Exercie, Par IV: THE LOG RU 4. The olow Growh Model onider he olow rowh model wihou echnoloy prore and wih conan populaion. a) Define he eady ae condiion and repreen i raphically. b) how he effec of chane in he avin rae and in he depreciaion rae of capial on he rowh rae of capial and oupu per capia in he hor and lon run. c) ume he producion funcion and δ 0.. If per capia producion i 5, wha i he equilibrium value of? 4. The olow Model wih ounrie onider wo counrie ocoloco () and ambapai (). They are characeried a follow: Producion funcion: y 0 k y 0 k avin rae: 0. 0. Depreciaion of capial: δ 0. δ 0. a) ompue he eady ae value of capial and oupu per capia in he wo counrie. b) ocoloco ciizen pend more han hoe in ambapai. In fac, hey have a lower avin rae. I i poible ha ocoloco ciizen have, neverhele, a hiher per capia income? hy? c) ambapai ciizen wan o have he ame per capia income a he one of ocoloco ciizen. To hi purpoe, how much hould hey chane heir avin rae? Repreen raphically he adjumen proce oward he new equilibrium. 4.3 apial ccumulaion and he Marinal Propeniy o ave onider he followin producion funcion: 3 / 4 / 4 Le he avin rae be 0. and he depreciaion rae of capial be δ 0.. a) rie down he law of capial accumulaion and ive i economic inerpreaion. b) ompue he eady ae level of per capia oupu and capial per worker.
c) how he effec of an increae in he propeniy o ave on he level of per capia oupu raphically and provide ome economic inuiion for your reul. 4.4 The eady ae In counry he producion funcion per worker i: f. a) rie down he lon run equilibrium condiion. ompue he eady ae value of capial per worker and oupu per worker, iven ha 0.5 and δ 0.. b) The overnmen of counry decide o ive incenive o ave. Becaue of hi, he marinal propeniy o ave increae o 0.6. ompue he new eady ae level of / and of /. how raphically he effec of uch a policy and explain he adjumen proce oward he new equilibrium. c) ompue he level of conumpion per worker before and afer he policy decribed in b). Do you hink uch a policy i uiable? Give an economic inerpreaion of your reul. 4.5 The olow Model wih Populaion Growh onider he olow rowh model wihou echnoloy prore ( 0) bu wih poiive populaion rowh, / n. a) nalyically derive he eady ae condiion and repreen i raphically. b) ompue he rowh rae of per capia oupu and areae oupu. 4.6 Growh Rae onider he olow rowh model wih poiive echnoloy prore and poiive populaion rowh. a) In eady ae, he rowh rae of per effecive worker oupu i zero. ompue he rowh rae of per capia oupu and oal oupu. b) Given he producion funcion α α ( ), compue he relaion beween olow reidual and echnoloy prore. c) onider he producion funcion a poin b) where: α 0.75, producion row a he annual rae of 7.5%, workin force increae a he annual rae of 5.6% and capial row a he annual rae of.4%. ompue he olow reidual (or rae of rowh of
oal facor produciviy, TFP), he rae of rowh of work produciviy and he rae of rowh of echnoloical prore. d) hy i he rae of rowh of he echnoloical prore reaer han he olow reidual? 4.7 The olow Model wih Populaion Growh and Technical Prore In counry he producion funcion i: and he rae of echnical prore and populaion rowh rae are boh poiive. a) Derive he expreion for he equilibrium level of capial and oupu per effecive worker. b) Derive he expreion for he equilibrium level of conumpion per effecive worker. hich i he value of ha maximie he level of conumpion per effecive worker? c) how raphically wha happen if here i an increae in he rae of echnoloical prore. ha will be he effec on he rae of rowh of oupu, per capia oupu and oupu per effecive worker? 4.8 The olow Model wih Populaion Growh and Technical Prore In counry F he producion funcion and he main rowh parameer are: ( ), 0%, δ %, 3% e 5%. a) ompue he eady ae value of: - capial per effecive worker; - oupu per effecive worker; - rowh rae of oal produc; - oupu per worker. b) The overnmen decide o increae axe. Becaue of hi he avin rae become 5%. ompue he effec on oupu per effecive worker and he rowh rae of oal oupu in he followin cae: - he overnmen ue he addiional reource (aumin ha overnmen balance remain balanced) o finance curren expene; - he overnmen ue he addiional reource o finance R&D (becaue of hi he rae of echnoloical prore become 5%). c) fer a lon period in which he economic rowh rae ha been poiive bu mall, counry F ha experienced a boom (bi increae in he economic rowh rae). ha are he poible caue? How can one dicriminae amon hem? 3
