University College Dublin, MA Macroeconomics Notes, 2014 (Karl Whelan) Page 1

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1 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 1 Growh Accouning The final par of his course will focus on wha is known as growh heory. Unlike mos of macroeconomics, which concerns iself wih wha happens over he course of he business cycle (why unemploymen or inflaion go up or down during expansions and recessions), his branch of macroeconomics concerns iself wih wha happens over longer periods of ime. In paricular, i looks a he quesion Wha deermines he growh rae of he economy over he long run and wha can policy measures do o affec i? As we will also discuss, his is relaed o he even more fundamenal quesion of wha makes some counries rich and ohers poor. In his se of noes, we will cover wha is known as growh accouning a echnique for explaining he facors ha deermine growh. Producion Funcions The usual saring poin for growh accouning is he assumpion ha oal real oupu in an economy is produced using an aggregae producion funcion echnology ha depends on he oal amoun of labour and capial used in he economy. For illusraion, assume ha his akes he form of a Cobb-Douglas producion funcion: Y = A K α L β (1) where K is capial inpu and L is labour inpu. Noe ha an increase in A resuls in higher oupu wihou having o raise inpus. Macroeconomiss usually call increases in A echnological progress and ofen refer o his as he echnology erm. As such, i is easy o imagine increases in A o be associaed wih people invening new echnologies ha allow firms o be more producive. Ulimaely, however, A is simply a measure of producive

2 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 2 efficiency and i may go up or down for oher reasons, e.g. wih he imposiion or eliminaion of governmen regulaions. Because an increase in A increases he produciveness of he oher facors, i is also someimes known as Toal Facor Produciviy (TFP), and his is he erm mos commonly used in empirical papers ha aemp o calculae his series. Usually, we will be more ineresed in he deerminaion of oupu per person in he economy, raher han oal oupu. Oupu per person is ofen labelled produciviy by economiss wih increases in oupu per worker called produciviy growh. Produciviy is obained by dividing boh sides of equaion (1) by L o ge which can be re-arranged o give Y L = A K α L β 1 (2) Y L = A ( K L ) α L α+β 1 (3) This equaion shows ha here are hree poenial ways o increase produciviy: Technological progress: Improving he efficiency wih which an economy uses is inpus, i.e. increases in A. Capial deepening (i.e. increases in capial per worker) Increases in he number of workers: Noe ha his only adds o growh if α + β > 1, i.e. if here are increasing reurns o scale. Mos growh heories assumes consan reurns o scale: A doubling of inpus produces a doubling of oupus. If a doubling of inpus manages o more han double oupus, you could argue ha he efficiency of producion has improved and so perhaps his should be considered an increase in A raher han

3 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 3 somehing ha sems from higher inpus. If, here are consan reurns o scale, hen α + β 1 = 0 and his erm disappears and producion funcion can be wrien as Y L = A ( K L ) α (4) The Deerminans of Growh Le s consider wha deermines growh wih a consan reurns o scale Cobb-Douglas producion funcion (so β = 1 α) Y = A K α L 1 α (5) and le s assume ha ime is coninuous. In oher words, he ime elemen evolves smoohly insead of jus aking ineger values like = 1 and = 2. How do we characerise how his economy grows over ime? Le s denoe he growh rae of Y by G Y. This can be defined as G Y = 1 Y dy (6) In oher words, he growh rae a any poin in ime is he change in oupu (he derivaive dy of oupu wih respec o ime, ) divided by he level of oupu. We can characerise he growh rae of Y as a funcion of he growh raes of labour, capial and echnology by differeniaing he righ-hand-side of equaion (5) wih respec o ime. Before we do his, you should recall he produc rule for differeniaion, i.e. ha dab dx = B da dx + AdB dx (7) For producs of hree variables (like we have in his case) his implies dabc dx = BC da dx + AC db dx + AB dc dx (8)

