Intermediate Macroeconomics

Transcription

1 Intermediate Macroeconomics Lecture 2 - The Solow Growth Model Zsófia L. Bárány Sciences Po 2011 September 14

2 Reminder from last week The key equation of the Solow model: k(t) = sf (k(t)) }{{} (δ + n)k(t) }{{} actual investment break-even investment i (δ + n)k sf (k) k k

3 In the steady state From any starting point the economy converges to k. In the steady state: variable growth rate k(t) = k 0 y(t) = f (k ) 0 K(t) = L(t)k = k L 0 e nt n Y (t) = L(t)f (k ) = f (k )L 0 e nt n No growth in output per worker in the long run. Only transitional growth, this is driven by growth in k, and is declining over time. Does the Solow model imply that all economies should be the same? NO.

4 Economies can still differ There are several reasons why economies could be different: Reason 1 They might not be at the steady state, but on the transition towards their steady state. Economies can be at a different stage of the transition. they might be at different distances from the same steady state, they might be converging to different steady states.

5 Economies can still differ II. Reason 2 The steady states can be different. The steady state capital, k, solves the following equation: 0 = sf (k ) (n + δ)k if the saving rates differ, s 1 s 2, if the depreciation rates differ, δ 1 δ 2, if the growth rate of the populations differ, n 1 n 2, if the production functions differ, f 1 f 2, then the steady state will be different. Which of these do you think policy-makers can affect?

6 Economies can still differ II. Reason 2 The steady states can be different. The steady state capital, k, solves the following equation: 0 = sf (k ) (n + δ)k if the saving rates differ, s 1 s 2, if the depreciation rates differ, δ 1 δ 2, if the growth rate of the populations differ, n 1 n 2, if the production functions differ, f 1 f 2, then the steady state will be different. Which of these do you think policy-makers can affect? directly the savings rate. Most

7 The role of the savings rate Consider an increase in the savings rate: from s A, to s B, where s A < s B Intuitively what does a higher savings rate imply?

8 The role of the savings rate Consider an increase in the savings rate: from s A, to s B, where s A < s B Intuitively what does a higher savings rate imply? More savings

9 The role of the savings rate Consider an increase in the savings rate: from s A, to s B, where s A < s B Intuitively what does a higher savings rate imply? More savings more investment

10 The role of the savings rate Consider an increase in the savings rate: from s A, to s B, where s A < s B Intuitively what does a higher savings rate imply? More savings more investment more capital.

11 The role of the savings rate II. i (δ + n)k k

12 The role of the savings rate II. i (δ + n)k s A f (k) k

13 The role of the savings rate II. i (δ + n)k s A f (k) k A k

14 The role of the savings rate II. i (δ + n)k s B f (k) s A f (k) k A k

15 The role of the savings rate II. i (δ + n)k s B f (k) s A f (k) k A k B k

16 The role of the savings rate II. i (δ + n)k s B f (k) s A f (k) k A k B k The steady state capital is higher, when the savings rate is higher. s A < s B k A < k B

17 A higher savings rate leads to A higher steady state per capita capital: s A < s B ka < k B A higher steady state per capita income: s A < s B f (ka ) < f (k B ) But how does the economy get to the new steady state?

18 Transition to the new steady state At time t 0, when the savings rate increased the economy was at its previous steady state, with ka capital per capita. What happens when the savings rate increases? k(t 0 ) = s B f (k A ) (n + δ)k A s A f (k A ) (n + δ)k A = 0

19 Transition to the new steady state At time t 0, when the savings rate increased the economy was at its previous steady state, with ka capital per capita. What happens when the savings rate increases? k(t 0 ) = s B f (k A ) (n + δ)k A > s Af (k A ) (n + δ)k A = 0

20 Transition to the new steady state At time t 0, when the savings rate increased the economy was at its previous steady state, with ka capital per capita. What happens when the savings rate increases? k(t 0 ) = s B f (k A ) (n + δ)k A > s Af (k A ) (n + δ)k A = 0 Therefore the per capita capital starts to increase, and it increases until the new steady state, kb is reached.

21 Transition to the new steady state II. s s B s A k t 0 t k k B k A t 0 t 0 t t

22 Transition to the new steady state III. y y B y A ẏ t 0 t c c A t 0 t t 0 t

23 The effects of a permanent increase in the savings rate temporary increase in the growth rate of per capita capital temporary increase in the growth rate of per capita output permanent increase in the level of per capita capital permanent increase in the level of per capita output A change in the savings rate has a long-run level effect and, but does not have a long-run growth effect. The economy s BGP changes, which implies a change in the level of output per worker at any point in time: y A (t) < y B (t) for all t > t 0

24 The impact on consumption What happens to per capita consumption?

25 The impact on consumption What happens to per capita consumption? on impact, income is unchanged, and the savings rate increases consumption drops

26 The impact on consumption What happens to per capita consumption? on impact, income is unchanged, and the savings rate increases consumption drops then, income increases, and the savings rate is constant consumption increases

