MCR_L1_Solow PDF - Macroeconomics, Solow-Swan Model

Macroeconomics The Solow-Swan Model Alejandro Cuñat  References These notes are based on: – R.J. Barro & X. Sala i Martin (2004): Economic Growth, 2nd edition, MIT Press – C.I. Jones (2016): “The Facts of Economic Growth,” in J.B. Taylor &...

Macroeconomics The Solow-Swan Model Alejandro Cuñat  References These notes are based on: – R.J. Barro & X. Sala i Martin (2004): Economic Growth, 2nd edition, MIT Press – C.I. Jones (2016): “The Facts of Economic Growth,” in J.B. Taylor & H. Uhlig (eds.), Handbook of Macroeconomics, 2A – D. Romer (2018): Advanced Macroeconomics, 5th edition, McGraw Hill 2  Outline Introduction The Solow-Swan model: assumptions Dynamics of k Steady state Level effects Golden rule Convergence Micro-foundations Appendix 3  Introduction 36 Handbook of Macroeconomics Growth rate, 1960–2011 7% South Korea 6% Botswana Taiwan 5% Romania Malta Japan Singapore China Thailand Cyprus 4% Egypt Portugal Spain Ireland Panama Argentina Hong Kong Italy 3% Cape Verde Brazil Israel Germany India Tunisia Malaysia Turkey France Luxembourg Morocco Chile United States 2% Lesotho Paraguay Australia Canada Switzerland Burkina Faso Philippines Norway Nepal Guatemala New Zealand 1% Ethiopia Mali Iran Gabon Malawi Namibia Togo Trinidad/Tobago 0% Comoros Cote dIvoire Jamaica Madagascar Nigeria Venezuela –1% C. Afr. Republic Congo Niger Guinea –2% 1/64 1/32 1/16 1/8 1/4 1/2 1 GDP per person (US = 1) in 1960 Fig. 26 The lack of convergence worldwide. Source: The Penn World Tables 8.0. the United States—grew more slowly. The pattern is quite strong in the data; a simple Source: C.I. Jones (2016): “The Facts of Economic Growth,” regression line leads to an R-squared of 75%.u in Handbook of Macroeconomics, Volume 2A 4 Lucas Fig. Relevance 26 showsofthat differences in growth a simplistic viewrates Convergence of convergence does not hold for the world as a  The Solow-Swan model Very influential theory of economic growth; simple dynamic model: – Current income equals current production, which is the result of current endowments of capital and labour, and the current state of technology. – Part of current income is saved and invested into additional capital, available for production tomorrow. – Tomorrow’s income, which is the result of tomorrow’s capital and labour endowments and state of technology, rises,… and so on. Helps us understand how certain variables (saving and investment, technology, technical progress) relate to economic growth. Delivers empirical predictions that we can contrast with the data. Like any model we will discuss in the course: some relevant successes, but also some shortcomings (both theoretical and empirical). 5 Solow  Assumptions L agents sell each 1 unit of labour services, and own and rent out capital K. Income is used for consumption C and saving S, invested in additional K. There is one (final) good only, output Y, with price P = 1 constant, which can be used for C and investment I. Firms hire L, rent K, and produce the final good with technology Y = F(K, AL); A: “labour-augmenting” productivity; AL: “effective” labour. All markets are perfectly competitive. Under constant returns to scale (assumed below), firms’ revenues PY = Y equal individuals’ (factor owners’) incomes. ⇒ Y = C + S Closed economy; no government ⇒ Y=C+I ⇒ I=S [Alternative setup excluding markets: Robinson Crusoe manages the technology that transforms inputs into outputs, consumes, and invests.] 6  Assumptions: neoclassical production function Constant returns to scale (CRS): F(λK, λAL) = λF(K, AL) for any λ > 0 Positive, diminishing marginal products: FK > 0, FKK < 0, FL > 0, FLL < 0 Inada conditions: lim FK = lim FL = ∞, lim FK = lim FL = 0 K→0 L→0 K→∞ L→∞ CRS allows us to work with the production function in intensive form f(k): F(K, AL)/AL = F (K /AL,1) ≡ F(k,1) ≡ f(k). Capital and output per effective labour: k ≡ K /(AL), y ≡ Y/(AL) = f(k) The properties of F(K, AL) assumed above imply f (0) = 0; f′(k) > 0; f′′(k) < 0; lim f′(k) = ∞; lim f′(k) = 0 k→0 k→∞ Y = K α(AL)1−α ⇔ y = k α. Under perfect competition, α is the (constant) capital share in income (and 1 − α is the labour share). 