Midterm. Wednesday, March hour, 30 minutes

Transcription

1 San raniso Sae Universiy Miael Bar ECON 7 Spring 3 Miderm Wednesday Mar or 3 mines Name: Insrions. Tis is losed boo losed noes exam.. No allaors of any ind are allowed. 3. Sow all e allaions. 4. If yo need more spae se e ba of e page. 5. lly label all graps. Good L

2 . 5 poins. Consider e Solow model briefly desribed as follows. Op is proded aording o Y K L were as all e properies of e Neolassial prodion fnion. Capial evolves aording o K K I were is e depreiaion rae and I is aggregae invesmen. People save a fraion s of eir inome. Tis fraion is exogenos. Ts e oal saving and oal invesmen in is eonomy is S I sy. Te poplaion of worers grows a a onsan rae of n wi is exogenos in is model. Ts L n L. K a. Le be apial per worer. Prove a op per worer an be L expressed as a fnion of only i.e. f. Y K L K L L L Te seond sep follows from e assmpion a as Consan Rerns o Sale CRS. Ten we define f. b. Te following grap plos erain flows for e Solow model. On e grap indiae e seady sae apial per worer ss op per worer y ss and onsmpion per worer ss. Op onsmpion saving ss y ss ss

3 . Sppose a e Cinese eonomy is desribed by e Solow model and e grap in e previos seion represens e alibraed Solow model o e Cinese eonomy. Based on is informaion an eonomis from e Ban of Cina laims a e saving rae in Cina is oo ig. Explain briefly wy e eonomis is orre based on e Solow model of orse. In yor explanaion se e grap from e las seion wi is reproded ere for yor onveniene. Op onsmpion saving ss GR GR Te saving rae is oo ig bease i is possible o inrease e seady sae onsmpion by lowering e saving rae. Te above figre sows e iges possible seady sae onsmpion per worer GR e golden rle onsmpion per worer wi is iger an e rren seady sae of e eonomy.

4 . poins. Sppose a we model e oal op in e eonomy wi e Cobb-Doglas prodion fnion: Y A K L. Sppose a in e daa e GDP an be broen down ino nambigos labor inome I L nambigos apial inome I K and ambigos inome I?. Assming a e fraion of ambigos inome paid o labor is e same as e labor sare in oal GDP alibrae e parameer for an eonomy wi I L I K GDP 4. irs noie a I 4.? GDP I L [ GDP I I L GDP I 3?? I? ] I L 4 3 3

5 3. poins. Te following Malab fnion allaes e probabiliy densiy of a normal random variable X ~ N for any realizaion x of X. Te formla of e densiy is f x e x fnion y = npdfxmsigma if nargin < m = ; end if nargin < 3 sigma = ; end xn = x - m/sigma; y = exp-.5 * xn.^./ sqr*pi.* sigma; a. How wold yo se is fnion in order o allae e probabiliy densiy of a normal random variable wi mean 5 and sandard deviaion 3 a e poin x? Wrie e ommand a yo wold need o ype in e ommand window. In e ommand window we need o ype: y = npdf-53 b. Sppose a yo wan e fnion o aep e variane of e normal random variable v insead of e sandard deviaion. Ts e firs line of yor fnion is: fnion y = npdfxmv. How wold yo modify e original fnion o aommodae is ange? Wrie e seond line of yor new fnion. sigma = sqrv. Te variable nargin sores e nmber of argmens a e ser passed o e fnion. Explain wa is e prpose of e if ondiions. Te ondiions provide e defal vales for and. In parilar if e ser does no provide e vale of e ode assmes and if e ser did no provide a vale for e ode assmes a. Ts y = npdfx will ompe e sandard normal densiy. d. Can is fnion aep a veor of vales for x and rern a veor of y vales? Explain wy or wy no. Te fnion aeps a veor of x s. Wen applying e formla e ode ses e.^ and./ wi applies e formla o ea elemen in a veor. 4

6 4. 3 poins. Consider e Neolassial Grow Model disssed in lass. Tere is a single represenaive oseold and a single represenaive firm a live forever. Te oseold s period iliy fnion l and e lifeime iliy is U { l} l were is e dison faor. Te oseold as ni of ime so e labor spply is l. Te oseold owns e apial so wi e law of moion x were x is invesmen and is given. Te oseold reeives a real wage w per ni of labor spplied o e firm a renal rae r per ni of apial so rened o e firm and - e profi dividend from e represenaive firm. Tere is a single represenaive firm a prodes e op in is eonomy wi prodion fnion y K L were K is apial and e L is labor. Assme a K L saisfies all e assmpions of a neolassial prodion fnion. Te eonomy is losed and ere is no governmen s e feasibiliy onsrain is: x y. a. Wrie e oseold s lifeime iliy maximizaion problem. { s.. max } w r Noie a e profi is zero sine e prodion fnion as onsan rerns o sale. Also noie a we sbsied e leisre from e ime onsrain: l and e oie variable now is - e labor spplied wor ime. 5

7 6 b. Wrie e Lagrange fnion orresponding o e oseold s problem in e las seion and derive e opimaliy ondiions for labor spplied and invesmen Eler eqaion. r w L irs order ondiions: ]: [ ]: [ : r w Combining e ondiions for and gives: w w Using e ondiion for onsmpion a ime and +: Combining wi e ondiion for gives e Eler eqaion: r To smmarize e opimaliy ondiions are: r w

