summer teaching -- micro principles midterm sol

Solution to Midterm Examination (International Business, WUCL)
Summer 2007 Instructor: 衛忠欣 (Jong-Shin Wei)
經濟學 (Principles of Economics -- Micro) (07)342-6031 ext.6201
93001@mail.wtuc.edu.tw http://www.wtuc.edu.tw/ib

請務必依序工整另卷作答。三日後公佈解答於網路學園，更歡迎以email函索
Open books/notes exam in 60 minutes; absolutely no talking nor borrowing items during exams. 可使用自己的資料、字典或翻譯機。行動電話若響起，該生扣十分、以強調基本禮貌。2:30 pm – 3:30 pm, Wednesday, July 11, 2007. Watch your time and good luck! 務必工整扼要、依序作答；如果字跡難以辨認，視同未答。畫蛇添足、答非所問、自曝其短，將不利得分。教師也有權公告具有特色之作答。Watch your time and have fun!

1.[10 points]
Suppose that the following conversation took place between Andy and Becky, who have not known each other well.
Andy: “I will love to visit your apartment tomorrow.”
Becky: “Not tomorrow. Maybe some other time, ok?”
Andy: ‘Why not? Didn’t you expect my visit?”
Becky: “No, do not take me wrong. I just need to get rid of cockroaches first.”

Was Becky signaling? Before you answer it, make sure that you understand what signaling means and the reason, if any, for her to send out the signal.
Ans: No. Her statement does not meet the two conditions required for being a signal. Here she might be unwilling to invite Andy. Period. Note that she did not signal to convey any information to Andy. She just responded to his presumption with a negative answer.

2.[20 points]
Recall the English auction and Vickrey auction presented during lectures. Suppose that in an English auction the bid started at $100. Bidder A bid $120 to kick off the competition. Then bidder B bid $130, followed by $140 made by bidder C. Bidder A then raised it to $160. When bidder B responded to it by making a bid at $180, no further bid was made. The auctioneer announced that bidder B won. In other words, the subject was sold to bidder B at the price as $180.
(a) Can we say that bidder B ended up with the so-called winner’s curse (得標者之遺憾)?
(b) If those bidders participate in the Vickrey auction selling the same item, what do we expect to be the final (or trading) price? Hint: compare it with 180, 160, 140, … and so on.
Ans:
(a) No. The reservation price for bidder B should be no less than 180, which is why B bid so. Therefore, when B got the item and paid $180, we cannot say that he or she got the winner’s curse.
(b) The reservation price for bidder B should be no less than 180; the reservation price for bidder A should be no less than 160. So, the second highest bid in Vickrey auction should be somewhere between 160 and 180. [Recall the truth-revealing feature in such auction.] Saying 160 or slightly above 160 will be just fine. In any case, cannot answer it by any number greater than or equal to 180!

3.[20 points]
(a) 一般大學的轉學考試，多採「闈場外命題」。常聞有熱心的第三者，探問命題的教科書、範圍、甚至題型。命題教授不堪其擾，但也無可奈何。欣賞『天之驕子』這部影片之後，命題教授有何對策呢？
(b) 由『天之驕子』這部影片中，找出一項訊號。Must carefully explain why you believe so.
Ans:
(a) Professors can write exam papers in a way that whoever got such information will not benefit from doing so. Of course, literally incorporating such scenario into the exam papers, as I am doing it here, is another resolution. No voice will ever be raised.
(b) The contest itself is a signal to the quality of that (high) school in the sense that history, school tradition, honor, and strive for excellence are valued. Note that many of you (with myself included) might say that the contest demands a lot of time in memorizing old stuffs and that only few students got selected to participate in it. Nevertheless, keep in mind that it is just a signal.

4.[30 points]
Suppose that each agent has 4 hours to spend on the production of goods X and Y, whose quantities are denoted by x and y respectively. The PPF (productivity possibility frontier) of agent A is given by x + y = 4; the PPF of agent B is given by x + y/4 = 4.
(a) In the production of good X, who has the absolute advantage?
(b) In the production of good Y, who has the absolute advantage?
(c) What do we know about the relative or comparative advantage?
(d) For producing one unit of good Y, how many hours does A need to spend?
(e) For producing one unit of good Y, how many hours does B need to spend?
(f) Before the trade takes place, suppose that A produces 2 units of good X and 2 units of good Y; B produces 2 units of good X and 8 units of good Y. When both are allowed to trade, what might be the relative price of good one in transaction?
Ans: First we should sketch x + y = 4 and x + y/4 = 4 on the x-y plane. Also mark (2, 2) on A’s PPF; mark (2, 8) on B’s PPF. Notice that both A and B are equally efficient in producing good X. Yet, from the slope of PPFs, we can see the comparative advantage.
(a) Both tie.
(b) B (due to 1/4 < 1)
(c) A should produce good X; B should produce good Y (in light of slopes being -1 and -4)
(d) 1 hour (see the coefficient of Y in x + y = 4)
(e) 1/4 hour (see the coefficient of Y in x + y/4 = 4)
(f) A’s production plan is (2, 2); B’s production plan is (2, 8). Then we can have B export good Y and import good X; have A export X in exchange for Y. The relative price of good X can be any number between 1 and 4. [Try 3 on your own.] One can also see from the picture that both can be better off after trading.

