game has 2 stages and proceeds as follows: Stage 1: The Legislator chooses a bill $B_L \in [0,10]$. Stage 2: After seeing $BL$, the Judiciary can choose to pass or reject the bill. If the bill is passed, the executed bill will be $B = BL$. If the bill is rejected, the society goes back to the status quo, $B = B_0 = 0$. Define player $i$'s payoff as $\pii = |Bi^* - B|$. Assume that the Judiciary will not reject the bill if rejecting the bill is indifferent to her. (a) Under the Subgame Perfect Nash Equilibrium, what is the equilibrium bill passed, \( B_{SPNE} \)?

(b) Fix \( B{L}^{} = 9 \), and make \( B{J}^{} \) variable. Express the passed bill under the SPNE as the function of the Judiciary's preference \( B(B_{J}^{}) \). Make sure you have discussed all possibilities \( B_{J}^{} \in [0,10] \).

3. (24 %) Consider a duopoly market that consists of two firms, OpenAI and Google. The competition proceeds as follows: Stage 1: Each firm simultaneously determines its own R&D input level, $x_i$. Stage 2: Knowing the R&D input level of each other, each firm simultaneously determines its own output level, $q_i$. 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 $Q = qO + q_G$. (a) Consider the Cournot duopoly equilibrium. Given \( x{O}, x{G} \) determined, find the equilibrium output chosen by each firm in Stage 2.

(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.

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