8 . Provide eonomi iniion for e opimal invesmen ondiion Eler eqaion afer rewriing i in e form a sows e marginal pain and gain from invesmen. Eqaion in e las seion an be wrien as: r Te lef and side is e pain deline in iliy as a resl of invesing exra ni of inome in pysial apial and erefore giving p ni of onsmpion in period. Reall a e marginal iliy of onsmpion is e iliy is e ange in iliy resling from ni ange in onsmpion. Te rig and side is e iliy gain from a invesmen. In period e rern on is invesmen in nis of onsmpion is eqal o r + e non-depreiaed ni of apial originally reaed. To onver is rern ino iliy we mliply by e marginal iliy from onsmpion and o onver o presen vale we mliply by e dison faor. Ts e opimal invesmen ondiion reqires balaning e marginal pain and e marginal gain from invesmen. Any model of invesmen ms ave a ondiion similar o is one. 7

9 8 d. Sow a ombining e opimaliy ondiions of e oseold and e firm wi e mare learing ondiions leads o e following ompeiive eqilibrim ondiions for e Neolassial Grow Model NGM in is seion: 3 rom e firm s problem we ge: r w and we already sbsied e apial mare learing ondiion K and e labor mare learing ondiion L. Sbsiing ese ino e oseold s opimaliy ondiions gives eqaions and presened ere. Combining e feasibiliy onsrain wi e law of moion of apial: x y x Tis gives feasibiliy onsrain in ondiion 3.

10 9 e. Prove a e ompeiive eqilibrim alloaion is Pareo Opimal. Ta is prove a e ompeiive eqilibrim alloaion solves e appropriae Soial Planner s problem. Te Soial Planner s problem is: s.. max } { Lagrange: L irs order neessary ondiions: irs order ondiions: ] : [ ] : [ : Combining e ondiions for and gives: Using e ondiion for onsmpion a ime and +: Combining wi e ondiion for gives e Eler eqaion:

11 Te las eqaion is e feasibiliy onsrain wi an be obained from differeniaing e Lagrange fnion wi respe o or simply by noing a e feasibiliy onsrain of e soial planner is e same as e feasibiliy in a ompeiive eqilibrim seing. 3 Ts e ompeiive eqilibrim alloaion solves e Soial Planner s problem and erefore i ms be Pareo Opimal reall a any Pareo Opimal alloaion is a solion o some Soial Planner s problem.

12 5. poins. Te general ompeiive eqilibrim ondiions for e Neolassial Grow Model were derived in lass: 3 Sppose a prodion fnion and period iliy fnion are: ln ln A a. Wrie e eqilibrim ondiion 3 for e fnional forms provided above. A CD A CD A CD 3

13 b. Sppose prodiviy grows a onsan rae A i.e. A A A and ere is a niqe balaned grow pa on wi y grow a e same onsan rae of and onsan. Terefore on a BGP y y e following raios are onsan: y y yo y 3 y are no ased o prove is. Sppose a in e daa. 3 Wa wold be in is eonomy? y y Wrie e eqilibrim ondiions 3 for a balaned grow pa expressed in erms of e raios and. y y BGP BGP BGP3 y y y

14 d. Sppose a in e U.S. and in Cina e apial sare is 3 and onsmpion onsies 3 of GDP. Sppose a e available disreionary ime per wee is ors. Te average worer in e U.S. wors 4 ors per wee wile e average worer in Cina wors 5 ors a wee. Sow ow yo wold alibrae e parameer for e U.S. and Cina denoe ese by U.S. and Cina. Using eqaion BGP U.S U. S. Cina Cina e. Provide e eonomi iniion for e resl from las seion explain iniively wy Cina U.S.. Te parameer is e weig on onsmpion in e iliy fnion wi indiaes ow imporan onsmpion relaive o leisre is e weig on leisre is. Sine in Cina people wor more i maes sense a leisre is less imporan. 3

15 6. 5 poins. Consider e neolassial grow model disssed in lass. Tere is a single represenaive oseold and a single represenaive firm a live forever. Te oseold s period iliy fnion l and e lifeime iliy is U { l} l were is e dison faor. Te oseold as ni of ime so e labor spply is l. Te oseold owns e apial so wi e law of moion x were x is invesmen and is given. Te oseold reeives a real wage w per ni of labor spplied o e firm a renal rae r per ni of apial so rened o e firm and - e profi dividend from e represenaive firm. Tere is a single represenaive firm a prodes e op in is eonomy wi prodion fnion y K L were K is apial and e L is labor. Assme a K L saisfies all e assmpions of a neolassial prodion fnion. Tere is a governmen a olles axes { x w } on onsmpion invesmen labor inome and apial inome. Te governmen spends ese axes on governmen onsmpion g and lmp-sm ransfers and we assme a e bdge is balaned in every period. a. Wrie e oseold s problem. No need o solve i. { x } s.. max x x x Hoseold s problem: w w r Sbsiing e law of moion of apial in e bdge onsrain gives { } s.. max x w w r x Noie a e profi is zero sine e prodion fnion as onsan rerns o sale. Also noie a we sbsied e leisre from e ime onsrain: l and e oie variable now is - e labor spplied wor ime. 4

16 5 b. Sppose a e ondiions for ompeiive eqilibrim are given: w g R 3 : : : Were x x x R Based on ondiion alone provide eonomi iniion abo ow axes on labor inome and onsmpion disor e opimal labor spply. Wa do yo expe o appen o ors wored wen w or go p? We see in eqaion a iger axes on labor or iger axes on onsmpion mae leisre eaper relaive o onsmpion. We erefore expe a based on only e onsmer wold wor less.. Consider a version of e NGM model in is qesion wi inelasi labor spply so a e lifeime iliy is given by } { U. Te res of e model is as desribed in is qesion. Wrie e ompeiive eqilibrim ondiions for is version of e model f g R : ' : Were ' x x x f R Noie a ere are only wo ondiions now bease labor spply is no longer a oie variable of e oseold and insead e will always spply.