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Models of Economic Growth A Outline: Because this area is complex and mathematical there are two files of slides for this topic Lecture A • Introduction – trends in growth • Neoclassical growth models Lecture B • Endogenous growth models • The convergence debate Below are slides for lecture A See next file for lecture B Introduction • Need to define ‘economic growth’ (in book this is growth in GDP per capita, not GDP growth). • Some background on history of economic growth – including own country data • Also, worthwhile to stress importance of small differences in growth rates e.g. 2% growth per year GDP p.c. increases 7.4 fold in 100 years 0.6% GDP per capita increase 1.8 times in 100 .... 72 / growth rate = no. of years to double, hence China’s 10% p.a. implies 7.2 years The very long run Growth of GDP per capita (average annual percentage changes) 1500-1820 OECD Non-OECD World 0.04 1820-1900 1.2 0.4 0.8 1900-2000 2.0 0.6 1.9 Source: Boltho and Toniolo (1999, Table 1) OECD refers to North America, Western Europe, Japan, Australia and New Zealand. 5000 15000 25000 35000 USA, UK and EIRE 1965 1970 1975 1980 1985 1990 1995 2000 year GBR USA IRL Growth of GDP p.c: USA=2.2%, GBR=2.0%, Ireland=3.7% (but post-93, 8.5%) GDP per capita is US$ 1996 constant prices. Source: Penn World Table 6.1 0 1000 2000 3000 4000 China and India 1965 1970 1975 1980 1985 1990 1995 2000 year CHN IND Growth: pre-90 China 3.7%, India 4.4%. 1990-2000: China 7.0%, India 4.4% Source: Penn World Table 6.1 0 5000 10000 15000 Brazil, S. Korea, Philippines 1965 1970 1975 1980 1985 1990 year BRA PHL KOR Source: Penn World Table 6.1 (http://pwt.econ.upenn.edu/aboutpwt.html) 1995 2000 Other data • Above are from Penn World Table 6.1, now 6.3 is available http://pwt.econ.upenn.edu/ Some further links at: http://users.ox.ac.uk/~manc0346/links.html GDP per capita growth not everything • Focusing on ‘economic growth’ does neglect health, the environment, education, etc • UN’s Human Development Index (HDI) gives equal weight to life expectancy, education and GDP per capita (http://hdr.undp.org/reports/global/2004/) • Ultimate interest ‘well-being’ or ‘happiness’. Layard, R. (2003). "Happiness: Has Social Science a Clue?" http://cep.lse.ac.uk/events/lectures/layard/RL030303.pdf. • GDP measures aggregate value added – whether coal power station or wind farm • Friedman, Ben (2005) The Moral Consequences of Economic Growth argues growth is important for ‘stable’ societies Neoclassical model • There are many ways to teach this. Book tends to use equations, but can do a great deal with intuition and few diagrams. • This model most often attributed to Robert Solow (1956) – US Nobel prize winner …. but Trevor Swan (1956) (a less well known Australian economist) published (independently) a very similar paper in the same year – hence refer to Solow-Swan model Neoclassical growth model • Model growth of GDP per worker via capital accumulation • Key elements: – Production function (GDP depends on technology, labour and physical capital) – Capital accumulation equation (change in net capital stock equals gross investment [=savings] less depreciation). • Questions: – how does capital accumulation (net investment) affect growth? – what is role of savings, depreciation and population growth? – what is role of technology? Solow-Swan equations Y Af (K , L) (production function) Y G D P , A technology, K capital, L labour dK sY K (capital accum ulation equation) dt s proportion of G D P saved (0 s 1) depreciation rate (as p roportion) (0 1) Solow-Swan analyse how these two equations interact. Y and K are endogenous variables; s, and growth rate of L and/or A are exogenous (parameters). Outcome depends on the exact functional form of production function and parameter values. Neoclassical production functions Solow-Swan assume: a) diminishing returns to capital or labour (the ‘law’ of diminishing returns), and b) constant returns to scale (e.g. doubling K and L, doubles Y). For example, the Cobb-Douglas production function 1 w here 0 1 