Global Ecoomy Chris Edmod Sources of Ecoomic Growth Revised: Jauary 12, 2007 You will hear may reasos why coutries differ so widely i their levels ad growth rates of output, most of it speculatio. We impose some disciplie o our thikig by usig data ad a productio fuctio. They allow us to allocate output ad its growth rate betwee iputs (quatity ad quality of capital ad labor) ad total factor productivity (everythig else), which ofte arrows dow the list of possible explaatios. Accoutig for output ad output per worker Earlier we itroduced the aggregate productio fuctio, which related (real) GDP to the quatity of iputs i productio ad the efficiecy with which those iputs are used. Specifically, our augmeted productio fuctio was Y = AF (K, HL) = AK α (HL) 1 α, (1) where (as before) A is total factor productivity (TFP), K is the physical capital stock, H is huma capital, ad L is the quatity of labor. For may coutries, we ow have reasoably good data for GDP, employmet, the capital stock, ad eve huma capital, makig the productio fuctio a practical tool, ot just a abstract idea. If we kow the productio fuctio, we ca compute total factor productivity A usig data o output ad iputs, ad use equatio (1) to explai differeces i Y across coutries. This exercise is called a level decompositio, sice it decomposes the level of output (Y ) ito compoets due to the various factors o the right side of the productio fuctio (A, K, H, L). We ca do a similar exercise with output per worker. If we divide both sides of the productio fuctio by L, we get Y/L = A(K/L) α H 1 α, (2) so that output per worker depeds o total factor productivity (A), capital per worker (K/L), ad huma capital (H). (Our ability to covert this ito a relatio i ratios follows from the costat-returs-to-scale property of the productio fuctio). The missig igrediet is the value of α. I Problem Set #1 we saw that a good rule of thumb is α = 1/3. For those who have t doe the problem set (yet), the short explaatio goes like this: for this productio fuctio, if capital ad labor are both paid their margial products, the a
Sources of Growth 2 Employmet Educatio Capital GDP Mexico 34.65 7.23 1,617 852 US 142.08 12.05 19,600 9,169 Table 1: Mexico ad US: Aggregate data for 2000 fractio α of output is paid to capital ad the complemetary fractio 1 α is paid to labor. I most coutries, the relevat fractios are roughly oe-third ad two-thirds, hece our choice. Example (Mexico ad US). Use the data i Table 1 to compute TFP for Mexico ad the US. Employmet is expressed i millios, educatio i years, ad capital ad GDP i billios of 2000 US dollars. We set approximate huma capital H with the umber of years of educatio. Sice we have all the elemets of the productio fuctio but A, we ca compute it from A = Y K α (HL) 1 α. A measure of TFP computed i this way is ofte called a Solow residual : it is a measure of the amout of output that is left uexplaied by iputs like capital ad labor. To be cocrete, for Mexico we have A M = 852/[1617 1/3 (7.23 34.65) 2/3 ] = 1.827. A similar calculatio for the US implies A US = 2.376. We ca also use the productio fuctio to make explicit comparisos across coutries. If we apply equatio (2) to two coutries ad take the ratio, we fid (Y/L) 1 (Y/L) 2 = [ A1 A 2 ] [ ] α [ ] 1 α (K/L)1 H1, (K/L) 2 H 2 where the subscripts 1 ad 2 idicate the two coutries. The ratio of output per worker is thus attributed to some combiatio of the ratios of TFP, capital per worker, ad huma capital. If we have data, we ca say which of these factors is the most importat. We ca do a similar aalysis of output. Example (Mexico ad US, cotiued). You occasioally hear people i the US say that Mexica workers are paid so much less that they pose a threat to America jobs. (I Mexico, you hear the same thig about Chiese workers). We ca t address that issue yet but we ca say somethig about the source of differeces i output per worker, which is closely related to differeces i wages. The data i Table 1 imply that output per worker is 2.62 times higher i the US, but why? From the compoets: the capital-labor ratio is 2.96 times higher, huma capital (educatio) is 1.67 times higher, ad total factor productivity is 1.30 times higher. The magitudes of the effects o
