Economic Growth I

1
Economic Growth I

• Chapter Seven

2
Introduction

• Having analyzed the overall production,
  distribution, and allocation of national income,
  we now consider the determinants of long-run
  growth.
• Stylized fact in developed economies, output
  grows over time (although irregular at times)
  the trend is upward
• Different countries also enjoy very different
  standards of living in terms of income per
  person standard of living means what?
• Our goal is to understand what causes these
  differences in income over time and across
  countries.
• What determines a countrys output at any point
  in time? So where must the differences across
  countries come from?
• Solow growth model shows how saving, population
  growth, and technological progress affect the
  level of an economys output and its growth over
  time

3
International Differences in the Standard of
Living 1999
4
Income and poverty in the world selected
countries, 2000
5
The Accumulation of Capital

• Starting with the production function, Y
  F(K,L), what are the 3 possible sources of
  long-run output growth
• Increase in capital stock, K
• Increase in labor force (population increase), L
• Increase in technology the production function F
  changes
• Our analysis of economic growth considers all of
  these factors, but focuses primarily on the
  determination of the capital stock.
• Assumption there is no technological progress
  and no growth in population we relax these later
• What is the fundamental difference between our
  analysis of economic growth and our previous
  analysis of income determination? Static vs.
  dynamic?

6
The Supply and Demand for Goods

• The Supply of Goods and the Production Function
• Supply of goods depends on production function, Y
  F(K,L)
• F exhibits constant returns to scale, zY
  F(zK,zL)
• z 1/L ? Y/L F(K/L,1)
• The amount of output per worker, Y/L, is a
  function of the amount of capital per worker,
  K/L.
• Does the size of labor force affect the
  relationship between output per worker and
  capital per worker?
• Write all variables in per-worker terms y Y/L,
  k K/L, yf(k)
• Example Y (KL)1/2 y f(k) f(k) ?

7
The Production Function

• What does the slope of this per-worker production
  function represent?
• MPK f(k1) f(k)
• Why is it that as the amount of capital per
  worker increases, the production function becomes
  flatter?
• When k is small (large), is MPK large (small)?
  Why?

8
The Supply and Demand for Goods

• The Demand for Goods and the Consumption
  Function
• The demand for goods in the Solow model comes
  from consumption and investment.
• Y/L C/L I/L ? y c i output per worker is
  divided between consumption per worker and
  investment per worker
• The Solow model assumes that each year people
  save a constant fraction s of their income and
  consume (1-s)
• c (1-s)y what assumptions have we made thus
  far about demand? G T NX 0
• What does this consumption function imply about
  investment? y (1-s)y i ? i sy investment
  equals saving, what is adjusting to ensure these
  two equate?
• For a given k, what determines per capita output?
  What determines the allocation of output between
  consumption and investment?

9
Growth in the Capital Stock and the Steady State

• Capital stock is a key determinant of output, if
  capital grows over time then so will output
• 2 forces that influence change in the capital
  stock
• Investment expenditure on new plant/equipment
  and causes capital stock to rise
• Depreciation wearing out of old capital, causes
  capital to fall
• i sy ? i sf(k) investment per worker is a
  function of capital stock per worker Figure 7-2
• What governs output? What governs output
  allocation?
• We assume that a certain fraction ? of the
  capital stock wears out each year ? -
  depreciation rate
• How much capital depreciates every year? ?k

10
Output, Consumption, and Investment
11
Depreciation
12
Capital accumulation
The basic idea Investment makes the capital
stock bigger, depreciation makes it smaller.
13
Growth in the Capital Stock and the Steady State

• The overall change in the capital stock is the
  net effect of new investment and depreciation
• ?k i - ?k sf(k) - ?k
• Figure 7-4, The higher the capital stock the
  higher the amount of output, investment, and
  depreciation
• At what level of capital is investment
  depreciation?
• If the economy finds itself at this capital
  stock, k, will the capital stock continue to
  change? Why or why not?
• The only investment being undertaken is
  replacement investment.
• At k, ?k 0, so k and yf(k) are steady over
  time.
• Thus, k is called the steady-state level of
  capital.