4 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 4 In our case, we have dy = da K α L 1 α = K α L 1 α da + A L 1 α dk α + A K α dl 1 α (9) We can use he chain rule o calculae he erms involving he impac of changes in capial and labour inpus: dk α dl 1 α = dkα dk dk = dl1 α dl dl = αk α 1 dk dl = (1 α) L α (10) (11) Plugging hese formulae ino he righ places in equaion (9) we ge dy = Kα L 1 α da + αa K α 1 L 1 α dk + (1 α) A K α L α dl (12) The growh rae of oupu is calculaed by dividing boh sides of his by Y which is he same as dividing by A K α L 1 α. ( 1 dy K α Y = L 1 α ) da A K α L 1 α + α ( A K α 1 L 1 α A K α L 1 α ) dk + (1 α) ( A K α L α ) dl A K α L 1 α (13) Cancelling he various erms ha appear muliple imes in he erms inside he brackes and we ge 1 dy Y = 1 da A + α 1 dk + (1 α) 1 dl K L (14) This can wrien in more inuiive form as G Y = G A + αg K + (1 α) G L (15) The growh rae of oupu equals he growh rae of he echnology erm plus a weighed average of capial growh and labour growh, where he weigh is deermined by he parameer α. This is he key equaion in growh accouning sudies. These sudies provide esimaes of how much GDP growh over a cerain period comes from growh in he number of workers,

5 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 5 how much comes from growh in he sock of capial and how much comes from improvemens in Toal Facor Produciviy. One can also show ha he growh rae of oupu per worker is he growh rae of oupu minus he growh in he number of workers, so his is deermined by G Y G L = G A + α ( G K G L ) (16) This is a re-saemen in growh rae erms of our earlier decomposiion of oupu growh ino echnological progress and capial deepening when he producion funcion has consan reurns o scale. Ideally, I d like you o be able o undersand how equaion (15) was derived bu cerainly you should know his formula and undersand is meaning. For example, remember he producion funcion is Y = A K α L 1 α. The reason an increase of x percen in A ranslaes ino an increase of x percen in oupu is because A muliplies he oher erms. In conras, K is aken o he power of α. An increase in K, say by replacing i wih (1 + x) K is equivalen o muliplying he exising level of oupu by (1 + x) α. Because α is assumed o be less han one, his is a smaller increase han comes from increasing A by a facor of (1 + x). To undersand why a 1% increase in boh K and L leads o a 1% increase in oupu, noe ha if we muliplied boh he inpus in K α L 1 α by (1 + x), we would ge A ((1 + x) K ) α ((1 + x) L ) 1 α = (1 + x) α (1 + x) 1 α A K α L 1 α = (1 + x) A K α L 1 α

6 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 6 How o Calculae he Sources of Growh: Solow (1957) For mos economies, we can calculae GDP, as well as he number of workers and also ge some esimae of he sock of capial (his las is a bi rickier and usually relies on assumpions abou how invesmen cumulaes over ime o add o he sock of capial.) We don direcly observe he value of he Toal Facor Produciviy erm, A. However, if we knew he value of he parameer α, we could figure ou he growh rae of TFP from he following equaion based on re-arranging (15) G A = G Y αg K (1 α) G L (17) Bu where would we ge a value of α from? In a famous 1957 paper, MIT economis Rober Solow poined ou ha we could arrive a an esimae of α by looking a he shares of GDP paid o workers and o capial. 1 To see how his mehod works, consider he case of a perfecly compeiive firm ha is seeking o maximise profis. Suppose he firm sells is produc for a price P (which i has no conrol over), pays wages o is workers of W and rens is capial for a renal rae of R (his las assumpion ha he firm rens is capial isn imporan for he poins ha follow bu i makes he calculaions simpler.) This firm s profis are given by Π = P Y R K W L (18) = P A K α L 1 α R K W L (19) Now consider how he firm chooses how much capial and labour o use. I will maximise profis by differeniaing he profi funcion wih respec o capial and labour and seing he 1 Technical Change and he Aggregae Producion Funcion, Review of Economics and Saisics.

