Intermediate Macroeconomics, EC2201 L2: Economic growth II Anna Seim Department of Economics, Stockholm University Spring 2017 1 / 64
Contents and literature The Solow model. Human capital. The Romer model. Institutions and growth. Literature: Jones (2014), Ch. 5-6. Klein (2016a). Acemoglu et al. (2005). Mankiw et al. (1992). Seim and Parente (2013). 2 / 64
The Solow model Draws on the production model from lecture 1. Capital accumulation endogenous. Following Jones (2014), we set up the model in discrete time. 1 1 See Klein (2016a) for a treatment of the Solow model in continuous time. 3 / 64
Notation Y t : output at time t. K t : the capital stock at time t. L t : labour input at time t. I t : investment at time t. A: total factor productivity (TFP). L: the labour force. K 0 : initial stock of capital. δ: the capital depreciation rate. s: the investment (saving) rate. n: the population growth rate. g: the labour-productivity growth rate. 4 / 64
Production Production is given by: Closed-economy resource constraint: Y t = F (K t,l t ). (1) C t + I t = Y t. (2) 5 / 64
Capital accumulation Capital in period t + 1 is determined by: Equation (3) can be re-arranged to read: where K t+1 K t+1 K t. K t+1 = K t + I t δk t. (3) K t+1 = I t δk }{{} t, (4) net investment In each period t > 0, K t results from previous investment and depreciation. To get the model started, an initial value for K 0 is needed. 6 / 64
Extracted from: Jones (2014). 7 / 64
Consumption and investment Assume: This implies: I t = sy t. (5) C t = (1 s)y t. (6) 8 / 64
Solving the model Assume that there is no population growth so that L t = L. Equations (4), (5) and (1) imply: K t+1 = sf (K t,l) δk t, (7) 9 / 64
Steady state Steady state when K t+1 = K t, i.e. K t+1 = 0. Equation (7) implies that the steady-state level of capital K must satisfy: sf (K,L) = δk, (8) 10 / 64
Extracted from: Jones (2014). 11 / 64
Extracted from: Jones (2014). 12 / 64
Steady state under Cobb-Douglas Assume that the production function is Cobb-Douglas: Y t = AKt α L (1 α). (9) Equation (7) suggests that K satisfies: sa(k ) α L (1 α) = δk. 13 / 64
Steady state under Cobb-Douglas Solving for K : K = ( ) 1 sa (1 α) L. (10) δ Plugging (10) into (9) yields steady-state output: ( ) α Y = A 1 (1 α) s (1 α) L (11) δ 14 / 64
Steady-state capital and output per worker Dividing (10) by L we obtain steady-state capital per worker: k K L = ( ) 1 sa (1 α), (12) δ Dividing (11) by L we obtain steady-state output per worker: y Y L = A 1 (1 α) ( ) α s (1 α), (13) δ where we note that (1 α) 1 > 1. Recall that in the simple production model of lecture 1, y = Ak α, suggesting that TFP plays a larger role in the Solow model. 15 / 64
Taking the model to the data Recall that equation (8) predicts sy = δk, so that: K Y = s δ. (14) Idea: plot capital-income ratios against investment rates. 16 / 64
Extracted from: Jones (2014). 17 / 64
Understanding the steady state Why does the economy reach a steady state? Diminishing returns. In the steady state, investment is offset by capital depreciation. 18 / 64
Extracted from: Jones (2014). 19 / 64
Extracted from: Jones (2014). 20 / 64
Extracted from: Jones (2014). 21 / 64
Extracted from: Jones (2014). 22 / 64
Transition dynamics No long-run growth in the Solow model: the economy settles down at K and Y. But countries can grow temporarily during transitions to steady states. If K t < K the growth rate will be positive. If K t > K the growth rate will be negative. The further from the steady state, the larger the (absolute value of) growth rates. The growth that we observe may be due to shocks that cause deviations from steady states, or to shifts in steady states. 23 / 64
Extracted from: Jones (2014). 24 / 64
Extracted from: Jones (2014). 25 / 64
Extracted from: Jones (2014). 26 / 64
Taking stock of the Solow model Predicts long-run GDP per capita. Helps us understand growth rates out of steady state. Shortcomings: Suggests TFP differences are key to explaining cross-country differences in income, but these are exogenous. Does not explain why investment rates vary across countries. Not a theory of long-run growth. Diminishing returns imply that capital accumulation cannot generate sustained growth. 27 / 64
Extensions 1. Population growth. 2. Labour-augmenting technological progress. Both extensions straightforward because of constant returns to scale. Need to modify the expression for capital accumulation. 28 / 64
Population growth Let n (L t+1 L t )/L t be the population growth rate, so that Dividing (3) by L t implies: k t+1 L t+1 L t }{{} 1+n L t+1 L t = 1 + n. = k t + s Ak α t }{{} y t δk t. We obtain the following approximation: k t+1 sak α t (δ + n)k t. (15) 29 / 64
