Process of convergence dr Joanna Wolszczak-Derlacz ecure 4 and 5 Solow growh model a
Solow growh model Rober Solow "A Conribuion o he Theory of Economic Growh." Quarerly Journal of Economics 70 February 1956: 65-94. Nobel Prize - 1987 Everyhing reminds Milon Friedman of he money supply. Everyhing reminds me of sex, bu I ry o keep i ou of my papers."
Solow neoclassical growh model Analysis of mainly medium erm deerminans and effecs of economic growh process Assumes ha capial o labor raio /=k is he key facor influencing he growh process Uses Cobb-Douglas producion funcion as a basis for he analysis The Solow Growh Model is designed o show how growh in he capial sock, growh in he labor force, and advances in echnology inerac in an economy, and how hey affec a naion s oal oupu of goods and services. goods and service Mankiw, 2013
Neoclassical producion funcion no echnology Y F, Y is he oal amoun of producion of he final good a ime is he capial sock is he labour force
F F,, 1. Diminishing reurnes o labour and capial separaely 0 0 F F 0 0 2 2 2 2 F F Basic chacacerisics of neoclassical producion funcion 2. Consan reurns o scale Y Facor of producion Y 0 Y 1 Y
The Cobb-Douglas producion funcion in hree dimensions
Cobb-Douglas producion funcion Y F, 1 Α elasisiy of produc in relaion o he capial Inerpreaion: If he capial rises by 1% producion increases by α%, ceeris paribus 1- α elasisiy of produc in relaion o he labor Inerpreaion: If he labor rises by 1% producion increases by 1-α %, ceeris paribus Coefficiens α and 1- α are respeceively capial s and labor s share of income For he consan reurns o scale: 1> α>0 [α +1- α=1]
Wha is he value of α capial s share of income? source: Bernanke i Gurkaynak 2002 afer Weil D. 2009 s.55
Assumpions 1. Facor of producion: labour and capial 2. The economy is closed wihou governmen inervanion 3. From he income flow : and if 4. Depreciaion rae is consan: 5. Populaion growh rae I S Depreciaio n Capial D S sy The change in he capial n Invesmen: expendiure on plan and equipmen. I sy Depreciaion: wearing ou of old capial; causes capial sock o fall e.g. If, say, he ypical machine lass for 5 years, hen he depreciaion rae is 20 percen I sy 1 Change in capial sock Invesmen Depreciaion
Y Neoclassical producion funcion Y Y = C + I C= 1-sY C S=sY Y I
Y Y D=δ C S=sY I Invesmen required o keep capial consan
Seady sae Y Y* Y D=δ C S=sY I Depreciaion * 0 sy
Seady sae A *, invesmen equals depreciaion and capial will no change over ime. Y Y* Y Y D=δ S=sY Above *, depreciaion exceeds invesmen, so he capial sock shrinks. * 0 sy sy sy 0 0 Capial sock grows Capial sock shrinks Below *, invesmen exceeds depreciaion, so he capial sock grows.
k The per worker producion funion: Y y Capial per worker 2 Oupu per worker 3 Trick ake logs and hen derivaives see appendix Taking logs and differeniaing expression 2 wih respec o ime, we obain: n k k d d d d d k d k ln ln ln ln ln ln ln ln, 4
Subsiuing for. n k sy n k sy n sy n sy k k, from equaion 1 we derive: k n sy k 5
The seady-sae is a condiion of he economy in which oupu per worker Produciviy of labour and capial per worker Capial inensiy do no change over ime. This is due o he rae of new capial producion from invesed savings exacly equalling he rae of exising capial depreciaion k We define seady sae by he condiion ha 0 hen seing equaion 5 o zero sy n k
Finding he seady sae value of capial per worker and oupu per worker sy n k y Y F, F, 1 k k ss s n 1/1 y ss s n /1
Per worker producion funcion y y n+δk c sy i ike depreciaion, populaion growh is one reason why he capial sock per worker shrinks. If n is he rae of populaion growh and δ is he rae of depreciaion, hen n+δk is break-even invesmen, which is he amoun necessary o keep consan he capial sock per worker k. k
y y* y Seady sae y n+δk sy k k* k k In he seady sae, he posiive effec of invesmen on he capial per worker jus balances he negaive effecs of depreciaion and populaion growh. Once he economy is in he seady sae, invesmen has wo purposes: 1 Some of i, k*, replaces he depreciaed capial, 2 The res, nk*, provides new workers wih he seady sae amoun of capial.
An increase in savings y y* y n+δk s y sy k* k k An increase in he saving rae causes he capial sock o grow o a new seady sae. Increase in s rises k* and y*
Populaion growh change y* y y n+δk sy k k* k Increase in populaion growh n lowers k* and y*
Change in he rae of populaion growh An increase in he rae of populaion growh shifs he line represening populaion growh and depreciaion upward. The new seady sae has a lower level of capial per worker han he iniial seady sae. Thus, he Solow model predics ha economies wih higher raes of populaion growh will have lower levels of capial per worker and herefore lower incomes. Change in he savings rae In he long run, an economy s saving deermines he size of k and hus y. The higher he rae of saving, he higher he sock of capial and he higher he level of y. An increase in he rae of saving causes a period of rapid growh, bu evenually ha growh slows as he new seady sae is reached. Conclusion: alhough a high saving rae yields a high seady-sae level of oupu, saving by iself canno generae persisen economic growh.
