2. Suppose that a consumer has the utility function

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1 中正大學 { 修正 }. Suose that you consume nothing ut milk and cake and your references remain constant through time. In 3, your income is $ er week, milk costs $ er ottle, cake costs $ er slice and you uy six ottles of milk and four slices of cake er week. In 4, your income rises to $ er week, the rice of milk rises to $.5 er ottle and the rice of cake rises to $.5 er slice. () In which year are you haier ( a)? () In which year do you eat more cake ( )? Justify and [ 修正答案 ] illustrate your answer with indifference curves..(a)4 ()4 (a) x x x + x M, x 6, x 4, M'.5x +.5x M',( )'

2 94-6 經濟學試題解析 () x M ( )' B A M ' x Homothetic C 4 U4 U3 6 ' x B A x U3 會提高到 U4, 故 4 的效用叫高 () 在 C 點時, 因為預算限的斜率大於無異曲險的斜率, 為達成新均衡, 則 x x x. Suose that a consumer has the utility function / 3 / 3 U ( x, x x x and an income of $ M. ) () When rices are $ for good and $ for good and income is 4, find the otimal consumtion of oth goods ( c). () Suose that the rice of good rises to $3, while the rice of good stays at $. With the urchasing ower held constant after the rice change, what is the sustitution effect (in numers)

3 中正大學 [ 修正答案 ] ( d)? (hint: calculate the new demand for good ). (c) (8, 8) / 3 / 3 L x x + λ ( 4 x x ) FOC * ( x, x 8 (d) ) ( 3, 3 ) ( 4, ) 3 3 ( + ) ( + ) * (8, 8) 4 x + x () M ' x + 3x () ()-() 4 M ' x 8, M ' 3 x (,, 4) x (, 3, 4) 依據 Slutsky aroach M x, SE x x This question relates to the relationshi etween demand and marginal revenue curves for a few functional forms. () For a constant elasticity demand curve of the form q a

4 94-64 經濟學試題解析 () find the vertical distance etween the demand and marginal revenue curves ( e). () For a non-linear demand curve ( q a + ), find the vertical [ 修正答案 ] distance etween the demand and marginal revenue curves f ) (. 3. (e) q a ( ) a (f) TR q AR ( + ) q ( ) q MR dtr dq a ( ) a ( + ) q the virtical distance etween D and MR curves is MR AR MR q q.5 ( ) ( ) a a.5 q a + ( q aq + a ) AR 3 TR q ( q aq + a q) MR (3q 4aq + a ) the virtical distance etween D and MR curves is ( q a) MR AR MR ( q + aq) q

5 中正大學 Consider TSMC s roduction function is / / 3 / 4 f (L, K, M) L K M, where L, K, M denote the amounts of laor, caital and material inuts, resectively. Let w, r, t reresent the rices er unit of laor, caital and material inuts, resectively. () What is the marginal rate of technical sustitution etween caital (K) and material (M) ( g)? () If there is a 5% increase in caital inut, how aout outut? If all inuts have increased y %, what haens to outut ( h)? (hint: Derive the elasticity of outut with resect to laor (L) and caital (K)) (3%; %) [ 補充解法 ] (g) / / 3 / 4 Q Y f ( L, K, M ) L K M 3/ / / 3 / 4 Y ' f ( λ L, λk, λm ) λ L K M Y ' λ /3 λ 3 生產彈性為 λ Y 3/ λ Y ' 3 λ Y (h) / 3 / 4. Q Y f ( L, K, M ) L K / M ln Y ln f ( L, K, M ) ln L + ln K + ln M 3 4 Y& L & K& M & 對時間微分, ln + ln + ln Y L 3 K 4 M + 5% % 3 4 Y & L& K& M&. ln + ln + ln Y L 3 K 4 M

6 94-66 經濟學試題解析 () % + % + %.83% 3 4. The Chung-Cheng insurance comany sells home fire insurance and rovides a discount to olicyholders who have smoke alarms (ownershi of smoke detectors is something that the comany can check in the event that a major fire occurs). Suose that it costs a homeowner $ to urchase and install a smoke detector. Moreover, suose that exected losses fall y $ to urchase and install a smoke detector in oeration. If the home fire insurance market is erfectly cometitive, Chung-Cheng insurance comany will give ( r) (how much of a discount) to someone who has installed a smoke detector. On the other hand, if Chung-Cheng insurance comany is a monooly rovider of fire insurance, it will offer ( s) (how much of a discount) to olicyholders who install smoke detectors. [ 答案補充 ] (r) D (s) D (r) D 假設 D: 購買火災險的折扣 (discount) 已裝置 fire alarm P: 發生火災的機率 A: 保險費收入 (insurance fee) L:Loss 理賠金額 (full insurance) 未裝置警鈴保險公司的期望利潤為 P( A L) + ( P)( A) 發生火災下理賠後的收入 + 未發生火災的收入裝置警鈴保險公司的期望利潤為

7 中正大學 P( A L D) + ( P)( + 發生火災下理賠後的收入 + 未發生火災的收入 + 當被保險 人裝置警鈴, 一但發生火災時, 被保險人的損失減少金額 ( 相 當於保險公司少理賠的金額 ) 因為完全競爭下, 保險公司利潤為, 則未裝置警鈴保險公司的期望利 潤為, 故 P( A L D) + ( P)( + P ( A L) + ( P) A D + 又未裝置警鈴保險公司的期望利潤為, 即 P ( A L) + ( P)( A) D (s) D 獨占者下, 保險公司的期望利潤為 P( A L D) + ( P)( + 發生火災下理賠後的收入 + 未發生火災的收入 + 當被保險 人裝置警鈴, 一但發生火災時, 被保險人的損失減少金額 ( 相 當於保險公司少理賠的金額 ) 因為獨占者下, P( A L D) + ( P)( + Eπ FOC P D + P D D