Long run economic growth, part 2. The Solow growth model

Transcription

1 Long run economic growth, part 2. The Solow growth model

2 The Solow growth model The seminal Solow growth model dates bac to 1950 s and belongs to the fundamentals of growth theory The Solow model is remarable for its simplicity The Solow model is a good starting point and a springboard for further models We will analyze it in continuous time, we mae the time units (the difference between t and t+1) as small as possible At the centre of the model is the neoclassical production function

3 The neoclassical production function Production function is noeclassical when is characterised by : 1. Constant returns to scale in capital and labour ay AF( ak, a) 2. Positive & diminishing marginal product of capital and labour: F K dy dk 0, FK dk 0 F dy d 0, F d 0 3. Inada conditions: limf K 0 K, lim K F K 0 lim 0 F, lim F 0

4 Other important assumptions of the Solow model One sector economy that produces a homogenous good that can be consumed or invested The economy is closed, there is no government sector : Y t = C t + I t ; G = 0, X = 0 The saving rate s is constant: S = sy, s>0 The population growth n rate is constant n The depreciation rate of capital d is constant

5 A temporary assumption (just for now) To mae our analysis easier, we will assume that technology is constant and equal to 1 We will change this assumption very soon For now, the (neoclassical) production function is: Y=A F (K, ) = F (K, )

6 Per capita production function Since the production function is characterised by constant returns to scale, we can write : Y F( We will use the following notation: y K, ) Y, K F( Hence, the production function can be written in intensive form as K,1) y f ()

7 Per capital production function is still characterised by: f lim 0 f df d 0, df,lim d f 0 0

8 A neoclassical per capita production function

9 otice that The per capita production function depends only on per capita capital stoc If we understand the dynamics of capital per capita we can understand economic growth!

10 The evolution of capital over time dk dt t K I t dk t ; where : I t S t sy t K sy t dk t However, we are looing for the dynamics of the small, that is capital per person: d dt t d( K ) dt

11 The accumulation of capital per person K 2 K K K sy dk n sy d n Fundamental equation of the Solow growth model: sy ( d n)

12 Saving and investment y f() y 1 c 1 sf() i 1 1

13 The effective deprecation or brea-even investment y (d+n) Investment needed to eep at a constant level

14 Moving toward the steady-state y (d+n) sf() investment depreciation 1 *

15 The steady state sy ( d n) If investment is just enough to cover effective depreciation: sy=(d+n) then capital per worer will remain constant: 0 This constant value, denoted *, is called the steady state capital stoc.

16 The steady-state

17 The steady-state y* y y=f() (+n) s f() *

18 The steady-state The only possible steady-state capital labor ratio is * Output at that point is y* = f(*); consumption is c* = f(*) (n + d)* If begins at some level other than *, it will move toward * For below *, saving > the amount of investment needed to eep constant, so rises For above *, saving < the amount of investment needed to eep constant, so falls

19 The steady-state We have established that in the steady-state, capital per person is constant. That, of course implies that output per person and consumption per person are also constant: y c 0

20 The Solow s suprise Investment in new capital (and the growth of population) cannot lead to continued growth in per capita income. What can lead to sustained growth of output per person? As we will see next wee technology!

21 The steady-state What about the growth rates of K (capital) and Y (output) in the steady-state? Recall that K=; Y=y Therefore: K K d( ) dt In the steady-state: K K 0 n then Exactly the same reasoning applies to Y Y

22 The steady-state - summary In the steady-state, the per worer values are stable: y y c c 0 In the steady-state, Y & K & C exhibit are characterized by steady growth rate equal to the rate of growth of population: K K Y Y C C n

23 Steady-state: a Cobb-Douglas production function Assume a Cobb-Douglas function: Intensive form (per worer): The steady-state: 1 K Y y 1 ) ( ) ( 0 ) ( 0 * n d s n d s n d s 1 1 ) ( * n d s

