1. To express the production function in terms of output per worker and capital per worker, divide by N: K f N

Transcription

1 THE LOG RU Exercise 8 The Solow Model Suppose an economy is characerized by he aggregae producion funcion / /, where is aggregae oupu, is capial and is employmen. Suppose furher ha aggregae saving is proporional o income wih a saving rae of s 0., he rae of depreciaion is δ 0., here is no echnological progress and populaion is consan. a) Calculae he seady-sae levels of oupu and capial per worker. b) Suppose he saving rae falls o s 0.. Calculae he effecs on oupu and capial per worker in seady sae. Show graphically how he equilibrium changes and explain he mechanisms a play. c) ow suppose he saving rae remains a s 0., bu he depreciaion rae falls o δ Calculae he effecs on oupu and capial per worker in seady sae. Show graphically how he equilibrium changes and explain he mechanisms a play. a) By he law of capial accumulaion: ( ) I + + δ, which in per capia erms reads as: ( ) I + + δ. To express he producion funcion in erms of oupu per worker and capial per worker, divide by : f. Since in equilibrium of closed economy: S I we have: f s δ + ) (. In seady sae oupu and capial per worker are consan. Hence he righ hand side is equal o zero ( ) + implying: f s δ, (savings/invesmencapial depreciaion) where we are using he assumpion +. Using: f, * * s δ,

2 δ s *, * δ s. Hence: * * δ s oe: he seady-sae level of / and / depend only on he raio δ s. Plugging in he values: *. Subsiuing back ino he oupu per worker: * *. b) Inuiion: a lower saving rae leads o lower invesmens and so o lower capial accumulaion. Hence, we will have lower levels of capial and oupu per workers. nalyically, using previous resul: f s δ Subsiuing he new value for s : * < smaller han he iniial value. Obviously: The decrease in he saving rae from s o s can be graphically represened by he shif in he invesmen curve from sf(/) o s f(/). Seady sae levels of boh capial and oupu per worker decrease. During he adjusmen o he new seady sae here is a recession wih he sock of capial decreasing and negaive growh. When he new seady sae levels are reached, savings are equal o he depreciaion of capial and he economy is in equilibrium.

3 c) Inuiion: a lower depreciaion rae implies a larger amoun of capial remains useful over ime. Since he saving rae has no changed, his implies ha capial accumulaes faser. Hence, we will have higher levels of capial and oupu per workers. nalyically, using previous resul: s f δ Subsiuing he new value for d : equal o he saring level. Obviously: The decrease in he saving rae is compensaed by he reducion in he depreciaion rae. Graphically: no only sf(/), bu also δ(/) shifs down. So, we are back o he original seady sae levels of capial and oupu per worker. oe: his is no unexpeced since how we noiced in poin a) seady-sae level of / and / depend only on he raio s. Being his raio he same in poin a) and c), he answer is he same. δ *

4 Exercise 9 The Solow Model The governmen of he counry, saring from a siuaion of balance, ends up in defici for a cerain percenage (d) of GDP. a) ssuming ha he rae of privae saving and he populaion growh do no vary, graphically show he impac of his defici on he per-capia income and on he per-capial capial in he growh model. b) Show he differences beween he previous saionary sae and he one occurring wih he defici, by explaining if his defici affecs he growh rae of his economy. a) Wih he defici he invesemen becomes: I s - d. The capial accumulaion is indeed given by: + ( δ) + I ( δ) + s - d By dividing boh side of he previous equaion by (number of workers), we will obain: + s d δ Puing f ( ) we will have ha: + ( s d) f ( ) δ This relaion is shown in he graph below:. δ ( /) 0 / I / E E f ( /) s f ( /) (s-d)f ( / ) I / 0 / 4

