Chapter 2. The Solow Growth Model

Transcription

1 Chapter 2. The Solow Growth Model Instructor: Dmytro Hryshko 1 / 56

2 Solow model Solow model is a starting point for more complex models. Abstracts from modeling heterogeneous households (in tastes, abilities, etc.), heterogeneous sectors in the economy, and social interactions. It is a one-good economy with simplied individual decisions. We'll discuss the Solow model both in discrete and continuous time. 2 / 56

3 Households and production Closed economy, with a unique nal good. Time is discrete: t = 0; 1; 2; : : :, 1 (days/weeks/months/years). A representative household saves a constant fraction s 2 (0; 1) of its disposable income (a bit unrealistic). A representative rm with the aggregate production function Y (t) = F (K(t); L(t); A(t)). 3 / 56

4 Production function Y (t) = F (K(t); L(t); A(t)); where Y (t) is the total production of the nal good at time t, K(t) the capital stock at time t, and A(t) is technology at time t. Think of L as hours of employment or the number of employees; K the quantity of \machines" (K is the same as the nal good in the model; e.g., K is \corn"); A is a shifter of the production function (a broad notion of technology: the eects of the organization of production and of markets on the eciency of utilization of K and L). 4 / 56

5 Assumption on technology Technology is assumed to be free. That is, it is publicly available and Non-rival consumption or use by others does not preclude an individual's consumption or use. Non-excludable impossible to prevent another person from using or consuming it. E.g., society's knowledge of how to use the wheels. 5 / 56

6 Assumptions on production function Assumption 1: The production function F (K; L; A) is twice dierentiable in K and L, (K; L; (K; L; A) F K (K; L; A) = > 0; F L (K; L; F KK F (K; L; 2 < 0; F LL F (K; L; 2 < 0: F is constant returns to scale in K and L. > 0 F K > 0; F L > 0: positive marginal products (the level of production increases with the amount of inputs used); F KK < 0; F LL < 0: diminishing marginal products (more labor, holding other things constant, increases output by less and less). 6 / 56

7 Constant returns to scale assumption F (K; L; A) is constant returns to scale in K and L if F (K; L; A) = F (K; L; A). In this case, F (K; L; A) is also said to be homogeneous of degree one. In general, a function F (K; L; A) is homogeneous of degree m in K and L if m F (K; L; A) = F (K; L; A). 7 / 56

8 Theorem 2.1 (Euler's theorem) Suppose that g(x; y; z) is dierentiable in x and y and homogeneous of degree m in x and y, with partial rst derivatives g x and g y (z can be a vector of variables). Then, (1): mg(x; y; z) = g x (x; y; z)x + g y (x; y; z)y; and (2): g x and g y are homogeneous of degree m 1 in x and y. Proof: By denition m g(x; y; z) = g(x; y; z) (2.2). Par 1: Dierentiate both sides wrt to obtain m m 1 g(x; y; z) = g x (x; y; z)x + g y (x; y; z)y, for any. Set 1. Thus, mg(x; y; z) = g x (x; y; z)x + g y (x; y; z)y. Part 2: Dierentiate (2.2) wrt x: m g x (x; y; z) = g x (x; y; z), or m 1 g x (x; y; z) = g x (x; y; z). 8 / 56

9 Endowments, market structure and market clearing Competitive markets. Households own labor, supplied inelastically: all the endowment of labor L(t) is supplied at a non-negative rental price. Market clearing condition: L d (t) = L(t). Rental price of labor at t is the wage rate at t, w(t). Households own the capital stock and rent it to rms. R(t) is the rental price of capital at t. Market clearing condition: K d (t) = K s (t). K d used in production at t is consistent with households' t endowments and saving behavior. 9 / 56

10 K(0) 0 initial capital stock is given. The price of the nal good, P (t), is normalized to 1 at all t. The nal good (within each date) is numeraire. Capital depreciates at the rate 2 (0; 1) (machines utilized in production lose some of their value due to wear and tear). 1 unit now \becomes" 1 units next period. R(t) + (1 ) = 1 + r(t), where r(t) is the real interest rate at t. 10 / 56