Sources of Growth 3 output per worker are (Y/L) US (Y/L) M = A US A M [ ] 1/3 [ (K/L)US HUS (K/L) M = (1.30)(2.96) 1/3 (1.67) 2/3 H M = (1.30)(1.44)(1.41) = 2.62. It seems, therefore, that total factor productivity, capital per worker, ad huma capital all play a role i accoutig for the 2.6 to 1 ratio of US to Mexica output per worker. [Note: differeces betwee the US ad Mexico are smaller with this data, which has bee PPP-adjusted, tha if we had simply multiplied Mexica GDP by the exchage rate to express it i dollars. The reaso: may goods ad services are cheaper i Mexico tha the US, so whe we apply the same prices to both coutries the differeces are smaller]. ] 2/3 Growth rates Our ext task is to apply similar methods to growth, but first we eed to be clear about what we mea by growth rates. For may purposes i this course, we will defie the growth rate of a variable x betwee dates t ad t + 1 as γ = log x t+1 log x t = log x t+1. The expressio log here meas the atural logarithm, the fuctio LN i Excel. We refer to γ as the cotiuously-compouded growth rate for reasos we will igore for ow. (If you re iterested, see the Mathematics Review ). Typically the dates are years, so γ is a aual growth rate. If we wat to express it as a percetage, we multiply by 100. You ca stop here if you like, but if we were i your place we d be askig: why ca t we use the traditioal defiitio? Most of you would probably defie the growth rate as 1 + g = x t+1 /x t g = (x t+1 x t )/x t. Why use γ rather tha g? Some thoughts: There s little differece if the growth rates are small. This is t a argumet i favor of our defiitio, but it s a useful poit. Suppose x t = 100 ad x t+1 = 110. The g = 0.100 ad γ = log 110 log 100 = 0.0953, so the growth rates are 10% ad 9.53%. If the growth rate was smaller, the differece would be smaller, too. If you re ot mathematically iclied, go immediately to the ext poit. If you are, we would tell you that γ is a first-order Taylor series approximatio to g. Note that γ is a fuctio of g: γ = log x t+1 log x t = log(x t+1 /x t ) = log(1 + g).
Sources of Growth 4 This follows from a property of logarithms: log x log y = log(x/y). A first-order approximatio of the fuctio aroud the poit g = 0 is γ log(1) + (1)(g 0) = g, where meas approximately equal to. which are very small if g is small. Higher-order terms are g 2 /2, g 3 /6, ad so o, Growth rates are additive. Suppose you re iterested i the growth rate of a product xy. For example, x might be the price deflator ad y real output, so that xy is omial output. With the traditioal measure, the growth rate of xy is 1 + g xy = x t+1y t+1 x t y t = (1 + g x )(1 + g y ). If g x = g y = 0.10, the g xy = 0.21. But ote what happes with our defiitio: ( ) ( ) ( ) xt+1 y t+1 xt+1 yt+1 γ xy = log = log + log = γ x + γ y. x t y t x t y t They add up! Thus the growth rate of a product is the sum of the growth rates. That s ot quite true for traditioal growth rates, because of the compoud iterest effect: (1 + g x )(1 + g y ) = 1 + g x + g y + g x g y. The last term is small if the growth rates are, but it s ot zero. This additive feature of growth rates is the primary reaso we use them. For similar reasos, the growth rate of x/y equals the growth rate of x mius the growth rate of y. Averages are easy to compute. Suppose we wat to kow the average growth rate of x over periods: γ = (log x t+1 log x t ) + (log x t+2 log x t+1 ) + + (log x t+ log x t+ 1 ). If you look at this for a miute, you might otice that most of the terms cacel. The term log x t+1, for example, shows up twice, oce with a positive sig, oce with a egative sig. If we elimiate the redudat terms, we fid that the average growth rate is γ = log x t+ log x t = log(x t+/x t ). We ca compute it, the, from the iitial ad fial value of x. Accoutig for growth We ca apply the methods of the first sectio to growth rates. As before, the startig poit is the productio fuctio. If we take the atural logarithm of both sides of the productio fuctio (1), we fid log Y t = log A t + α log K t + (1 α) (log H t + log L t )