4.9 Growh Facor onider he olow model wih echnical prore and populaion rowh. Dicu wheher he followin elemen have a raniory or permanen effec on he rae of economic rowh, and, if ye, in which direcion he effec oe. - an inflow of new immiran who become par of he labor force; - a one-ime exoenou reducion in he level of echnoloy (for example, due o he adopion of a iher environmenal leilaion); - a birh conrol proram; - incenive o reearch and developmen which permanenly increae he rae of echnical prore. Explain hrouh which mechanim hee elemen affec he economy. 4.0 Produciviy and he Labor Marke In counry he wae and price equaion are repecively e P ) ( P. + e a) umin ha, derive he expreion for. In which relaion are he unemploymen rae and produciviy? e b) umin ha, derive an expreion for. ha i he relaion beween he unemploymen rae and produciviy? ha i he difference o your anwer iven in a)? c) ume ha 5 % and 0. ompue he equilibrium level of unemploymen and real wae. e d) ume ha and ha increae by 0%. arin from he equilibrium compued in c), compue he equilibrium value of unemploymen and real wae in he wo followin period. e) If increae by 0% each year, which i he rae of chane of real wae? Explain your anwer. 4
OLUTIO 4. The olow Growh Model a) f δ In eady ae per capia capial i uch ha per capia avin (lef hand ide) i equal o capial depreciaion (rih hand ide). / f(/) /* δ (/) f(/) /* / b) The eady ae rowh rae doe no depend neiher on he avin rae nor on he depreciaion of capial. everhele, hi variable can influence he rowh rae in he hor run. Reducion in or incremen in δ imply lower eady ae level of per capia capial and of per capia oupu (ee he raph). Hence, durin he adjumen proce oward he new eady ae he rowh rae of per capia oupu ha o be neaive. The adjumen proce i hown in he raph. In he oppoie cae of incremen in or reducion in δ he adjumen proce i he oppoie wih a emporary poiive rowh rae. 5
/ / / δ(/) f(/) f(/) f(/) / / / / / / δ (/) δ (/) f(/) f(/) / / / (/) 6
c) ow compue he pre worker producion funcion (divide by ): ince: 5 we e: 5 ubiuin in he eady ae condiion: 5 0. 5 from which: 0.5 4. The olow Model wih ounrie a) The eady ae condiion i: f δ ocoloco cae: 0. 0 k 0. k Divide by k c : k 0 k y 400 0 k 400 ambapai cae: 0. 0 k 0. k Divide by k : k 0. 0 0. 0 k y 400 0 k 00 7
b) ocoloco ciizen earn a reaer per capia income y > y. In fac, per capia income depend on 3 facor: he avin rae and he producion funcion (echnoloy) which have a poiive effec on i, and he depreciaion rae of capial which ha a neaive effec on i. In he exercie he depreciaion rae i he ame for boh counrie. They differ in he avin rae (reaer in ambapai), difference ouweihed by he difference in he producion funcion. c) ambapai ciizen have o chooe a uch ha k aifie: y 0 k 400 from which: 400 k 600 0 In order o have k 600 we need: 0 k 0. k ubiuin: 0, 600 0.4 400 The increae in implie an upward hif in he avin funcion. ow, in he iniial equilibrium poin ( 400, 00) per capia avin exceed per capia depreciaion. Hence, per capia capial ar accumulain ill we reach he new eady ae ( 600, 400 ). Durin he adjumen proce he rowh rae of per capia oupu will be poiive. Once we are a he new eady ae he rowh rae will become zero. / ambapai δ (/) 400 f (/) 00 f(/) f(/) 400 600 / 8
9 4.3 apial ccumulaion and he Marinal Propeniy o ave a) δ + f δ + ) ( The capial per worker row unil invemen [ ] / i reaer han he depreciaion of capial [ ]. / δ. Due o decreain reurn o capial he rae of rowh of capial per worker i decreain in ime and will convere o a eady ae value of zero. b) In eady ae we have: 0 + δ Pluin ino he producion funcion: 0 4 4 3 / / δ 0 4 3 / δ 4 / 4 3 / δ 6 4 4 δ 8 4 3 / c) The increae in he lon run of make i poible o increae / and /. everhele, hi will no affec heir rae of rowh which remain zero. he beinnin, he rae of rowh of capial i poiive becaue / > δ /. o, he rae of rowh of / i poiive oo. Then, becaue of he decreain reurn o capial, / will row more han /. Thu, a he end we will have /δ /. The rae of rowh convere o zero and he yem convere o he eady ae.