Y AK L y Y L 1 AK L L AK L K A A k L Hence, now have y = output (GDP) per worker as function of capital to labour ratio (k) GDP per worker and k output per worker Assume A and L constant (no technology growth or labour force growth) y y=Af(k)=Ak concave slope reflects diminishing marginal product of capital dY/dK=dy/dk=Ak-1 k (capital per worker) Accumulation equation If A and L constant, can show* dk sy k dt This is a differential equation. In words, the change in capital to labour ratio over time = investment (saving) per worker minus depreciation per worker. Any positive change in k will increase y and generate economic growth. Growth will stop if dk/dt=0. *accu m u latio n eq u atio n is: dK sY K , d ivid e b y L yield s dK dt A lso n o te th at, dK K d / d t /L dt L d t dk dt sin ce L is a co n stan t. / L sy k Graphical analysis of dk (Note: s and constants) dt sy k output per worker y k (depreciation) sy net investment (savings = gross investment) k* k (capital per worker) Solow-Swan equilibrium y=Ak output per worker y y* consumption per worker k sy k* k GDP p.w. converges to y* =A(k*). If A (technology) and L constant, y* is also constant: no long run growth. What happens if savings increased? • raising saving increases k* and y*, but long run growth still zero (e.g. s1>s0 below) • call this a “levels effect” • growth increases in short run (as economy moves to new steady state), but no permanent ‘growth effect’. y=Ak y y1 * y0 * k s1y s0y k0* k1* What if labour force grows? dk Accumulation eqn now sy ( n ) k w here n dt /L (m ath note 2) dt y y output per worker Population growth reduces equilibrium level of GDP per worker (but long run growth still zero) if technology static dL (n)k k sy Population growth (n>0) pivots the ‘depreciation’ line upwards, and reduces k and y steady state kn* k Analysis in growth rates Distance between lines is growth rate of capital per worker Can illustrate above with graph of gk and k net investment dk dk sy ( n ) k dt dt g s y ( n ) k k k Distance y between lines represents growth s per s average of capital in capital worker (gproduct k) k n net disinvestment k* k capital per worker Rise in savings rate (s0 to s1) gk y s0 k s1 y k B C A n k k* g Y, g k gY=(MPK/APK) gk = sk gk (sk = in Cobb-Douglas case) NB: This graph of how growth rates change over time 0% Saving change y s1 k Time Golden rule • The ‘golden rule’ is the ‘optimal’ saving rate (sG) that maximises consumption per head. • Assume A is constant, but population growth is n. • Can show that this occurs where the marginal product of capital equals ( n) P roof: dk sy ( n ) k 0 at steady state, dt hence sy ( n ) k , w here * indicates steady state equilibrium value T he problem is to: m ax c y sy y * ( n ) k k First order condition : 0 dy * dk * ( n ) hence M Pk = dy * dk * n Graphically find the maximal distance between two lines y output per worker y slope=dy/dk=n+ y** (n)k maximal consumption per worker sgold y k** k capital per worker … over saving y=Ak output per worker y slope=dy/dk=n+ y** sovery maximal consumption per worker (n)k k** sgoldy k* k capital per worker Economies can over save. Higher saving does increase GDP per worker, but real objective is consumption per worker. Golden rule for Cobb Douglas case • • • • Y=KL1- or y = k Golden rule states: MPk = (k*)-1 =(n + ) Steady state is where: sy* = ( +n)k* Hence, sy* = [(k*)-1]k* or s = (k*) / y* = Golden rule saving ratio = for Y=KL1- case Assuming perfect competition, and factors are paid marginal products, is share of GDP paid to capital (see C&S, p.481). Expect this to be 0.1 to 0.3. Solow’s surprise* • Solow’s model states that investment in capital cannot drive long run growth in GDP per worker • Need technological change (growth in A) to avoid diminishing returns to capital • Easterly (2001) argues that “capital fundamentalism” view widely held in World Bank/IMF from 60s to 90s, despite lessons of Solow model • Policy lesson: don’t advise poor countries to invest without due regard for technology and incentives * This is title of Chapter 3 in Easterly (2001), which