3. 針對 X、Y、Z 三種財貨的四組不同價格水準為(1,2,1)、(2,1,1)、(1,2,2)、(2,1,2)時，選擇的消費組合分別對應為(10,5,5)、(3,5,6)、(13,3,3)、(15,3,1)，請證明是否符合 SARP(強性顯示性偏好公理)(10)？

二、Four consumers make statements about their preferences or (utility maximizing) purchasing behaviors on movies ($$x1$$) and pizza ($$x2$$). According to each statement, write down a utility function, if possible, that could represent their preferences. 1. (3%) Anna says that she always prefers a consumption bundle with more movies and pizza.

2. (3%) Bess says that she always (weakly) prefers a consumption bundle with more movies or more pizza (or both).

3. (3%) Cathy says that she always spends 1/3 of her income on movies, and 2/3 on pizza.

4. (3%) Debra says that more bonus will not make her eat more pizza (given that prices of movies and pizza remain the same).

三、Emma wants to buy a dress from Francesca, whose cost of making the dress is $300. The benefit of Emma to wear the dress is $500. 1. (2%) A “fair price” would let Emma and Francesca equally enjoy the gains from the transaction. What is the fair price in this case?

In fact, $500 is Emma’s expected benefit from the dress. With probability 1/2, Emma would like the dress very much, and obtain a benefit $800; and with probability 1/2, Francesca’s design doesn’t really impress Emma, whose benefit from the dress is only $200. The true benefit of the dress is Emma’s private information, and does not affect the transaction price. Suppose that Emma and Francesca first sign a contract and agree to trade at a price $p \geq 300$ (not necessarily the fair price). Then, Emma learns the true benefit of the dress ($800 or 200). Finally, if they trade, Francesca spends the cost to make the dress, and Emma pays the price $p and gets the dress. Both Emma and Francesca are risk neutral. An efficient trading rule maximizes social surplus (the sum of consumer surplus and producer profit). Here, it requires Francesca make, and Emma get the dress if and only if the cost is smaller than the benefit. 2. (3%) What is the expected social surplus under the efficient trading rule?

3. (3%) What is the expected social surplus if Emma cannot breach the contract, namely, she must always pay the price \( p \) and trade with Francesca?

In practice, if a party breaches the contract, the court may order that party to pay the "expectation damages" to the trading partner (the victim of the breach). By definition, expectation damages compensate the victim an amount that would give her the same payoff had the contract been carried out. For the remaining questions, suppose that when Emma decides to breach the contract (i.e., refuse to trade with Francesca), Francesca has not yet incurred the cost, and Emma will be ordered to pay expectation damages. 4. (3%) What is the amount of expectation damages Emma should pay in case of breach?

5. (3%) What is the price $p, if any, that would induce the efficient trading rule? If there are more than one, find out all such prices; and if there is none, explain why.

Lastly, suppose that either Emma or Francesca can make extra efforts to improve Emma's consumption experiences. By making such efforts (at a cost $1), the probability of a benefit of $800 becomes $\frac{1}{2} + x$, and the probability of a benefit of $200 becomes $\frac{1}{2} - x$, where $0 < x < \frac{1}{2}$. 6. (3%) What is the social benefit of such efforts?

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