Gowth - lectue note fo ECON1910 Jøgen Heibø Modalsli Mach 11, 2008 This lectue note is meant as a supplement to the cuiculum, in paticula to Ray (1998). In some of the lectues I will use slightly diffeent figues and calculations fom Ray, and the main pupose of this note is to give you that in witten fom. The main ole is to act as a supplement on the Haod-Doma and Solow models. I have explicitly used a lot of intemediate calculations to each the esults, to mae the deivations easy to follow. It also acts as an oveview of the eading list, gouping the chaptes in Ray and Banejee into what I thin is elevant headings. The infomation in this text coesponds to the thee lectues labelled gowth on the lectue plan. Text in Ray should be ead even if it is not efeed hee - impotant things such as section 3.5 in Ray is not coveed hee. Univesity of Oslo, Depatment of Economics and ESOP. Email: j.h.modalsli@econ.uio.no Phone: 22841174 Office: ES1038 1
Contents 1 What you should ead and what you do not need to ead 3 2 Intoduction 3 3 The Haod-Doma model 3 3.1 How to deive it........................................... 3 3.2 Policy implications.......................................... 4 3.3 Haod-Doma with population gowth............................... 4 4 The Solow model 4 4.1 Poduction function......................................... 4 4.2 Baseline Solow model........................................ 5 4.3 Solow with technological gowth.................................. 6 5 Gowth model extensions 9 5.1 Othe poduction factos....................................... 9 5.2 Human capital............................................ 9 5.3 Endogenous gowth.......................................... 10 5.4 Extenalities............................................. 10 5.5 Total facto poductivity gowth.................................. 11 6 The Big Push model 11 6.1 Gowth coodination......................................... 11 7 Institutions and the ole of histoy 11 8 Appendix: Some calculations 12 8.1 Human capital............................................ 12 2
1 What you should ead and what you do not need to ead The cuiculum is Ray (1998), chapte 3, 4 and 5 Banejee et al. (2006), chapte 2, 3 and 5 Hand-out on the Big Push Model (fom a textboo by Todao and Smith), which I will tal about in the thid lectue on gowth, elated to Ray ch. 5. Available fom the couse webpage. You will pobably have some use of eading this note Howeve, you can sip advanced algeba: Section 3.5.6 in Ray is not equed eading (but you will pobably have good use fom eading page 87 anyway) The appendices to the chaptes in Ray ae not equied eading As I will say in the lectues, if you have seious poblems following algeba (lie it is laid out in this note), it is not stictly mandatoy. But I do not thin it necessaily is any easie to undestand the models using only othe foms of intuition. You should at the vey least be able to intepet equations lie equation (32) below, and undestand the figues. 2 Intoduction Facts, histoy and statistics - see Ray (1998), especially chapte 3. Also teated at the lectue. 3 The Haod-Doma model Ray ch. 3.3.1 3.1 How to deive it This follows Ray chapte 3.3.1, I just added some intemediate calculation. See Ray fo the full details. Y t = C t + S t (1) Y t = C t + I t (2) S t = I t (3) (4) Capital next peiod: K t+1 = (1 δ)k t + I t (5) K t+1 = (1 δ)k t + S t (6) Now intoduce savings ates and capital-output atios to get s = sy (7) K = θy (8) Y = 1 θ K (9) θ = K Y (10) 3
We then deive the Haod-Doma esults as (see Ray fo moe discussion) 3.2 Policy implications θy t+1 = (1 δ)θy t + sy t (11) θy t+1 = θy t δθy t + sy t (12) θy t+1 θy t = sy t δθy t (13) θy t+1 θy t Y t = s δθ (14) Y t+1 Y t Y t = s θ δ (15) g = s θ δ (16) s θ = g + δ (17) Capital-output atio is seen as exogenous, but technology-diven. Savings ates can be affected by policy. Implicitly, thee is a poduction function with fixed facto shaes 1 with a labo eseve in the nondeveloped secto. The cental issue fo the planne is how can we get enough capital to get eveyone into the industialized secto? The Haod-Doma model was witten afte the Geat Depession, with geat suplus of labo. Moe tal about this at the lectue 3.3 Haod-Doma with population gowth The gowth equation becomes s θ g + n + θ (18) whee g is pe capita gowth. The esult is quite staightfowad: as thee ae moe people each peiod, we must invest moe fo a given level of pe capita gowth. 4 The Solow model Ray ch. 3.3.3 4.1 Poduction function In the Solow model, poduction is explicitly a esult of two poduction factos: Labo and Capital. Denoting total output as Y, and using P fo labo and K fo capital: 2 Y = F (K, P ) (19) We will assume that the poduction function has constant etuns to scale. By this I mean that if we incease both factos by the same faction, total output will incease by the same. Fo example, if we double the numbe of poduction factos fom above, we get twice the output: 2Y = F (2K, 2P ) (20) 1 This can be efeed to as a Leontief poduction function. I might tal moe about this at the lectue 2 It is moe common (and logical, pehaps) to use L fo labo. Howeve, I will follow the lettes used by Ray (1998). 4