11 Firm optimization and equilibrium Firms' objective, at each t, is to maximize prots; rms are competitive in product and rental markets, i.e., take w(t) and R(t) as given. max = F (K d ; L d ; A(t)) R(t)K d w(t)l d ; K0;L0 where superscript d stands for demand, and is prot. Note that maximization is in terms of aggregate variables (due to the representative rm assumption), and R(t) and w(t) are relative prices of labor and capital in terms of the nal good. 11 / 56

12 Since F is constant returns to scale, there is no well-dened solution (K d ; L d ) ( means optimal, or prot-maximizing). If (K d 0; L d 0) and > 0, then (K d 0; L d 0) will bring. Thus, want to raise K and L as much as possible. A trivial solution K = L = 0, or multiple values of K and L that give = 0. When prots are zero, would rent L and K so that L = L s (t) and K = K s (t). 12 / 56

13 Factor prices w(t) = F L (K(t); L(t); A(t)); R(t) = F K (K(t); L(t); A(t)): Using these two equilibrium conditions and Euler's theorem, we can verify that rms make zero prots in the Solow growth model. Y (t) = F (K(t); L(t); A(t)) = F K (K(t); L(t); A(t)) {z} LHS {z } RHS K(t) + F L (K(t); L(t); A(t)) L(t) = R(t)K(t) + w(t)l(t). Factor payments (RHS) exhaust total revenue (LHS), hence prots are zero. 13 / 56

14 Assumption 2: Inada conditions We'll assume that F satises Inada conditions: lim F L (K; L; A) = 1; and lim L!0 L!1 F L(K; L; A) = 0 for all K > 0; all A lim K!0 F K (K; L; A) = 1; and lim K!1 F K(K; L; A) = 0 for all L > 0; all A Capital is essential in production (can be relaxed): F (0; L; A) = 0 for all L and A: The conditions say that the \rst" units of labor and capital are highly productive, and that when labor and capital are suciently abundant, the use of a marginal unit does no add anything to the existing output. 14 / 56

15 Solow model in discrete time The law of motion of K(t) is given by K(t + 1) = I(t) + (1 )K(t); where I(t) is investment at time t. Recall the national income accounts identity for a closed economy: Y (t) = C(t) + I(t): Also, S(t) = I(t) = Y (t) C(t). By assumption, S(t) = sy (t), and so C(t) = Y (t) sy (t) = (1 s)y (t). Supply of capital resulting from households' behavior is K s (t + 1) = (1 )K(t) + S(t) = (1 )K(t) + sy (t). Equilibrium conditions are: K s (t + 1) = K(t + 1) and L(t) = L(t). The fundamental law of motion of the Solow model is: K(t + 1) = (1 )K(t) + sy (t) = (1 )K(t) + sf (K(t); L(t); A(t)) : 15 / 56

16 Denition of Equilibrium Equilibrium is dened as an entire path of allocations, C(t), K(t), Y (t), and prices, w(t) and R(t), given an initial level of capital stock K(0), and given sequences fl(0); L(1); L(2); : : : ; L(t); : : : g and fa(0); A(1); A(2); : : : ; A(t); : : : g. 16 / 56

17 Equilibrium without population growth and technological progress In this case, L(t) = L, and A(t) = A. Example. The Cobb-Douglas production function. Y (t) = F (K(t); L(t); A(t)) = AK(t) L 1 ; 0 < < 1: 17 / 56

18 Note that F (K; L; A) = A(K(t)) (L) 1 = A 1 K(t) L 1 = AK(t) L 1 = Y (t). Let 1 L. Then Y (t) L = A K(t) L Dene k(t) K(t) L L L 1 = A K(t) L 1 1 = A K(t). L Y (t) and y(t) L. Then y(t) = Ak(t). R(t) = F K = AK(t) 1 L 1 = A( K(t) L ) 1 = Ak(t) 1. w(t) = F L = A(1 )K(t) L = A(1 )( K(t) L ) = A(1 )k(t). Note also that w(t)l = Y (t) R(t)K(t), and so w(t) = Y (t) L R(t)k(t) = Ak(t) Ak(t) 1 k(t) = Ak(t) Ak(t) = A(1 )k(t) = (1 )y(t). 18 / 56