/ f(/) / / δ(/) f(/) f(/) / / / 4.4 The eady ae a) In eady ae we have: δ 0.5 0. 5 and 5 b) / 36, / 6 The increae in he avin rae make i poible ha in period 0 avin per worker are reaer han capial demand per worker. Thi lead o accumulaion of capial per worker unil we e back o equilibrium. / and / increae and he new eady ae i. / / 6 /5 f(/) δ(/) f(/) f(/) 0 /5 / 36 0 /
c) Per capia conumpion i: δ ( ). δ ih 0.5 (olden rule value which maximie per capia conumpion) we have.5. ih 0.6,. 4. The acion aken by he overnmen lead o a reducion in he level of per capia conumpion becaue he marinal propeniy o ave in hi cae i no loner a he olden rule level. 4.5 The olow Model wih Populaion Growh a) By definiion, in eady ae capial and oupu per capia are conan. If populaion row a he rae > 0, in eady ae capial and oupu have o increae a he ame rae. In eady ae: n If hi condiion i aified each new worker will be endowed wih an amoun of per capia capial equal o ha of he old worker. In order o mainain conan per capia oupu i i neceary ha per capia avin depreciaion of capial. Hence, in eady ae: f δ ( + ) f are equal o per capia δ plu he per capia capial amoun for he new worker
/ / * f(/) ( +δ)(/) (/) / * / b) In eady ae he rowh rae of per capia oupu will be zero while he rowh rae of oal oupu will be. 4.6 Growh Rae a) In order for he raio o row a a zero rae, denominaor and numeraor mu row a he ame rae: + rowh rae of oal oupu (he rowh rae of he denominaor can be compued uin loarihm). The rowh rae of per capia oupu, i: / (he rowh rae of a raio i equal o he rowh rae of he numeraor minu he rowh rae of he denominaor). ince in eady ae +, we have: / + rowh rae of per capia oupu b) By he producion funcion α α α α α ( ), he rowh rae of oal oupu i equal o he um of he rowh rae of i componen each muliplied by i exponen: ( α ) + α + α from which: α ( α) α reidual
c) The olow reidual can be compued a: [ α + ( α ] reidual ) Uin he iven daa: [ 0.75 0.056 + 0.5 0.04] 0.07.7% reidual 0.075 The rowh rae of labor produciviy i: 7.5% 5.6%.9% The rowh rae of echnoloical prore i reidual. Hence: α 0.07 0.75 0.036 3.6% d) The reidual meaure he acual conribue of echnoloical prore o economic rowh. The reidual i maller han he echnoloical prore becaue he impac of echnoloical prore on economic rowh i limied by i weih in he producion funcion α, which i maller han. 4.7 The olow Model wih Populaion Growh and Technical Prore a) Recall ha i he amoun of effecive labor, a meaure of how echnoloy () increae labor produciviy. The producion funcion per effecive worker i The equilibrium condiion in eady ae (where oupu per effecive worker and capial per effecive worker are conan) i: f ( δ + + ) ubiuin he previou producion funcion: δ + + δ + + b) onumpion per effecive worker i equal o oupu minu avin: f ( ) 3
ubiuin: ( ) δ + + ( ) δ + + In order o compue he olden rule value of which maximie conumpion per effecive worker, we derive he expreion wih repec o : ( ) 0 δ + + 0 c) Becaue of an increae in he rae of echnoloical prore he line ( δ + + ) hif up. In he new equilibrium oupu per effecive worker and capial per effecive worker will be lower. / ( + +δ)(/) f(/) / 0 / ( + +δ)(/a) (/) /( ) /( 0 ) /() Due o capial depreciaion, he rowh of populaion and he level of echnoloical prore, here i a reducion in capial per effecive worker. In he new equilibrium avin are no enouh o compenae hi reducion. Therefore, he level of capial and oupu per effecive worker ar o decreae unil hey reach heir new eady ae value. Durin he raniion, he rae of rowh of oupu per effecive worker will be neaive. I will be zero aain once he new equilibrium i reached (ee he raph). The rae of rowh of per capia oupu ( ), will increae o he new level. naloouly he rae of rowh of oal oupu increae o he new equilibrium level +. 4
