Macroeconomics I, UPF Professor Anonio Ciccone SOUTIONS PROBEM SET. (from Romer Advanced Macroeconomics Chaper ) Basic properies of growh raes which will be used over and over again. Use he fac ha he growh rae of a variable equals he ime derivaive of is log o show: (a) The growh rae of he produc of wo variables equals he sum of heir growh raes. Z() Z () X () Y () ) _ d ln Z() d ln[x()y ()] d ln X() d ln Y () Z() d d d + d _ X() X() + _ Y () Y () (b) The growh rae of he raio of wo variables equals he di erence of heir growh raes. Z () X() Y () Z() ) _ X() d ln Z() d[ln Y () ] d[ln X()+( ) ln Y ()] d[ln X()] Z() d d d d d[ln Y ()] d _ X() X() _Y () Y ().2 (from Romer Advanced Macroeconomics Chaper ) Implicaions of change in he growh rae for he evoluion of a variable. Suppose ha he growh rae of some variable X is consan and equal o a 0 from ime 0 o ime ; drops o 0 a ime ; rises gradually from 0 o a from ime o 2 ; and is consan and equal o a afer ime 2. (a) Skech a graph of he growh rae of X as a funcion of ime.
Figure (Noe ha he graph beween and 2 may as well be a sraigh line or convex shaped) (b) Skech a graph of ln X as a funcion of ime. Figure 2.3 Properies of consan reurns funcions. The funcion F (; ) is homogenous of degree d if F (; ) d F (; ) for all 0. If d, he funcion is said o be linear homogenous (or, alernaively, subjec o consan reurns o scale if i is a producion funcion). (a) (TBG) If producion facors are paid heir margial producs, all oupu is paid ou o producion facors. Show ha if F (; ) @F (;) @F (;) linear homogeneous, hen F (; ) @ + @ (his resul is called Euler s heorem). Wha does his imply abou he pro s of a perfecly compeiive rm producing subjec o consan reurns o scale? F (; ) d F (; ) ; d @F (;) @ + + @[F( ;)] @ @F (;) @ @[F( ;)] @ + @[F( ;)] @ @F( ;) @ F ; + F ; 2 + F ; F ; + F ; ( ) + F ; F ; F ; + F ; F (; ) A rm wih a consan reurns o scale producion funcion acing in a perfecly compeiive environmen has zero economic (or pure) pro, i.e. every inpu is paid exacly is marginal conribuion o he oupu, implying ha income is exhaused enirely. There are no pro s from speculaion, chance or he exploraion of for example marke inequaliies. One migh refer o his as fair facor prices. 2
(b) Consan reurns means ha all measures of produciviy are independen of he scale of producion. Show ha if F (; ) is @F (;) @F (;) linear homogenous, hen @ and @ are homogenous of degree 0. Suppose now ha a rm producing subjec o consan reurns o scale doubles is capial and is labor. Wha happens o h marginal produc of capial and he marginal produc of labor as a resul? F (; ) @F (;) @ @F( ;) @ F ; F ; ) F (; ) F ; F ; 0 F (; ) F 2 (; ) @F (;) @ @F(; ) @ F 2 ; F 2 ; ) F 2 (; ) F 2 ; F2 ; 0 F 2 (; ) alernaively: @F (;) @ lim! F (+;) F (;) (+) ) @F (;) @ lim! F ((+);) (+) F (;) lim! F (+;) F (;) ((+) ) lim! F (+;) F (;) (+) @F (;) @ @F (;) @ lim! F (;+) F (;) (+) ) @F (;) @ lim! F (;(+)) (+) F (;) lim! F (;+) F (;) ((+) ) lim! F (;+) F (;) (+) @F (;) @ Due o hese resuls, a rm ha chooses 2 does no in uence is marginal produc of capial and is marginal produc of labor a all. There is no reason ha an addiional employee equiped wih he same amoun of capial as all oher employees would be less or more producive han any earlier hired employee. 3