Sources of Growth 5 for ay date t. This follows from two properties of logarithms: log(xy) = log x + log y ad log x a = a log x. If we take the differece betwee two adjacet periods, we get log Y t = log A t + α log K t + (1 α) ( log H t + log L t ), whose compoets should be recogizable as cotiuously-compouded growth rates. If we cosider differeces over several periods, we ca divide each term by the umber of periods to get ( ) ( ) ( ) log Yt+ log Y t log At+ log A t log Kt+ log K t = + α ( ) [log Ht+ log H t ] + [log L t+ log L t ] + (1 α) or γ Y = γ A + αγ K + (1 α)(γ H + γ L ), (3) where γ X is the average cotiuously-compouded growth rate of the variable X. I short, the growth rate of output ca be attributed to growth i productivity, capital, labor, ad huma capital. Moreover, the terms add up. As with levels, we ca do the same for the growth rate of output per worker: γ Y γ L = γ A + α(γ K γ L ) + (1 α)γ H. (4) Exercises based o (3) ad (4) are referred to as growth accoutig. Employmet Educatio Capital GDP 1965 2,751.4 4.77 65,192.41 34,988.5 2000 6,019.1 7.89 258,200.21 150,896.0 Table 2: Chile: Aggregate data for 1965 ad 2000. Example (Chile i 1965 ad 2000). GDP icreased by almost a factor of five betwee 1965 ad 2000. Ca we say why? The relevat data are reported i Table 2. I the table, employmet is the umber of workers i millios, educatio is the average years of schoolig for people older tha 25, ad the capital stock ad real GDP are measured i millios of 2000 US dollars. The first step is to compute the growth rates. Over this period, the average aual growth rate of real GDP was γ Y = log(y 2000) log(y 1965 ) 35 = (11.93 10.46)/35 = 0.0417, or 4.17%. Usig the same method, we fid that the growth rates of the other variables i the table are γ K = 3.93%, γ H = 1.44%, ad γ L = 2.24%. We fid the growth rate of total factor productivity (A) from equatio (3), which implies γ A = 0.42%. (You could also fid A for each period ad compute the growth rate directly). So why did output grow? Our umbers idicate that of the 4.17% growth i output, 0.42% was due to TFP, 1.31% [= 3.93/3] was due to icreases i capital, 1.49% [= 2.24 (2/3)] was due to icreases i labor, ad 0.96% [= 1.44 (2/3)] was due to icreases i educatio.
Sources of Growth 6 Executive summary 1. Recall: a productio fuctio liks output to iputs. 2. Therefore: the growth rate of output depeds o growth rates of (i) iputs ad (ii) total factor productivity. Review questios 1. Frace ad the UK. I 2000, the data were Employmet Educatio Capital GDP Frace 27.497 7.86 3,852 1,351 UK 29.697 9.42 2,873 1,326 Which coutry had higher output per worker? Why? Aswer. Ratios were as follows: ( ) (Y/L)F (Y/L) UK = ( AF A UK ) ( ) 1/3 ( (K/L)F HF (K/L) UK H UK ) 2/3 1.10 = (1.10)(1.45) 1/3 (0.83) 2/3. I additio, Frace wo the World Cup (i 1998). 2. US ad Japa. Explai why output grew faster i Japa betwee 1970 ad 1985. Data: Uited States Japa 1970 1985 Growth 1970 1985 Growth GDP 2083 3103 2.66 620 1253 4.69 Capital 8535 13039 2.83 1287 3967 7.50 Labor 78.6 104.2 1.88 35.4 45.1 1.61 Employmet is measured i millios of workers, GDP ad capital i billios of 1980 US dollars. Growth rates are cotiuously-compouded average aual percetages. Aswer. We do t have educatio data here, so we use the o-augmeted productio fuctio (set H = 1, so that the effects of educatio o labor quality shows up implicitly i A). I levels (as opposed to growth rates) we see that the US had much greater output per worker i 1970: 26.5 (thousad 1980 dollars per worker) vs 17.5. Where did this differetial come from? Oe differece is that America workers i 1970 had three times more capital to work with: K/L was 108.6 i the US, 36.4 i Japa. If we use our productio fuctio, we fid that total factor productivity A was also slightly higher i the US i 1970: 5.64 vs 5.35.
Sources of Growth 7 Thus, the major differece betwee the coutries i 1970 appears to be i the amout of capital: America workers had more capital ad therefore produced more output, o average. By 1985, much of the differece had disappeared. It s obvious from the umbers that the biggest differece betwee Japa ad the US over the 1970-85 period is i the rate of growth of the capital stock. For the US, the output growth rate of 2.66% per year ca be divided ito 0.93% due to capital ad 1.26% due to employmet growth. That leaves 0.47% for productivity growth. For Japa the umbers are 2.48% for capital, 1.08% for labor, ad 1.13% for productivity. Evidetly the largest differece betwee the two coutries was i the rate of capital formatio: Japa s capital stock grew much faster, raisig its capital-labor ratio from 36.4 i 1970 to 88.0 i 1985. Further readig McKisey has studied extesively the sources of total factor productivity differeces, with reports o coectios to govermet regulatio, competitio, ad maagemet practices aroud the world. Much of this is available o the McKisey website (subscriptio oly). Former parter William Lewis has writte a terrific overview: The Power of Productivity (Uiversity of Chicago Press, 2004). c 2007 NYU Ster School of Busiess