Suggested Solutions to Assignment 4 (REQUIRED) Submisson Deadline and Location: March 27 in Class

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1 EC 450 Advanced Macroeconomics Insrucor: Sharif F Khan Deparmen of Economics Wilfrid Laurier Universiy Winer 2008 Suggesed Soluions o Assignmen 4 (REQUIRED) Submisson Deadline and Locaion: March 27 in Class Toal Marks: 50 Par A Shor Quesions [0 marks] Explain why he following saemen is True, False, or Uncerain according o economic principles Use diagrams and / or numerical examples where appropriae Unsuppored answers will receive no marks I is he explanaion ha is imporan Each quesion is worh 0 marks A According o he growh heory, an increase in he invesmen rae in physical capial in a closed economy does no have any impac on he longrun growh rae of oupu per worker [Diagrams required] Uncerain The answer depends on he growh heory we are considering In he basic Solow model, he general Solow model, he Solow Model wih human capial and he semi-endogenous growh model, an increase in he invesmen rae in physical capial in a closed economy does no have any impac on he long-run growh rae of oupu per worker However, in he AK growh model, an increase in he invesmen rae in physical capial leads o a rise in he long-run growh rae of oupu per worker In he general Solow model wih exogenous echnological progress, an increase in he invesmen rae in physical capial ( ie, he savings rae) will have no effec on he longrun growh rae of oupu per worker, y I will cause a shif in he seady-sae growh pah of y from one level o a new and higher level, wih he long-run growh rae being he same before and afer as he rae of exogenous echnological progress, namely g See Figure 53, Figure 54, Figure 55, Figure 56, and pages 35 o 38 of he exbook for a graphical explanaion In he AK model, i can be shown ha on he balanced growh pah oupu per worker and capial per worker grows a a common consan endogenous growh rae g e sa δ, where s is he invesmen rae in physical capial, A is a consan defined as L, where L is he fixed amoun of labor in he economy, δ is he consan depreciaion rae An increase in he invesmen rae in physical capial (s) will lead o an increase in sa δ, which can be illusraed by lefward roaions in he ( sa + δ ) k and sak curves, Page of 9 Pages

2 and a verically upward shif in he sa curve in Figure 83 of he exbook Figure 83 hus shows ha an increase in he invesmen rae in physical capial will lead o a permanenly higher growh raes in capial per worker and GDP per worker The growh brake from he Solow models and he semi-endogenous growh model, diminishing reurn o capial, is simply no longer presen in he aggregae producion funcion of he AK model The source of growh in he AK model is hus aggregae consan reurns o he reproducible facor, capial See pages of he exbook for a deailed explanaion Page 2 of 9 Pages

3 Par B Problem Solving Quesions [40 marks] Read each par of he quesion very carefully Show all he seps of your calculaions o ge full marks B [40 Marks] Consider an economy wih he following Cobb-Douglas aggregae producion funcion ( A L ) Y K, 0 < <, where Y is aggregae oupu, and K is he sock of aggregae capial, L is oal labor L grows exogenously a A is he effeciveness of labor a period Assume ha consan raes n Capial depreciaes a a consan rae δ The evoluion of aggregae capial in he economy is given by K K sy δk +, 0 < < s, 0 < δ <, where s is a consan and exogenous saving rae Assume ha he labour produciviy variable, A, depends on aggregae oupu because of learning-by-doing producive exernaliies which arise from workers being involved in producion The following equaion shows his exernal learning by doing effec φ A Y, 0 < φ < Show ha he aggregae producion funcion in his model is: [ ( )] ( ) [ ( )] Y K L Why have we assumed φ <? Show ha he aggregae producion K, whenever φ > 0 Show also ha if funcion has increasing reurns o ( ) L φ i has consan reurns o K alone [5 marks] 2 Show ha in his model: A A + [3 marks] K K + φ [ ( )] [ φ ( )] [ ( )] L L + Page 3 of 9 Pages

4 Assume now ha φ < Le y~ and ~ y y A Y A L and ( ) 3 Show ha he ransiion equaion for k ~ is: k ~ be defined as usual: k k A K ( A L ) ϕ ( ) ~ + k ~ k + n + n ϕ ( ) ~ [ ( )] φ sk + δ ( ) ~ ~ sk + ( δ ) k ~ φ ( ) ( ) Find he seady sae values for k ~ and y~, and show ha hese are meaningful whenever ( ) ( + n ) > ( δ ) (which we assume) Show also ha he ransiion equaion implies convergence o seady sae [6 marks] 4 Find he expression for he growh rae, g se, of oupu per worker in seady sae Commen wih respec o wha creaes growh in oupu per worker in his model (when φ < ) [6 marks] Now assume φ and n 0, so L is equal o some L for all 5 Show ha he model can be condensed o he wo equaions: Y AK, A L, K sy + δ K ( ) + Find he growh rae, g e, of oupu per worker Commen wih respec o wha creaes growh in oupu per worker in his model (when φ ) [0 marks] This quesion is same as Exercise 7 of Chaper 8 of he exbook Page 4 of 9 Pages

