Final Exam. Tuesday, December hours

Transcription

1 San Francisco Sae Universiy Michael Bar ECON 560 Fall 03 Final Exam Tuesday, December 7 hours Name: Insrucions. This is closed book, closed noes exam.. No calculaors of any kind are allowed. 3. Show all he calculaions. 4. If you need more space, use he back of he page. 5. Fully label all graphs. Good uck

2 . (0 poins). The nex figure shows he naural log of real GDP per capia in wo counries, and B, over he 40 year period a. Based on he figure (circle he correc answer), i. Counry is growing faser han counry B before 990 ii. Boh counries grow a he same rae before 990 iii. Boh counries grow a he same rae afer 990 iv. Counry s real GDP per capia is growing a acceleraing rae before 990. b. Based on he figure, he approximae annual growh rae of GDP per capia, in counry, over he enire period is (circle he correc answer), i. % ii..5% iii. 4% iv. 5% c. Based on your answer o par b, and given ha populaion in counry grows a % per year, find he approximae annual growh rae of real GDP in counry. RGDP RGDP POP POP.5% RGDP % RGDP 3.5%

3 . (0 poins). Consider he Solow model discued in cla, and described as follows. Oupu is produced according o Y K, 0. Capial evolves according o K K ( ) I, where is he depreciaion rae and I is aggregae invesmen. People save a fracion s of heir income. This fracion is exogenous. Thus, he oal saving and oal invesmen in his economy are S I sy. The populaion of workers grows a a consan rae of n, which is exogenous in his model. Thus, ( n). a. Solve for he seady sae capial per worker, oupu per worker, and consumpion per worker (i.e. derive he expreions for k, y, c ). Deriving he law of moion of capial per worker: K K ( ) sk ( n) ( n) sk k k n n Using he definiion of a seady sae: k k k k k k n ( n) ( ) k ( n ) sk k c s n y k sk n s n ( s) y sk k k

4 b. Derive he golden rule saving rae, s GR (he saving rae which maximizes he seady sae consumpion per worker). We sar by deriving he opimal capial per worker: max c s k k k s.. ( n ) k sk The consrain means ha indeed he capial per worker is a is seady sae level. Plugging he consrain ino he objecive, gives: sk maxc k ( n ) k k The firs order condiion is: kgr ( n ) 0 ( n ) kgr kgr n Comparing his o he seady sae capial per worker, from he previous secion, s k, implies ha he golden rule saving rae is: n s GR 3

5 3. (0 poins). Suppose ha you wan o measure he produciviy level,, in a given economy, under he aumpion ha aggregae oupu is Cobb-Douglas, i.e. Y K, 0. a. Wrie he formula ha you would use, and describe wha daa you would need for obaining a ime series on. K Y I would need daa on real GDP, Y, physical capial, workers or oal hours),, and capial share,. K, labor inpu (eiher number of b. Suppose ha you ploed ln( ) as a funcion of ime, and he resuling graph, wih he fied linear rend and is equaion, are presened below. ln(tfp) ln(tfp) in U.S. y = 0.0x Years ln() inear (ln()) Based on he above graph, approximaely, wha is your esimaed average growh rae of produciviy (in %)? Round your answer o he neares enh of a percen. pproximaely,.% per year. 4

6 4. (0 poins). Suppose ha he echnology producion funcion is ˆ where is he level of echnology, 0,, is he number of researchers in he economy and is he consan cos of echnological improvemen in erms of researchers. a. ccording o his model, higher level of echnology, all else being he same, leads o slower growh rae of echnology. True/false, circle he correc answer, and briefly explain. The erm is a decreasing funcion of, which means ha if he sock of echnology already in exisence is large, i is harder o develop new echnology because scieniss need o cover a lo of maerial. Therefore, he erm is referred o as he "fishing-ou effec" of echnology producion. The analogy o fishing is ha afer caching all he big and lazy fish in he lake, i is harder o cach new fish. b. Suppose ha is growing a consan rae. Derive he approximae relaionship beween he growh rae of produciviy ( Â ) and he growh rae in he number of researchers ( ˆ ). Show your derivaions.,, ( ˆ ) ( ˆ ) Taking logs: 0 ln( ˆ ) ln( ˆ) This approximaely: 0 ˆ ˆ ˆ ˆ 5