Exercise 1 A counry is described by he solow model, wih a producion funcion y k 1/ 2 Suppose ha k=400. The fracion of oupu invesed is 50%. Is he counry a is seady sae level of oupu per worker, above he seady sae or below? Show how you reach your conclusion.
EXERCISE 2 In counry A he rae of invesemen is 5% and in counry B i is 20%. The wo counries have he same rae of depreciaion and he same populaion growh rae. Assuming, ha he value of α is 1/3, wha is he rao of seady-sae oupu per worker in counry A o seady-sae oupu per worker in counry B? Wha would be he raio if he value of α was 2/3?
EXERCISE 3 The following ables show he daa on invesmens raes and oupu per worker for wo pairs of counries. For each pair calculae he raio of GDP per worker in seady sae ha is prediced by Solow model all counries have he same value of δ and α=1/3.then calculae he acual raio of GDP per worker for which pairs of counries does he Solow model do a good job of predicing relaive income? Weil, ch.3, problem 5 Counry Invesmen rae Oupu per worker average 1975-2009 in 2009 Thailand 35.2% 13 279 USD Bolivia 12.6% 8 202 USD Counry Invesmen rae Oupu per worker average 1975-2009 in 2009 Nigeria 6.4 % 6 064 USD Turkey 16.3 % 29 699 USD
Source: Weil 2013 p.85 Tes of Solow model s predicions
EXERCISE 4 In counry A he populaion grows a 0% per year and in counry B populaion grows a 4% per year. Assuming, ha he wo counries have he same he same rae of depreciaion 5%, he same savings rae, and ha he value of α is 1/3, wha is he rao of seady-sae oupu per worker in counry A o saedy-sae oupu per worker in counry B? Wha would be he raio if he value of α was 2/3?
Source: Weil 2013 p.103 GDP per capia and populaion growh
Savings in Solow model y y* y n+δ k s y sy k* k k Change in he savings rae In he long run, an economy s saving deermines he size of k and hus y. The higher he rae of saving, he higher he sock of capial and he higher he level of y. An increase in he rae of saving causes a period of rapid growh, bu evenually ha growh slows as he new seady sae is reached. Conclusion: alhough a high saving rae yields a high seady-sae level of oupu, saving by iself canno generae persisen economic growh.
Spource: Weil D. 2009 p. 70 Savings rae by decile of income per capia
GDP per capia in 2009 in logs Savings raes and GDP per capia 14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.00-150.00-100.00-50.00 0.00 50.00 100.00 Naional savings rae in %, 2009 Source: own based on daa from Weil 2013
Average annual growh rae 1975-2009 Saving raes and growh of GDP per capia 10.00 8.00 6.00 4.00 2.00 Serie1 0.00-150.00-100.00-50.00 0.00 50.00 100.00-2.00-4.00-6.00 Naional savings rae in 2009 Source: own based on daa from Weil 2013
The relaionship beween invesmens and savings According o closed Solow model I=S, invesmen raes = savings rae Bu in open model: free movemen of capial Capial moves oward is higher reurns Increase savings rae increase in capial sock decrease in he marginal produc of capial rens ransfer of capial abroad Invesmens do no depend on he savings
Savings and Invesmens Source: Weil 2013, p. 335 based on Feldsein and Horioka 1980
Wha should be he saving rae? The effec of increse in s is he rise of k and y a he seady sae The effec of increse in s is ha lower par of income is consumed Y=S+C If we save all income consumpion =0, If we consume all income, saving=0 Wha is he savings rae s * maximizes he seady sae level of aggregae consumpion C * per uni of effecive labour The answer is so called golden rule
Golden saving rule Condiion for seady sae k 0 k 0 sy k sy k sy k 1 2 3 4 C dc dk y sy dc dk 0 k 1 k k 5 6 7 k 1 0 8 k 1 9 Coming back o he seady sae condiion 4 and puing 9 sy k k divide by y 1 k k s y
Golden saving rule c c_max s=α s
Appendix some Mahemaics X X 1 X X X X X 1 percenage change X X X dx d lim0 X X X d / d / 0.02 he labor force is growing a 2 percen per year
Naural logs z xy ln z ln x ln y z x y ln z ln x ln y z x ln z ln x y y dy f x ln x, dx dy ln x, dx 1 x dy dx dx d 1 x x x x Derivaive wih respec o ime of he log of some variable is he growh rae of ha variable d ln X / d 1/ X dx / d X X
Take logs and derivaives Y Y d d d d d Y d Y Y Y 1 ln 1 ln ln ln 1 ln ln ln ln ln 1 1
Sources: Weil D. 2009, 2012. Economic Growh, Chaper 2&3. Pearson Addison Wesley Acemoglu D. Growh Theory Since Solow and he Povery of Naions. World Bank, April 26, 2006hp://econ-www.mi.edu/files/970 Mankiew N.G. 2012, Macroeconomics, 8h Ediion, Worh Publishers