7 Neoclassical production function       Assumptions: diminishing marginal productivity f (k) f (k) f′′(k) < 0: the higher k, the smaller the increases in f(k) when k rises (by one unit); f(k)/k decreases in k. f′(k) > 0: an increase in k leads to an increase in f(k). O k 8     Assumptions: dynamic behaviour [In general, variables are a function of t. We avoid notation “x(t)” and write “x” except for when the time argument is needed to avoid confusion.] Define x· ≡ dx /dt and assume the following dynamic behaviour: – L· /L = n ⇔ L(t) = L(0)e nt, n: constant growth rate of population – A· /A = g ⇔ A(t) = A(0)e gt, g: constant growth rate of productivity – K· = I − δK = (Y − C) − δK, δ: constant depreciation rate n+g+δ >0 s: constant saving rate ⇒ constant investment rate: I/Y = s · C = (1 − s)Y ⇒ K = sY − δK 9  Dynamics of k K Differentiating k = w.r.t. t, AL · · · · 1 A L + L A k=K −K = AL (AL)2 · · AL ( L A ) sY − δK K L A = − + = AL Y K =s − (n + g + δ) = AL AL = sf (k) − (n + g + δ)k sf(k): gross investment per unit of effective labour (n + g + δ)k: break-even investment per unit of effective labour 10  Steady state · sf (k*) = (n + g + δ)k* ⇔ k* = 0 f(k) f (k) y* c* = f(k*) − sf(k*) = (n + g + δ)k c* = f(k*) − (n + g + δ)k* sf(k) sf(k*) O k* k 11  Steady state Steady state (or balanced growth path): situation in which all variables grow at constant rates. Our assumptions about technology ensure there is a unique steady state with k* > 0 constant. We ignore steady state k* = y* = 0 due to its lack of interest. · k* = sf(k*) − (n + g + δ)k* = 0 ⇔ sf(k*) = (n + g + δ)k* · = 0. ⇒ K /Y constant in steady state Since y = f(k), y* When k = k* (and y = y*) constant, – K = ALk* and Y = ALy* grow at constant rate g + n. K Y – = Ak* and = Ay* grow at constant rate g. L L 12  Steady state & phase diagram f (k) f (k) y* (n + g + δ)k sf (k) k* k k < k* ⇒ sf (k) > (n + g + δ)k k > k* ⇒ sf (k) < (n + g + δ)k · k For any starting point k(0) > 0, the economy converges to steady state k*. 0 k* k · k · · k < k* ⇒ k > 0 k > k* ⇒ k < 0 13  Steady state For any starting point k(0) > 0, the economy converges to steady state k*. ⇒ One can think of the steady state as the situation towards which the economy tends in the long run. The long-run growth rate of Y/L is determined solely by the rate of technological progress g,… …which is exogenous to the model! Diminishing marginal productivity of capital (f′′(k) < 0) prevents capital accumulation from being an engine of long-run growth: – Capital accumulation leads to ever smaller increases in output and saving (= gross investment sf(k)). – Eventually, in steady state, sf(k) just covers (n + g + δ)k. 14    Long-run growth of GDP per capita The Facts of Economic Growth 5 Log scale, chained 2009 dollars 64,000 32,000 16,000 2.0% per year 8000 4000 2000 1880 1900 1920 1940 1960 1980 2000 Year Fig. 1 GDP per person in the United States. Source: Data for 1929–2014 are from the U.S. Bureau of Economic Analysis, NIPA table 7.1. Data before 1929 are spliced from Maddison, A. 2008. Statistics on world population, GDP and per capita GDP, 1-2006 AD. Downloaded on December 4, 2008 from http://www.ggdc.net/maddison/. The vertical logarithmic scale makes the reading of growth rates easy: y(t) = e gt y(0) d ln y dy/dt ⇔ ln y(t) = gt + ln y(0); the growth rate is the slope g = =. The paper is divided broadly into two parts. First, I presentdtthe facts related y to the growth of the “frontier” over time: what are the growth patterns exhibited by the richest Source: C.I. countries inJones the (2016): world?