5 The effec of a defici consan and proporional o he income is similar o he one given by he reducion of he rae of he oal saving (privae plus public). In seady sae we will have a lower level of capial and of produc per worker. b) The defici modifies he rae of oal saving for his economy, ha however does no affec he growh rae. The assupions made wih he Solow Model hold: ) The rae of saving doesn affec he rae of growh of he producion in he long run; ) The rae of saving deermines he level of producion per worker in he long run; ) Changes in he rae of saving produce change in growh only in he shor run. Exercise 0 The Solow Model Consider he equaion describing he seady sae equilibrium in an economy wih echnological progress. a) Suppose he saving rae falls. Show graphically wha happens o he seady sae levels of capial per effecive worker and oupu per effecive worker. Explain wha happens o he growh rae of oupu per effecive worker, he growh rae of oupu per worker and he growh rae of oupu. b) ow consider ha he saving rae did no change bu he rae of echnological progress increased. Show graphically wha happens o he seady sae levels of capial per effecive worker and oupu per effecive worker. Explain wha happens o he growh rae of oupu per effecive worker, he growh rae of oupu per worker and he growh rae of oupu. a) The saving rae decreases. We use s o refer o he new saving rae. In general, lower saving leads o lower invesmen and a lower capial accumulaion. In he seady sae we have ha: s' f ( δ + g + g ) 5

6 / (d+g+g)(/) f(/) (/) (/) sf(/) s f(/ ) (/) (/) / The decrease in he saving rae from s o s implies ha invesmen is now lower. Graphically, he saving funcion shifed downward. Corresponding o he old seady sae level of capial per worker now we have a required invesmen higher han he saving; consequenly capial per effecive worker decreases unil he economy reaches is new seady sae where he required invesmen is again equal o saving. During he adjusmen period he economy undergoes a period of negaive growh. In he new equilibrium capial per effecive worker and oupu per effecive worker are lower and heir growh rae is again zero. The growh rae of oupu in seady sae is no affeced by he decrease in saving: i sill grows a a rae equal o he sum of he growh rae of echnological progress and populaion growh rae. lso he seady sae growh rae of oupu per worker is no affeced by he decrease in saving: i sill grows a he same rae as echnological progress. b) The growh rae of echnological progress grows from g o g. This implies ha saving is below he required invesmen. Since he following condiion mus hold in seady sae: s f ( δ + g + g ') here will be a emporary reducion in he level of capial per effecive worker, unil he economy reaches is new seady sae, where capia per effecive worker and oupu per effecive worker are again consan. 6

7 / (d+g+g )(/) (d+g+g)(/) f(/) (/) sf(/) (/) (/) (/) / Despie he fac ha he levels of capial and oupu per effecive worker are lower, in he new seady sae oupu and oupu per capia grow a a higher rae because: g g + g ' y d( / ) g / e g > g. ' 7

8 EXPECTCIOS Exercise Consumpion and wealh worker earns euro a year and expecs his real salary o grow by 0% per year for he nex wo years. he end of his period he worker will reire. ssume ha he worker has euro in financial wealh, and ha here is a proporional income ax, wih an average ax rae of 5%. Suppose furher ha he real ineres rae remains consan and equal o zero. a) Define and compue he oal wealh and he human wealh of his worker. b) Suppose he worker expecs o live 0 more years afer reiring. How much could he spend each year o mainain his consumpion consan? a) The oal wealh of an individual is equal o he sum of his human wealh, defined as he presen value of labour income and his nonhuman wealh, which includes financial wealh and housing wealh. The worker has: human wealh equal o he presen discouned value of he flow of oal labour income ha he expecs o earn when employed ne of income axes. The sequence of ne labour income is: 50,000 ( ) 50,000 ( 0.5) 50, , , , , ( ) ( ) 750 ( + 0.) ( 0.5) 50, , 5 50,000 0 per 4 Since he real ineres rae is equal o zero by assumpion we have ha he expeced discouned value of he flow of income is equal o he sum of income earned in each period. Therefore, human wealh is equal o n, , ,5 07,575 onhuman wealh is equal o 0,000 Toal wealh is equal o 07,575+0,000 7,575 euro b) If he individual expecs o live for anoher 0 years and if he wishes o mainain consumpion consan in each period, hen in each period he could spend: 7,575,757.5 euro. 0 Exercise - Invesmen and profis consrucor mus decide wheher o buy a drill a he price of euro. The drill depreciaes by 0% each year and generaes real profis of.000 euro saring from he curren year, becoming unproducive a he beginning of he hird year. a) Suppose he real ineres rae is 5%. Deermine wheher i is convenien o purchase he drill. b) Suppose he real ineres rae is 5%. Deermine wheher i is convenien o purchase he drill. Explain he differences compared wih a). a) Referring o Tobin s q heory, we have: Purchase cos of V 0,

