EPP - Macroeconomics 1

Transcription

1 EPP - Macroeconomics 1 Lecture 1 - Growth facts & the Solow model Zsófia L. Bárány Sciences Po 2014 September

2 About the course I. 2-hour lecture for 12 weeks, Mondays from 10:15-12:15 3 big topics covered: 1. economic growth 2. long-run inequality 3. labor markets office hour by appointment, in office E.404 send me an if you want to come zsofia.barany@sciencespo.fr lecture notes will become available weekly on my website: macro 1

3 About the course II. 2-hour tutorial every week, on Wednesdays tutorials in 2 groups, with Aseem Patel or Vanda Almeida you will have a problem set related to each lecture go to the tutorial having solved the problem set at home during the tutorials you will cover the problem sets, and other extra material (for example on mathematical methods) you should also raise any questions you have related to the lectures or the problem sets

4 What is expected from you? attend all the lectures attend all the tutorials, this is compulsory solve the problem sets at home before the tutorial this will help you to understand the material, and also in the exam read the additional readings, this will help you develop your intuition your grade will be the weighted average of: one small group presentation of an assigned article (20%) one problem set to be handed in (20%) the final exam (60%)

5 Let s start!

6 Economic growth one of the most important questions in economics (especially in macro and in development): why are some countries much richer than others? empirical questions were there always income differences? if not, when did these income differences emerge? what are the empirical correlates of income differences? theoretical questions under what conditions does an economy feature growth? how can we model technological progress?

7 Course plan first half of the course: look at the most important data and cover the main models Solow neoclassical growth model OLG endogenous growth models today look at some country level data Solow model discrete time - canonical model continuous time - with population growth and technological progress

8 Cross country income differences Introduction to Modern Economic Growth Density of coutries gdp per capita Figure 1.1. Estimates of the distribution of countries according to PPPadjusted GDP per capita in 1960, 1980 and time, x 1 (t) x 2 (t) will also grow, while log x 1 (t) log x 2 (t) will remain constant). Figure

9 the spreading out might be partly due to the increase in average incomes useful to look at the log of income per capita if the growth rate is constant if x(t) grows at a constant rate, i.e. x(t + 1) = (1 + g) x(t), ln y(t + 1) ln y(t) g the slope of the log-transformed series gives the growth rate of the original series if x 1 (t) and x 2 (t) both grow at the same rate, then x 1 (t) x 2 (t) also grows, while ln x 1 (t) ln x 2 (t) is constant

10 Cross country income differences log the distributionintroduction of log income to Modern per Economic capita is Growth spreading out, but less Density of coutries log gdp per capita Figure 1.2. Estimates of the distribution of countries according to log GDP per capita (PPP-adjusted) in 1960, 1980 and United somestates, of it andmight Russia receive be driven greater weight by because employment they have larger differences? populations. The

11 Introduction to Modern Economic Growth Cross country income differences per worker Density of coutries log gdp per worker Figure 1.4. Estimates of the distribution of countries according to log GDP per worker (PPP-adjusted) in 1960, 1980 and * large inequality in income per worker/per capita across countries * slight but noticeable increase in inequality across countries ( inequality increase across individuals in the world) when one compares an advanced, rich country with a less-developed one, there are striking differences in the quality of life, standards of living and health.

12 Economic growth and income differences how can one country Introduction be 30 to Modern times richer Economicthan Growth another? Density of coutries average growth rates Figure 1.7. Estimates of the distribution of countries according to the growth rate of GDP per worker (PPP-adjusted) in 1960, 1980 and Africa was not beneficial. South Africa is still one of the richest countries in sub-saharan

13 Examples of different trajectories Introduction to Modern Economic Growth log gdp per capita USA UK Spain Singapore Botswana Brazil India South Korea Guatemala Nigeria year Figure 1.8. The evolution of income per capita in the United States, United Kingdom, Spain, Singapore, Brazil, Guatemala, South Korea, Botswana, Nigeria and India, these are the patterns we would like to understand richer than Nigeria because it has grown steadily over an extended period of time, while Nigeria has not (and we will see that there is a lot of truth to this simple calculation; see

14 Introduction to Modern Economic Growth Origins of income differences? log gdp per capita Latin America Western Offshoots Western Europe Asia Africa year Figure The evolution of average GDP per capita in Western Offshoots, Western Europe, Latin America, Asia and Africa, US, UK has not always been much richer than India, Ghana does not include observations for all countries going back to Finally, while these data include a correction for PPP, this is less reliable than the price comparisons used to construct but some countries managed to grow steadily since 1800 the price indices in the Penn World tables. Nevertheless, these are the best available estimates