“The Facts ofIEconomic Second, focus onGrowth,” in Handbook the spread of Macroeconomics, of economic Volume 2A growth throughout 15 the world. To what extent are countries behind the frontier catching up, falling behind,  Level effects f(k) f (k) y* 1 (n + g + δ)k y* 0 s1 f(k) s0 f(k) s0 < s1 O k* 0 k* 1 k 16  Level effects (n1 + g + δ)k f(k) f (k) y* 0 (n0 + g + δ)k y* 1 sf(k) n0 < n1 O k* 1 k* 0 k 17  Level effects, no growth effects y = kα ⇔ Y/L = Ay = Ak α, α ∈ (0,1) · k = sf(k) − (n + g + δ)k = 0 ⇒ s(k*)α = (n + g + δ)k* 1 α (n + g + δ) (n + g + δ) s 1−α α s 1−α k* = , y* = (k*) = Mankiw et al. (1992): empirical evidence supporting the predicted effects of s and n on income per capita. In steady state, K /L = Ak* and Y/L = Ay* grow at constant rate g. Changes in parameters (s, n, δ) lead to: – “level effects”: changes in the steady-state values of k*, y*, but… – …no “growth effects”: the long-run growth rate remains constant at g. 18  Golden rule f(k) f (k) y* yGR (n + g + δ)k sf(k) cGR sGR f(k) sGR f(kGR) c* = f(k*) − (n + g + δ)k* cGR = max{f(k) − (n + g + δ)k} s dk ⇒ [ f′(kGR) − (n + g + δ)] =0 ds O kGR k* k 19   Golden rule c* = f(k*) − (n + g + δ)k*; the saving rate sGR for which the maximum (“golden-rule”) level of steady-state consumption is attained is such that dk [ f′(kGR) − (n + g + δ)] =0 ⇒ f′(kGR) = (n + g + δ) ds s > sGR ⇒ “dynamic inefficiency” or over-saving: lowering s would raise c at all points in time (during the transition to, and at the new steady state). s < sGR: raising s implies an inter-temporal trade-off between consuming less initially and consuming more at the new steady state. – Whether this outcome is “good” or “bad” depends on how households weigh today’s consumption against the path of future consumption. – We cannot judge its desirability unless we make specific assumptions about how agents discount the future. 20    Absolute convergence · k sf(k) = − (n + g + δ) k k · kP kP · kR n+g+δ kR sf(k)/k kP kR k* k 21  Conditional convergence · kP sP f(kP) = − (n + g + δ) kP kP · kR sR f(kR) · = − (n + g + δ) · kR kR kR kP kR kP sP < sR n+g+δ sR f(kR)/kR sP f(kP)/kP kP kR k* P k* R k 22  Convergence Absolute convergence: countries with identical parameter values have the · · same k*; kP < kR ⇒ kP /kP > kR /kR (due to f′′(k) < 0). Conditional convergence: – Differences in parameter values lead to different k*. If k* ≠ k*, then i P R · · kP < kR need not imply kP /kP > kR /kR. – k· i /ki depends on the distance of ki to the country’s own steady state k*: i · · ki < k′i ⇒ ki /ki > k′i /k′i (due again to f′′(k) < 0). · = αk· /k. ⇒ Similar convergence patterns for k and y y = k α: y/y If countries differ in parameter values, they need not converge. In the data, poor countries do not grow faster than rich countries. Within a country, regions are similar in parameters. In the data, poor regions grow faster than rich regions of the same country. 23 Absolute convergence?         470 Chapter 11 Conditional convergence 0.025 0.02 Annual growth rate, 1880–2000 0.015 0.01 0.005 !0.4 !0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Log of 1880 per capita personal income Figure 11.2 Convergence of personal income across U.S. states: 1880 personal income and 1880–2000 income growth. The average growth rate of state per capita income for 1880–2000, shown on the vertical axis, is negatively related to the log of per capita income in 1880, shown on the horizontal axis. Thus, absolute β convergence exists for the U.S. states. The estimated Source: R.J. Barro coefficient β and X. Sala iisMartin significantly positive—indicating (2004): Economic β convergence—for Growth, 2nd edition, MIT Press 24 seven of the ten subperiods. The coefficient has the wrong sign (β < 0) for only one of the Japanese prefectures  Micro-foundations Assume production has pollution as a side effect. According to the Solow model: – Is this good, bad, or irrelevant? – What should the government do? What does the Solow model predict about the effects of: – A reduction of today’s income tax on the saving rate? – News that productivity will have a permanently higher level in the future on today’s economic variables? 