9 The presen value of profis, wih r : VR e ( Π ) Π + ( ) Π +,000 + ( 0.),000 0, r δ + r. We mus consider he curren period () and he nex period (+) since he presen value of profis will be equal o zero in period +. The drill should be purchased only if: VR ( Π) VR ( Π) V > 0 or equivalenly, > V Purchasing he drill is herefore convenien. b) In his case: Purchase cos of V 0, 000. The presen value of profis, wih r 0. 5 : VR e ( Π ) Π + ( ) Π +,000 + ( 0.),000 9, r δ + r. Purchasing he drill is no convenien ye. The real ineres rae is higher. This increases he discoun facor puing less weigh on fuure profis relaive o curren cos. Exercise - The yield curve Suppose ha he acual ineres rae is equal o %. Financial markes expec ha he ineres rae will increase by % nex year and hen i will remain consan. a) Compue he yield of a -year bond, a -year bond and a -year bond. b) Draw he yield curve and provide an inerpreaion of is slope. a) The ield o mauriy on a n-year bond is defined as ha consan annual ineres rae ha makes he bond price oday equal o he presen value of fuure paymens on he bond. Given he definiion i is possible o prove ha he yield o mauriy is approximaely he average of he curren -year ineres rae and he -year ineres raes expeced for he nex (n-) years. Therefore: i % e i + i+ 0,0 + 0,04 i % e e i + i+ + i+ 0,0 + 0,04 + 0,04 i,% b) The erm srucure of ineres raes is he relaionship beween yield and mauriy of a bond. On a given day, we observe he yields on bonds of differen mauriies, and so we can race graphically how he yield depends on he mauriy of a bond. The curve ha conveys he relaionship beween yield (in he y axis) and mauriy (on he x axis) is he yield curve. The yield on a -year bond is %, he yield on a -year bond is %, he yield on a -year bond is,%. i,% 9

10 The yield curve is upward sloping. This means ha he long-erm ineres raes are higher han shor-erm ineres raes. Moreover, he fac ha he yield curve is upward sloping ells us ha financial markes expec shor-erm raes o be higher in he fuure. Exercise 4 The IS-LM wih expecaions The equilibrium of a small open economy characerized by perfec capial mobiliy and a flexible exchange rae can be represened by a radiional IS-LM diagram. The governmen of his counry decides o adop a conracionary fiscal policy in order o improve is naional accouns. a) Illusrae graphically and economically he effecs of such a policy on he equilibrium income, he ineres rae and he exchange rae. b) The cenral bank agrees wih he objecives of he governmen and would like o reduce he recessive effecs of he fiscal policy. Wha ype of policy should i adop? Describe he new equilibrium and he adjusmen process graphically and economically. c) Would he effecs of he fiscal policy be differen if he counry insead had a fixed exchange rae? Could he cenral bank aenuae he recessive effecs of he fiscal policy? Moivae your answer. The governmen of a counry wans o apply differen kinds of economic policies o sock exchange markes. Discuss he impac of he following unexpeced economic policies on sock prices. Price of a sock equals (π e 0 r i) e e $ D+ $ D+ $Q e + i ( + i )(+ i + a) n unexpeced moneary expansion policy s M When an unexpeced moneary expansion occurs (wihou any change in fiscal policy), P, so he LM curve shifs righ causing a decrease in ineres raes, i, invesmen increases and oupu increases by he muliplier effec. So he new equilibrium becomes E, and i i IS LM E E' LM' 0