15 When did the divergence begin? Introduction to Modern Economic Growth log gdp per capita Asia Western Europe Western Offshoots Latin America Africa year Figure The evolution of average GDP per capita in Western Offshoots, Western Europe, Latin America, Asia and Africa, Figure 1.12 shows the evolution of income per capita for United States, Britain, Spain, Brazil, China, India and Ghana. This figure confirms the patterns shown in Figure 1.10 there isn t a full consensus of what happened before

16 Income and welfare Introduction to Modern Economic Growth life expectancy TZA BDI YEM TGO MDG ETH MLI BFA GNB NGA TCD MOZ NER UGA MWI ZMB RWA JPN AUS CHE HKG CAN CRI GRCISR ESP ITAISL SWE FRA MAC NZLGBR AUT BEL FIN NLDNOR CHL IRL DNK USA CZE SVN PRT KOR BRB LKA PAN HRV MEX URY ALB ARG ECU MKD BLZ JAM ARM GEO COL BGR LBN LCA POL SVK SYR VEN TUN MYS LTU HUN TTO CHN JOR EST MUS ROM SLV PRY VCT CPVDZA LVA BRA NIC PHL MAR EGY PER IRN THA TUR BLR HND MDA AZE UKRDOM KGZ RUS IDN GTM TJK KAZ INDBOL COM PAK NPLBGD ZAF GAB GHA BEN GMB KEN SEN COG LSO CIV CMR ZWE GIN GNQ log gdp per capita 2000 SWZ LUX Figure 1.6. The association between income per capita and life expectancy at birth in * GDP brought about welfare, by the introduction but it isof anewstrong technologies predictor and creation of ofwelfare new firms, measures which replaces existing firms and technologies. This creates a natural social tension, even in a * for example life expectancy is an important part of welfare

17 Understanding modern growth important distinction between the drivers of growth proximate reasons fundamental reasons proximate reasons for UK being richer than Ghana it has more physical capital, machines, human capital, better technology better infrastructure firms in the UK invest more, households save more fundamental reasons what is the reason for these differences? geography institutions culture history mechanics of growth is about the proximate causes

18 The Solow growth model

19 The Solow model first general equilibrium model of growth mechanism people save a fraction of their income future capital produce output save capital... result: a theory of growth Solow model sheds light on is there sustained growth in the lon-run, or does growth run out of steam eventually? what is the relationship between growth and investment/saving? why do countries have different experiences? are countries inherently different? are they just at different points on the same path?

20 Assumptions time is discrete: t = 0, 1, 2,... closed economy, single good (one sector model) used for consumption and as new capital (gross investment) two actors in the economy representative firm: rents capital and labor for production representative household: - receives labor and capital income, and profits from owning the firms - consumes and saves in the form of capital three markets labor market capital market final good market

21 Households do not optimize own all the labor, supply it inelastically (no labor-leisure choice, as in most growth models) own all the capital and rent it out to firms receive income Y (t) = w(t)l s (t) + R(t)K s (t) + Π(t) }{{}}{{}}{{} labor inc capital inc profit consume and save according to the Keynesian consumption function S(t) = sy (t) C(t) = (1 s)y (t) they save s fraction of their income, s is a parameter no preference specified we cannot make welfare statements neoclassical growth model: consumers choose consumption optimally (we ll look at it next week)

22 Technology technology to produce output Y (t) = F (K(t), L(t), A(t)) K(t) capital: equipment and structure, L(t) labor A(t) technology, production shifter it is free: publicly available, non-excludable, non-rival good no intermediate inputs in production value added F ( ) is neoclassical constant returns to scale in K and L positive, diminishing marginal returns to K and L Inada conditions

23 Market structure, endowments and market clearing all markets are perfectly competitive labor market clearing: endowment of labor in economy = labor demanded by the representative firm given the wage rate L d (t) = L s (t) = L(t) capital market clearing: supply of capital = demand of capital given the rental price K d (t) = K s (t) = K(t) goods market clearing (no government): Y (t) = C(t) + I (t) S(t) = I (t) law of motion for capital K(t + 1) = (1 δ)k(t) + I (t) δ: depreciation rate, this fraction depreciates during prod K(0): initial capital endowment given