19 Steady-state equilibrium Recall the law of motion of the capital stock: K(t + 1) = (1 )K(t) + sf (K(t); L; A); where we utilized our assumptions that technology and population are not growing. Divide the equation by L, to obtain, k(t + 1) = (1 )k(t) + s Y (t) L = (1 )k(t) + sf K(t) L ; 1; A : Normalize A to one, and dene F ( K(t) L ; 1; 1) = f (k(t)). Thus, the law of motion in per worker terms is: k(t + 1) = (1 )k(t) + sf (k(t)): A steady-state equilibrium without population and technological growth is an equilibrium path so that k(t) = k for all t. 19 / 56

20 Steady-state equilibrium contd. For a steady state, Also, k = sf(k ) + (1 )k sf(k ) = k or f(k ) k = s : (2.18) y = f(k ) c = (1 s)f(k ): Notes: since k = K L is constant and L is not growing, K is not growing in the steady state either; to keep capital per worker, k, constant investment per worker should be equal to the amount of capital per worker that needs to be \replenished" because of depreciation; k will be a function of s and, i.e., k = k (s; ). 20 / 56

21 Proposition 2.2 Assume that Assumptions 1 and 2 hold. Then there exists a unique equilibrium where k 2 (0; 1) satises (2.18). Proof. Existence: from Assumption 2, lim k!0 f(k) k = 1 and f(k) lim k!1 = 0, and f(k) is continuous. By the intermediate k k value theorem, there exists k that satises f(k) k = f 0 (k)k f(k) k 2 = w k 2 < 0: Thus, f(k) is everywhere strictly decreasing, and there must k exist one k that satises (2.18). 21 / 56

22 Steady-state predictions We normalized A to 1 before. Redene our per worker production function as f(k) = A ~ f(k). The steady-state equilibrium will now be obtained from sa ~ f(k ) = ~ f(k ), and k = k (s; A; ). Thus, sa ~ f(k (s; A; )) = k (s; A; ). It can be shown (s; A; ) (s; A; ) (s; A; ) (s; A; ) (s; A; ) (s; A; ) < Intuition: economies with higher saving rates and better technologies will accumulate higher capitals per worker and will be richer; higher depreciation rates lead to lower standards of living and lower capitals per worker. 22 / 56

23 In the steady state: ~f(k ) k = sa ; where k = k (A; s; ). Dierentiate both sides wrt s to obtain: k f ~ 0 (k (k = ) 2 As 2 = (k ) 2 As 2 f(k ) k f 0 (k = (k ) 2 ) As 2 w > 0: 23 / 56

24 Golden rule steady-state In steady state, c = f(k (s)) k (s): Maximizing wrt s, we need to set: or f 0 = 0; f 0 = 0: > 0, the golden rule savings rate must f 0 (k gold ) =. 24 / 56

25 Corollary Let x(t), a and b be scalars. If a < 1, then the unique steady state of the linear dierence equation x(t + 1) = ax(t) + b is globally (asymptotically) stable: x(t)! x = b 1 a. 2. Let g : R! R be dierentiable in the neighborhood of the steady state x, dened as g(x ) = x, and suppose jg 0 (x )j < 1. Then the steady state x of the nonlinear dierence equation x(t + 1) = g(x(t)) is locally (asymptotically) stable. Moreover, if g is continuously dierentiable and satises jg 0 (x)j < 1 for all x, then x is globally (asymptotically) stable. 25 / 56