5.[20 points]
True or False. [判斷對或錯] No explanation is necessary. 每答對一題，得4分。每答錯一題，倒扣1分。
(a) “Economics is a study of mankind in the ordinary business of life,” was first written by Adam Smith.
Ans: False. [See page xxiii, text]
(b) Mary was wondering whether she should take economics or English in the summer school. Both courses are offered in the afternoons from Monday thru Friday. Eventually she decided to take economics, not English. We might say that the opportunity cost of taking economics is less than the opportunity cost of taking English.
Ans: True. [See page 6, text]
(c) “The whole of economics as a social science is nothing more than the refinement of everyday thinking,” was first said by Albert Einstein.
Ans: False. [See page 20, text]
(d) Every point on the PPF is Pareto efficient.
Ans: False. [Recall lectures]
(e) 五個攤位一列排開，鄰接攤位皆等距。消費者均勻分佈在閉區間 [0, 6]。五個攤位之位置，分別以數字 1, 2, 3, 4, 5標示於實數線上。依據 Hotelling 的思維，最中央的攤位之市場佔有率是1/6，離最中央最遠的攤位之市場佔有率則是1/4.
Ans: True. [(1/6) + (1/6)(1/2) = 3/12 = 1/4]

(放棄智慧財產權；歡迎全貌下載流傳。)

summer teaching -- math for mgmt midterm sol

Solution to Midterm Examination (International Business, WUCL)
Summer 2007 Instructor: 衛忠欣 (Jong-Shin Wei)
管理數學 (Mathematics for Management) (07)342-6031 ext.6201
93001@mail.wtuc.edu.tw http://www.wtuc.edu.tw/ib

姓名： 學號：

請務必依序工整另卷作答。三日後公佈解答於網路學園，更歡迎以email函索
Closed books/notes exam in 60 minutes; absolutely no talking nor borrowing items during exams. 可使用自己的字典或翻譯機。行動電話若響起，該生扣十分、以強調基本禮貌。09:30 am – 10:30 am, Wednesday, July 11, 2007. Watch your time and good luck!
務必原卷工整扼要作答、不得超出作答空間；如果字跡難以辨認，視同未答。畫蛇添足、答非所問、自曝其短，將不利得分。教師也有權公告具有特色之作答。Watch your time and have fun!

1. [20 points] Recall the example given in class, regarding three workers (A, B, and C) and a (labor-intensive) task to be completed. As before, assume perfect divisibility and fixed productivity. Furthermore, assume that
if A and B work together while C does not show up, it takes 20 days to complete it;
if A and C work together while B does not show up, it takes 20 days to complete it;
if B and C work together while A does not show up, it takes 40 days to complete it.
How many days does A need for the task if A works alone?
Ans: Without the loss of generosity, we assume that the amount of task (to be carried out) is k > 0. Each worker’s productivity is characterized by k divided by the number of days for this worker to contribute alone. Let x, y, and z represent the productivity of workers A, B, and C respectively.
Hence,
k = 20x + 20y + 0z = 20x + 20y,
k = 20x + 20z,
and
k = 40y + 40z.
From 20x + 20y = 20x + 20z = k we have y = z, hence k = 40y + 40z leads to y = k/80, yielding y = (1/80)k.
It follows at once that x = (k - 20y)/20 = (3/80)k.
To conclude, A needs 80/3 days to complete the task if A works alone.

2. [20 points] Choose non-negative ordered pair (x, y) to maximize x + 2y
subject to x - y ≤ 4, x + y ≤ 8, x ≤ 2, and y ≤ 4. Must show key steps.
Ans: First we depict lines x - y = 4 and x + y = 8. Next, we identify closed half-spaces x - y ≤ 4, x + y ≤ 8, x ≤ 2, and y ≤ 4. The intersection of all these four half-spaces is the region with four vertex points (0, 0), (2, 0), (2, 4), and (0, 4). [In formal language, we shall say that constraints x - y ≤ 4 and x + y ≤ 8 are not binding.] We then examine the family of lines characterized by x + 2y = k. Visual inspection leads to (2, 4).

3. True or False. No explanation is necessary. 每答對一題，得4分。每答錯一題，倒扣1分。
(a) A line on the plane can not have more than two horizontal intercepts.
Ans: False.
(b) A line on the plane must have at least one vertical intercept.
Ans: False.
(c) A line on the plane must have the unique slope.
Ans: False.
(d) Let a, b, c, d, e, and f be parameters (that are real numbers). Equations ax + by = c and dx + ey = f may have no solution in (x, y).
Ans: True. [兩條不相交的平行線]
(e) Behind the equation xy = 1 we can find two functions.
*Ans: True. [One function is f(x) := 1/x; another is g(x, y) := xy.]

4. [20 points] 在電影 A Beautiful Mind 裡，數學家 John Nash 的治學精神，與你(妳)的學習此門課，最大的不同為何？你(妳)的未來雇主，又將如何透過學校，了解此點呢？
Ans: skipped.

5. [20 points] 下圖是一場五子棋目前賽況。你(妳)持白子待攻，對手持黑子。請小心以1, 2, 3, 4, …在圖上，分別依序標示接下來你(妳)與對手的棋路。當然，贏棋是你(妳)的目標。


Ans: (to be solved on July 12, 2007)
(放棄智慧財產權；歡迎全貌下載流傳。)

「國際文化商務推展員認證班」 flyer