o, moe geneally ay = F (ak, ap ) (21) whee a is any constant. Note, howeve (and this is impotant) that if you incease only one of the factos, poduction inceases by less. Thin of a factoy consisting of ten machines and one hunded woes that poduces one thousand cas a yea. Now, if you double both the numbe of machines and woes (to 20 and 200), it is not uneasonable to thin that you get twice as many cas. Howeve, if we only double the numbe of machines (so we have 20 machines and 200 woes), we will no doubt get moe than one thousand cas, but as the woes have to be spead moe thinly acoss the machines output will not double. Fo ou application: If we eep the numbe of people constant, adding capital will incease poduction, but with smalle and smalle inceases fo a given amount of capital. Because of the constant etuns to scale assumption above, we can show this by multiplying both factos by 1 P (setting a = 1 P in equation (21)) to get 1 P Y = F ( 1 P K, 1 P P ) (22) Y P = F (K P, P (23) P y = F (, 1) = f() (24) 4.2 Baseline Solow model Intoducing dot notation, I now declae to mean the change in fom one peiod to the next peiod. In othe wods, t t+1 t. We fist declae that change in total capital comes fom saving (which adds to capital) and depeciation (which destoys capital), and saying that both saving (s) and depeciation (δ, the Gee lette delta, you can wite d in you own notes if you lie): K t = sy t δk t (25) We then intoduce the vaiable capital pe peson, and as in Ray, use lowe case lette fo pe-capita vaiables. The definition of is then, of couse, K L. Now how does change ove time? Fist assume that thee is constant population. Then the change in capital pe peson will be the same as the elative change in total capital. Howeve, if population gows, thee has to be highe total capital gowth fo a given capital pe peson gowth. It is eally quite logical. This is expessed in the following equation (whee we divide by K and L and because we want elative gowth): = K K L L We then use the equations (25) and (37) to deive the following: (26) (27) sy δk = L K L (28) = sy K δk K L L (29) = sy δ n (30) = sy δ n (31) = sy (δ + n) (32) 5
Equation (32) tells us how capital pe woe gows (given what we poposed in equations (25) and (37)). Inceased savings incese capital, while high depeciation ates and high population gowth slows capital gowth. Also note that thee is a level sy = (δ + n) (33) whee capital gowth is zeo. Remembe fom 4.1 that if we incease capital while holding population constant, output inceases (but less as we each highe levels). This can be witten as whee f( ) is a function having these popeties. We can then ewite (32) as y = f() (34) = s f() (δ + n) (35) This elationship can be illustated gaphically. Figue 1 shows on the hoizontal axis and y on the vetical axis. The geen line is y = f(). Note how, as we said, it always inceases, but less so as we each highe levels of. The blue line is s y. Now, as we have said that s is constant, this line will always be a constant faction of the geen line. The ed line is (n + δ). This elationship is linea. Now, fo the level of capital shown in Figue 1 we can see the level of poduction, the level of saving, and how much capital pe peson is educed because of depeciation and population gowth. We see that fo this paticula situation capital will incease, we will move towads the ight in the figue ove time. These moves will depend on the distance between the savings cuve and the population-depeciation cuve, and we will move in the diections given in figue 2 As discussed above, thee is a level whee f() = (δ +n), meaning capital gowth is zeo. This is efeed to as the steady state level, and is shown in Figue 3. Fo futhe discussion of the steady state, see the textboo. 4.3 Solow with technological gowth Ray ch. 3.4 Obviously, the baseline Solow implication that thee is no gowth does not fit eality. We theefoe postulate some exogenous ( outside-the-model ) gowth, which we do not ty to explain at this point - we just state that technology gows by some constant faction each yea. The technology-augmented poduction function is then witten Y = F (K, EP ) (36) that is, we multiply the labo foce by the technology. 3 In shot, it means that ove time, labo becomes moe and moe poductive, so that the labo of one peson mattes moe in the poduction. We now scale all vaiables by this technology gowth (Ray denotes 3 Thee possible ways of doing this would be to modify equation (19), adding an A to epesent technology: (a) Y = A F (K, P ), (b) Y = F (A K, P ), (c) Y = F (K, A P ) Ray uses the thid altenative, but you should not be vey concened about the diffeences hee. 6