11 From he heory we know ha: - profis dividends $Q - i $Q b) fiscal expansion policy accompanied by a moneary policy ha keeps he level of producion unchanged. In his case IS curve will shif righ (because of he expansionary fiscal policy) which will increase boh he oupu and he ineres rae, a he same ime LM curve will shif lef (because of conracionary moneary policy) o be able o keep oupu unchanged. ew equilibrium will be E wih unchanged level of oupu and higher ineres rae. LM' i E' LM E IS' IS Increase in ineres rae has an negaive impac on sock prices. s does no change i will no have any effec: - profis dividends $Q - i $Q The ne effec of he combinaion of wo policies is a reducion in sock prices. Exercise 5 - Expecaions in he IS-LM model The newly eleced prime miniser of a European counry promised o decrease income axes during his elecoral campaign. The ciizens of his counry believe he elecoral promises ha were made.

12 a) Using an IS-LM model wih only wo periods ( oday; + omorrow) wih expeced inflaion equal o zero (boh oday and omorrow), deermine he impac on curren producion and on he curren level of he ineres rae in he wo following scenarios: a) The ECB does no change is moneary policy. a) The ECB enacs a moneary policy o preven fuure producion from varying. b) Wha is he impac of he policy indicaed in poin a) on he yield curve? Moivae your answer. a) Le s assume ha axes are acually decreased in (+). T ' The IS and LM in ime + will be: IS : ' ( ', T ', r') + G' M ' LM : ' L( r') P' Then, he IS curve will shif o he righ while he LM curve will no move if he cenral bank does no aler moneary policy. Graphically: r IS' IS E' LM E ' The equilibrium moves from E o E and as a consequence ineres rae and oupu increase: r,. From oday s viewpoin, he expeced increase in oupu and he expeced decrease in axes will shif he IS curve o he righ, bu he expeced increase in he ineres rae will shif he IS curve o he lef. The ne effec of hese wo conrasing forces canno be deermined a priori: in order o know wheher he IS curve will shif righ or lef, i is necessary o know he precise specificaion of he IS and LM curves and he exac variaion in axaion.

13 Hence in ime we will have: r IS LM a) In +, he IS curve will shif righ, leading o higher oupu and a higher ineres rae. If he ECB inervenes o keep oupu consan, i will conrac moneary policy, so he LM curve shifs up, and he equilibrium moves form poin E o poin E. r' IS' LM' IS E' LM E In his case, oupu is unaffeced, he real ineres rae is higher han in a) and axes have been reduced. gain, his implies ha in we will have differen forces ha influence he posiion of he IS curve, while he LM curve says fixed: in paricular he expeced ax decrease shifs he IS curve o he righ, while he ineres rae increase shifs he IS curve o he lef. '

14 So in ime we will have: r IS LM lso in his case he ne effec of he forces is indeerminae, bu he IS curve is more likely o shif lef han in a). b) In + he policy mix keeps he oupu a is previous level. However he resricive moneary policy will increase r a lo. The financial markes will expec an increase of he expeced ineres e rae a lo, hence he yield curve will move up in +. i + The slope of he yield curve will depend on he hypohesis on he shifs of IS in ime. The more plausible hypohesis is ha he curren IS doesn move (because he increase of he ineres rae is offse by he effec due o he decrease of axaion). In his case he yield curve won move in ime and is slope is going o be seeper (look he graph below). i + 4