7 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 7 resuling derivaives equal o zero. This gives wo condiions These can be simplified o read Π = αp A K α 1 L 1 α R = 0 K (20) Π = (1 α) P A K α L α W = 0 L (21) Solving hese we ge Π = α P Y R = 0 (22) K K Π = (1 α) P Y W = 0 (23) L L Take a close look hese equaions. α = R K P Y (24) 1 α = W L P Y (25) P Y is oal nominal GDP (he price level imes real oupu) W L is he oal amoun of income paid ou as wages (he wage rae imes number of workers). R K is he oal amoun of income paid o capial (he renal rae imes he amoun of capial). These equaions ell us ha we can calculae 1 α as he fracion of income paid o workers raher han o compensae capial. (In real-world economies, non-labour income mainly akes he form of ineres, dividends, and reained corporae earnings). Naional income accouns come wih various decomposiions. One of hem describes how differen ypes of incomes

8 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 8 add up o GDP. In mos counries, hese saisics show ha wage income accouns for mos of GDP, meaning α < 0.5. A sandard value ha ges used in many sudies, based on US esimaes, is α = 1. I would noe, however, ha some sudies do his calculaion assuming 3 firms are imperfecly compeiive if his is he case (as i is in he real world) hen he shares of income earned by labour and capial depend on he degree of monopoly power. So one needs o be cauious abou growh accouning calculaions as hey rely on heoreical assumpions ha could poenially be misleading. Solow s 1957 paper concluded ha capial deepening had no been ha imporan for U.S. growh for he period ha he examined ( ). In fac, he calculaed ha TFP growh accouned for 87.5% of growh in oupu per worker over ha period. The calculaion became very famous i was one his papers ha was cied by he Nobel commiee when awarding Solow he prize for economics in TFP is someimes called he Solow residual because i is a backed ou calculaion ha makes hings add up: You calculae i as he par of oupu growh no due o inpu growh in he same way as regression residuals in economerics are he par of he dependen variable no explained by he explanaory variables included in he regression. Example: The BLS Mulifacor Produciviy Figures Mos growh accouning calculaions are done as par of academic sudies. However, in some counries he official saisical agencies produce growh accouning calculaions. In he U.S. he Bureau of Labor Saisics (BLS) produces hem under he name mulifacor produciviy calculaions, (i.e. hey use he erm MFP insead of he erm TFP bu concepually hey are he same hing.) Many of he sudies add some bells and whisles o he basic calcula-

9 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 9 ions jus described. For example, he BLS ry o accoun for improvemens in he qualiy of he labour force by accouning for improvemens in he level of educaional qualificaions and work experience of employees. In oher words, hey view he producion funcion as being of he form Y = A K α (q L ) 1 α (26) where q is a measure of he qualiy of he labor inpu. Figure 1 shows a summary of he BLS s calculaions of he sources of growh in he US from 1987 o They conclude ha average growh of 2.2 percen in he U.S. privae nonfarm economy can be explained as follows: 0.9 percen comes from capial deepening, 0.3 percen comes from changes in labour composiion and 0.9 percen comes from changes in wha hey call mulifacor produciviy. Looking a differen samples, however, we can see large changes beween differen periods in he conribuion of MFP. From , produciviy growh averaged only 1.5 percen and MFP growh was weak, conribuing only 0.5 percen per year o growh. During his period, here was a lo of discussion abou he slowdown in growh relaive o previous eras, wih much of he focus on he poor performance of TFP growh. Paul Krugman s firs popular economics book was called The Age of Diminished Expecaions because people seemed o have acceped ha he US economy was doomed o low produciviy growh. From , produciviy growh averaged a very respecable 2.65 percen, wih MFP growh conribuing 1.35 percen. During his period, here was a lo ofdiscussion of he impac of new Inerne-relaed echnologies ha improved efficiency. While he peak of his enhusiasm was around he do-com bubble of he 2000s when here was

10 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 10 los of alk of a New Economy, pos-ech-bubble produciviy performance was also prey good. From , produciviy growh has been weaker han in he previous decade, averaging only 1.3 percen. MFP growh has been paricularly weak, averaging only 0.6 percen over his period. New Economy opimism has receded, a opic ha we will reurn o laer. In addiion o he poor performance of U.S. produciviy growh, anoher facor ha is weighing on he poenial for oupu growh is a slow growh rae of he labour force. Afer years of increasing numbers of people available for work due o normal populaion growh, immigraion and increased female labour paricipaion, he US labour force has flaened ou (see Figure 2). This is being driven by long-run demographic rends as he large baby boom generaion sars o reire. This rend is se o coninue over he nex few decades. Figure 3 shows ha he dependency raio (he raio of non-working o working people) is projeced o increase significanly as he populaions grows older on average. See my blog pos Is he U.S, Se for an Era of Slow Growh?