14
Investment, Depreciation, and the Steady State
15
Growth in the Capital Stock and the Steady State

• The steady state level of capital is significant
  for two reasons
• An economy at the steady state will stay there.
• An economy not at the steady state will
  eventually go there regardless of the level of
  capital with which the economy begins.
• The steady state represents the long-run
  equilibrium of the economy.
• Suppose economy starts with k1 lt k, why will the
  capital stock rise?
• Suppose economy start with k2 gt k, why will the
  capital stock fall?

16
Approaching the Steady State A Numerical Example

• Production Function Y (KL)1/2
• Derive the per-worker production function f(k).
• Assume 30 of output is saved and the capital
  stock depreciates at a rate of 10.
• Assume the economy starts off with 4 units of
  capital per worker.
• See Excel Worksheet
• Over time what level of capital stock, output,
  consumption, investment, and depreciation does
  the economy approach?
• Is there another way to derive the steady-state
  without so many calculations?

17
Case Study The Miracle of Japanese and German
Growth

• Japan and Germany experienced rapid economic
  growth following World War II.
• The war destroyed a large portion of their
  capital stocks (plants, equipment, heavy
  machinery).
• Between 1948 and 1972 real GDP per capita grew at
  8.2 per year in Japan and 5.7 per year in
  Germany while the U.S. experienced a meager 2.2
  per year in comparison.
• Does this make any sense from the standpoint of
  the Solow growth model? What happens to output
  after a collapse in the capital stock? What
  happens to saving and investment? Should output
  begin to grow at a faster rate? Why or why not?

18
How Saving Affects Growth

• Low levels of initial capital is not the only
  thing that affects the rate of economic growth
  the fraction of output devoted to
  saving/investment affects economic growth
• Consider an increase in the saving rate from s1
  to s2
• What happens to the investment schedule?
• At the initial saving rate s1, and the initial
  capital stock k1, the amount of investment just
  offsets what?
• What happens immediately after the saving rate
  rises?
• Where will the new steady-state end up?
• The Solow model shows that the saving rate is a
  key determinant of the steady-state capital
  stock. If the saving rate is high (low), the
  economy will have a large (small) capital stock
  and a high (low) level of output.

19
How Saving Affects Growth

• What does the Solow model say about the
  relationship between saving and growth?
• Is the relationship permanent or temporary?
• Can we more fully explain the impressive
  performance of Japan and Germany after WWII?

20
Case Study Saving and Investment Around the World

• Revisit why are some countries rich and some
  poor?
• What answer does the Solow model provide?
• Does international data support this theoretical
  result?
• The data clearly show a positive relationship
  between the fraction of output devoted to
  investment and the level of per capital income.

21
The Golden Rule Level of Capital

• The Solow model shows how the rate of saving and
  investment determines the long-run levels of
  capital and income. Is higher saving always a
  good thing since it always leads to higher
  income?
• What amount of capital accumulation is optimal
  from the standpoint of economic well-being?
• Assume that we can set our nations savings rate,
  what rate should we choose? What should be our
  goal?
• Policymakers should aim for a savings rate that
  delivers a steady state with the highest level of
  consumption possible.
• The steady-state value of k that maximizes
  consumption is called the Golden Rule level of
  capital, kg.

22
Comparing Steady States

• Where is the golden rule level of capital?
• Steady-state consumption y c i ? c y i
• Substitute steady-state values for output and
  investment c f(k) - ?k
• Increase in steady-state capital has two opposing
  effects, what are they?
• Steady-state consumption is the gap between
  output and depreciation (investment)
• kg is the capital level that maximizes s.s.
  consumption

23
Comparing Steady States

• What happens to output and depreciation when we
  increase the capital stock when capital is below
  the golden rule level? (Figure 7-7)
• Do the relative slopes of the production function
  and the depreciation schedule tell us anything?
• What does this imply about consumption?
• What happens when the capital stock is above the
  golden rule level?
• Again, do the relative slopes give us any
  information?
• What happens to consumption?
• At the golden rule level of capital what is the
  relationship between the slopes of the production
  function and the depreciation schedule?

24
Comparing Steady States

• Because the 2 slopes are equal at kg, the golden
  rule is described by MPK ?
• Suppose s.s. capital is k and we are considering
  increasing capital to k1
• How much extra output is produced?
• How much extra depreciation?
• What is the net effect on consumption?
• What should we do if MPK-? lt 0? MPK-? gt0?

