The Facts of Economic Growth and the Introdution to the Solow Model

Transcription

1 The Facts of Economic Growth and the Introdution to the Solow Model Lorenza Rossi Goethe University

2 Course Outline FIRST PART - GROWTH THEORIES Exogenous Growth The Solow Model The Ramsey model and the Golden Rule Introduction to Endogenous Growth models The AK model - Romer (1990) Two sector model of Endogenous growth SECOND PART - BUSINESS CYCLE Introduction to NK model The BMW model as a static approximation of a forward-looking NK model The BMW model in a closed economy: in ation targeting versus Taylor rules The BMW model in an open economy: comparisons with the Mundel Fleming modelchristiano, Eichenbaum and Evans model (2005)

3 TODAY Brief Review Growth Stylized Facts Introduction to the Solow Model Derivations The model performance and stylized facts

4 STYLIZED FACT 1 There is an enormous variation in the per capita income across economies. The poorest countries have per capita income that are less than 5 percent of per capita incomes in the riches countries.

5 The Rich and the Poor

6 The Rich and the Poor

7 World Population by GDP per worker

8 PER CAPITA GDP 1960

9 PER CAPITA GDP 2000

10 The world distribution of income in 1970

11 The world distribution of income in 2000

12 STYLIZED FACT 2 Rates of Economic Growth vary substantially across countries

13 STYLIZED FACT 3 Growth rates are not generally constant over time. For the world as whole, growth rates were close to zero over most of the history but have increased sharply in the twentieth century. For individual countries, growth rates also change over time.

14 STYLIZED FACT 4 A country relative position in the world distribution of per capita incomes is not immutable. Countries can move from being poor to being rich, or viceversa. EXAMPLE: Venezuela vs Italy.

15 Fifteen Growth Miracles

16 Fifteen Growth Disasters

17 OTHER STYLIZED FACTS. FACT 5 In the US over the last century, 1 the real rate of return to capital, r, shows no trend upward or downward; 2 the shares of income devoted to capital, rk /Y, and the share of income devoted to labor, wl/y, show no trend; 3 the average growth rate of output per person has been positive and relatively constant over time, i.e. the US exhibits steady, sustained per capita income growth.

18 Real Per Capita GDP in the US

19 OTHER STYLIZED FACTS: FACT 6 Growth in output and growth in the volume of international trade are closely related

20 Growth in trade and GDP

21 OTHER STYLIZED FACTS: FACT 7 Both skilled and unskilled workers tend to migrate from poor to rich countries or regions ROBERT LUCAS: this movements of labor tell us something about real wages. The returns of both skilled and unskilled

22 Kaldor Stylized Facts and the Solow Model

23 The Solow Model Solow seminal paper (1956). "A Contribution to the Theory of Economic Growth" Check for the ability of the model to explain the stylized facts Neoclassical model Countries produce and consume one single good (units of GDP); There is no international trade (since there is only one good) Technology is exogenous Perfect competition in all markets

24 The Solow Model The basic model is characterized by two equations 1 a production function; 2 a capital accummulation equation

25 The Solow Model The Neoclassical Aggregate Production Function Y (t) = F (A (t), K (t), L (t)) where K (t) is physical capital, L (t) is labor and A (t) is a exogenous technology shift (TFP) Technology is free; it is publicly available as a non-excludable, non-rival good.

26 The Solow Model: Key Assumption Assumptions F exhibits constant return to scale in K and L =) F is linear homogeneous (homegeneous of degree 1)

27 The Solow Model: Key Assumption Capital accumulation K = I d K. I = sy where s represents a constant savings rate. d is capital depreciation rate

28 The Solow Model The Inada Conditions lim K () K!0 = and lim K () = 0 for all L > 0 and all A K! lim L () L!0 = and lim F L () = 0 for all K > 0 and all A L! Important in ensuring the existence of interior equilibria.

29 The Solow Model Firms pro ts maximization FOCs From the assumption of homogeneity of degree 1 and thus rms pro ts are zero!

30 The Solow Model The production function can be speci ed as follows: Y t = K α t (A t L t ) 1 α with 0 < α < 1 it is Cobb-Douglas production function with constant return to scale, where A is a technology variable labour augmenting. AL are the e cient units of labor. The rate of growth of technological progress is exogenous and de ned as da 1 dt A = d Ȧ ln (A) = dt A = g

31 The Solow Model

32 The Solow Model The demand for capital implies and thus the capital share is R t = αk α 1 t (A t L t ) 1 α RK Y = 1 (AL) 1 α K αkα Y The labor demand implies = α constant w t = (1 α) K α t (A t L t ) α A t and thus the labor share is wl Y = (1 α) K α (AL) α AL Y = 1 α constant The labor share and the capital share are constant in the long run, in accordance with Kaldor stylized facts.