24 Firm optimization firms face a static problem (why?) optimality requires max F (K, L, A(t)) w(t)l R(t)K {K,L} R(t) = F K (K, L, A(t)) and w(t) = F L (K, L, A(t)) First important consequence of CRS: zero profit Euler theorem: F (K, L, A) = F K (K, L, A)K + F L (K, L, A)L proof: differentiate λf (K, L, A) wrt λ and evaluate at λ = 1

25 Firm optimization continued the firm s profit at the optimal capital and labor is Π(t) = F (K, L, A(t)) w(t)l R(t)K = F (K, L, A(t)) [F L (K, L, A(t))L + F K (K, L, A(t))K] }{{} = F (K, L, A(t)) F (K, L, A(t)) }{{} = 0 so CRS Euler theorem zero profit

26 Equilibrium in the Solow model An equilibrium in the Solow model is a series of prices {w(t), R(t)} t=0 and quantities {C(t), S(t), Y (t), K(t)} t=0 given {A(t), L(t)} t=0 and K(0) such that firms maximize profits given prices factor markets clear C(t) = (1 s)y (t) and S(t) = sy (t) for all t 0 capital evolves according to K(t + 1) = (1 δ)k(t) + I (t) for all t 0 Note: an equilibrium is a whole sequence of objects this is true in general for growth models, as these are dynamic problems

27 Fundamental law of motion Household s income w(t)l(t) + R(t)K(t) + Π(t) = Y (t) = F (K(t), L(t), A(t)) using this and I (t) = S(t): K(t + 1) = I (t) + (1 δ)k(t) = S(t) + (1 δ)k(t) = sy (t) + (1 δ)k(t) = sf (K(t), L(t), A(t)) + (1 δ)k(t) the equilibrium of the Solow model is described by this non-linear difference equation and the path of A(t), L(t) and K(0)

28 The canonical Solow model Extra assumptions: no population growth, no technological progress L(t) = L and A(t) = A we do a useful transformation: everything transformed to per capita terms (= per worker here) capital-labor ratio: k(t) = K(t) L output (income) per capita using the CRS property can be expressed as y(t) = Y (t) L = F (K(t), L, A) L = F (k(t), 1, A) f (k(t)) LF = ( K(t) L, 1, A ) L

29 the fundamental equation of the Solow model: k(t + 1) = sf (K(t), L, A) + (1 δ)k(t) L = sf (k(t)) + (1 δ)k(t) other equilibrium var can also be expressed as a function of k(t) rental rate wage rate R(t) = F K (K(t), L, A) = f (k(t)) w(t) = F L (K(t), L, A) = f (k(t)) f (k(t))k(t) second important consequence of CRS: marginal product of capital and labor only depend on the capital-labor ratio

30 Dynamic evolution of the economy we characterized everything in terms of the only state variable k(t): k(t + 1) = sf (k(t)) + (1 δ)k(t) with k(0) = K(0) L = k 0 given this difference equation implies a sequence {k(t)} t=0 what happens in the economy over time? what does this sequence look like?

31 Introduction to Modern Economic Growth The steady state of the Solow model k(t+1) 45 k* sf(k(t))+(1 δ)k(t) 0 k* k(t) Figure 2.2. Determination of the steady-state capital-labor ratio in the Solow model without population growth and technological change. steady state: a k s.t. if k(t) = k, then k(t + 1) = k Note: another steady state at k = 0, we ignore this which corresponds to the case where capital is an essential factor, meaning that if K (t) =0, then output is equal to zero irrespective of the amount of labor and the level of technology.

32 Does a steady state exist and is it unique? the way we drew the curve, yes it exists if the two curves cross it is unique if they only cross once the Inada conditions and the decreasing marginal returns guarantee that these two hold these were assumed of F (K, L, A) are these inherited by f (k)?