26 Global stability jx(t + 1) x j = jg(x(t)) g(x )j = Z x(t) x g 0 (x)dx < jx(t) x j: Thus for any x(0) < x, x(t) is an increasing sequence. The distance between x(t + 1) and x becomes smaller than the distance between x(t) and x. In the Solow model, x(t + 1) k(t + 1), and g(x(t)) sf(k(t)) k(t) + k(t). Note that g 0 (x) = sf 0 (k(t)) is not always less than 1 in absolute value. For ex., it tends to a large number when k(t) is small. Need some dierent arguments. 26 / 56

27 Local stability in the Solow model Since f(k) is strictly concave by assumption: f(k) > f(0) + f 0 (k)(k 0) = f 0 (k)k: (2.29) In steady state, k = g(k ) = sf(k ) k + k, and g 0 (k ) = sf 0 (k ) + 1. k {z } SS (2.29) implies that sf(k ) = > sf 0 (k ), or < sf 0 (k ), or sf 0 (k ) < 0 or sf 0 (k ) + 1 < 1 which is required for local stability. 27 / 56

28 Global Stability in the Solow Model For all k(t) 2 (0; k ), k(t + 1) k = g(k(t)) g(k ) = Z k k(t) g 0 (k)dk < 0; since g 0 (k) > 0 for all k. Thus, k(t + 1) also lies below k. Also, k(t + 1) k(t) k(t) = s f(k(t)) k(t) > s f(k ) k = 0; where inequality follows from the fact that f(k) k k. Thus, k(t + 1) 2 (k(t); k ). is decreasing in 28 / 56

29 The Solow model in continuous time Consider the following (dierence) equation x(t + 1) x(t) = g(x(t)), where g( ) is some function. In terms of our law of motion, e.g., x(t) = k(t), and g(x(t)) sf(k(t)) k(t). This equation species the change in x between two discrete time points. What happens inside of the interval [t; t + 1] is \not known." Assume that we can partition time very nely. If t and t + 1 are \close enough," the following approximation is reasonable x(t + t) x(t) tg(x(t)): Dividing both sides by t and taking the limits gives x(t + t) x(t) lim = _x(t) g(x(t)); t!0 t where _x(t) = dx(t) the time derivative of x(t). dt 29 / 56

30 The law of motion of capital in continuous time Now we refer to w(t) and R(t) as instantaneous rental rates (the ows received at instant t). Assume that the labor force grows at a constant rate n, L(t) = exp(nt)l(0). Note that log L(t) = nt + log(l(0)). Thus, _L(t) L(t) = n. Dene k(t) = K(t). Taking natural logs from both sides gives L(t) log(k(t)) = log(k(t)) log(l(t)). Taking time derivatives from both sides gives _k(t) = K(t) _ L(t) _ = K(t) _ sf (K(t);L(t);A) K(t) n = n = k(t) K(t) L(t) K(t) K(t) sf (K(t);L(t);A) sf (K(t);L(t);A)=L(t) sf (k(t)) (n + ) = (n + ) = (n + ). K(t) K(t)=L(t) k(t) Thus, _k(t) = sf (k(t)) (n + )k(t): Again, the steady-state occurs when _ k(t) = 0, and sf (k ) = (n + )k. 30 / 56

31 Add-on to the steady-state predictions Since we introduced population growth into the model, k = k (s; A; ; n), and y = y (s; A; ; n). We can show (s; A; ; n) (s; A; ; n) < Economies with higher population growth rates have lower capital-per-worker ratios and lower incomes per worker. 31 / 56

32 Factor shares in total income Consider again the Cobb-Douglas production function Y = AK L 1, 0 < < 1. The share of capital costs in total income is dened as R(t)K(t). Y (t) In competitive factor markets R(t) = F K (K(t); L(t); A(t)) and w(t) = F L (K(t); L(t); A(t)). Thus, R(t)K(t) = AK(t) 1 L(t) 1 K(t) = AK(t) L(t) 1 Y (t) = =. Y (t) Y (t) Y (t) Y (t) From Euler theorem, Y (t) = w(t)l(t) + R(t)K(t). Thus, w(t)l(t) = 1. For the Cobb-Douglas function, the shares of Y (t) capital and labor costs in total income are constant. For this function, y(t) = Y (t) = AK(t) L(t) 1 = A K(t) L(t) L(t) L(t) = Ak(t). Thus, _k(t) = sak(t) (n + )k(t), and k occurs when sa(k ) = (n + )k, or when (k ) 1 = sa n+. Hence, 1 1. k = sa n+ 32 / 56