27 The impact on consumption What happens to per capita consumption? on impact, income is unchanged, and the savings rate increases consumption drops then, income increases, and the savings rate is constant consumption increases The overall, steady state effect, depends on which of the above forces is stronger. cb = } (1 {{ s B) } decreases f (k B ) }{{} increases c A

28 The impact on consumption II. c = (1 s)f (k ) = f (k ) sf (k ) i f (k) sf (k) k

29 The impact on consumption II. c = (1 s)f (k ) = f (k ) sf (k )= f (k ) (n + δ)k i f (k) (δ + n)k sf (k) k k

30 The impact on consumption II. c = (1 s)f (k ) = f (k ) sf (k )= f (k ) (n + δ)k i f (k) (δ + n)k c k k Need to find the biggest distance between f (k) and (δ + n)k.

31 Consumption maximising savings rate I. On the BGP, since actual investment equals the break-even investment, c = f (k ) (n + δ)k. We are looking for the savings rate, s, that maximises this difference. But how does s affect this equation? Remember that the steady state per capita capital is determined by the following equation: sf (k ) = (n + δ)k This implies that we can write the steady state per capita capital as a function: k (s, n, δ).

32 Consumption maximising savings rate II. Take the derivative of with respect to s: c s = f (k ) k c = f (k ) (n + δ)k k (s,n,δ) s (n + δ) k (s,n,δ) s = ( f (k ) (n + δ) ) k (s, n, δ) = 0 }{{}} s {{} 0 >0 We know that an increase in s increases k. Whether consumption rises or not in the long-run depends on whether the marginal product of capital, f (k ) is more or less than (n + δ).

33 Thus, for every n and δ there is an s such that consumption is maximised, and this s gr is defined by: f (k (s gr, n, δ)) = (n + δ) s gr is the golden-rule savings rate, and k gr is the golden-rule level of per capita capital, the one that maximises the level of consumption. if s A < s B < s gr, then the increase in the savings rate increases long-run consumption if s gr < s A < s B, then the increase in the savings rate decreases long-run consumption if s A < s gr < s B, then the effect is not clear

34 Summary of an increase in the savings rate An increase in the savings rate: 1. does not have growth effects, it does not affect the long-run growth rate 2. has level effects, it unambiguously increases per capita capital and output its effect on consumption depends on the relation of the savings rate to the golden-rule of savings If the level effects are large enough, maybe the lack of growth effects is not that important. But how big are the level effects? And how fast does the economy converge to the steady state?

35 Quantitative implications on the level of output I. Long-run effect of the saving rate on output (y = f (k )): y s = f (k ) k (s, n, δ) s We need k / s to get y / s. k is defined by k = 0: sf (k (s, n, δ)) = (n + δ)k (s, n, δ) This holds for all values of s, hence the derivatives wrt to s of the two sides are equal: sf (k ) k s + f (k (s, n, δ)) = (n + δ) k s

36 Quantitative implications on the level of output II. By rearranging: k s = f (k ) (n + δ) sf (k ) Substituting this into the equation of y / s we get: y s = f (k )f (k ) (n + δ) sf (k ) Multiplying both sides with s/y, we get the elasticity of output wrt the savings rate: s y y s = s f (k )f (k ) f (k ) (n+δ) sf (k ) (n+δ)k = f (k ) f (k )[(n+δ) (n+δ)k f (k )/f (k )] = k f (k )/f (k ) 1 k f (k )/f (k ) Using that sf (k ) = (n + δ)k.

37 Quantitative implications on the level of output III. Thus s y y s = k f (k )/f (k ) 1 k f (k )/f (k ) Where k f (k )/f (k ) is the elasticity of output wrt to capital at k = k, denoting this by α K (k ): s y y s = α K (k ) 1 α K (k ) In competitive markets capital is paid its marginal product, and the share of total income that goes to capital is: k f (k ) f (k ) = α K (k ) 1 3 Hence the elasticity of output wrt the saving rate is about 1/2. Thus a 10 % increase in the savings rate increases the level of output by about 5 %.

38 The Solow model with technological progress The production function is: Y = F (K, AL) where A is technology, which increases the efficiency of labour. AL is then the available efficiency labour. The normalisation of output is not with the unit of labour; but with efficiency units of labour (λ = 1/AL): Y AL = F ( K AL, AL AL ) y = f (k) = F ( K AL, 1) We assume that technology grows at a fixed, exogenous rate: Ȧ(t) = ga(t) The change in the stock of capital is the same: K(t) = I (t) δk(t) = sy (t) δk(t)

39 The evolution of capital per efficiency unit of labour is: k(t) = K(t) A(t)L(t) ( t = K(t) t A(t)L(t) K(t) A(t) L(t) t (A(t)L(t)) 2 K(t) L(t) A(t)L(t) L(t) K(t) Ȧ(t) A(t)L(t) A(t) = K(t) A(t)L(t) Substituting L(t) = nl(t), Ȧ(t) = ga(t) and K(t) = sy (t) δk(t): k(t) = Using that sy (t) A(t)L(t) δk(t) A(t)L(t) K(t) A(t)L(t) n Y (t) A(t)L(t) = y(t) = f (k(t)) we get: k(t) = sf (k(t)) (δ + n + g)k(t) ) +L(t) A(t) t K(t) A(t)L(t) g This is the key equation of the Solow model with technology.

