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Mankiw Economic Growth

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					Lecture 12
Economic Growth
Economic Growth

Explain improvements in standards of
 living (GDP per capital) along time
Explain differences across countries
learn how our own growth rate is
 affected by shocks and our
 government’s policies
Solow Growth Model


                                        slide 1
    some statistics
   In Uganda, 96% of people live on
    less than $2/day. (data link)
   2.8 billion people live on less than $2/day
    (1.1 billion under $1/day)
   GDP per capita
       Chad in 1960: $1212, in 2000: $908
       Venezuela in 1960: $7840, in 2000: $6420
       Korea in 1960: $1495, in 2000: $15875
       H.K. in 1960: $3090, in 2000: $26698


                                                   slide 2
   Huge effects from tiny
   differences
  annual            percentage increase in
growth rate        standard of living after…
 of income
 per capita …25 years …50 years …100 years

 2.0%       64.0%       169.2%     624.5%

 2.5%       85.4%       243.7%    1,081.4%




                                             slide 3
Long term growth effect

 Rule of 72: 1% growth rate,
  approximately takes 72 years to
  double GDP
 What will happen if China keeps 10%
  growth rate and US keeps 3% growth
  rate (US per capita GDP $42,000
  China $6800)



                                        slide 4
    World Distribution of Income




5
                               slide 5
World Income Map




6
                   slide 6
South vs. North




7
                  slide 7
8
    slide 8
Real GDP per capita, 1975–2003




9
                                 slide 9
Life Expectancy and Income (Preston, 1976)




10
                                             slide 10
11
     slide 11
                Heights of Males and Females in China

                             1                                               2
165
160
155




         1920        1940         1960            19801920        1940           1960   1980
                                              cohort
                                     95% CI                  Fitted values
                12
      Graphs by 1=URBAN SITE(U) 2=RURAL SITE(R)
                                                                                           slide 12
Happiness and Income




13
                       slide 13
    The Solow Model
   due to Robert Solow,
    won Nobel Prize for contributions to
    the study of economic growth
   a major paradigm:
     widely used in policy making
     benchmark against which most
      recent growth theories are compared
   looks at the determinants of economic
    growth and the standard of living in the
    long run

                                               slide 14
   How Solow model is different
   from Chapter 3’s model
1. K is no longer fixed:
   investment causes it to grow,
   depreciation causes it to shrink.

2. L is no longer fixed:
   population growth causes it to grow.

3. The consumption function is simpler.

4. No G and T


                                          slide 15
Production

   Initially assume constant population (L)
    and no technology change

   Production of goods and services:
                Y  F (K , L)
   Constant Returns to Scale:
               zY  F (zK , zL)

                                          slide 16
       Production
   Letting z = 1/L, we get the production function in
    per capita terms:
              Y / L  F (K / L,1)  y  f (k )
    y = Y/L = output per worker
    k = K/L = capital per worker

   Constant Returns to Scale  size of the
    economy does not affect the relationship
    between capital per worker and output per
    worker

                                                         slide 17
  Production


MPK  F / K  f (k )

     Decreasing MPK: f (k )  0

     This implies the following shape for
     the production function:


                                            slide 18
     Production

 y
               Low MPK        f(k)


High MPK




                                 k
      MPK is the slope of this curve.

                                        slide 19
Production

   Cobb-Douglas case:

    Y  AK L1


y Y  AK  L1  AK L  A(K / L)  Ak
   L      L
      y Ak

                                               slide 20
Demand

   Assume a closed economy with no
    government: NX = G = 0

     Y C  I Y / L C / L I / L
              y  c i
   Assume that people save a fraction s of
    their income (and therefore consume 1 – s),
                                       0 s 1
       C  (1 s)Y  c  (1 s) y

                                                 slide 21
Demand

   Substituting:
         y  c i  (1 s) y i
                i sy
   In equilibrium:
                 y f (k )

                 i sf (k )

                                  slide 22
    Capital Accumulation

   Two elements determine how the
    capital stock changes over time:

