Lecture 2: Intermediate macroeconomics, autumn 2014

Transcription

1 Lecture 2: Intermediate macroeconomics, autumn 2014 Lars Calmfors Literature: Mankiw, chapters 3, 8 and 9.

2 1 Topics Production Labour productivity and economic growth The Solow model (neoclassical growth model) Endogenous growth Important determinants of growth Solow and Piketty

3 2 Basics about production functions 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

4 Figure 3-3: The production function 3

5 Figure 3-4: The marginal product of labour schedule 4

6 5 Profit maximisation General: suppose y = f (x,z). The first-order conditions (FOCs) for maximum of y are: = = 0 = = 0 Profit maximization =, = 0 MPL = = 0 MPK =

7 6 Production function YY = AAAA(KK, LL) YY = AAKK αα LL 11 αα AA = total factor productivity Cobb-Douglas production function It holds that: YY YY AA AA + αα KK KK + (11 αα) LL LL αα = capital income share 11 αα = 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: AA AA YY - YY αα KK LL - (11 αα) KK LL

8 Figure 3-5: The ratio of labour income to total income 7

9 8

10 9

11 Rules of differentiation 10 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) and for polynomials: (2)

12 Table 9-3:Accounting for economic growth in the United States 11

13 12 Profit maximisation with Cobb-Douglas production function the labour share

14 Table 3-1: Growth in labour productivity and real wages: The U.S. Experience 13

15 Table 8-1: International differences in the standard of living 14

16 15

17 16 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 Fk (, 1) f ( k) Output per capita is a function of capital intensity

18 17 The Cobb-Douglas case Suppose that Y K L 1 : Y KL1 K L L L y K L k Including total factor productivity (A) so that Y AK L 1 : Y AKL1 K L L L y AK L A Ak

19 18 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 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

20 Figure 8-1: The production function 19

21 Figure 8-2: Output, consumption, and investment 20

22 Figure 8-3: Depreciation 21

23 Figure 8-4: Investment, depreciation, and the steady state 22

24 Convergence of GDP per capita 23 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) to the same GDP per capita The catch-up factor Strong empirical support for the hypothesis that GDP growth is higher the lower is initial GDP per capita - when controlling for other factors - conditional convergence - convergence rate: 2 % per year

25 Figure 8-5: An increase in the saving rate 24

26 Figure 8-6: International evidence on investment rates and income per person 25

27 The golden rule level of capital 26 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 = k Hence: c = f(k) - k Consumption is maximised when the marginal product of capital equals the rate of depreciation, i.e. MPK = Mathematical derivation The first-order condition for maximisation of the consumption function: c/ k f 0 f k = k

28 Figure 8-7: Steady-state consumption 27

29 Figure 8-8: The saving rate and the golden rule 28

30 Figure 8-9: Reducing saving when starting with more capital than in the golden rule steady state 29

31 Figure 8-10: Increasing saving when starting with less capital than in the golden rule steady state 30

32 31 A steady state with population growth n L L population growth k i k nk Change in capital intensity (k = K/L) = Gross investment Depreciation Reduction in capital intensity due to population growth In a steady state: k = i δk nk = 0, i.e. i = (δ+n)k

33 32

34 Figure 8-11: Population growth in the Solow model 33

35 Figure 8-12: The impact of population growth 34

36 Figure 8-13: International evidence on population growth and income per person 35

37 36 A steady state with population growth Y F( K, L) Y K (1 ) L Y K L In a steady state, k = K/L is constant. Because k K L k K L 0, We have K K L L n är Y K (1 ) L n (1 ) n n Y K L GDP growth = Population growth

38 37 Golden rule with population growth c = y i = f(k) ( + n)k Consumption per capita is maximised if MPK = + 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 ) should equal population growth (n) Mathematical derivation Differentiation of c-function w.r.t k gives: c/ k f ( n) 0 f k = + n k