7 5. (0 poins). Suppose ha he oal oupu, as a funcion of labor, in indusries / and is given by Y and Y, and he oal labor available for hese wo indusries is 0. lso suppose ha he produciviies are given by, 4. a. Find he efficien allocaion of labor beween he wo indusries. Efficien allocaion equalizes he marginal producs in he wo secors: MP / * 4, * MP / 6 b. Suppose ha he curren allocaion is 3, 7. Deermine wheher his allocaion is (i) efficien, (ii) overallocaion o secor, or (iii) overallocaion o secor, and illusrae your answer graphically. MP MP Oupu lo due o overallocaion o secor. Efficien allocaion: * * 6,

8 6. (0 poins). John and Melia work as waiers in a resauran, and earn heir income from ips. Suppose ha each of hem can choose one of hree effor levels: {0,, }. The individual coss (in $) of hese effor levels are {0, 49, 00}. The income from ips ha each of hem receives, depends on heir individual effor levels as follows: Individual effor level 0 Tips earned in $ (individual) a. How much effor will each of hem choose, and how much ips will each of hem earn, if ips earned are privae (no shared wih oher waiers)? Explain briefly. Individual effor level 0 Tips earned in $ (individual) Individual cos Individual profi (Revenue Cos) The highes individual profi earned is wih effor level of, and each of hem will earn $00 in ips. b. Now suppose ha he resauran policy is ha all waiers share he ips earned equally, regardle of how much ips each of hem earned. How much effor will each of hem choose, and how much ips will each of hem earn, under his policy? To answer his quesion you are required o consruc he payoff marix and use he concep of Nash Equilibrium. John Melia 0 0 0, 0 50, 00, 0, 50 5, 5 0, 50 0, 00 50, 0 00, 00 There is only one Nash Equilibrium in he above game: {, }, which means ha each of hem will work a effor level. Noice ha is a dominan sraegy for boh players, and herefore i is always a bes response. s a resul, each of hem will earn $00 worh of ips, insead of $00 under he policy where ips are no shared. 7

9 c. Wha do we learn from he above example abou he inefficiencies embodied in communal (socialis) indusry? Wih shared ips, workers have incenives o shirk (free ride). If John works hard (a effor level ), Melia has an incenive o only work a effor level. nd vise versa. Each worker in a communal indusry (e.g. "The Grea eap Forward" plan by Mao), benefis from effors of ohers, and herefore has a negaive incenive o work hard. 8

10 7. (0 poins). Suppose ha he quaniy of fish ha grows each year in ake S S Superior depends on he exising sock of fish as follows: G. 000 a. Calculae he opimal sock of fish in he lake. The opimal sock maximizes growh, and hus allows for maximal harves. To find i, we solve: The firs order condiion is: max S S S 500S 0.5S 500 S 0 S * 500 b. Calculae he maximum susainable yield. The maximum susainable yield: * * S S G 000 *

11 c. Suppose ha he curren sock is S 00 and he curren harves (fish caugh) is H 00. Calculae he nex period sock of fish in he lake, S. The growh in he curren period is: S S G The nex period sock is herefore: S S G H Thus, he nex period sock is smaller han his period. 80 d. Suppose ha due o overfishing, he governmen wans o reduce he number of fishermen in he lake. Sugges a policy ha will achieve his goal. Fishing permi. I increases he cos of fishing, and reduces he number of people fishing in he lake. e. Wha is he difficuly wih applying similar policy you suggesed in he las secion o comba global warming? mosphere, as opposed o mos lakes, does no belong o a single counry. Thus, a permi ha reduces polluion and abuse of amosphere requires coordinaion and agreemen of many counries. 0