40 As before k(t) = sf (k(t)) }{{} (δ + n + g)k(t) }{{} actual investment break-even investment i (δ + n + g)k sf (k) k k

41 In the steady state From any starting point the economy converges to k. In the steady state: variable growth rate k(t) = k 0 y(t) = f (k ) 0 K(t) L(t) = A(t)k(t) = A 0e gt k g Y (t) L(t) = A(t)f (k(t)) = A 0e gt f (k ) K(t) = A(t)L(t)k = k L 0 A 0 e (n+g)t Y (t) = A(t)L(t)f (k ) = f (k )L 0 A 0 e (n+g)t g n+g n+g The only source of growth in output per worker is growth in the effectiveness of labour.

42 Implications of the Solow model if a country is far from its steady state, it will grow fast in the beginning, then slow down, and the growth rate will tend asymptotically to zero transitional growth driven by the growth in k countries with high s high A tend to have higher k low n, low δ pretty successful implications

43 Implications II. - Convergence Unconditional/absolute convergence: poorer countries grow faster than rich ones this is not supported by the data (see Romer and Barro book) this is also not implied by the model Conditional convergence: countries further away from their own possibly different steady states grow faster within OECD countries and US states this is supported can assume same/similar steady states this is implied by the model example: A can be richer than B, with k A > k B, and A can be proportionally further from its steady state than B

44 Implications III. - Growth rates The Solow model predicts that growth rates decline and asymptote towards zero. hard to reconcile with the data one solution: technological change (as we have already seen) good, as it implies constant growth rates within countries the distribution of incomes between countries are roughly constant this is true in the BGP however, we assume the growth and do not explain it other ways of long-run growth without technological change: 1. drop decreasing returns - parameters have growth effects 2. drop Inada conditions

45 Growth accounting In the Solow model long-run growth in output per worker only depends on technological progress. However, short run growth can come from capital accumulation (if the economy is not in the steady state) and from technological progress. Determining the source of any observed growth in output per worker is an empirical issue GROWTH ACCOUNTING Ẏ (t) = Denote by Y (t) K(t) K(t) Y (t) + (A(t) A(t)L(t) L(t) ) + L(t)Ȧ(t) Y (t) A(t) = Y (t) Y (t) A(t)L(t) L(t) and L(t) = Y (t) A(t)L(t) A(t).

46 Growth accounting II. Divide both sides by Y (t): Ẏ (t) Y (t) = K(t) Y (t) K(t) Y (t) K(t) K(t) + L(t) Y (t) Y (t) L(t) = α K (t) K(t) K(t) + α L(t) L(t) L(t) + R(t) L(t) L(t) + A(t) Y (t) Y (t) Ȧ(t) A(t) A(t) Remember that α K is the elasticity of output wrt capital, similarly α L is the elasticity of output wrt labour, and R(t) denotes the last term. Due to homogeneity of degree one and marginal product pricing: α K + α L = 1. Subtracting L(t)/L(t) from both sides we get: Ẏ (t) Y (t) L(t) ( K(t) L(t) = α K (t) K(t) L(t) ) + R(t). L(t)

47 Growth accounting III. Ẏ (t) Y (t) L(t) ( K(t) L(t) = α K (t) K(t) L(t) ) + R(t). L(t) The growth rates of Y, L and K are measurable. As before, α K can be measured as the share of income that goes to capital. R(t) is the Solow residual: it captures all sources of growth besides capital accumulation Applications 1. understanding the rapid growth in East Asia 2. understanding the productivity growth slowdown and catch up in the US

48 Growth accounting III. Ẏ (t) Y (t) L(t) ( K(t) L(t) = α K (t) K(t) L(t) ) + R(t). L(t) The growth rates of Y, L and K are measurable. As before, α K can be measured as the share of income that goes to capital. R(t) is the Solow residual: it captures all sources of growth besides capital accumulation Applications 1. understanding the rapid growth in East Asia Young (1995): growth almost only due to capital accumulation, labour force participation and improving labour quality Hsieh (1998): growth accounting from factor returns - fall in returns to capital not large enough technology must have played a larger role 2. understanding the productivity growth slowdown and catch up in the US

49 Growth accounting III. Ẏ (t) Y (t) L(t) ( K(t) L(t) = α K (t) K(t) L(t) ) + R(t). L(t) The growth rates of Y, L and K are measurable. As before, α K can be measured as the share of income that goes to capital. R(t) is the Solow residual: it captures all sources of growth besides capital accumulation Applications 1. understanding the rapid growth in East Asia 2. understanding the productivity growth slowdown and catch up in the US slowdown: reduced growth in workers skills, effect of oil prices, reduced innovations, government policies catch-up: increased use of computers and other types of IT