15 If, in ime, he effec of he increase of he ineres rae will prevail, he IS should move o he lef, he curren ineres rae decreases and he yield curve will sar from a lower value in ime, hence i will cross he original one and i will be also seeper han he original one. In oher words, if he curren ineres rae falls, hen i will probably fall by less han he large increase in he expeced ineres rae, so he -year ineres rae will increase. Then he yield curve roaes: he shor end falls bu he long end increases. + If, in ime, he effec of he decrease in axaion will prevail, he IS should move o he righ, he curren ineres rae is increasing and he yield curve simply will sar from a higher value in (in his case he new yield curve will be parallel bu above he original one). + THE OPE ECOOM Exercise 6 - Invesmen and he rade balance Consider a counry ha is open o foreign rade where he exchange rae and prices are fixed and in which only he goods marke exiss. In equilibrium, he counry has a rade surplus. a) Graphically illusrae he effecs on equilibrium income and on he rade balance of an exogenous increase in invesmen. b) Does his increase in invesmen influence he level of oupu a which rade is balanced? 5

16 a) Saring from poin (, X >0), an exogenous increase in invesmens leads o an increase in auonomous expendiure, and herefore of aggregae demand, and finally, hrough he muliplier, of equilibrium oupu. Boh ZZ and DD curves shif upwards by an amoun equal o ΔI (in fac, ZZC+I+G-εIM+X and DDC+I+G). The new equilibrium income is B. For sure rade balance deerioraes (I IM X ), bu i is uncerain wheher a he end X will be greaer, less han or equal o zero. Z DD B ZZ B B DD ZZ X TB B X X B X b) TB does no vary because ZZ and DD boh shif upwards by he same amoun (ΔI), herefore hey inersec in he same poin as before TB. This can also be deduced from he fac ha X does no shif because he variaion in I does no influence ne expors for a given value of. Exercise 7 - The goods marke in an open economy 6

17 Consider an open economy, wihou financial markes, wih fixed prices and exchange raes. In equilibrium, here is a rade defici. a) ssume ha he governmen adops an economic policy ha causes a real depreciaion of he domesic currency. ssuming ha he Marshall-Lerner condiion is saisfied, discuss he effecs of such a policy on equilibrium income, and on he rade balance b) ow consider an increase in foreign income. Discuss he differences beween his case and he one analyzed in poin a) a) ε, hen by Marshall-Lerner condiion: X. Z DD ZZ B B ZZ 45 TB B X X B X X B For sure he rade defici is reduced bu i is uncerain wheher a he end X will be greaer, less han or equal o zero. I all depends on how much he ZZ curve shifs upwards. b) * X X 7

18 Z DD ZZ B B ZZ 45 TB B X X B X X B In boh cases (a and b) we obain increases in oupu and in ne expors (rade balance). everheless in poin a) we obain redisribuive effecs (ciizens ha rely heavily on foreign goods are wors off); in poin b) insead here are no redisribuive effecs. 8

19 Exercise 8 - Compeiiveness in an open economy Due o a srong depreciaion of he euro, he Unied Saes have become less compeiive leading o a rade defici (X<0). a) Saring from his siuaion, which economic policy can allow he Unied Saes o increase producion and improve is balance of rade? b) ssuming ha he governmen auhoriies believe ha he level of producion is oo high, which policy would lead o an improvemen in he rade balance while reducing producion? c) To avoid a recession (in oher words o mainain a consan level of producion), how should his policy be modified? a) depreciaion of he real exchange rae (ε ). oe ha a depreciaion of he dollar implies an appreciaion of he euro. If ε, by he Marshall-Lerner condiion, X Z DD ZZ B B ZZ 45 TB B X X B X X B X 9

20 We canno deermine ex ane by how much X and increase. everheless, he Marshall-Lerner condiion guaranees ha he increase in income will no generae an increase in impors such ha hese may offse he increase in expors; herefore X will improve (rade defici will be reduced). b) conracionary fiscal policy (e.g. G ). In fac, G IM X Z DD DD B ZZ ZZ B B 45 TB B X X B X X c) combinaion of a conracionary fiscal policy (e.g. G ) and a depreciaion of he real exchange rae (ε ) 0