.4 Suppose ha he producion funcion F (; ) is subjec o consan reurns o scale o capial and labor. Suppose also ha he producion funcion is subjec o decreasing reurns o capial 2 F (;) @ @ @MP 2 @ 0, and ha he producion funcion is coninuosly di ereniable (an imporan propery of hese funcions is explained in (a) below). (a) Decreasing reurns o capial imply ha marginal produciviy of capial falls as he rm employs more capial per worker (by de niion) and ha marginal produciviy of labor increases (less obvious). Show ha his implies ha he marginal produc of labor is @ 2 F (;) @@ @MP an increasing funcion of he sock of capial: @ 0. @F (;) (Hins: Firs show ha he fac ha 0 is homogenous of degree 0 implies ha F (; ) @F (;) @ @ + @F (;) @ ha for coninuosly di ereniable funcions @2 F (;) @@. Then use he fac @2 F (;) @@ ). PART I @ 2 F (;) @ 2 + @2 F (;) @@ @[ @F (;) @ ] @ + @[ @F (;) @ ] @ @F(;) @ + @F(;) @ @F( ;) @ + @F( ;) @ F + F 2 0 0 PART II @MP @ @2 F (;) @@ @2 F (;) @@ from PART I @2 F (;) @ 2 0 @MP @ 0 (b) Implicaions of (a) above for labor demand. Assume ha a perfecly compeiive rm endowed wih a capial sock produces wih he producion funcion above. Show ha he labor demand of his rm is sricly increasing in is capial sock. If we increase he capial sock, (a) ells us ha he marginal produc of labor, MP, increases. Thus, on he labor marke we have MP w which will increase demand for labor as long as he wage rae remains consan. 4
Formally: F (; ) F ; ) @F (;) @ F ; + F ; 2 F ; F (; ) ) @2 F (;) @@ F @ @ 2 h F @ @ 2 + F i @ @ 2 @ @ 2 F F 0 F # 0 from(a) ) @ @ 0 ) @ @ 0.5 Consider a household who values consumpion C and leisure F according o he following uiliy funcion: U (C; F ) C + ( ) F where 0 and 0. e C be he numeraire (i.e. se is price equal o ). (a) Wha is an elasiciy? And he elasiciy of subsiuion? Show he relaionship beween he parameer 0 and he elasiciy of subsiuion beween consumpion and free ime. Recall ha he elsaiciy of @ ln( subsiuion is de ned as: F C ) @ ln w @( F C ) w @w F where F and C are he C uiliy maximizing demand for leisure and consumpion and w is he real wage. I.e. he elasiciy of subsiuion is he percenage change in he demand for leisure relaive o consumpion in response o a one-percen increase in he relaive price of leisure (noe ha w can be inerpreed as he price of buying leisure as he opporuniy cos of leisure is equal o he wage ha could be earned working). maximizing uiliy means o nd F and C such ha - in Equilibrium - he Marginal Rae of Subsiuion beween leisure and consumpion is equal o he relaive price (wih he price of consumpion normalized o ). 5
w @u @F @u @C ) w C +( )F + ( ) F C +( )F + C( ) C F, F C w ) @( F C ) @w w ( F C ) ( ) w w ( ) w Hence is he elasiciy of subsiuion beween consumpion and free ime. I.e. an increase of he real wage by % leads o a decrease of he demand for leisure relaive o he demand for consumpion by %. In general he elasiciy describes he percenage change in demand for a good due o a % change of he price of ha good, whereas he elasiciy of subsiuion may be described as he percenage change in relaive demand beween wo goods due o a change of heir relaive price. (b) Wha makes Cobb-Douglas uiliy and producion funcion so special? Show ha when, hen U (F; C) ln C+( ) ln F [and hence ha he uiliy funcion is of he Cobb-Douglas form U (C; F ) C F. (Hin: Wrie he uiliy funcion as C +( )F ln U (C; F ) ln and apply l Hospial s rule.) ln U (C; F ) ln C +( )F simply plugging in yields: ln U (C; F ) ln 0 0 0 which is no de ned. Thus we look a he limi: lim [ln U (C; F )] lim!! l 0 Hospial lim! 0 B @ 2 4 ln C +( )F C +( )F # 2 3 5 ( ) ln C 4e ln C 2 +( )e( ) ln F ln F 2 2 3 5 C A 6
[ln C+( )ln F ] ln C + ( ) ln F As shown in (a) he elasiciy of subsiuion equals in he case of a Cobb- Douglas funcion, i.e. he demand of an inpu facor relaive o he second inpu facor decreases by % if i s price relaive o he price of he second inpu facor increases by %. (c) (TBG) Can labor supply be independen of he real wage? Suppose ha he oal ime endowmen of he households is E. Wha is he household s budge consrain? Deermine he household s labor supply as a funcion of he ime endowmen E and he real wage w (he money wage relaive o he money price of consumpion). Show ha he supply of labour