Lecture 2: Intermediate macroeconomics, autumn Lars Calmfors
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1 Lecture 2: Intermediate macroeconomics, autumn 2009 Lars Calmfors
2 1 Topics Production Labour productivity and economic growth The Solow Model Endogenous growth Long-run effects of the current recession Literature: Mankiw and Taylor, Chapters 3, 7 and 8; OECD Economic Outlook, Chapter 4, pp
3 2 GDP per capita, percent of OECD average, PPP-adjusted Position 1970 Index Position 1980 Index 1 Switzerland USA USA Switzerland Luxembourg Canada Sweden 113 (105*) 4 Luxembourg Canada Iceland Denmark France France Norway Australia Sweden 107 (98*) 9 Netherlands Denmark New Zeeland Belgium Great Britain Australia Belgium Netherlands Germany Austria Italy Italy Austria Germany Norway Japan Japan Great Britain Finland Finland Iceland New Zeeland Spain Spain Ireland Greece Greece Ireland Portugal Portugal Mexico Mexico Turkey Turkey 27 * If Mexico and Turkey are excluded.
4 3 GDP per capita, percent of OECD average, PPP-adjusted Position 1990 Index Position 1998 Index 1 Luxembourg Luxembourg USA USA Switzerland Norway Canada Switzerland Japan Denmark Norway Iceland France Canada Iceland Belgium Denmark Japan Sweden 105 (94*) 10 Austria Belgium Netherlands Austria Australia Finland Germany Italy Ireland Australia France Germany Finland Netherlands Italy Great Britain Great Britain New Zeeland Sweden 96 (85*) 20 Spain New Zeeland Ireland Spain Portugal Portugal Greece Greece Mexico Mexico Turkey Turkey 30 * If Mexico and Turkey are excluded.
5 4 GDP per capita, US dollars, PPP-adjusted, percent of OECD average, ranking by country Luxembourg Norway United States Ireland Switzerland Netherlands Canada Australia Austria Sweden Iceland Denmark United Kingdom Belgium Germany Finland Japan Euro area France Spain Italy Greece New Zealand Korea Czech Republic Portugal Hungary Slovak Republic Poland Mexico Turkey 40 Source: OECD
6 5 Y F( K, L) MPL F( K, L + 1) F( K, L) MPL dy df( K, L) dl dl F L MPK F( K + 1, L) F( K, L) MPK dy df( K, L) dk dk F K
7 Figure 3-3: The production function 6
8 Figure 3-4: The marginal product of labour schedule 7
9 8 Profit maximisation General: suppose y f (x, z). The first-order conditions (FOCs) for maximum of y are: dy dx dy dz f x f z 0 0 Profit maximisation π PY RK WL PF( K, L) RK WL dπ PFL W 0 F dl L W P dπ PFK R dk 0 F K R P
10 9 Production function Y AF( K, L) A total factor productivity It holds that: ΔY ΔA ΔK ΔL + α + (1 α) Y A K L α capital income share 1- α labour income share GDP growth total factor productivity growth + contribution from growth of the capital stock + contribution from growth of the labour force Growth accounting The Solow-residual: Δ A ΔY α ΔK (1 α) ΔL A Y K L
11 Figure 3-5: The ratio of labour income to total income in the US and the UK 10
12 Mathematical preliminaries: the natural logarithm 11 Recall that nx is the natural logarithm of x. By definition: x ea a n x Properties: n ( xy) n x+ n y n x n x n y y nx β β nx
13 Rules of differentiation 12 If y f( g) and g g( x) so that then y dy dx f( g( x)) f dg g dx f g g x (1) Moreover, the derivative of the n-function is given by: d( n x) 1 dx x (2) and for polynomials: γ d( x ) dx γ x γ 1
14 Cobb-Douglas production function 13 Y AF( K, L) AK L α 1 α K, L and Aand thus also Y are functions of time (continuous-time formulation). Yt () AtKt () () αlt () 1 α Taking logarithms: ny( t) nat ( ) + nkt ( ) α + nlt () 1 α ny() t nat () + α nkt () + (1 α) nlt () Differentiation w.r.t. time gives: d n Y() t d n A() t + α d n K() t + (1 α) d n L() t dt dt dt dt dy i 1 da i 1 + α dk dt Y dt A dt i 1 + (1 α) dl i 1 K dt L Call dy i i i i Y, da A, dk K och dl L dt dt dt dt i i i i Y A + α K + (1 α) L Y A K L α profit share 1- α wage share The discrete-time equivalent is: Δ Y Δ A + α Δ K + (1 α) Δ L Y A K L Δ Y Y t Y t 1 etc.