25
Comparing Steady States

• At the golden rule level of capital, the marginal
  product of capital net of depreciation (MPK - ?)
  equals zero.
• Will the economy naturally gravitate towards the
  golden rule steady-state level of capital?
• If a policymaker wants a specific steady-state
  capital stock, such as the golden rule, the
  appropriate savings rate must be used to support
  it.
• What happens when the saving rate is set below
  (above) the one required to support the golden
  rule?
• What happens to the steady-state capital stock?
• What happens to the steady-state consumption?

26
Finding the Golden Rule Steady State A Numerical
Example

• Per-worker production function y k1/2, ? 0.1
• The policymaker chooses s in order to maximize
  consumption.
• In the steady-state sf(k) ?k
• k/f(k) s/? ? k/(k)1/2 s/0.1 ? k
  100s2
• What happens to steady-state capital, output, and
  depreciation as the savings rate climbs?
• What happens to consumption? What is the golden
  rule savings rate? What is net marginal product
  of capital?
• Why does net marginal product of capital
  eventually become zero?
• Is there an easier way to find the golden rule
  level of capital, consumption, and saving?
  Perhaps using calculus

27
The Transition to the Golden Rule Steady State

• So far we have assumed that the policymaker can
  choose any savings rate and the economy will jump
  directly to the golden rule steady state
  unrealistic assumption
• Rather, suppose economy has reached a steady
  state other than golden rule
• What happens to consumption, investment, and
  capital when the economy transitions between
  steady states?
• Are there any undesirable consequences in the
  transition process that might deter policymakers?
• Consider two cases the economy begins with more
  capital than in the golden rule steady state, or
  with less
• The two cases offer very different problems for
  policymakers.

28
Starting With Too Much Capital

• With too much capital, what should policymaker do
  to approach the golden rule?
• What happens immediately following a reduction in
  the savings rate? Why?
• What happens to c, y, and i over time?
• Is there anything noticeable about the path of
  consumption?
• When k gt kg, reducing saving is a good policy
  it increases consumption at every point in time

y
c
i
t0
29
Starting With Too Little Capital

• With too little capital, what should policymaker
  do to approach the golden rule?
• What happens immediately and over time to y, c,
  and i?
• Is there anything noticeable about the path of c?
  
• Does there appear to be any tradeoff between
  current and future economic well-being?
• When k lt kg reaching the golden rule requires an
  initial reduction in consumption which will rise
  over time.

y
c
i
t0
time
30
Population Growth

• Does the basic Solow model explain sustained
  (permanent) growth in output?
• To explain permanent growth in output we must
  augment the basic model with population growth.
• Assume population grows at a constant rate n
• Example n .01 ? population grows at 1 per
  year
• What now are the 3 forces acting on the stock of
  capital to drive it towards a steady-state? How
  does population growth specifically change
  capital per worker?
• Derive ?k i (?n)k what do i, ?, and n do
  to k?
• (?n)k break even investment the amount of
  investment needed to keep the capital stock per
  worker constant

31
The Steady State With Population Growth

• Why does break even investment include the term
  nk?
• How does population growth reduce k as opposed to
  depreciation?
• k satisfies ?k 0 ? sf(k) (?n)k ? s/(?n)
  k/f(k)
• What happens if k lt (gt) k?
• Once the economy is in the steady state,
  investment has 2 purposes. What are they?

32
The Effects of Population Growth

• Population growth alters the basic Solow model 3
  ways
• It explains sustained economic growth, but does
  it explain sustained growth in the standard of
  living?
• It gives another reason for why some countries
  are rich and some are poor. How?
• It affects the criterion for determining the
  golden rule level of capital. What is the new
  golden rule condition?

33
Case Study Population Growth Around the World

• Does the Solow model tell us anything about the
  correlation between high population growth and
  low steady-state income per worker?
• Why would high population growth tend to
  impoverish a country?
• Does the international data support this theory?

34
Chapter Summary

• The Solow growth model shows that, in the long
  run, a countrys standard of living depends
• positively on its saving rate.
• negatively on its population growth rate.
• An increase in the saving rate leads to
• higher output in the long run
• faster growth temporarily
• but not faster steady state growth.
• If the economy has more capital than the Golden
  Rule level, then reducing saving will increase
  consumption at all points in time, making all
  generations better off. If the economy has less
  capital than the Golden Rule level, then
  increasing saving will increase consumption for
  future generations, but reduce consumption for
  the present generation.