Thi apparen conradicion can be explained a follow. n increae in he rae of echnoloical prore imply implie an increae in he number of uni of effecive labor. Thi implie a reaer level of oal oupu. Bu, ince, a poined ou before, capial per effecive labor uni decreae, oupu per effecive labor uni decreae a well. / 0 4.8 The olow Model wih Populaion Growh and Technical Prore a) ee previou exercie: δ + + δ + + ubiuin: 0. 4 0.0 + 0.03 + 0.05 0. 0.0 + 0.03 + 0.05 The rowh rae per worker i 4%, he rowh rae of oal oupu i 8%, + while he rowh rae of oupu per effecive worker i zero. c) In he fir cae he only effec i hrouh he chane in he avin rae which reduce oupu per effecive worker and, emporarily, he rae of rowh of oal oupu: 0.5.5 0.0 + 0.03 + 0.05 In hi cae he rowh rae of eady ae oupu remain: 8%. + 5
In he econd cae we have an addiional effec oin hrouh echnoloical prore. Oupu per effecive worker will reduce even more: 0.5.5 0.0 + 0.05 + 0.05 while he eady ae rowh rae of oal oupu i: 0%. + d) The boom can be due o an incremen in he rowh rae of echnoloical prore. In fac, in eady ae he economic rowh rae i equal o he echnoloical rowh rae. lernaively, he boom can b due o he adjumen proce oward a hiher level of per capia oupu and per capia capial. Thi can be due o an increae in he avin rae or o a reducion in he depreciaion rae. In order o dicriminae beween he wo poible explanaion i i neceary o verify wheher he rowh rae of echnoloical prore ha ame behaviour of he rowh rae of per capia oupu. If i i o, hen he fir explanaion i he correc one. 4.9 Growh Facor The rae of rowh of eady ae i + Permanen effec Hence, a permanen chane in he rae of rowh can be obained only chanin one of hee wo facor. Thi can be achieved by birh conrol proram, which reduce 6, and incenive o reearch and developmen, which permanenly increae. Temporary effec The oher wo proram have only emporary effec on he rae of rowh. The inflow of new immiran who become par of he labor force i equivalen o an increae in leadin o a reducion in capial per effecive worker and oupu per effecive worker. In he new hor run equilibrium ) and echnoloical rowh (a rae, here will be an exce of avin, rowh of populaion (a rae ). Thu, capial and oupu per effecive worker will * * increae unil hey are back o heir iniial equilibrium level. Durin he adjumen/raniion proce he rae of rowh of per capia producion (uually equal o zero) i poiive and he rae of rowh of oal producion will be reaer han +. In he cae of a one-ime exoenou reducion in he level of echnoloy (differen from a chane in he rae of rowh of ), he proce will be oppoie o he one decribe above.
reducion in implie an increae in and o in. The hor run equilibrium level will be. arin from here, capial and oupu per effecive worker will decreae unil hey are back o heir iniial equilibrium level * *. Durin he adjumen/raniion proce he rae of rowh of per capia producion (uually equal o zero) i neaive and he rae of rowh of oal producion will be maller han +. /() (/)) /() (/()) f(/()) ( + +δ)(/()) (/()) (/()) /() * (/()) /() 4.0 Produciviy and he Labor marke a) e can rewrie he wae and price equaion a: P P + ( u ) ( u ) + u + + from which: The unemploymen rae doe no depend on he echnoloy level, bu only on he markup. 7
e b) If, by he wae and price equaion, we have: + u ( u ) + The unemploymen rae depend on produciviy. If decreae (increae), u will be reaer (maller) han i equilibrium level. If remain conan, u will be equal o i equilibrium level. c) Uin he formula in poin a), unemploymen i: 0.5 u 0. 0% + 0.5 Uin he price equaion, real wae will be: P + 0.5 8 d) arin from a iuaion of equilibrium ( ) where u 0 % e 8, increae by P 0%. Thu:. In, uin he price equaion: P +.5 In 3,. 8.8 The unemploymen rae oe back o i naural level becaue he wae equaion hif up. Uin he equaion a poin a): 0.5 u 3 0. 0% + 0.5 Real wae remain a heir previou level: P 3 8.8 8
/P E 3 E 3 8,8 3 8 E % 0% u e) From he price equaion we can ee ha real wae are proporional o. Thi implie ha real wae will increae a he ame rowh rae a ( ). oe: a conan coefficien. Thi can be proved a follow: In period real wae will be: P + + i conidered a ince he rowh rae of produciviy i P ( + ) + + The rae of rowh of real wae P P P P + ( + + ) +, in he econd period we will have: beween he wo period i: + Hence, if produciviy row a a rae of 0%, and real wae will row a he ame rae. 9