Figue 1: The Solow model Figue 2: Paths of movement in the Solow model 7
Figue 3: Steady state this by adding a hat on the vaiables). Then, a steady state ˆ - that does not change ove time - means that actually inceases ove time at the same ate as technology gowth. To get the ˆ, we divide the big K by effective labo foce - that is, dividing by EP. This means that the vaiables we use no longe diectly elate to the numbe of people - a given level of ˆ o ŷ may coespond to diffeent levels of pe-capita income depending on what the technology level is. We declae that technology gows by some pe-detemined value π. 4 By the definition of ˆ, we can show that ˆ ˆ = K K L L Ė E (37) whee Ė E = π as stated above. Using the same algeba as in the pevious section, we get the equation fo capital change ˆ = s f(ˆ) (δ + n + π)ˆ (38) and the steady state equation sf(ˆ) = (δ + n + π)ˆ (39) The steady state level of ˆ now denotes a situation whee total poduction gows by π. Why? Obseve that in the steady state ŷ is zeo. That means that the gowth of poduction measued pe effective 4 Note fo people new to economics: this has nothing to do with the mathematical constant 3.14. π is simply the gee lette pi, hee used to denote a numbe that may be, fo example, 2 o 3 pecent. 8
woe is zeo. But the effective woe becomes moe and moe poductive. Theefoe, output pe peson is steadily inceasing. With some algeba: ŷ = Y EP ŷ E = Y P (40) (41) We then see that even though ŷ is constant, Y P, pe capita income, is gowing at the same ate as E. We can daw diagams simila to Figues 1-3 by having ˆ and ŷ on the axes. Fist lectue ends about hee 5 Gowth model extensions To some extent, all of these can be seen as extensions of the Solow model. This section coesponds to Ray (1998) chapte 4. 5.1 Othe poduction factos In the pevious chapte, the factos of poduction wee capital and labo. We can add moe, o use something else instead, based on what we want to model. In eality, thee is an infinite numbe of poduction factos (and often the diffeences will not be clea-cut), such as cas, computes, the labo people with maste s degees in development economics, the labo of haidesses, steel, land, etc). We have to do some gouping of these. In the following we will loo at human capital, an extended view on labo. 5.2 Human capital Ray ch. 4.2 and 4.3 High and low sill - complementaities o substitutes? In the following we will assume they ae substitutes (it taes X pesons to do the wo of one peson with high human capital, whee X > 1). If H is total human capital, h is human capital pe peson - you can thin of this of the aveage yeas of schooling. H = h P (42) H P = h (43) (Note that thee is some depeciation hee as well - obviously one has to eep a cetain amount of investment just to eep the human capital level constant - people die and new people ae bon, so you have to school people all the time. We abstact fom this hee) Modify the Solow equation to ead y = f(, h) (44) We can now invest both in capital,, and human capital, h. Income is divided into Investment in physical capital s y Investment in human capital q y Consumption (1 s q) y 9
The detailed exposition is shown in section 8.1 below fo those especially inteested. complicated, but some calculation is involved. This gives seveal implications: (Ray p. 102-104) It is not vey Thee can be deceasing etuns to physical capital yet constant etuns in total Policy can influence total gowth ates and not only levels Convegence not necessaily the case Retun to capital may be highe in ich counties because of high levels of human capital Note that this is possible because we assumed such a poduction function. poduction (such as unsilled labo ), this might not be the case. If we add moe factos of 5.3 Endogenous gowth Ray ch. 4.4 Baseline: We can use labo to discove new things, in addition to poducing things. Ray 4.4.2: The diffeence between inventions and diffusions of nowledge. Ray 4.4.3: A model of technical pocess, shown in equation (4.5) to (4.7): A shae u of human capital is used in poduction of goods (fo consumption and investment), while a shae (1 u) is used fo eseach (impoving technology). This gives a tade-off between poduction today and bette technology tomoow. How is such poduction of technology financed? In a planned economy, the state could just dictate that some people should do eseach. In a mixed economy, lie Noway, a lage pat is financed via taxes. But if you want pivate initiatives fo eseach, you need some ind of intellectual popety ights - in essence, a (limited) monopoly fo those who discove things. 