24 The Solow Model To summarize: With no productivity growth, the economy reaches a steady state, with constant capital per person (or capital labor ratio), output per person, and consumption per person In the steady-state, the amount of capital per person output per person and consumption per person depend, among others on: the saving rate, the rate of population growth and on the depreciation rate

25 The effect of an increased saving rate on the steady-state capital and output per person

26 An increase in the saving rate y y 2 * y 1 * (+n) y=f() s 2 f() s 1 f() 1 * 2 * An increase of s increases *, and y*.

27 Prediction: Higher s higher *. And since y = f(), higher * higher y *. Thus, the Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income per worer in the long run.

28 The effect of a higher population growth rate on the steady-state capital labor ratio

29 Population growth rate and GDP per capita 29

30 The Solow Model Should a policy goal be to reduce population growth? Doing so will raise consumption per worer ote however that the Solow model also assumes that the proportion of the population of woring age is fixed (exactly: population = worers) But when population growth changes, this may change the % of the woring-age population Changes in cohort sizes may cause problems for social security systems and areas lie health care

31 The Golden Rule: Introduction Different values of s lead to different steady states. How do we now which is the best steady state? The best steady state has the highest possible consumption per person: c* = (1 s) f(*). An increase in s leads to higher * and y*, which raises c* reduces consumption s share of income (1 s), which lowers c*. So, how do we find the s and * that maximize c* (in the steady-state)?

32 The Golden Rule capital stoc * gold the Golden Rule level of capital per worer: the steady state value of that maximizes steady-state consumption per person. To find it, first express c * in terms of * : c * = y * i * = f ( * ) sy * = f ( * ) (d+n) * In the steady state: sy * = (d+n) *

33 The Golden Rule capital stoc Then, differentiate with respect to, to find the value of * that maximises c*: c* f ( *) ( d dc* 0 d * dc* f '( *) ( d d * f '( *) ( d n) n) * n) 0

34 The Golden Rule capital stoc Graph f( * ) and (d+n) * ; loo for the point where the gap between them is biggest. y f ( ) * * gold gold steady state output and depreciation * gold * c gold i * * gold gold (d+n) * f( * ) steady-state capital per worer, *

35 The Golden Rule capital stoc c * = f( * ) (d+n) * is biggest where the slope of the production function equals the slope of the depreciation line: MPK = (d+n) * gold * c gold * f( * ) steady-state capital per worer, *

36 The golden rule: an example using C-D p.f. Let s assume a Cobb-Douglas production function: In intensive form: Consumption in the steady-state is: Maximise it with respect to capital 1 K Y y n d c ) ( * 1 1 * 1 1 ) ( ) ( 0 ) ( * * n d n d n d d dc GOLD GOLD GOLD 1 1 * ( n) d GOLD

37 How much do we need to save, to achieve GOLD? Recall that in the steady-state: Since the golden-rule level of capital per person is also the steady-state capital per person, then we now that 1 1 ) ( * n d s * * ) ( ) ( * n d s n d GOLD GOLD GOLD s GOLD

38 The transition to the Golden Rule steady-state The economy does OT have a tendency to move toward the Golden Rule steady state. Achieving the Golden Rule requires that policymaers adjust s. This adjustment leads to a new steady state with higher consumption. But what happens to consumption during the transition to the Golden Rule?

39 Starting with too much capital If * * gold then increasing c * requires a fall in s. In the transition to the Golden Rule, consumption is higher at all points in time. y c i t 0 time

40 Starting with too little capital If * * gold then increasing c * requires an increase in s. Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption. y c i t 0 time

41 Summary The economy will reach a steady-state, where the values of capital & output & consumption per worer will be constant Investment in new capital (and the growth of population) cannot lead to continued growth in per capita income. GDP per worer in the steady-state depends on (among others): the saving rate and the population growth rate

42 What s ahead? Discussing convergence (next wee) Adding technology to the Solow s model (next wee) A very quic glimse at different growth models (in two wees)