TECHNOLOGICAL PROGRESS AND GROWTH: THE GENERAL SOLOW MODEL

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Chapter 5 first lecture
Introducing Advanced Macroeconomics Growth and
business cycles
TECHNOLOGICAL PROGRESS AND GROWTH THE GENERAL
SOLOW MODEL
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The general Solow model

• Back to a closed economy.
• In the basic Solow model no growth in GDP per
  worker in steady state. This contradicts the
  empirics for the Western world (stylized fact
  5). In the general Solow model
• Total factor productivity, , is assumed to
  grow at a constant, exogenous rate (the only
  change). This implies a steady state with
  balanced growth and a constant, positive growth
  rate of GDP per worker.
• The source of long run growth in GDP per worker
  in this model is exogenous technological growth.
  Not deep, but
• its not trivial that the result is balanced
  growth in steady state,
• reassuring for applications that the model is in
  accordance with a fundamental empirical
  regularity.
• Our focus is still
• what creates economic progress and prosperity

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The micro world of the Solow model

• is the same as in the basic Solow model, e.g.
• The same object (a closed economy).
• The same goods and markets. Once again, markets
  are competitive with real prices of 1, and
  , respectively. There is one type of output (one
  sector model).
• The same agents consumers and firms (and
  government), essentially with the same behaviour,
  specifically one representative profit
  maximising firm decides and given and
  .
• One difference the production function. There is
  a possibility of technological progress The
  full sequence is exogenous and for
  all . Special case is (basic Solow
  model).

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The production function with technological
progress

• with a given sequence,
• with a given sequence,
• With a Cobb-Douglas production function it makes
  no difference whether we describe technical
  progress by a sequence, , for TFP or by the
  corresponding sequence, , for labour
  augmenting productivity.
• In our case the latter is the most convenient.
  The exogenous sequence, , is given by
• Technical progress comes as manna from heaven
  (it requires no input of production).

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• Remember the definitions and
  .
• Dividing by on both sides of
  gives the per capita production function
• From this follows
• Growth in can come from exactly two
  sources, and is the weighted average of
  and with weights and .
• If, as in balanced growth, is constant,
  then !

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The complete Solow model

• Parameters . Let .
• State variables and .
• Full model? Yes, given and the model
  determines the full sequences

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• Note That is capitals share ,
  labours share , pure profits . Our
  should still be around .
• Also note defining effective labour input as
  The model is matematically equivalent
  to the basic Solow model with taking the
  place of , and taking the place of ,
  and with !We could, in principle, take
  over the full analysis from the basic Solow
  model, but we will nevertheless be

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Analyzing the general Solow model

• If the model implies convergence to a steady
  state with balanced growth, then in steady state
  and must grow at the same constant rate
  (recall again that is constant under
  balanced growth). Remember also Hence if
  , then . If there is
  convergence towards a steady state with balanced
  growth, then in this steady state and will
  both grow at the same rate as and
  hence and will be constant.
• Furthermore from the above mentioned equivalence
  to the basic Solow model,
  and
  converge towards constant steady state values.
• Each of the above observations suggests analyzing
  the model in terms of

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1. and
2. From we get
3. From and we get
4. Dividing by on
   both sides gives
5. Inserting gives the transition
   equation
6. Subtracting from both sides gives the Solow
   equation

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Convergence to steady statethe transition
diagram

• The transition equation is
• It is everywhere increasing and passes through
  (0,0).
• The slope of the transition curve at any is
• We observe .
  Furthermore,
  . We assume that the latter very plausible
  stability condition is fulfilled.

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• The transition equation must then look as
  follows

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• Convergence of to the intersection point
  follows from the diagram. Correspondingly
  .
• Some first conclusions are
• In the long run, and
  converge to constant levels, and ,
  respectively. These levels define steady state.
• In steady state, and both grow at the same
  rate as , that is, at the rate and the
  capital output ratio, ,
  must be constant.

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Steady state

• The Solow equationtogether with
  gives
• Using and we get the
  steady state growth paths

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• Since ,
• It also easily follows from
• that
• There is balanced growth in steady state
  and grow at the same constant rate, , and
  is constant.
• There is positive growth in GDP per capita in
  steady state (provided that ).
  

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Structural policies for steady state

• Output per capita and consumption per capita in
  steady state are
• Golden rule the , that maximises the entire
  path, . Again .
• The elasticities of wrt. and
  are again and ,
  respectively.
• Policy implications from steady state are as in
  the basic Solow model encourage savings and
  control population growth.
• But we have a new parameter, ( corresponds
  to ). We want a large , but it is not easy
  to derive policy conclusions wrt. technology
  enhancement from our model ( is exogenous).

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Empirics for steady state

• Assume that all countries are in steady state in
  2000!
• Its hard to get good data for , so make the
  heroic assumption that is the same for all
  countries in 2000.
• Set (plausibly)
• If is GDP per worker in 2000 of country ,
  the above equation suggests the following
  regression equationwith and measured
  appropriately (here as averages over 1960-2000),
  and where

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• An OLS estimation across 86 countries gives

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• High significance! Large R2! Even though we have
  assumed that is the same in all countries!
• But always remember the problem of correlation
  vs. causality.
• Furthermore the estimated value of is not in
  accordance with the theoretical (model-predicted)
  value of . Or
• The conclusion is mixed the figure on the
  previous slide is impressive, but the figures
  line is much steeper than the model suggests.