21 Z DD DD B ZZ ZZ C C ZZ B B 45 TB C X X C X B X X C X In fac, he wo policies have an opposie effec on producion bu a posiive effec on he rade balance:. By he conracionary Fiscal Policy: G IM X. By he depreciaion: ε, by he Marshall-Lerner condiion, X. e Effec: X

22 Exercise 9 - The inernaional bond marke The ineres rae in Japan is % and he nominal exchange rae beween he yen and he euro (price of a yen in erms of euro) is 0.0. Following a credibiliy crisis, he financial markes expec a 0% depreciaion of he euro in he following year. a) How many euro would a European invesor earn, if oday he invess in a one-year Japanese bond? b) Leaving aside all addiional coss (exchange rae risk and ransacion coss), if he ineres rae in Europe is 7%, would i be preferable for a European invesor o inves in European or Japanese bonds? In his case, is uncovered ineres pariy saisfied? a) E 00 exchange rae In he case of a depreciaion E e + E e +00(-0%)90% yen Reurn of an European invesor ha decides o inves in Japanese bonds: Invess 00 yen a he Japanese ineres rae (%) obaining on mauriy: 00 ( + 0.0) 0 yen Reconvers 0 yen in euro, by muliplying i for he inverse of he expeced exchange rae, /E e +: 0/ E e + 0/90. he value in euro of he invesmen in yen Expeced Reurn:.-0. in euro b) Reurn of an European invesor ha decides o inves in domesic bonds: Invess euro a he European ineres rae (7%) obaining on mauriy: ( ). 07 invesmen in euro Reurn: in euro I is more profiable o inves in he Japanese marke (0.>0.07) Subsiuing he values of he exercise in he uncovered ineres pariy formula i can be shown ha he condiion is no saisfied: * E + i ( + i ) e E+.07 is differen from 00*.0(/90) There are arbirage opporuniies beween he wo forms of invesmen because expeced reurns are no equalized. Exercise 0 - Fiscal policy in he Mundell-Fleming model The equilibrium of a small open economy characerized by perfec capial mobiliy and a flexible exchange rae can be represened by a radiional IS-LM diagram. The governmen of his counry decides o adop a conracionary fiscal policy in order o improve is naional accouns. a) Illusrae graphically and economically he effecs of such a policy on he equilibrium income, he ineres rae and he exchange rae.

23 b) The cenral bank agrees wih he objecives of he governmen and would like o reduce he recessive effecs of he fiscal policy. Wha ype of policy should i adop? Describe he new equilibrium and he adjusmen process graphically and economically. c) Would he effecs of he fiscal policy be differen if he counry insead had a fixed exchange rae? Could he cenral bank aenuae he recessive effecs of he fiscal policy? Moivae your answer. a) Conracionary Fiscal Policy: G or T Goods Marke: G or T Z Money Marke: M d i (if he money supply says consan we have an excess supply of money) Currency Marke: i domesic securiies are less convenien (ouflow of capial: sale of domesic currency for foreign currency) E (depreciaion) The effecs on he Money marke and on he Currency marke imply ha he recessive effecs of he reducion in public expendiure (reducion of ) are parially compensaed by an increase in invesmens (due o he reducion of i) and an increase in ne expors (due o he depreciaion of he exchange rae E ). IS i IS LM i i B B B B E B E E

24 b) The CB can reduce he recessive effecs of he conracionary fiscal policy wih an expansionary moneary policy ha brings income back o he iniial level. The expansionary moneary policy implies ha he previous excess supply of money increases leading E o decrease furher. Boh hese variaions have an expansionary effec on income compleely compensaing he reducion in due o he reducion in G or he increase in T. i IS IS LM LM i i i B B B i C C C E B C E C E B E c) In a fixed exchange rae regime he CB mus mainain a consan ineres rae in order for he exchange rae no o vary. To eliminae he effecs of he fiscal policy on he ineres rae, a conracionary moneary policy mus be enaced. Such a policy will however amplify he recessive effecs generaed by he fiscal policy. i IS IS LM LM i i i C i B C B B C E C B E B E E C 4