25  Micro-foundations The lack of micro-foundations limits the model’s usefulness: – No welfare evaluation of different outcomes and policies – No behavioural responses to the economic/political/social environment or to changes in policies and expectations Two main workhorses to micro-found dynamic behaviour: – Overlapping generations model (OLG): discrete time, age heterogeneity (young and old), finite lives, possibility of dynamic inefficiency – Ramsey model: usually modeled in continuous time, representative agent (no heterogeneity), infinite time horizon, dynamic efficiency Both models are similar to the Solow model in terms of their predictions on the source of long-run growth, convergence, etc. 26  Appendix: Do differences in growth rates matter? A country has income per capita Y(0)/L(0) = 1 today. If it grows at rate g: – How much will the country’s income per capita be in t = 20 years? – How many years t will it need to double its income per capita? g [ Y(0)/L(0) ] Y(t) Y(0) 1 Y(t)/L(t)] = e gt t= ln L(t) L(0) Growth rate g Income after 20 years Years to double income 0,01 1,2 70 0,02 1,5 35 0,03 1,8 23 0,04 2,2 18 0,05 2,7 14 0,06 3,2 12 0,07 3,9 10 0,08 4,7 9 27 Back  Appendix: Lucas on growth “I do not see how one can look at figures like these without seeing them as representing possibilities. Is there some action a government of India could take that would lead the Indian economy to grow like Indonesia’s...? If so, what exactly? If not, what is it about the ‘nature of India’ that makes it so? The consequences for human welfare involved in questions like these are simply staggering: Once one starts to think about them, it is hard to think about anything else.” Robert E. Lucas Jr. 28 Back  Appendix: Solow on economic theory “All theory depends on assumptions that are not quite true. That is what makes it theory. The art of successful theorizing is to make the inevitable assumptions in such a way that the final results are not very sensitive. A “crucial” assumption is one on which the conclusions do depend sensitively, and it is important that crucial assumptions be reasonably realistic. When the results of a theory seem to flow specifically from a special crucial assumption, then if the assumption is dubious, the results are suspect.” Robert M. Solow 29 Back  Appendix: Neoclassical production function ( AL ) K From our definitions, Y = F(K, AL) = AL f = AL f(k). f′(k) – FK (K, AL) = AL = f′(k) > 0 AL ( AL 2 ) K – FL(K, AL) = AL f′(k) − + A f(k) = A[ f(k) − kf′(k)] f′′(k) – FKK (K, AL) = ⇔ f′′(k) = ALFKK (K, AL) < 0 AL Under CRS, if firms maximise profits and markets are competitive, inputs are paid their marginal products and factor payments exhaust all output: – FK (K, AL) = r = f′(k), FL(K, AL) = w = A[ f(k) − kf′(k)] – wL + rK = Y – Y = K α(AL)1−α ⇔ y = k α: rK = αY, wL = (1 − α)Y 30      Back       Appendix: Conditional convergence 0.066 9 7 13 0.062 17 12 36 3 37 16 46 10 0.058 395 6 2111 4541 824 18 Annual growth rate, 1930–90 20 25 32 4 0.054 31 44 23 4238 19 29 33 34 30 43 0.05 2 35 40 22 15 0.046 1 0.042 26 0.038 28 27 14 0.034 !2.6 !2.2 !1.8 !1.4 !1 !0.6 !0.2 Log of 1930 per capita income Figure 11.5 Convergence of personal income across Japanese prefectures: 1930 income and 1930–90 income growth. The growth rate of prefectural per capita income for 1930–90, shown on the vertical axis, is negatively related to the log of per capita income in 1930, shown on the horizontal axis. Thus absolute β convergence exists for the Japanese prefectures. The numbers shown identify each prefecture; see table 11.10. is 0.0125 (0.0032). Source: Aand R.J. Barro testX. forSala thei equality of coefficients Martin (2004): Economic over time2nd Growth, is edition, stronglyMIT rejected; Press the 31 p value is 0.000. US states  Food for thought (beyond this course) Dr. A. Brausmann addresses the links between growth, natural resources, pollution, etc. in her Environmental Economics course. Barro, R.J., & X. Sala-i-Martin (1992): ‘‘Convergence,’’ Journal of Political Economy, 100, pp. 223-251 Easterly, W. (2001): The Elusive Quest for Growth: Economists’ Adventures and Misadventures in the Tropics, MIT Press Karabarbounis, L., & B. Neiman (2014): “The Global Decline of the Labor Share,” Quarterly Journal of Economics, 129(1), pp. 61-103 Kremer, M., J. Willis & Y. You (2021): “Converging to Convergence,” NBER Macro Annual 2021 Mankiw, N.G., D. Romer & D.N. Weil (1992): “A Contribution to the Empirics of Economic Growth,” Quarterly Journal of Economics, pp. 407-437 Rodrik, D. (2015): Economics Rules: The Rights and Wrongs of the Dismal Science, Norton 32