39 38 Alternative perspectives on population growth 1. Malthus ( ) - population will grow up to the point that there is just subsistence - man will always remain in poverty - futile to fight poverty 2. Kremer - population growth is a key driver of technological growth - faster growth in a more populated world - the most successful parts of the world around 1500 was the old world (followed by Aztec and Mayan civilisations in the Americas; hunter-gatherers of Australia)

40 39 Labour-augmenting technical progress Steady state L grows by n % per year E grows by g % per year k = 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

41 Figure 9-1: Technological progress and the Solow growth model 40

42 41 Growth and labour-augmenting technological progress Y K ( LE) 1 Y K (1 )( L E ) Y K L E In a steady state K/LE is constant ( L/ L E/ E) n g K/ K n g. Y Y ( n g) (1 )( n g) n g GDP growth = population growth+ technological progress y Y L n g n g y Y L Growth in GDP per capita = rate of technological progress

43 Table 9-1: Steady-state growth rates in the Solow model with technological progress 42

44 43 Golden rule with technological progress c = f(k) - ( + n + g)k Consumption per efficiency unit is maximised if MPK = + 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 - ) should equal GDP growth (n + g). Mathematical derivation Differentiation w.r.t. k: c/ k f ( ng) 0 k Golden rule with technological progress, cont. f k = + 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.

45 44

46 Endogenous or exogenous growth 45 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

47 46 A two-sector growth model Business sector Education sector Y = FK, (1-u)EL E = g(u)e K = sy - K Production function in business sector Production function in education sector Capital accumulation u = share of population in education E/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

48 47 More about growth Human capital is crucial for growth - strong empirical regularity - educational level - R & D expenditure Free trade appears to promote growth - comparison between open and closed economies - effects after trade liberalisations - other factors? - impact of geography (instrumental approach) Industrial policy - good if technological externalities and if the government can identify them - but can governments do this? Institutions - legal protection for shareholders and creditors leads to better functioning capital markets - quality of government: protection of property rights or grabbing hand

49 48 The importance of geography Direct negative effect of tropical conditions on productivity Impact of geography on institutions (Acemoglu, Johnson and Robinson) During colonial times Europeans settled in non-tropical areas - legal systems protecting individual Instead extractive institutions in tropical areas Strong path dependence of institutions

50 49 Thomas Piketty and the Solow model Recent book by French economist Thomas Piketty: Capital in the Twenty-First Century. Claim that capital is becoming more and more important, so that k/y is increasing when growth falls. Take Solow model with population growth only (constant saving rate s) out of gross income y = f(k). kk = ssss(kk) (δδ+n)k kk = 00 ssss(kk) (δδ + nn)kk = 00 kk = ssss(kk) δδ + nn kk ff(kk) = kk yy = Piketty model ss δδ + nn Constant savings rate ss out of net income yyy = f(k) δk kk = ss [ff(kk) δδk] nnnn = ss yy nnnn kk = 00 ssssss nnnn = 00 kk = ssssss nn kk yyy = sss nn (A)

51 Thomas Piketty and the Solow model, cont. 50 Piketty model kk yyy = sss nn nn 00 kk yyy Exploding capital stock Solow model kk yy = ss nn+δδ nn = 00 kk yy = ss δδ ss = aaaaaa δδ = kk yy = 44 Piketty model is unrealistic (Krusell/Smith critique) Look at: cc = (11 ss )yy yy yy (B)

52 (A) gives nk = ssssss 51 nk = sss [y δk] nk = sssss sssssss According to (A): yyy = nnnn sss y = nn+sssss sss kk (C) (D) (C) and (D) together give yyy yy = nnnn sss nn + ssδδ ss = nn nn + sssss (E) (E) and (B) together give: cc (11 sss)nn = yy nn + sssss Zero population growth, i.e. n = 0 thus gives cc yy = 00 All output would be devoted to saving/investment, i.e. there would be zero consumption if there is zero growth. There is always so much saving that the capital stock is increasing. This requires

53 more and more saving out of gross income when more saving is needed to make up for higher and higher depreciation. This is not reasonable. 52