     Investment: addition of new plants and
      equipment (makes capital stock rise)
     Depreciation: wearing out of existing
      capital stock (makes capital stock fall)



                                                 slide 23
 Capital Accumulation

    In other words:

Change in
               Investment  Depreciation
Capital Stock

     k               sf (k )   
                                     k



                                          slide 24
Capital Accumulation

                 f(k)
            c
                  sf(k)
       y
           i
                    k


                          slide 25
Capital Accumulation

     sf (k ) k  k  0
    Investment higher than depreciation
     capital stock increases

     sf (k ) k  k  0
    Depreciation higher than investment
     capital stock increases



                                          slide 26
Capital Accumulation

   Steady-state capital stock (k*):

          sf (k*) k *

   Steady state output, consumption,
    investment:
        y* f (k*)
       c* (1 s) y* (1 s) f (k*)
       i* sy* sf (k*)
                                        slide 27
Determining the capital–labor ratio in the
steady state




28
                                             slide 28
   Capital Accumulation
k1  k* sf (k1) k1  k  0
Low k  high MPK  high returns from
investment  capital stock grows

 k2  k* sf (k2) k2  k  0
High k  low MPK  low returns from
investment  capital stock decreases

In both cases, the economy converges to the
steady state (long-run equilibrium)

                                              slide 29
Capital Accumulation

   Cobb-Douglas:           f (k )  Ak

   In steady state:

           sf (k*)  A(k*) k *
                                 1/( 1)            1/(1 )
    (k*) 1      k* 
                        
                        
                        
                                 
                                 
                                 
                                            
                                             
                                             
                                             sA     
                                                     
                                                     
                                                  

                 sA     
                        
                           sA   
                                 
                                 
                                             
                                             
                                                   
                                                     
                                                     




                                                                slide 30
Increase in Savings Rate
                   k
                   s2f(k)

                   s1f(k)



                        k
      k1*    k2*

                            slide 31
Increase in Savings Rate

   Higher s means that more resources
    will be dedicated to investment 
    higher capital stock in steady state

   Therefore, output per capita will be
    also higher



                                           slide 32
     Golden Rule
   What is the relationship between steady-
    state consumption and savings rate?
             c* (1 s) f (k*)
   Two conflicting forces:
     Higher s  higher output  higher the
      amount of resources available for
      consumption  c* 
     Higher s  lower the proportion of
      income allocated to consumption  c* 

                                               slide 33
Golden Rule

   For low values of s, c* increases with
    s

   For high values of s, c* decreases
    with s

   Golden Rule: capital stock implied by
    the savings rate such that c* is
    maximized
                                            slide 34
Golden Rule

   More formally:

      c* (1 s) f (k*)  f (k*) sf (k*)

    But in steady state: sf (k*) k *

          c* f (k*)k *



                                            slide 35
Golden Rule

 Golden Rule: find k* such that c* is
 maximized

    max c* f (k*)k *
      k*

     dc*  f (k*)  0  f (k*) 
                g               g
     dk *

            MPK 

                                        slide 36
Golden Rule
              k
              f(k)
 MPK = 




                     k
    kg *

                         slide 37
Golden Rule

 sg is the savings rate that implies kg*:
                           k
                           f(k)
 MPK = 
                            sgf(k)



                                  k
      kg *                                  slide 38
The relationship of consumption per worker to
the capital–labor ratio in the steady state




39
                                        slide 39
Golden Rule

 Cobb-Douglas case: f (k )  Ak

     MPK  f (k )  Ak 1

 Golden Rule: MPK = 
                          1/( 1)            1/(1 )
A (k*) 1   k*                 A
                                           
                                           
                                           
     g            g                        
                   
                   
                    A   
                          
                          
                                      
                                      
                                            
                                              
                                              




                                                         slide 40
Golden Rule

 In steady state:   sg A(k*) k*
                          g      g


                               1
                     1/(1 ) 
     (k*)1   A 
               


                                       A
                             

sg  g        
                            


              A  
                            

        A      
               
                    
                             
                             
                             