33 The Solow Model Production function in terms of output per worker: y = k α A 1 α where y = Y /L and k = K /L Production funtion in terms of e ecient unit of labor per worker is ỹ = k α where ỹ = Y / (AL) and k = K / (AL).

34 The Solow Model To see the growth implications of the model, we take the log and then di erentiate y = Ak α, we nd ẏ y = (1 α) Ȧ A + α k k notice however that k k = then K L L K. which is equal to K L L K = L K L K k k = K K where the labor force growth rate is exogenous and given by L L = n. L L L 2 LK

35 The Solow Model We can express capital in terms of e cient unit of labor per worker k = K AL then k k = K K L L A A

36 The Solow Model Remember that K = I as, if Y K K is constant, then While from k k = K K then K L L d K. This means that K K K K = s Y K d is also constant. we can rewrite, K K = k k + L L = s Y /L K /L k = sy (d + n) k = sk α A 1 α (d + n) k is the law of motion of capital per worker, while k = sỹ (g + d + n) k = s k α (g + d + n) k is the law of motion of capital per unit of e cient labor. d can be written

37 The Solow Model An economy start with a given stock of capital per worker k 0, a given population growth rate, n, a given technology growth rate, g, and a given investment rate. 1 How does output per worker, y, (or output per e cient units of labor per worker, ỹ) evolve over time? 2 How does the economy growth? 3 How does output per worker compare in the long-run between two economies that have di erent investment rate? 4 How does output per worker compare in the long-run between two economies that have di erent technology growth rate?

38 The Solow Model With Ȧ A = 0, then capital per worker is k = sk α in the steady state k = 0 and thus k = (d + n) k s d + n y = (k ) α = 1 1 α s d + n α 1 α In the Solow model countries with higher savings/investment rate will tend to be richer, ceteris paribus. Such countries have more capital per worker and thus more output per worker. Countries with high population growth rate, in contrast will tend to be poorer according to the Solow model.

39 The Solow Model The Solow diagram determines the steady state value of output per worker The dynamics converges to the steady state value of capital per workers k.

40 The Solow Model The Solow diagram and household consumption

41 The Solow Model. The Golden rule Consumption is At the steady state consumption is c t = F (k t ) sf (k t ) c = F (k ) sf (k ) = (k ) α (d + n) k consumption is maximum if c k = 0 : F 0 (k) (d + n) = 0

42 The Solow Model. The Golden rule c k = 0 implies α (k) α 1 (d + n) = 0 and thus, solving for k k GR = α d + n 1 1 α we call k GR the Golden rule capital stock, that is the value of k such = 0 and consumption is maximum. that c k

43 The Solow Model. The Golden rule

44 The Solow Model. The Golden rule DYNAMIC EFFICIENCY. If k GR > k 1 =) s 1 < s GR and c 1 < c GR. Increasing savings increases also the steady state per capita consumption DYNAMIC INEFFICIENCY. If k GR < k 2 =) s 2 > s GR and c 2 < cgr. The economy is oversaving and is said to be dynamically ine cient.

45 The Solow Model An increase in the investment rate s the steady state value of capital per worker increases.

46 The Solow Model An increase in population growth

47 The Solow Model Consider now Ȧ A 6= 0 = g. In the steady state k = 0 and thus while output per worker is k = ỹ = k α = y (t) = s d + n + g 1 1 α s d + n + g s d + n + g α 1 α α 1 α A (t) where t is included just to remind that A is an exogenous growing variable.

48 The Solow Model Consider now Ȧ A 6= 0 = g. The Solow diagram becomes

49 The Solow Model An increase in the savings rate

50 The Solow Model An increase in the savings rate and conditional convergence.

51 The Solow Model Output per worker along the balanced growth path is determined by technology, investment rate and the population growth rate Changes in the investment rate and the population growth rate a ect the long-run level of output per worker, but do not a ect the long-run growth rate of output per worker. Policy changes do not have long-run growth e ects. Policy changes can have level e ects, that is a permanent policy change can permanently raise (or lower) the level of per capita output. Conditional convergence.