The steady state with population growth Steady state when k t+1 = k t. Equation (15) implies that k satisfies: sa(k ) α = (δ + n)(k ). (16) Solving for k : Using y = Ak α, we obtain y : ( ) 1 sa k (1 α) =. (17) δ + n ( ) α y = A 1 (1 α) s (1 α). (18) δ + n 30 / 64
Labour-augmenting technological progress Let E denote labour efficiency, so that L t E t is efficiency units of labour at time t. Under Cobb-Douglas, the production function is now given by Y t = K α t (L t E t ) 1 α. (19) We let lower-case letters denote variables in terms of efficiency units of labour, i.e. k K/LE and y t Y t = K t α (L t E t ) 1 α = kt α. (20) L t E t L t E t 31 / 64
Labour-augmenting technological progress Let g (E t+1 E t )/E t be the labour-efficiency growth rate, so that E t+1 E t = 1 + g. Dividing (3) by L t E t implies: k t+1 L t+1 L t }{{} 1+n E t+1 E t }{{} 1+g We obtain the following approximation: = k t + sk α t δk t. k t+1 sk α t (δ + n + g)k t. (21) 32 / 64
The steady state with technological progress Steady state when k t+1 = k t. Equation (21) implies that k satisfies: s(k ) α = (δ + n + g)(k ). Solving for k : ( k = Using y = k α, we obtain y : ( y = s δ + n + g s δ + n + g ) 1 (1 α). (22) ) α (1 α). (23) 33 / 64
Steady-state growth rates in the Solow model with poulation growth and technological progress Variable Notation Steady-state growth rate Capital per effective worker k = K/(L E) 0 Output per effective worker y = Y /(L E) 0 Output per worker Y /L = y E g Total output Y = y L E n + g 34 / 64
Mankiw, Romer and Weil (1992) Taking logs of (23) we obtain: ln(y ) = α (1 α) ln(s) α ln(δ + n + g) (1 α) The model thus predicts that the elasticity of income per capita with respect to the saving rate is α/(1 α). If the capital share, α, is 1/3 this elasticity should be 1/2. Mankiw, Romer and Weil (1992) find that the elasticity is around 1.4, implying α 0.59. 35 / 64
Maniw, Romer and Weil (1992) add human capital to (19): Y t = K α t H β t (L t E t ) 1 α β, (24) where H t is the stock of human capital and α + β < 1 implies decreasing returns to all capital. As before, we let lower-case letters denote quantities per effective unit of labour and obtain y t K t α H β t (L t E t ) 1 α β ( ) α ( ) β Kt Ht = = kt α h β t. (25) L t E t L t E t L t E t 36 / 64
Let s k be the fraction of income invested in physical capital and s h the fraction invested in human capital. Human capital also depreciates at rate δ, and evolves according to: H t+1 = H t + s h Y t δh t. Following the same steps as before and some algebra implies that the two types of capital evolve according to k t+1 s k k α t h β t (n + g + δ)k t. (26) h t+1 s h k α t h β t (n + g + δ)h t. (27) 37 / 64
Steady state Imposing k t+1 = h t+1 = 0 on (26) and (27) implies: Combining (28) and (29) implies: s k k α h β = (n + g + δ)k. (28) s h k α h β = (n + g + δ)h. (29) k h = s k s h (30) 38 / 64
Plugging in h from (30) in (28) implies: Collecting terms: ( s k k α k s ) β h = (n + g + δ)k. s k s (1 β) k s β h = (n + g + δ)k(1 α β). If and only if: k = ( (1 β) s k s β ) 1 (1 α β) h. (31) n + g + δ 39 / 64
Plugging in k from (31) in (30) and some algebra implies: ( s h α = k s (1 α) ) 1 (1 α β) h. (32) n + g + δ Using (25), output converges to: y = ( (1 β) s k s β ) α h n + g + δ ( (1 α β) sk αs(1 α) h n + g + δ ) β (1 α β). (33) 40 / 64
Which can be simplified to read: ( y = s α k sβ h (n + g + δ) (α+β) ) 1 (1 α β). (34) Taking logs of (34), we obtain: ln(y α ) = (1 α β) ln(s β k) + (1 α β) ln(s h) (α + β) ln(n + g + δ). (1 α β) This suggests that the elasticities of income per capita with respect to s k and s h are, respectively, α (1 α β) and β (1 α β). 41 / 64
Why is this interesting? Since α 1 α β > α 1 α, the effect of investment in physical capital on output per capita is greater when human capital is taken into account. Indirect effect on human capital amplifies the effect of investment in physical capital. 42 / 64
Mankiw, Romer and Weil (1992): conclusions Mankiw, Romer and Weil take the model with human capital to the data and report highly significant elasticities with respect to investment that translate to α and β around 1/3. Moreover, the model with human capital can explain almost 80 percent of the cross-country variation in income per capita in feasible samples. 43 / 64
Towards endogenous growth The Solow model offers valuable insights into cross-country differences in income per capita but fails to explain sustained growth. Next: endogenous growth The AK model. The Romer model. 44 / 64