                                       A
               
                            
                             




            sg 


                                              slide 41
       Transition to Golden Rule
   Case 1: s > sg, i.e., steady-state capital too
    high. Decrease s in order to reach sg
                                   k
                                   s2f(k)

                                   sgf(k)



                                        k
                kg *        k*                   slide 42
       Transition to Golden Rule
k                     y

k*                    y*

kg *                  yg *

                  t                   t
c                     i

cg *                  i*
c*
                      ig *

                  t                   t
                                   slide 43
       Transition to Golden Rule
   Case 2: s < sg, i.e., steady-state capital too
    low. Increase s in order to reach sg
                              k
                                    sgf(k)

                                    sf(k)



                                            k
                 k*         kg *                 slide 44
       Transition to Golden Rule
k                    y
kg *                 y*
                     yg *
k*

                 t                    t
c                    i
cg *                 ig*

c*                   i*

                 t                    t
                                   slide 45
Transition to Golden Rule

   If the economy begins above the
    golden rule (s too high), consumption
    increases in all future periods 
    decrease in s leads to welfare
    improvement
   If the economy begins below the
    golden rule (s too low), consumption
    falls during transition  there is a
    tradeoff between consuming today or
    in the future
                                            slide 46
         International Evidence on Investment
         Rates and Income per Person
Income per
person in 1992
(logarithmic scale)
           10 0,00 0


                                                                             Canada
                                                                        U.S.      De nm ark Ge rmany           J apan


             10 ,000                                                                                       Finland
                                                     Me xic o                                    U.K.
                                                                    Brazil                              Singapore
                                                                                  Israel
                                                                                       Franc eItaly
                                           Pak istan
                                  Egy pt          Ivory
                                                Coast                 Pe ru

                                                                   Indonesia
              1,0 00
                                                 India                Zim babwe
                                                                Ke nya
                                  Uganda
                           Chad            Came roon



                10 0
                       0            5       10            15          20          25          30          35            40
                                                                                   Investment as percentage of output
                                                                                   (average 1960 –1992)

                                                                                                                         slide 47
    Population Growth
   Assume that the population--and labor
    force-- grow at rate n. (n is exogenous)
                  L
                        n
                   L
   EX: Suppose L = 1000 in year 1 and the
    population is growing at 2%/year (n = 0.02).
    Then L = n L = 0.02  1000 = 20,
    so L = 1020 in year 2.

                                               slide 48
    Break-even investment
( + n)k = break-even investment,
    the amount of investment necessary
    to keep k constant.

Break-even investment includes:
    k to replace capital as it wears out
   n k to equip new workers with capital
    (otherwise, k would fall as the existing capital
    stock would be spread more thinly over a
    larger population of workers)
                                                   slide 49
    The equation of motion for k

   With population growth, the equation of
    motion for k is

       k = s f(k)  ( + n) k



       actual
    investment                   break-even
                                 investment

                                              slide 50
The Solow Model diagram
Investment,
break-even
              k = s f(k)  ( +n)k
 investment
                             ( + n ) k

                                  sf(k)




                       k*   Capital per
                            worker, k
                                          slide 51
        The impact of population
        growth
              Investment,
              break-even            (  + n2 ) k
               investment
                                          (  + n1 ) k
An increase in n
causes an                                          sf(k)
increase in break-
even investment,
leading to a lower
steady-state level
of k.

                            k 2*   k1* Capital per
                                       worker, k
                                                     slide 52
    Prediction:
   Higher n  lower k*.

   And since y = f(k) ,
    lower k*  lower y* .

   Thus, the Solow model predicts that
    countries with higher population growth
    rates will have lower levels of capital and
    income per worker in the long run.