15 Profit maximisation with Cobb-Douglas production function 14 π PY RK WL PAKα L1 α RK WL dπ dk PAKα 1L1 α R α 0 dπ dl (1- α) PAKαL α W 0 Re-arranging these equations implies: α R RK RK PAKα 1L1 α PAKαL1 α PY 1 α W WL WL PAKαL α PAKαL1 α PY
16 Growth in labour productivity 15 ΔY α ΔK + (1- α) ΔL + ΔA Y K L A (A) GDP growth contribution from growth of capital stock + contribution from growth of labour + total factor productivity growth Labour productivity: Y/L Δ Y L Y L Δ Y Y Δ L L Subtracting ΔL L from both sides of equation (A) gives: ΔY ΔL α ΔK + (1- α) ΔL + ΔA ΔL Y L K L A L Δ Y Δ L Δ Y L K K Δ L L Δ A A α + Growth in labour productivity contribution from capital deepening + total factor productivity growth Capital deepening: Increase in capital intensity (capital relative to labour) Capital deepening can be decomposed into ICT capital deepening and non-ict capital deepening ICT Information and Communications Technology
17 16
18 17 Annual growth of total factor productivity in OECD countries, Irland ( ) Finland ( ) Grekland ( ) Sverige ( ) USA Australien ( ) Storbritannien ( ) Portugal ( ) Frankrike Japan ( ) Tyskland Kanada Österrike ( ) Nya Zeeland ( ) Belgien ( ) Nederländerna ( ) Danmark ( ) Italien ( ) Spanien ( )
19 18 Explanations of high productivity growth in Sweden Large contributions from both ICT-producing and ICT-using sectors Encompassing deregulations of product and service markets - low level of regulation - early deregulations High educational level (complementarity between ICT technology and high-skilled labour) High R&D expenditures (Research and development) Creative destruction in the 1990s
20 19 Constant returns to scale Y F(K, L) zy zf(k, L) F(zK, zl) 10 % larger input of capital and labour raises output also by 10 %. z 1 L Y L F K L (, 1) Y L y output per capita K L k capital intensity (capital stock per capita) y F(, k 1) f() k Output per capita is a function of capital intensity
21 20 The Cobb-Douglas case Suppose that Y K L α 1 α : Y KαL1 α α α α K α L L L y K L k Including total factor productivity (A) so that Y AK L α 1 α : Y AKαL1 α α α α K α L L L y AK L A Ak
22 21 The Solow model (1) y c + i Goods market equilibrium (2) c (1-s) y Consumption function, s is the savings rate (3) y f(k) Production function (4) d δk Capital depreciation, δ is the rate of depreciation (5) k i δk Change in the capital stock Change in the capital stock Gross investment Depreciation
23 22 The Solow model (cont.) Substituting the consumption function (2) into the goods market equilibrium condition (1) gives: y (1-s)y + i i sy Investment Saving Substitution of the production function into the investmentsavings equality gives: i sf(k) k i δk sf(k) δk In a steady state, the capital stock is unchanged from period to period, i.e. k 0 and thus: sf(k) δk
24 23 Convergence of GDP per capita Countries with different initial GDP per capita will converge (if they have the same production function, the same savings rate and the same depreciation rate). The catch-up factor Strong empirical support for the hypothesis that GDP growth is higher the lower is initial GDP per capita
25 Figure 7-2: Output, consumption and investment 24
26 Figure 7-3: Depreciation 25
27 Figure 7-4: Investment, depreciation and the steady state 26
28 Figure 7-5: An increase in the saving rate 27
29 Figure 7-6: International evidence on investment rates and income per person 28
30 29 Golden rule of capital accumulation Which savings rate gives the highest per capita consumption in the steady state? y c + i c y i In a steady state, gross investment equals depreciation: i δk Hence: c f(k) - δk Consumption is maximised when the marginal product of capital equals the rate of depreciation, i.e. MPC δ Mathematical derivation The first-order condition for maximisation of the consumption function: c/ k f δ 0 f k δ k
31 Figure 7-7: Steady-state consumption 30
32 Figure 7-8: The saving rate and the golden rule 31
33 Figure 7-9: Reducing saving when starting with more capital than in the golden rule steady state 32
34 Figure 7-10: Increasing saving when starting with less capital than in the golden rule steady state 33
35 34 A steady state with population growth n ΔL L population growth Δ k i δ k nk Change in capital intensity (k K/L) Gross investment Depreciation Reduction in capital intensity due to population growth In a steady state: Δ k i δ k nk 0, i.e. i ( δ + n) k 0
36 35 Derivation of the capital growth equation K capital stock, I gross investment, L population k K/ L capital stock per worker (capital intensity) i I/ L gross investment per worker Δ K I δ K ΔK K I K δ Use that: Δk ΔK ΔL and k K L Δ L L n Δk I δ k K n Hence: Δk I i L δ k L K n Δk i δ k k Multiplying by k n gives: Δk i δk nk i ( δ + n) k
37 Figure 7-11: Population growth in the Solow model 36
38 Figure 7-12: The impact of population growth 37
39 Figure 7-13: International evidence on population growth and income per person 38
40 39 A steady state with population growth Y F( K, L) ΔY α ΔK + (1 α) ΔL Y K L In a steady state, k K/L is constant. Because Δk ΔK ΔL k K L 0, We have ΔK K ΔL L n är ΔY α ΔK + (1 α) ΔL αn + (1 α) n n Y K L GDP growth Population growth