33 Transitional dynamics Given k(0) > 0 need to solve for k(t) from _k(t) = sak (t) (n + )k(t): Let x(t) k(t) 1. Thus, _x(t) = (1 )k(t) k(t). x(t) = (1 )sa (1 )(n + ) x(t). {z } b {z } a Dene y(t) = x(t) + b=a. Then _y(t) = _x(t) = b + ax(t) = b + a(y(t) b=a) = ay(t). The solution is y(t) = c exp(at). Thus, x(t) = b=a + c exp(at), x(0) = b=a + c, and x(t) = b=a + (x(0) + b=a) exp(at). So, k(t) = x(t) 1 1 sa = n + + k(0) 1 sa n + exp( (1 )(n + )t) 1 1 : 33 / 56

34 Solow model with technological progress Introduce sustained growth by allowing technological progress in the form of changes in A(t). How to model technological growth and its impact on Y (t)? Introduce technological progress so that the resulting allocations (Y (t), K(t), and C(t)) are consistent with balanced growth, as dened by the Kaldor facts constant K, stable r(t) Y and factor income shares, constant growth in Y =L. 34 / 56

35 Types of technological progress Dene dierent types of \neutral" technological progress. Hicks-neutral: ~F (K(t); L(t); A(t)) = A(t)F (K(t); L(t)): Solow-neutral (capital-augmenting): ~F (K(t); L(t); A(t)) = F (A(t)K(t); L(t)): Harrod-neutral (labor-augmenting): ~F (K(t); L(t); A(t)) = F (K(t); A(t)L(t)): Balanced growth is possible in the long-run only if technological progress is Harrod-neutral or labor-augmenting. 35 / 56

36 Theorem 2.6 (Uzawa's Theorem I) Consider a growth model with aggregate production function Y (t) = F ~ (K(t); L(t); A(t)), ~ CRS in K and L ( A ~ can be a vector). The aggregate resource constraint _K(t) = Y (t) C(t) K(t); population grows at a constant rate, L(t) = exp(nt)l(0), and there exists T < 1 such that for all t T, Y _ (t)=y (t) = gy > 0, _K(t)=K(t) = g K > 0, and _C(t)=C(t) = g C > 0. Then 1 g Y = g K = g C. 2 For any t T, there exists a function F : R+ 2! R + homogeneous of degree 1 in two arguments, such that the aggregate production function can be represented as Y (t) = F (K(t); A(t)L(t)), where A(t) 2 R + and A(t)=A(t) _ = g = g Y n. 36 / 56

37 Example. can be represented as Y (t) = (A K (t)k(t)) (A L (t)l(t)) 1 where A(t) = A K (t) 1 A L (t). Y (t) = K(t) (A(t)L(t)) 1 ; 37 / 56

38 The Solow model with technological progress: continuous time Let Y (t) = F (K(t); A(t)L(t)) be constant returns to scale in K(t) and L(t), and _ A(t) A(t) = g > 0. Dene the aggregates as ratios to eective labor, A(t)L(t). Thus, k(t) = is capital per eective labor, and y(t) = Y (t) A(t)L(t) K(t) A(t)L(t) is output per eective labor. Note that _ k(t) k(t) = _ K(t) K(t) g n; y(t) = Y (t) L(t) Y (t) = F (K(t);A(t)L(t)) = F A(t)L(t) A(t)L(t) = y(t)a(t) = A(t)f(k(t)). K(t) A(t)L(t) ; 1 = f(k(t)), and 38 / 56