11 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 11 Figure 1: Growh Accouning Calculaions for he U.S.

12 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 12 Figure 2: The U.S. Labour Force

13 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 13 Figure 3: The Raio of Non-Working o Working People in U.S.

14 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 14 Example: The Euro Area Longer-erm growh prospecs in Europe appear o be worse han in he Unied Saes. I am currenly working on a paper wih Kieran McQuinn ha does a growh accouning analysis for he euro area and consrucs longer-erm growh projecions. The following discussion is based on his work. Table 1 shows ha growh in oupu per worker in he counries ha make up he euro area has gradually declined over ime. In paricular, TFP growh has collapsed. From 2.7 percen per year over , TFP growh has fallen o an average of 0.2 percen per year over he period Table 2 shows ha weak performances for TFP growh can be seen widely across differen European counries. Europe is also on he cusp of a significan demographic change ha will reduce he poenial for GDP growh: See Figure 4. Populaion growh is slowing and oal populaion is se o peak in before he middle of his cenury. The populaion is also ageing significanly. Indeed he oal amoun of people aged beween 15 and 64 (i.e. he usual definiion of work-age populaion) has peaked and is se o decline subsanially over he nex half cenury. While Europe has many shor-erm macroeconomics problems due o weak aggregae demand and high levels of public and privae deb, i is also he case ha i faces severe challenges in relaion o long-erm growh. Mainaining growh raes a close o hose experienced hisorically will likely involve reforms o raise he size of he labour force (such as raising reiremen ages and immigraion) and boos produciviy.

15 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 15 Table 1: The Euro Area s Growh Performance

16 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 16 Table 2: Counry-by-Counry Growh Performance

17 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 17 Figure 4: Demographic Projecions for he Euro Area from Eurosa

18 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 18 Example: A Tale of Two Ciies Alwyn Young s 1992 paper A Tale of Two Ciies: Facor Accumulaion and Technical Change in Hong Kong and Singapore is an ineresing example of a growh accouning sudy. He compares he growh experiences of hese wo small Asian economies from he early 1970s o Young explained his moivaion for picking hese wo economies in erms of heir similariies and heir differences: In he prewar era, boh economies were Briish colonies ha served as enrepo rading pors, wih lile domesic manufacuring aciviy... In he poswar era, however, boh economies developed large expor-dependen domesic manufacuring secors. Boh economies have passed hrough a similar se of indusries, moving from exiles, o clohing, o plasics, o elecronics, and hen, in he 1980s, gradually moving from manufacuring ino banking and financial services... The poswar populaion of boh was composed primarily of immigran Chinese from Souhern China... While he Hong Kong governmen has emphasized a policy of laissez faire, he Singaporean governmen has, since he early 1960s, pursued he accumulaion of physical capial via forced naional saving. Boh economies were successful: Hong Kong had oal growh of 147% beween he early 1970s and 1990 and Singapore had growh of 154%. Bu Young was ineresed in exploring he exen o which TFP conribued o growh in hese wo economies. The resuls of his growh accouning calculaions are shown on he nex page. He found ha Singapore s approach did no produce any TFP growh while Hong Kong s more free marke approach lead o srong TFP growh wih his elemen accouning for almos half of he growh in oupu per worker. One can argue his was a beer oucome because Hong Kong achieved he growh wihou

19 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 19 having o diver a huge par of naional income owards invesmen raher han consumpion. As we will see in he nex lecure, however, TFP-based growh has an advanage over growh based on capial accumulaion because i is more susainable. Table from Alwyn Young s 1992 Paper

20 Universiy College Dublin, MA Macroeconomics Noes, 2014 (Karl Whelan) Page 20 Things o Undersand from hese Noes Here s a brief summary of he hings ha you need o undersand from hese noes. 1. The sources of growh in oupu per worker. 2. How o derive he growh rae of oupu under consan reurns as a funcion of he growh raes of capial, labour and TFP. 3. Solow s mehod for calculaing TFP growh. 4. Evidence from he BLS on US produciviy growh. 5. Evidence on growh in Europe. 6. Young s Tale of Two Ciies.