The AK model What happens when we relax the assumption of diminishing returns to capital? Y t = AK t (35) Capital accumulation: K t+1 = sak t + (1 δ)k t. (36) Equations (35) and (36) imply that the growth factor of GDP is: Y t+1 Y t = K t+1 K t = sa + 1 δ. (37) 45 / 64
Since the growth rate of GDP is: γ Y t+1 Y t Y t = Y t+1 Y t 1, γ = sa δ. (38) An assumption of constant returns to scale in production thus generates endogenous growth. 46 / 64
The Romer model Distinguish between objects (rivalrous) and ideas (non-rivalrous). Workers can either produce consumption goods or new ideas. A single idea can be used by many people simultaneously. Non-rivalry generates increasing returns. Omit capital for simplicity. 47 / 64
Notation Y t : output of the consumption good. A t : the existing stock of knowledge. L: the (constant) population. L Y : number of workers employed in goods production. L A : number of researchers (producing ideas). λ: fraction of labour force employed in research. ζ : productivity parameter. A 0 : the initial stock of knowledge. 48 / 64
Production of the consumption good: Y t = A t L Yt. (39) Production of new ideas: A t+1 = ζ A t L At. (40) Resource constraint for labour: L Yt + L At = L. (41) Assume that labour is allocated according to: L At = λl, (42) L Yt = (1 λ)l. (43) 49 / 64
Solving the model Dividing (39) by L gives output per capita: Re-arranging (40) and using (42): y t Y t L = A t(1 λ). (44) A t+1 A t = ζ L At = ζ λl g. (45) 50 / 64
Equation (45) implies: A t+1 = (1 + g)a t. Applying the constant growth rule from lecture 1, we obtain: A t = (1 + g) t A 0. (46) Finally, combining (44) and (46): y t = (1 λ)(1 + g) t A 0. (47) where g ζ λl. 51 / 64
Balanced growth In the Romer model, the growth rate of GDP per capita is constant and equal to g. This economy is on a balanced growth path, where all endogenous variables grow at rate g. Since g ζ λl, the growth rate is increasing in the productivity parameter, ζ, the share of researchers in the economy, λ, and the labour force, L. 52 / 64
Interpreting the Romer model Able to generate sustained growth in per capita GDP. Nonrivalry of ideas means that y t depends on the total stock of ideas according to (47). No diminishing returns to the existing stock of knowledge, cf. equation (40). 53 / 64
Extracted from: Jones (2014). 54 / 64
Extracted from: Jones (2014). 55 / 64
Extracted from: Jones (2014). 56 / 64
The role of institutions The factors we have listed (innovation, economies of scale, education, capital accumulation, etc.) are not causes of growth; they are growth. -North, D.C. and Thomas, R.P. (1973), The Rise of the Western World: A New Economic History, Cambridge University Press, Cambridge, UK. 57 / 64
Proximate versus fundamental causes of growth If growth is about factor accumulation and innovation, why do not poor countries invest and innovate? Why do countries provide different incentives to firms and households? Proximate causes of growth: factor accumulation and investment. Fundamental causes of growth: institutions (?). 58 / 64
Institutions and growth (Acemoglu et al., 2005) Institutions Humanly devised set of rules. Place constraints on individual behaviour. Economic institutions Property rights. The presence and functioning of markets. Contractual opportunities available to firms and households. Entry barriers. Political institutions Polity and electoral laws. Checks and balances on political leaders. 59 / 64
Democracy and income Is democracy conducive to high income? All the old democracies of the world are rich. Growth performance of autocracies highly heterogeneous. Dictatorships excel or fall behind. Many of the East Asian growth miracles started out autocratic. Complex relationship between polity and income. 60 / 64
Extracted from: Seim and Parente (2013). 61 / 64
Extracted from: Seim and Parente (2013). 62 / 64
Seim and Parente (2013) 3 types of autocrats: good (laissez-faire), landed elite or bad (kleptocrat). Type randomly drawn under autocracy. Economy goes from Malthusian production (involving land) to using a Solow technology (industrialising). Vested interest groups (landowners) seek to delay industrialisation in a democracy. Democracy a middle ground in terms of income: autocracies may do better or worse. Autocracies democratise as they grow sufficiently rich. 63 / 64
What we did The Solow model. Human capital. The Romer model. Institutions and growth. Literature: Jones (2014), Ch. 5-6. Klein (2016a). Acemoglu et al. (2005). Mankiw et al. (1992). Seim and Parente (2013). 64 / 64