                                              slide 53
            International Evidence on Population
            Growth and Income per Person
Income per
person in 1992
(logarithmic scale)
          100,000

                                  Ge rmany
                          De nm ark        U.S.
                                                    Canada

                                                                                          Israel
            10,000                               J apan   Singapore            Me xic o
                          U.K.
                                     Finland   Franc e
                             Italy
                                                                          Egy pt      Brazil

                                                                                                   Pak istan         Ivory
                                                                                     Pe ru                           Coast
                                                             Indonesia
             1,000                                                                           Came roon
                                                                                                               Ke nya
                                                                             India
                                                                                                         Zim babwe
                                                                      Chad                         Uganda



               100
                      0                        1                      2                       3                4
                                                                                       Population growth (percent per year)
                                                                                       (average 1960 –1992)

                                                                                                                             slide 54
55
     slide 55
   Clark 2005, p1308 Fig 1
                              slide 56
 The Golden Rule with Population
 Growth
To find the Golden Rule capital stock,
we again express c* in terms of k*:
 c* = y*         i*
    = f (k* )    ( + n) k*
c* is maximized when
      MPK =  + n
or equivalently,
     MPK   = n

                                         slide 57
    Technology Progress
   Rewrite the production function to
    incorporate technology change:
           Y  F (K , L E)
 E = efficiency of labor
 E  L = effective workers
Assume:
 Technological progress is labor-
 augmenting: it increases labor
 efficiency at the exogenous rate g:
                                         slide 58
Technology Progress

    Assume that E grows at rate g

    Therefore E  L grows at rate n + g

    Redefine all variables in terms of
     effective workers:
     k = K/EL = capital per effective
     worker
59
                                           slide 59
  Technology Progress

      Then y = Y/EL (= output per effective
       worker) is given by:
       Y / EL  F (K / EL,1)  y  f (k)
      Similarly for consumption and
       investment:
Y C  I Y / EL C / EL  I / EL  y  c i
C  (1 s)Y  C / EL  (1 s)Y / EL  c  (1 s) y
        I  sY  I / EL  sY / EL  i  sy
  60
                                                     slide 60
Technology Progress

    Therefore, the equations are the same as
     before
    The only change is in the law of motion for
     k. Capital per effective worker:
        Increases with investment
        Decreases with physical depreciation
        Also decreases because there are more
         effective workers to share the existing capital
         (higher L and E)

61
                                                      slide 61
  Technology Progress

      Then:
k i (  n  g)k  sf (k )(  n  g)k

      In steady-state, capital per effective
       worker is fixed:
            k 0
           sf (k*)  (  n  g)k *

  62
                                                slide 62
Technology Progress
                       ( +n+g)k
                       sf(k)




                           k
        k1   k*   k2
63
                                   slide 63
Technology Progress

    In steady state, income, consumption
     and investment per effective worker
     are also constant over time:

      y* f (k*)
     c* (1 s) y* (1 s) f (k*)
     i* sy* sf (k*)

64
                                            slide 64
Technology Progress

    Therefore capital, income,
     consumption and investment per
     worker grow at the rate g in steady-
     state:
      k* K */ EL  K */ L  Ek *
      y*Y */ EL Y */ L  Ey*
      c*C*/ EL  C*/ L  Ec*
      i* I */ EL  I */ L  Ei*
65
                                            slide 65
     Steady-State Growth Rates in the
     Solow Model with Tech. Progress
                                     Steady-state
     Variable          Symbol
                                     growth rate
 Capital per
                    k = K/ (L E )        0
 effective worker
 Output per
                    y = Y/ (L E )        0
 effective worker

Output per worker   (Y/ L ) = y E        g

  Total output      Y = y E L         n+g
                                               slide 66

                                                    slide 66
Technology Progress

    This follows since steady-state
     variables are constant and E is
     growing at the rate g

    Therefore, the inclusion of technology
     progress in the Solow model can
     generate sustained long-run growth

67
                                          slide 67
Technology Progress

    Moreover, total capital, output,
     consumption and investment grow at
     the rate n+g in steady state:
      K* ELk*, Y* EL y*
      C* ELc*, I* ELi*
     Given that steady-state variables are
     constant and EL is growing at the rate
     n+g
68
                                          slide 68
       Golden Rule
     Consumption per effective worker in
      steady state:
 c* (1 s) f (k*)  f (k*) sf (k*)  f (k*)(  n  g)k *
     Golden Rule: find k* s.t. c* is maximized:
       max c* f (k*)(  n  g)k *
            k*
dc*  f (k*)(  n g) 0  f (k*)  MPK   n g
           g                       g
dk *
       69
                                                       slide 69
         Government Policies to raise the rate
         of productivity growth