can be independen of he real wage, when. Why is he labor supply inelasic wih respec o he real wage in his las case? [Hin: hink abou income and subsiuion e ecs] In he las decades, real wages have risen srongly bu labor supply has no. Wha does his ell you abou he uiliy funcion of households? from uiliy maximizaion, we know: (see (a)) F C w, E w w, (E; w) E ( ) w + + given he budge consrain: F E w C ) @(E;w) @w 8 < E (( ) w + +) 2 ( + ) w 0 {z } : pivoal 0; 0 0; 0; The increase in he wage gives wo possible incenives, as far as labor supply is concerned. One would be o increase labor supply because paymen for every uni of work supplied is higher. The oher would be o work less because he same level of consumpion can be obained wih less e or in erms of supplied labor. In he case of he Cobb Douglas funcion boh e ecs, he income and he subsiuion e ec of an increase in he wage cancel ou (propery of addiive log-uiliy funcions), which is why labor supply is inelasic o changes in he real wage. 7
If i is empirically rue ha real wages have risen bu labor supply has no, we can conclude ha he (approximae) Cobb Douglas uiliy funcion is a good esimae of he rue uiliy funcion of households as far as heir preferences concerning leisure and consumpion are concerned. (d) Consider he case where in he household s uiliy funcion. Suppose ha here is a perfecly compeiive rm endowed wih a capial sock ha produces consumpion goods using a consan-reurn-o-scale producion funcion F (; ). Suppose also ha he producion funcion is subjec o decreasing reurns o capial, @2 F (;) @ @MP 2 @ 0, and ha he producion funcion is coninuously di ereniable. Find labor supply, labor demand, and he labor marke equilibrium boh graphically and analyically. Wha happens o employmen and real wages in his economy as he capial sock of he rm increases (hin: recall.4 (a))?! U(F; C) C F abour supply From.5 (c), we know: S (w) E ( ) w + + E ( E )w 0 + + E (noe: abour supply is inelasic w.r.. he real wage) abour demand Firm: max [F (; ) w MP ] ) max F ; w MP ) F ; w + F ; 2! 0 ) F (; ) w + F (; ) ( ) 0 ) D F (;) F(;) w 8
(noe: decreases as w increases) Equilibrium S D, E F (;) F(;) F (;) F(;) w ) w E Increasing he capial sock (@ 0): @w @ E @F (;) @ @F (;) @ @2 F (;) @ 2 0 # 0 Meaning, he wage rae increases, while employmen remains xed because labour supply is inelasic w.r.. changes in. However, we proved in.4 ha labour demand increases This is due o he increase in he wages. Graphically: Figure 3.6 (from Romer Advanced Macroeconomics Chaper ) Describe how, if a all, each of he following developmens a ecs he break-even and acual invesmen lines in our basic diagram of he Solow model: (a) The rae of depreciaion falls. Figure 4 The slope of he line represening Break-Even-Invesmen (BEI) decreases, i.e. BEI becomes less seep. This means ha for any ~ k less invesmen is required o susain he iniial value. Thus, a decrease in he rae of depreciaion increases he BGP-value of ~ k. (b) The rae of echnological progress rises. The slope of BEI increases, i.e. he line becomes seeper. Hence, an increase in a leads o an acceleraion of A lowering ~ k and hus a any iniial level more invesmen is required o keep ~ k consan. ~ k BGP decreases. 9