39 _k(t) sf (K(t); A(t)L(t)) K(t) = k(t) K(t) (n + g) sf (K(t); A(t)L(t))=(A(t)L(t)) = K(t)=(A(t)L(t)) ( + n + g) = sf(k(t)) k(t) (n + g + ): An equilibrium is dened as before, now in terms of the constancy of k(t) = K(t) K(t). The growth rate of, however, A(t)L(t) L(t) will be equal to g. 39 / 56

40 Equilibrium with technological progress Consider the Solow growth model with labor-augmenting technological progress at the rate g and population growth at the rate n. Then there exists a unique balanced growth path where the eective capital-labor ratio is constant and given by sf(k ) = (n + g + )k, where k = K. Output per worker, AL capital per worker and consumption per worker will grow at the rate g. 40 / 56

41 Proposition 2.12 Let A(0) be the initial level of technology. Denote the BGP level of eective capital-labor ratio by k (A(0); s; n; g; ) and the level of output per capita by y (A(0); s; ; g; n; t). (A(0); s; ; (A(0); s; ; n; = 0; < (A(0); s; ; (A(0); s; ; n; > 0 < (A(0); s; ; (A(0); s; ; n; > 0; < (A(0); s; ; (A(0); s; ; n; > 0 < 0: 41 / 56

42 What dierence does technological progress bring? The model now generates the growth in output per capita, and so can be mapped to the data better. The disadvantage is that this growth is driven exogenously. 42 / 56

43 Chapter 3. The Solow Model and the Data. Growth Accounting. Production function: Y (t) = F (K(t); L(t); A(t)); competitive factor markets. Dierentiate wrt t: Thus, _Y = F K _K + F L L _ + FA A K = F K K K + _L F LL L + A_ F AA A : _Y {z} Y g {z } {z} K = F KK Y " K _K g K + F L L _L {z} Y {z} L " L g L + F A A A_ ; Y A {z } where x is the contribution of technological growth (total factor productivity) to economic growth. x 43 / 56

44 In competitive markets, w = F L and R = F K, and so " l = F LL = wl Y Y = L, and " k = K the income shares of labor and capital. Thus, the contribution of TFP to growth can be calculated as: x = g K g K L g L : An estimate of TFP growth at t is ^x(t) = g(t) K (t)g K (t) L (t)g L (t): 44 / 56

45 Solow (1957): developed the growth accounting framework and applied to U.S. data for assessment of the sources of growth during the early 20th century. Conclusion: a large part of of the growth was due to technological progress. Possible biases in estimation of ^x: 1 Inated ^x if g L or g K underestimated (e.g., how to account for changes in human capital, changes in the prices of machines?). 2 Mismeasurement in g (e.g., due to changes in relative prices and the quality of products). 45 / 56

46 The Solow model and regression analysis Equilibrium is described by y(t) = A(t)f(k(t)) (3.6) _k(t) k(t) = sf(k(t)) k(t) (n + g + ): (3.7) Take natural logs from (3.6) and dierentiate wrt t: _y(t) y(t) = A(t) _ A(t) + f 0 (k(t)) k(t) f(k(t)) _ = g + f 0 (k(t))k(t) f(k(t)) {z } =" k (k(t)) _k(t) k(t) : (3.8) 46 / 56

47 Note that (3.7) can be written as _k(t) sf(exp log k(t)) = k(t) exp log k(t) (n + g + ): Taking a rst-order Taylor expansion around log k, we obtain 0 1 _k(t) k(t) B@ sf(k ) k n g {z } CA + =0 since SS sf 0 (k )k k sf(k )k (k (log k(t) ) 2 log(k )) f 0 (k )k = f(k ) 1 sf(k ) k (log k(t) log k ) = (" k (k ) 1)(n + g + ) (log k(t) log k ) : 47 / 56