   Improving infrastructure
   Would increased infrastructure spending increase
    productivity?
    • There might be reverse causation: Richer countries
       with higher productivity spend more on infrastructure,
       rather than vice versa
    • Infrastructure investments by government may be
       inefficient, since politics, not economic efficiency, is
       often the main determinant



                                                             slide 70
      Government Policies to raise the rate of
      productivity growth

   Building human capital
     • There’s a strong connection between productivity and
       human capital
     • Government can encourage human capital formation
       through educational policies, worker training and
       relocation programs, and health programs
     • Another form of human capital is entrepreneurial skill
     • Government could help by removing barriers like red
       tape
   Encouraging research and development
     • Government can encourage R and D through direct aid
       to research

                                                            slide 71
        Why is technological breakthroughs progress
        so unequal across countries?

   What determined whether/when new
    technology adopted?
     Geography view: importance of ecology, climate,
      disease environment, geography, in short,
      factors outside human control.
     Institutions view: importance of man-made
      factors; especially organization of society that
      provide incentives to individuals and firms.
     History’s accidents: some countries are unlucky
      and trapped in underdevelopment.
         72
                                                   slide 72
The Geography Factor




73
                       slide 73
The Institutions Factor




74
                          slide 74
Institutions and Economic
Performances




75
                            slide 75
Institutions and Economic
Performances




76
                            slide 76
           But institutions are complicated:
           identification problem
   Good institutions are correlated with many other good things.
    Theories about institutions are thus very difficult to test.

   The study of the causal role of institutions on economic growth
    is therefore complicated by concerns about endogeneity.

   For example, the United States is rich; it has good institutions;
    it has high levels of education; it has a common law heritage; it
    has a temperate climate.

   Good institutions are difficult to pin down precisely. We want to
    be very careful to disentangle different causal effects and
    isolate the effect of interest.


          77
                                                                  slide 77
      But institutions are also
      endogenous
 Institutions could vary because underlying
  factors differ across countries: Geography,
  ecology, climate
 Montesquieu’s story:

  – Geography determines “human attitudes”
  – Human attitudes determine both economic
  performance and political system.
  – Institutions potentially influenced by the
  determinants of income
      78
                                             slide 78
Factor Prices

    So far, we solved the model without
     any reference to wages and rental
     rates (factor prices)
    We just focused on how income is
     generated, but not on how it is
     distributed

    Assume that a competitive firm hires
     capital and labor to generate output
79
                                            slide 79
Factor Prices

    Assuming Cobb-Douglas technology:

        Y  F (K , EL)  AK (EL)1

    Then the problem for this firm is given
     by:
     max  PAK  (EL)1  w.L R.K
     K, L


80
                                           slide 80
Factor Prices

    First-order condition for K implies that:

      R / P  MPK  F / K  AK 1(EL)1
      R / P  A (K / EL) 1  Ak 1
    In steady-state, the real rental rate is
     fixed (since k is fixed)


81
                                                slide 81
Factor Prices

    First-order condition for L implies that:

       w/ P  MPL  F / L  AK (EL) E
       w/ P  A(1 )(K / EL) E  A(1 )k E

    In steady-state, the real wages
     increase at the rate g (since k is fixed
     and E grows at the rate g)

82
                                                 slide 82
 Factor Prices

     Assume that capital is initially below
      the steady-state. Then k will evolve
      according to the following path:

k
k*




 83                      t
                                               slide 83
  Factor Prices

      Rental rate:
        initially high (low k implies high MPK)
        decreases over time as capital
         accumulates and MPK decreases
R/P




  84                       t
                                                   slide 84
Factor Prices

    Define wage in terms of efficiency units as:

            w  w/ P  A(1 )k
            ~
                 E
         ~
     Then w:
        initially low (low k implies low MPL  labor
         abundant relative to capital)
        increases over time as capital accumulates
         and MPL increases
        constant in steady state