(c) The producion funcion is Cobb-Douglas, f (k) k, and capial s share,, rises The line represening acual invesmen (AI) moves upward i.e. i asigns a greaer value of s f ~k o any value k ~. If he capial share rises more invesmen is needed a any value of capial per e ecive worker. k ~ BGP increases. (d) Workers exer more e or, so ha oupu per uni of e ecive labor for a given value of capial per uni of e ecive labor is higher han before. Same e ec as (c). AI moves upward because for any k ~, s f ~k is greaer. As his does no resul in a consan increase of labor e eciviy, BEI remains unchanged.7 Can an increase in he savings rae end up increasing long-run income bu decreasing long-run consumpion per worker? Consider a Solow economy wihou echnological progress ha is on is balanced growh pah. Now suppose here is a permanen increase in he savings rae. Show he evoluion of consumpion per worker over ime (from he ime of he increase in he savings rae o he new balanced growh pah). In a Solow Economy wihou echnological progress, we have a condiion for he BGP: s f (k) (n + ) k; where : k Furhermore C Y s Y y s y f (k) s f (k) f (k) (n + ) k This means, ha in he basic diagramm of he Solow model, consumpion per worker is displayed as he di erence beween he producion funcion and he Break-Even-ine (if he economy is on is BGP). Maximizing consumpion per worker, we ge: f 0 (k) n + which gives he so-called golden-rule value of k ha goes along wih he maximum consumpion level. If an increase of he savings rae migh lead o a decrease of long-run consumpion per worker depends on he quesion if he economy is iniially endowed wih a level of capial per worker ha is lower, equal o, or higher han he golden rule level. If iniially k is smaller han k GR a marginal increase in he savings rae increases boh, income and consumpion, if k is equal o k GR a marginal increase in s 0
leaves c unalered. While if k k GR he change in s increases y bu decreases c (see illusraion). Figure 5 Figure 6 The inuiive explanaion says, ha for k k GR he increase in f (k) caused by he increase in s does no su ce o mainain k a he higher BGP level, such ha consumpion needs o be reduced in order o be able o keep k a he higher level..8 Solow model wih governmen. Consider a Solow economy wih governmen. The governmen axes households and consumes all of he ax revenue. Households consume a consan fracion c of heir disposable income Y T, where T is axes paid o he governmen. Show ha naional savings Y C G (savings by households and he governmen), where G denoes governmen consumpion, is equal o ( c) (Y T ). How does, herefore, an increase in axes a ec oupu per worker in he shor and in he long run (assume ha here is no echnological progress and ha he economy is on a balanced growh pah when axes are increased)? Y C G Y c (Y T ) T (Y T ) c (Y T ) ( c) (Y T ) E ec on oupu graphically: Figure 7 The inroducion of axes immediaely decreases acual invesmen which is a ow variable. This leads o a siuaion where Break-Even-Invesmen is greaer han acual invesmen. Hence, k decreases unil i reaches is new Balanced Growh Pah value. The sock variable k adjuss slower han invesmen. Oupu per worker does no aler in he shor run because he share of invesmen ha now goes o axes is used enirely for governmen consumpion. However, as k decreases in he long-run he missing invesmen leads o a decrease in oupu per worker. The new sable equilibrium is in E new. Formally: Balanced Growh Pah: s [f (k) T ] (n + ) k
) s f 0 (k) @k @T s (n + ) @k @T, @k @T s sf 0 (k) (n+) s sf 0 s (k) k [f(k) T ] 0 This is rue, because disposable income is always greaer han he share of oupu ha goes o capial. Moreover, because: @y @k 0, we have: @y @T 0.9 (TBG)(from Romer Advanced Macroeconomics Chaper ) Consider an economy wih echnological progress bu wihou populion growh ha is on is balanced growh pah. Now suppose here is a one-ime jump in he number of workers. (a) A he ime of he jump, does oupu per uni of e ecive labor rise, fall, or say he same? Why? A he ime of he jump oupu per uni of e ecive labor insanly falls, as well as capial per uni of e ecive labor does. While he capial sock is unable o change insanly and echnological progress does no aler oher han before ~k A mus decrease. The same argumen applies for ~y Y A. Inuiively, i is less capial per e