CAPITAL ACCUMULATION AND GROWTH: THE BASIC SOLOW MODEL

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Chapter 3 first lecture
Introducing Advanced Macroeconomics Growth and
business cycles
CAPITAL ACCUMULATION AND GROWTH THE BASIC SOLOW
MODEL
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The basic Solow model

• How can a nation become rich, i.e., initiate a
  growth process leading to higher GDP/consumption
  per capita in the long run.
• The basic Solow model provides some first
  answers It predicts how the evolution and the
  long run levels of GDP and consumption per capita
  depend on structural parameters such as the rate
  of investment and the growth rate of the labour
  force.
• Key elements of the Solow model
• In each period output is determined by the
  supplies of capital and labour through the
  production function
• Exogenous savings/investment rate, s, exogenous
  growth rate of labour force, n, and exogenous
  depreciation rate, d.
• Explicit description of capital accumulation
• Accumulation of capital is the main driving force
  for wealth. Basic model No technological
  progress.

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The micro world of the Solow model

• Object Closed economy.
• Time A sequence of periods/years,
• Agents Households and firms (and government).
• Commodities and markets Output, capital services
  and labour services (one asset physical
  capital).
• The market for output Supply firms output,
  . Demand from households for consumption and
  investment . Relative price 1.
• One-sector model Output can be used either for
  consumption or for investment.

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• The market for capital services Consumers own
  the capital stock, , and rent its services to
  firms.Supply of capital services . Firms
  demand
• Relative price (in units of output) for renting
  one unit of capital for one period real
  rental rate for capital.
• Real interest rate , where is
  the rate of depreciation. Or .
• (Alternative interpretation The firms own the
  capital, borrow for the purchase of capital at an
  interest rate of and bear the cost of
  depreciation themselfes).
• User cost .
• Labour market Households supply ( the
  labour force). Demand from firms . Relative
  price the real wage rate.
• Competitive markets and adjust to equate
  supply and demand in all markets full (or
  natural) utilization of ressources.

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The production side

• is modelled as if all production (all of GDP)
  comes from one profit maximizing firm that
  produces value added, , from of capital
  services, (machine-years), and labour
  services, (man-years), according to the
  production function
• Constant returns to scale
  . The replication argument!
• Positive marginal products
  

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• Marginal products are decreasing in the amount of
  the factor used(diminishing returns) and
  growing in the amount of the other factor used
• Profit maximization Given and , the firm
  chooses and toThe ususal
  necessary conditions for an optimum(These two
  equations do not determine and from
  given and , they only determine )

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• Competitive market clearing and
  , where and are the supplies in
  period tSince and are predetermined in
  any given period, and are determined this
  way. and predetermined what does that
  mean?

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The income distribution

1. No pure profits. Eulers rule
2. The functional income distributionThe income
   share of each factor is the elasticity of the
   production function with respect to the factor in
   question.

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• According to empirics, labours share is
  relatively constant around 2/3 over long periods
  except for short run fluctuationsLabours
  share of domestic factor incomes

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• Is there a production function that fulfills all
  of our assumptions and has fixed output
  elasticities independently of and ? Yes,
  the Cobb-Douglas production function
• where is total-factor-productivity (TFP).
• Check It follows thatThe Cobb-Douglas
  function seems as a realistic long run
  assumption. We even have reason to believe that
  .

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Households

• The number of households in period is ,
  which is predetermined. Household behaviour
• Each supplies one unit of labour inelastically.
  Total supply .
• Own the capital stock, , which is
  predetermined in period . Supply (as
  long as ).
• The representative household decides given
  , and hence
  . The intertemporal budget constraint We
  assume that the result of the consumers
  considerations is
• Biology

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The complete model

• Parameters and . NB No subscript
  on Basic Solow model.
• Endogenous variables
  andof which and are state
  variables

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• Given and the model determines
• Government in the model? Yes, simply interpret
  as private plus government savings.
• We viewed the capital accumulation equationas
  the households budget constraint.

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• Alternatively By definition we have that
  
• The condition for equilibrium in the output
  market, or the national accounting identity, is
  , and by definition .
  HenceCombining and gives the
  capital accumulation equation again is the
  savings rate and the investment rate.

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Analysing the basic Solow model

• Define and .
• From we get the per capita
  production function
• Note

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• Insert into
  to get
• Divide by on both sides to
  find that
• Insert to get the transition
  equation
• Subtracting from both sides of the transition
  equation gives the Solow equation

Replacement investment to compensate for
depreciation and growth of labour force
technical term appearing because of discrete
time
savings per capita
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The transition diagram

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• About the slope,
• It is everywhere strictly positive
• It goes to infinity as goes to zero
• It goes to as , and
  .
  Indeed we assume that (realistic!)
• The figure shows that in the long run.
  Hence ,
  and etc.
• The values etc. define the steady state.

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The Solow diagram

• Why does growth in and have to stop?
  Diminishing returns!

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Steady state

• The long run levels , etc. depend on
  parameters. How? What makes a nation rich?
• Look at the Solow equationIn steady state
  . Insert this to find

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• Some sharp predictions of the Solow model
• The elasticity of wrt. is
  (since we believe that ) an increase in
  of 10, e.g. from 20 to 22, should give an
  increase in of 5!
• The elasticity of wrt. and wrt.
  are and , respectively. Why is
  the latter not one? Capital accumulation!
• We have reached empirically testable hypotheses!
  Empirics

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Real GDP per worker against the average
investment share across 85 countries
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• Real GDP per worker against the average annual
  labour force growth rate across 85 countries