5 Endogenous Growh: Producive Exernaliies 6 Average annual growh rae of GDP per worker, (s K ) ½ (s H ) ½ Exercise 87 Semi-endogenous growh and endogenous growh if he producive exernaliy arises from Y raher han from K Noe he ypo in his exercise in he firs prin of he book The ransiion equaion in Quesion 3 should read: k + µ +n µ +n k ³s k +( δ) µ s k +( δ) k Tha is, he k on he lef hand side should be replaced by k +,andhes k [ φ( )]/() inside he parenhesis in he second line should be replaced by s k /() The aggregae producion funcion is found by insering A Y φ ino (6): Y (K ) ³ Y φ L

6 Endogenous Growh: Producive Exernaliies 7 Y Y K K L L Assuming ha φ</ ( ) ensures ha he exponens on K and L are posiive The sum of he exponens in he aggregae producion funcion is: + >, so his funcion exhibis increasing reurns o K and L Whenφ he producion funcion reduces o Y K L, which has consan reurns o K alone 2 From A Y φ and he aggregae producion funcion one ges: µ φ φ( ) K+ A + A Y φ + Y φ K µ L + L 3 Parallel o he chaper s analysis of he model of semi-endogenous growh (Secion 2) one derives: k + k K + K A + A L + L ³ K+ K ³ L+ L µ +n ³ K+ φ K φ φ( ) + µ K + K K + K ³ L + L ³ K+ K ³ L+ L Now using he capial accumulaion equaion gives: µ k µ + sy +( δ) K k +n K µ sỹ +( δ) +n k φ( ) L + L Insering ỹ k, which follows from he producion funcion, gives: µ k + h i s k k +( δ) +n

7 Endogenous Growh: Producive Exernaliies 8 k + µ +n µ +n k hs k +( δ) µ s k +( δ) k i The las formula gives wo alernaive expressions for he ransiion curve By definiion, in seady sae k + k k Inser his in he ransiion equaion (firs expression) o find he seady sae value of k : µ µ ³ k s +( δ) +n ( + n) µs ³ k +( δ) ( + n) s ³ k +( δ) Ã k s ( + n) ( δ) Using ỹ k gives he seady sae value of ỹ : Ã ³ k ỹ Under he saed condiion, ( + n) heexpressionsfor k and ỹ are meaningful s! ( + n) ( δ)! > ( δ), he denominaors above are posiive and I has hus been esablished ha he ransiion equaion has a unique sricly posiive inersecion wih he 45 -line Furhermore, he ransiion curve passes hrough (0, 0) and is everywhere sricly increasing, as can be verified direcly from inspecion of he second formula for he ransiion curve (noe ha φ< implies ha all he exponens are posiive) If he slope of he ransiion curve a he inersecion wih he 45 -line is smaller han one, convergence o seady sae follows from sair case ieraion in he usual ransiion diagram We compue he derivaive (using he firs of he expressions for he ransiion curve:

8 Endogenous Growh: Producive Exernaliies 9 h i s k +( δ) + µ h s k +( δ) +n µ d k + d k +n ( φ)( ) φ ( ) i h s k +( δ) s k +( δ) Insering here our expression for k in place of k givesheslopea k : d k + d k k k µ i h( + n) ( + n) +n µ µ +n +n " ( φ)( ) φ ( ) ( φ)( ) φ ( ) This is posiive and smaller han one since: ( + n) i ( φ)( ) φ ( ) φ µ +n Ã!# δ ( + n) Ã! δ ( + n) s k ( φ)( ) φ ( ) s k ³( + n) ( δ) > ( δ) implies ha he parenhesis is posiive and smaller han one The facor in fron of he parenhesis is iself posiive and smaller han one, so he produc is posiive and smaller han one One minus he produc mus hen also be posiive and smaller han one 4 Since ỹ y /A is consan in seady sae, y mus grow a he same rae as A, ha is, y + /y A + /A FromA Y φ A + A µ Y+ Y : φ µ y+ y φ ( + n) φ Insering A + /A y + /y (which holds in seady sae) and rearranging gives: y + y µ y+ y φ ( + n) φ y + y (+n) φ

9 Endogenous Growh: Producive Exernaliies 20 y + y y (+n) φ When φ < he exponen on +n is posiive Hence a larger populaion growh rae gives a higher growh rae of oupu per worker in seady sae 5 A he end of Quesion we found ha: Y K L, in case of φ Seing n 0removes he ime subscrip on L Define A (L) ( )/ Then: Y AK Since here is no populaion growh and he producive exernaliy is already embedded in he aggregae producion funcion, he equaion above and he capial accumulaion equaion (8), K + sy +( δ)k, makeupheeniremodelinseringy AK ino he capial accumulaion equaion and rearranging gives he growh rae of capial: K + sak +( δ) K K + K K sa δ, which, since Y AK and here is a consan labour force, is also he growh rae of Y, k,andy We have, of course, found he same growh rae as in he chaper s AK-model, only wih a sligh difference in he definiion of A All feaures wih respec o policy implicaions, scale effecs ec herefore bear over from he chaper s model o he one considered here Exercise 88 Taxaion and producive governmen spending: endogenous growh wihou producive exernaliies