ecive worker available and oupu has o be shared by more e ecive workers. Formally: @ ~ k @ @ ~y @ A 2 0 Y A 2 0 (b) Afer he iniial change (if any) in oupu per e ecive labor when he new workers appear, is here any furher change in oupu per uni of e ecive labor? If so, does i rise or fall? Why? The one-ime jump of may be inepreed as a new iniial siuaion for he economy ha does no maer in he long run. The decrease in k ~ leads o a siuaion where acual invesmen exceeds Break-Even-invesmen. Inuiively a decrease in k ~ increases MP and hus leads o MP R. Firms herefore need o pay less for heir capial han he capial yields for hem which makes invesmen lucraive. Hence, he capial sock and hereby k ~ and ~y increase unil he BGP-value of k ~ ha was valid before he increase in is obained again. k ~ and ~y end up a he iniial siuaion again. 2
(c) Once he economy has again reached a balanced growh pah, is oupu per uni of e ecive labor higher, lower, or he same as i was before he new workers appeared? Why? Due o he argumenaion in (b) he economy ends up in is iniial siuaion. This resul refers o he fac ha a jump in does no aler any growh rae, savings rae or producion funcion bu simply e ecs an iniial condiion. As one of he basic resuls of he Solow-Swan Analysis is ha iniial condiions of, A and do no maer, i is obvious ha he economy moves o i s balanced growh pah again. A comparable siuaion migh be he desrucion of capial (e.g. in a war) because his would also aler ~ k wihou a ecing growh raes of, A and. Figure 8.0 (from Romer Advanced Macroeconomics Chaper ) Consider a Solow economy ha is on is balanced growh pah. Assume for simpliciy ha here is no echnological progress. Now suppose ha he rae of populaion growh falls. (a) Wha happens o he balanced growh pah values of capial per worker, oupu per worker and consumpion per worker? Skech he pahs of his variables as he economy moves o is new balanced growh pah. Figure 9 BGP value of k : s f (k) (n + ) k ) s f 0 (k) @k @n k + n @k @n + @k @n ) @k @n k sf 0 (k) (n+) k 2 s(kf 0 (k) f(k)) 0 BGP value of y : @f @k @n 0 @k ) 0 @f @n @y @n 0 BGP value of c : 3
c BGP f (k) s f (k) ( s) f (k) ) @c @n ( s) f 0 (k) @k @n 0 This gives us: Figure 0 (b) (TBG) Describe he e ec of he fall in populaion growh on he pah of oupu (ha is, oal oupu, no oupu per worker). We know ha in he seady sae, i.e. if he economy is on is BGP, in an economy where here is no echnological progress k and herefore y Y are consan. Tha means, ha in his case oal oupu has o grow a he same rae as he equilibrium populaion. Hence, oal oupu grows a he growh rae n in he iniial equilibrium and a he - decreased - growh rae n 2 in he new seady sae. Furhermore, we know ha y Y increases if populaion growh decreases (par (a)). This resuls from he fac ha he populaion insanly grows slower as he growh rae of he populaion drops. However, adjusmen of he growh rae of oal oupu is gradually as compared o he one ime drop of he growh rae of he populaion, because as y Y increases oal oupu has o grow a a rae n 2 n n during he adjusmen process. 4
Figure ( ) ( ) X& X Figure 2 ln X ( ) Figure 3 w D D 2 S Equilibrium Increase in capial sock. Higher abor demand a any wage rae. eads o increase in equilibrium wage. α E
Figure 4 ( n + a + δ ) k % BEI s f ( k% ) AI k % BGP k % Figure 5 Oupu per worker increases f ( k % ) ( n + a + δ ) k % s ( ) f k% 2 s ( ) f k% engh of he doed line, i.e. seady-sae consumpion per worker decreases < k % GR 2 k % k % k %
Figure 6 c(s) s GR s Figure 7 Y y f ( k ) ( n + δ ) k ( c) f ( k ) old E new E ( ) ( ) ( ) c f k T k ( c)( T ) k BGP 2 k BGP
Figure 8 y% Y A ( a δ ) + k % f ( k % ) s f ( k% ) k% A k % 2 k % BGP Figure 9 Y y ( n δ ) + k ( n δ ) 2 + k f ( k ) c 2 c s f ( k ) k k BGP k BGP 2
Figure 0 k BGP y BGP c BGP