33 Examples of what can go wrong k(t+1) 45 k(t+1) sf(k(t))+(1 δ)k(t) 45 k(t+1) sf(k(t))+(1 δ)k(t) to Modern Economic Introduction Growth to Modern Economic Growth k(t) 0 0 Panel A Panel B k(t) 0 (t+1) k(t+1) sf(k(t))+(1 δ)k(t) 45 k(t+1) 45 sf(k(t))+(1 δ)k(t) 45 k(t+1) k(t+1) Figure 2.5. Examples of nonexistence sf(k(t))+(1 δ)k(t) and nonuniqueness of interior s 45 sf(k(t))+(1 states when Assumptions 1 and 2 are not satisfied simple way, and assume that f (k) =a f (k), where a > 0, so that a is a shift parameter, with greater values correspondin k(t) k(t) 0 productivity Panel A of factors. This Panel type C of Panel productivity B is referred to as Panel Hicks-neutra C Panel B sf(k(t))+(1 δ)k(t) 0 see below, but for now it is just a convenient way of looking at the impact of onexistence Figure anddifferences nonuniqueness 2.5. Examples acrossof countries. ofinterior nonexistence steady Sinceand f (k) nonuniqueness satisfies the regularity of interiorconditions steady impo and 2 arestates not satisfied. when does f Assumptions 1 and 2 are not satisfied. (k). k(t) k(t) 0 k(

34 Introduction to Modern Economic Growth A different representation output δk(t) f(k(t)) f(k*) consumption sf(k*) sf(k(t)) investment 0 k* k(t) Figure 2.4. Investment and consumption in the steady-state equilibrium. (see Theorem A.3 in Appendix Chapter A) there exists k such that (2.17) is satisfied. To see the uniqueness, amount differentiate of actual f (k) /k with investment respect to k, which is exactly gives such that it compensates [f for (k) /k] depreciation, (2.20) = f 0 (k) k f (k) i.e. k k 2 = w for the break-even k investment 2 < 0,

35 Comparative statics In the steady state k = sf (k ) + (1 δ)k }{{} δk = sf (k ) }{{} depreciation investment re-arranging: f (k (s, δ, A)) k (s, δ, A) = δ s k (s,δ,a) δ k (s,δ,a) s k (s,δ,a) A < 0: higher depreciation lower k, y > 0: higher savings rate higher k, y > 0: higher productivity higher k, y (multiplier effect on output)

36 The golden rule saving rate steady state consumption per capita c = c(k (s, δ, A)) = (1 s)f (k (s, δ, A)) s has two opposing effects: higher s higher k higher y higher s lower 1 s, lower fraction of output consumed consumption-maximizing saving rate: s gr arg max[(1 s)f (k (s, δ, A))] = arg max[f (k (s, δ, A)) δk (s, δ s s s gr implicitly defined by f (k (s gr, δ, A)) = δ saving is excessive if the depreciation of capital is higher than the marginal product of capital by saving less, consumption would increase

37 Transitional dynamics equilibrium path is not only the steady state, but the entire path of capital (and all other variables of interest) key questions: will the economy tend to the steady state? how will it behave along the transition path? concepts of stability: local stability: the economy converges to the steady state once it is close to it global stability: the economy converges to the steady state regardless of the starting point

38 Introduction to Modern Economic Growth Global stability k(t+1) 45 k* 0 k(0) k* k (0) k(t) Figure 2.7. Transitional dynamics in the basic Solow model.

39 Growth in the Solow model the dynamics of the model are fully determined by the evolution of capital the growth rate of capital implications k(t + 1) k(t) g k (t) = k(t) = s f (k(t)) k(t) = δ g k (k(t)) sf (k(t) δk(t) k(t) 1. no growth in the long run, as k(t + 1) = k(t) = k 2. growth during the transition, i.e. as long as k(t) < k 3. growth rate is declining over time

40 The Kaldor facts Nicholas Kaldor (1958) characterized modern economic growth by the following facts: 1. The rate of growth of the capital stock per worker is roughly constant over long periods of time 2. The rate of growth of output per worker is roughly constant over long periods of time 3. The capital/output ratio is roughly constant over long periods of time 4. The rate of return to capital is roughly constant over long periods of time 5. The shares of national income received by labor and capital are roughly constant over long periods of time 6. The real wage grows over time

41 The augmented Solow model positive, constant, exogenous population growth and technological progress to have a model that matches the Kaldor facts, we need labor-augmenting technology F (K(t), L(t), A(t)) = F (K(t), A(t)L(t)) model in continuous time: L(t) = L 0 exp(nt) n = L L A A(t) = A 0 exp(gt) g = A most growth models are in continuous time it is easier to solve them this way