41 40 Golden rule with population growth c y i f(k) (δ + n)k Consumption per capita is maximised if MPC δ + n, i.e. if the marginal product of capital equals the sume of the depreciation rate and population growth Alternative formulation: The net marginal product of capital after depreciation (MPK δ) should equal population growth (n) Mathematical derivation Differentiation of c-function w.r.t k gives: c/ k f ( δ + n) 0 f k δ + n k
42 41 Labour-augmenting technical progress Y F( K, L i E) E labour efficiency L i E efficiency units of labour y Y F( K, 1) F( k, 1) f ( k) LE LE k K LE Steady state L grows by n % per year E grows by g % per year Δk sf(k) (δ + n + g)k 0 Gross investment Depreciation + Reduction in capital intensity because of population growth + Reduction in capital intensity because of technological progress
43 Figure 8-1: Technological progress and the Solow growth model 42
44 43 Growth and labour-augmenting technological progress Y Kα ( LE) 1 α ΔY α ΔK + (1 α)( ΔL + ΔE ) Y K L E In a steady state K/LE is constant ( Δ L/ L + Δ E/ E) n + g Δ K/ K n + g. ΔY Y α( n + g) + (1 α)( n + g) n + g GDP growth population growth+ technological progress Δy ΔY ΔL n + g n g y Y L Growth in GDP per capita rate of technological progress
45 44 Table 8-1: Steady-State growth rates in the Solow model with technological progress Variable Capital per effective worker Output per effective worker Output per worker Total output Symbol k K/ (L E ) y Y/ (L E ) (Y/ L ) y E Y y E L Steady-state growth rate 0 0 g n + g
46 45 Golden rule with technological progress c f(k) - (δ + n + g)k Consumption per efficiency unit is maximised if MPK δ + n + g The marginal product of capital should equal the sum of depreciation, population growth and technological progress Alternative formulation: The net marginal product (MPK - δ) should equal GDP growth (n + g). Mathematical derivation Differentiation w.r.t. k: c/ k f ( δ + n+ g) 0 k f k δ + n +g Real world capital stocks are smaller than according to the golden rule. The current generation attaches a larger weight to its own welfare than according to the golden rule.
47 46 Endogenous or exogenous growth In the Solow model growth is exogenously determined by population growth and technological progress Recent research has focused on the role of human capital A higher savings rate or investment in human capital do not change the rate of growth in the steady state The explanation is decreasing marginal return of capital (MPK is decreasing in K ) The AK-model Y AK ΔK sy - δk Assume A to be fixed! ΔY/Y ΔK/K ΔK/K sak/k δk/k sa - δ ΔY/Y sa - δ A higher savings rate s implies permanently higher growth Explanation: constant returns to scale for capital Complementarity between human and real capital
48 47 A two-sector growth model Business sector Education sector Y F[K, (1-u)EL] ΔE g(u)e ΔK sy - δk Production function in business sector Production function in education sector Capital accumulation u share of population in education ΔE/E g(u) A higher share of population, u, in education raises the growth rate permanently (cf AK-model here human capital) A higher savings rate, s, raises growth only temporarily as in the Solow model
49 48 Human capital in growth models 1. Broad-based accumulation of knowledge in the system of education 2. Generation of ideas and innovations in research-intensive R&D sector 3. Learning by doing at the work place Policy conclusions 1. Basic education incentives for efficiency in the education system incentives to choose and complete education 2. Put resources in top-quality R&D 3. Life-long learning in working life Technological externalities / knowledge spillovers
50 49 Role of institutions Quality of institutions determine the allocation of scarce resources Legal systems secure property rights - helping hand from government (Europe) - grabbing hand from government Acemoglu / Johnson /Robinson - European settlers in colonies preferred moderate climates (US, Canada, NZ) - European-style institutions - Earlier institutions strongly correlated with today s institutions
51 50 Will the current recession have long-run growth effects? Traditional view: a recession only represents a temporary reduction in resource utilisation Modern view a recession can have permanent effects on potential output growth Effects on potential growth Slower growth of capital input - lower investment because of lower output and credit crunch in the short run and because of higher risk premia (higher interest rates and thus higher capital costs) in the medium run - capital becomes obstacle Higher structural unemployment Slower growth in total factor productivity - lower R&D expenditure - but also closing down of least efficient firms
52 51 Temporary effects of a recession Output trend Time Permanent effects of a recession Output Time
53 52
54 53 Contributions to changes in potential output growth, Percentage point pa differences in the potential growth rate from Potential Employmnet from Total Factor Productivity from Capital Total from Potential Employmnet from Total Factor Productivity from Capital Total Cumulative Contribution Ireland Spain Sweden US Simple OECD average Weighted OECD average Source: OECD Economic Outlook 85
55 54
Lecture 2: Intermediate macroeconomics, autumn Lars Calmfors
Lecture 2: Intermediate macroeconomics, autumn 2008 Lars Calmfors 1 GDP per capita, percent of OECD average, PPP-adjusted Position 1970 Index Position 1980 Index 1 Switzerland 154 1 USA 140 2 USA 147 2
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