5.4 Extenalities Ray 4.4.4 An extenality is an unintended consequence of an action. If you invent the steam engine because it is pofitable fo you factoy, and you don t have any legal system to potect intellectual popety, a liely extenality is that eveyone else can also un thei factoy by copying you steam engine. In fact, even if you have intellectual popety ights it is liely that some of what you have done cannot be potected, and thus gives ideas to othes. (Note that thee may also be negative extenalities: pollution in the neighbouing village, fo example). Positive extenalities between economic activities may lead to inceasing etuns to scale on the maco level. Complementaities: You actions depend on what eveyone else does. See paagaph on complementaities and Figue 4.3 in Ray. Aghion and de Aghion (2006) has an impotant discussion elated to extenalities and complementaities (see the paagaph New Gowth Theoies in a nutshell ). The authos ague that the following factos ae impotant to gowth: A legal envionment that allows inventos a shae of the evenues geneated by thei innovation Expectations of macoeconomic stability inceases expected pofits (and theeby would lead moe people to innovate?) Thee ae also seveal othe impotant points in the aticle elated to othe topics we have discussed (o will discuss in the thid lectue), such as human capital accumulation, financial maets and competition envionment. 10
Figue 4: Big push 5.5 Total facto poductivity gowth I will use TFP to mean Total Facto Poductivity. TFP gowth is gowth that is not due to any change in factos of poduction. If you have highe GDP because you save moe, that is not TFP gowth. TFP is simila to the technology paamete in Section 4.3. See Ray 4.4.5 TFP and the East Asian Miacle: See Ray ch. 4.5. Second lectue ends about hee 6 The Big Push model See handout fom Smith/Todao available at the couse page hee: http://www.uio.no/studie/emne/sv/oeonomi/econ1910/v08/undevisningsmateiale/todaosmith.html (UiO usename and passwod equied) I will use Figue 4 at the lectue. It shows that pofits fom a (potentially) industialized secto depends on the numbe of sectos that aleady ae (o ae expected to be) industialized. 6.1 Gowth coodination Chapte 5.2 in Ray. The Big Push model is an example of a model whee coodination plays an impotant ole. Complementaities and coodination (QWERTY/DVORAK as example) Linages: Ray 5.2.3 Inceasing etuns: Ray 5.3.2 7 Institutions and the ole of histoy Ray ch. 5.5 ( Othe oles fo histoy ) 11
Acemoglu et al. (2006) - discussed at the lectue Engeman and Sooloff (2006) - discussed at the lectue This has some ovelap with Section 5.4 above Thid lectue ends hee Refeences Acemoglu, D., S. Johnson, and J. Robinson (2006). Undestanding pospeity and povety: Geogaphy, institutions, and the evesal of fotune. In A. V. Banejee, R. Bénabou, and D. Moohejee (Eds.), Undestanding Povety, Chapte 2. Oxfod Univesity Pess. Aghion, P. and B. A. de Aghion (2006). New gowth appoach to povety alleviation. In A. V. Banejee, R. Benabou, and D. Moohejee (Eds.), Undestanding Povety, Chapte 5. Oxfod Univesity Pess. Banejee, A. V., R. Benabou, and D. Moohejee (2006, Apil). Undestanding Povety. Oxfod Univesity Pess, USA. Engeman, S. and K. Sooloff (2006). Colonialism, inequality, and long-un paths of development. In A. V. Banejee, R. Benabou, and D. Moohejee (Eds.), Undestanding Povety, Chapte 3. Oxfod Univesity Pess. Ray, D. (1998, Januay). Development Economics. Pinceton Univesity Pess. 8 Appendix: Some calculations 8.1 Human capital This efes to section 5.2. Capital gows as shown in (32), except that we (fo simplicity) assume that δ = n = 0 while human capital gowth depends on the schooling investment q: = sy (45) ḣ = qy (46) which, just lie the egula savings, is assumed to be a constant faction of total poduction. This says, fo example, that a county uses ten pecent of total GDP on education (we do not distinguish between public and pivate oganization hee). We now intoduce a new vaiable - the atio of human capital to physical capital - and call it. Then note that all these thee equations hold: = h = h h = (47) (48) (49) 12
Using this, we can manipulate the poduction function a bit. Total poduction is given by We then multiply by the constant 1, as we showed in equation (21) Y = f(k, hp ) (50) y = f(, h) (51) (52) y = 1 f(, h) (53) y = 1 f(h, h) (54) y = 1 f(h, h) (55) y = h f(1, ) (56) to get the poduction function in tems of h and. It can be shown that in steady state all types of capital and total output gow at the same ate (see Appendix to Ray ch. 4). (You should just accept this and not loo at the appendix). This means that ẏ y = = ḣ h (57) This also means that in the long un, is constant. We now stat again at (45), inseting fo poduction fom (56). = s h f(1, ) (58) = sf(1, ) (59) Similaly fo h fom (46): = sf(1, ) (60) ḣ = q h f(1, ) (61) Remembe that these must be equal in the long un: ḣ h = q f(1, ) (62) ḣ h = (63) f(1, ) q = sf(1, ) (64) q = s (65) = q s Inseting in (45) o (46), we then get the long-un gowth ate: (66) = sf(1, ) (67) 13
= sf(1, q s ) (68) = s 1 s f(s, q s) (69) s = f(s, q) (70) (To get the same expessions as in Ray, use the functional fom f(, h) = α h 1 α.) 14