85
                                                        slide 85
Factor Prices
     ~
     w




                           t
    This means that real wages (w/P):
      Grow faster than g during the
       transition
      Grow at the rate g in steady-state
86
                                            slide 86
Growth Accounting

 Want to be able to explain why and
  how countries grow
 There are many sources of growth

 First step is to decompose aggregate
  growth into its components:
      Growth in the labor force
      Growth in capital
      Growth in productivity

87
                                         slide 87
Sources of Economic
Growth
    Assume Cobb-Douglas Production
     Function
                          1
               Y  zK L
    Take log and differentiating
      dY dz  dK            dL
              (1   )
      Y   z   K             L


88
                                      slide 88
Computing TFP

 z: Total Factor Productivity (TFP) or
  “Solow Residual”
 Y is GDP, K is aggregate capital, N is
  number of workers
 Need to know α




89
                                           slide 89
What is “z”

 Human Capital (Education)
 Technological Progress

 Externality: environmental Issues

 Institutional Effect
      Firm Organization
      Patent Protection
      Corruptions


90
                                      slide 90
Labor Share in the Cobb-
Douglas Production Function
    Firm optimization:
            max( zK  N 1  wN  rK )
    First-order condition with respect to N:
               w  (1   ) zK  N 
    Labor share is wN/Y. Here:
        wN / Y  (1   ) zK  N 1 / Y  1  


91
                                                   slide 91
Result

    Can use average labor share as
     measure of 1  
      Labor share (total wages divided by
       GDP) in the U.S. is about 64%
      Estimate α to be 0.36

    Can now compute TFP as:
              z  Y / (K N )
                       .36   .64




92
                                             slide 92
Total Factor Productivity in
the U.S.




93
                               slide 93
Decomposing Growth Rates

    Taking logs of production function:
       log Y  log z  0.36log K  0.64log N
    The same applies to log differences:
      log Y   log z  0.36 log K  0.64 log N
    Log differences are approximately
     equal to percentage changes:
 %Y  %z  0.36%K  0.64%N

94
                                                slide 94
Growth Decomposition for
the U.S.




95
                           slide 95
Growth Decomposition for
the Asian Tigers




96
                           slide 96
Growth Accounting for China




                              slide 97
Human Capital in China




98
                         slide 98
Human Capital in China




99
                         slide 99
   Technology in China
Innovation
   New Goods
   Patents




   100
                         slide
                         Management and Productivity
                     6
 Patents 1996-2004
                     4
                     2
                     0




                         1      1.5    2         2.5         3          3.5   4   4.5
                             Management Score                                           101
Note: European firms only as uses the European Patent Office database                         slide
    Policies to promote growth

    Saving Rate
    Human capital investment
    Encouraging technological progress
    Right Institutions




    102
                                          slide
    Growth empirics: Confronting
    the Solow model with the facts
Solow model’s steady state exhibits
balanced growth - many variables grow
at the same rate.
   Solow model predicts Y/L and K/L grow at
    same rate (g), so that K/Y should be constant.
    This is true in the real world.
   Solow model predicts real wage grows at same
    rate as Y/L, while real rental price is constant.
    Also true in the real world.
    103
                                                   slide
    Convergence
   Solow model predicts that, other things
    equal, “poor” countries (with lower Y/L and
    K/L ) should grow faster than “rich” ones.
   If true, then the income gap between rich &
    poor countries would shrink over time, and
    living standards “converge.”
   In real world, many poor countries do NOT
    grow faster than rich ones. Does this
    mean the Solow model fails?
    104
                                                  slide
     Convergence
   No, because “other things” aren’t equal.
     In samples of countries with similar savings
      & pop. growth rates,
      income gaps shrink about 2%/year.
     In larger samples, if one controls for
      differences in saving, population growth, and
      human capital, incomes converge by about
      2%/year.


     105
                                                      slide
Convergence

     What the Solow model really predicts is
      conditional convergence - countries
      converge
      to their own steady states, which are
      determined by saving, population
      growth, and education.
      And this prediction comes true in the real
      world.


106
                                                   slide

				
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