is worth a quick read for controversy surrounding growth models and development issues What if technology (A) grows? • Consider y=Ak, and sy=sAk, these imply that output can go on increasing. • Consider marginal product of capital (MPk) MPk=dy/dk =Ak1, if A increases then MPk can keep increasing (no ‘diminishing returns’ to capital) • implies positive long run growth …. graphically, the production function simply shifts up y Technology growth: A2 > A1 > A0 2 y=A1k 1 output per worker y=A2k y=A0k 0 k k Capital to labour ratio …. mathematically Y K ( AL) E asier to use 1 w here 0 1 (T his assum es A augm ents labour (H arrod-neutral technological change) C an re-w rite dA A ssum e dt K ( AL) / A gA 1 A 1 1 K L (for reference this sa m e as A t A o e g At ) T rick to solving is to re-w rite as y Y K ( AL) AL AL 1 K = ( k ) A L w here y = output per 'effective w orker', an d k capital per 'effective w orker' C an show dk / k s(k ) (n a )k dt T h is can be solved (plotted) as in sim pler S olow m odel. Output (capital) per effective worker diagram output per effective worker Y/AL (Y/AL)* Y/AL NOTE: ‘dilution’ line now includes technology growth (a) (na)k s(Y/AL) (K/AL)* K/AL capital per effective worker If Y/AL is a constant, the growth of Y must equal the growth rate of L plus growth rate of A (i.e. n+a) And, growth in GDP per worker must equal growth in A. Summary of Solow-Swan • Solow-Swan, or neoclassical, growth model, implies countries converge to steady state GDP per worker (if no growth in technology) • if countries have same steady states, poorer countries grow faster and ‘converge’ – call this classical convergence or ‘convergence to steady state in Solow model’ • changes in savings ratio causes “level effect”, but no long run growth effect • higher labour force growth, ceteris paribus, implies lower GDP per worker • Golden rule: economies can over- or under-save (note: can model savings as endogenous) Technicalities of Solow-Swan • Textbooks (Jones 1998, and Carlin and Soskice 2006) give full treatment, in short: • Inada conditions needed ( “growth will start, growth will stop”) dY dY lim 0, lim , K dK K 0 dK • It is possible to have production function where dY/dK declines to positive constant (so growth declines but never reaches zero) • Exact outcome of Solow model does depend on precise functional forms and parameter values • BUT, with standard production function (Cobb-Douglas) Solow model predicts economy moves to steady state because of diminishing returns to capital (assuming no growth in technology A) Endnotes M ath note 1: y t y 0 e gt can be used to analyse i m pact of grow th over tim e L et y= G D P p.w ., g= grow th (e.g. 0.02 2% ), t= tim e. H ence, for g 0.02 and t 100, y t / y 0 e 7.39 2 M ath N ote 2: S tart w ith dK sY K , divide by L yields dt N ote that sim plify to hence dk dt hence dk dt dK dt dL 2 K dK d / dt L K /L dt dt L dt dk dL K /L /L dt dt L dK + nk = / L sy k dK / L sy k dt = sy ( n ) k or dK dt / L nk (quotient rule) (since n is labour grow th and K / L k ) Questions for discussion 1. What is the importance of diminishing marginal returns in the neoclassical model? How do other models deal with the possibility of diminishing returns? 2. Explain the effect of (i) an increase in savings ratio (ii) a rise in population growth and (iii) an increase in exogenous technology growth in the neoclassical model. 3. What is the golden rule? Can you think of any countries that have broken the golden rule? References Boltho, A. and G. Toniolo (1999). "The Assessment: The Twentieth Century-Achievements, Failures, Lessons." Oxford Review of Economic Policy 15(4): 1-18. Easterly, W. (2001). The Elusive Quest for Growth: Economists’ Adventures and Misadventures in the Tropics. Boston, MIT Press. Swan, T. (1956). "Economic Growth and Capital Accumulation." Economic Record 32: 344-361. Jones, C. (1998) Introduction to Economic Growth, (W.W. Norton, 1998 First Edition, 2002 Second Edition). Carlin, W. and D. Soskice (2006) Macroeconomics: Imperfections, Institutions and Policies, Oxford University Press.