42 nothing has changed on the production side, wage and rental rates are given as before, profit is zero people still save s fraction of their income, so S(t) = sy (t) the capital accumulation equation is now K(t) = sf (K(t), A(t)L(t)) δk(t) let k(t) now be capital per efficiency units of labor: k(t) = K(t) A(t)L(t) differentiating with respect to time we get k(t) k(t) = K(t) K(t) g n

43 output per unit of efficiency labor can be expressed as ŷ(t) Y (t) ( ) K(t) A(t)L(t) = F A(t)L(t), 1 f (k(t)) putting the above together k(t) sf (K(t), A(t)L(t)) δk(t) = g n k(t) K(t) sf (K(t), A(t)L(t)) = (δ + g + n) K(t) sf (k(t)) = (δ + g + n) k(t) k(t) = sf (k(t)) (δ + g + n)k(t) exactly the same structure as the baseline model, with effective depreciation δ + g + n

44 Balanced growth and steady state steady state is given by f (k ) k = δ + g + n s existence, uniqueness, stability same as before in this steady state y(t) = Y (t) L(t) = A(t)f (k ) w(t) = A(t)[f (k(t)) f (k(t))k(t)] wages and income per capita grow at rate g Y (t) = L(t)y(t) grows at rate g + n this steady state is also called a balanced growth path

45 Summary of the Solow model canonical model no long-run growth capital accumulation alone does not make an economy grow in the long run conditional convergence: if countries have the same parameters, s, δ, A and the same production function F, they will converge to the same steady state moreover, the countries further from the steady state (poorer countries) will grow faster richer countries should grow slower crucial mechanism for all of these result: the decreasing marginal product of capital extended model, with productivity and population growth long run growth possible if productivity grows: g > 0 with constant growth it is consistent with the Kaldor facts s and g, which are the most interesting, are exogenous simple theory, can be confronted with the data

46 Growth and convergence in the Solow model the Solow model contains two sources of growth 1. transitional growth due to capital accumulation endogenous 2. balanced growth due to technology exogenous it is important to distinguish these two when interpreting cross-country data answer questions such as: will China catch up to the US?

47 Decomposing growth consider the Cobb-Douglas framework income per capita is then Y (t) = K(t) α (A(t)L(t)) 1 α y(t) = A(t)k(t) α growth of per capita income is given by ẏ(t) y(t) = A(t) + α k(t) A(t) k(t) }{{}}{{} TFP convergence

48 Solow and the rate of accumulation Solow model gives a theory of k(t) k(t) production function) (assuming Cobb-Douglas k(t) k(t) = s f (k(t)) (δ+n+g) = sk(t) α 1 (δ+n+g) g k (k) k(t) growth in k depends only on k(t) and not on time Taylor approximation wrt ln k(t) around k g k (k) = g k (k ) + g(k) ln k k=k (ln k ln k ) = 0 + (α 1)(k ) α 1 s(ln k ln k ) = (1 α)(δ + g + n) (ln k ln k) }{{} speed of convergence

49 Speed of convergence the Solow model predicts that close to the steady state the speed of convergence is (1 α)(δ + n + g) convergence is fast if α is low: returns to capital decrease quickly δ + g + n is high: high effective depreciation past investments matter less the speed of convergence is around (1 α)(δ + n + g) = 2 (5% + 2% + 2%) = 6% 3 if we start from half the steady state capital stock: k(t) = 0.5k (t) g k = 6% ln(0.5) = 4.2% the growth rate of the aggregate capital stock should be K(t) K(t) = g k + g + n = 8.2%

50 Convergence of income per capita we want to see how fast China s per capita income should grow problem: we don t know how far k(t) is from k per capita income ln y(t) = ln A(t)+α ln k(t) ln k(t) ln k = ln y(t) ln y (t) α are we better off? do we know ln y(t) ln y (t)? assume US in steady state: ln y(t) ln y (t) = ln y CH (t) y US (t) growth of per capita income is then = ln = 2 ẏ(t) y(t) = g+(1 α)(δ+g+n)(ln y (t) ln y(t)) = 2%+12% = 14% transitional growth should be very quick not clear why China is still poor

51 Summary simple and tractable framework, which allows us to discuss capital accumulation and the implications of technological progress Solow model shows us that if there is no technological progress, there will be no sustained growth growth comes from capital accumulation, this ends if there are diminishing returns to scale generate per capita output growth, but only through exogenous technological progress capital accumulation: determined by the saving rate, the depreciation rate and the rate of population growth; all are exogenous need to understand what drives saving behavior and technological progress