48 Using (3.8), _y(t) y(t) g " k(k )(1 " k (k ))(n + g + ) (log k(t) log k ) : We can write (3.6) as log y(t) = log A(t) + log f(exp log k(t)): Approximating around log k, and so {z } log y(t) = log y + f 0 (k )k f(k (log k(t) log k ) ; ) = k (k ) {z } _y(t) y(t) g (1 " k(k ))(n + g + ) (log y(t) log y ) : (3.10) convergence 48 / 56

49 Cobb-Douglas example: Y (t) = A(t)K(t) L(t) 1 _y(t) y(t) g (1 )(n + g + ) (log y(t) log y ) (3.11) 1 = 0:02 1 (0:01 + 0:02 + 0:05) (log y(t) log y ) 3 = 0:02 0:054 (log y(t) log y ) The implied rate of convergence is too high: the gap in incomes between similar countries should be halved in little more than 10 years. g i;t;t 1 = b 0 + b 1 log y i;t 1 + i;t (3.12) OECD countries: ^b 1 is negative. The whole world: no such evidence. 49 / 56

50 More realistically, y diers across countries and depends on their characteristics (institutions, investment behavior). This motivates to run g i;t;t 1 = b i 0 + b 1 log y i;t 1 + i;t (3.13) Likely, cov(b i 0 ; log y i;t 1) > 0 so that ^ 1 is biased upward. Conditional convergence: g i;t;t 1 = X 0 i;t + b1 log y i;t 1 + i;t ; (3.14) where X i;t is a column vector (constant, schooling rates, fertility rate, investment rate, openness, rule of law and democracy, etc.) These regressions tend to deliver a negative estimate of b 1. Do they deliver causal eects of X's on growth as well? 50 / 56

51 Problems with (3.14) 1 X's and y i;t 1 are endogenous. ^'s will not measure causal eects and ^b 1 is inconsistent. 2 If X's include openness and schooling/investment rates, on the openness will not measure its full eect on growth since it will also likely aect schooling/investment rates. 3 Regressions assume closed Solow economies, or noninteracting \islands." 51 / 56

52 The Solow model with human capital Y (t) = K(t) H(t) (A(t)L(t)) 1 ; (3.21) 0 < ; < 1, + < 1. Output per eective unit of labor is ^y t = k(t) h(t). Laws of motion of physical and human capital per eective labor are: _k(t) = s k f (k(t); h(t)) ( k + g + n)k(t) _h(t) = s h f (k(t); h(t)) ( h + g + n)h(t): 52 / 56

53 The implied steady-state value of ^y is ^y = s k 1 n + g + k s k (s h ) is more important if () is larger. s h 1 : (3.23) n + g + h 53 / 56

54 A wold of augmented Solow economies. Mankiw, Romer and Weil (1992) Y j (t) = K j (t) H j (t) (A j (t)l j (t)) 1 ; j = 1; : : : ; J. Output per worker is then y s j = A j (t) k;j 1 n j + g j + k s h;j 1 : (3.24) n j + g j + h Assume A j (t) = Aj exp(gt). Then, the BGP income for country j in logs is log y j (t) = log Aj + gt + 1 log s k;j n j + g + k + 1 log s h;j : (3.25) n j + g + h 54 / 56

55 MRW estimated: log y j (t) = const + 1 log(s k;j) Table 3.1, column (1): ^ 2/3 too large. log y j (t) = const log(s k;j) 1 log(s h;j) 1 log (n j + g + k ) + j 1 log (n j + g + k ) 1 log (n j + g + h ) + j : Table 3.2, column (1): ^ 1/3 reasonable. Adj. R 2 0:80 human and physical investments explain more than 3/4 of the cross-country dierences in per capita incomes. 55 / 56

56 Calibrating productivity dierences Start with the production function: Y j = K j (A j H j ) 1 ; estimate K j, and H j. For any year, construct predicted series ^Y j based on the estimated A US ^Y j = K 1=3 j (A US H j ) 2=3 ; and compare with observed Y j. Fig. 3.2: Dierences in physical and human capital still matter a lot. There are signicant productivity dierences: the Solow model overpredicts the incomes in poor countries. The empirical t of the Solow model deteriorates with time. 56 / 56