Teaching Convergence: what should undergrads know

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Teaching Convergence what should undergrads know?

• Federico Guerrero
• Department of Economics
• University of Nevada

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Relevant questions

• Will poor countries eventually catch up to the
  levels of GDP per head characterizing the rich
  nations?
• Will the rich nations of, say, 2050 be the same
  as the ones that are wealthy today?
• Will the world distribution of income continue to
  worsen on a cross-country basis --as it has since
  the early 1800s? Or will it get better?

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Two concepts of convergence

• Beta convergence We say that there is
  beta-convergence if poor countries tend to grow
  faster than rich ones
• Sigma convergence We say that there is
  sigma-convergence if the cross-sectional standard
  deviation of real GDP per head for a group of
  economies --the ones in our sample-- is falling
  over time.

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Three numerical examples (1)

• Example 1
• Group R Group P
• Time 0 10,000 2,000
• Time T 5,000 4,000
• Conclusion 1 the rate of growth for group R is
  negative -50 the rate of growth for group P is
  positive 100. There is Beta-convergence.
• Conclusion 2 the distance between groups R and P
  has shrunk over time from 8,000 as of time 0 to
  1,000 as of time T (the standard deviation fell
  from 5,657 to 707). There is Sigma convergence

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Three numerical examples (2)

• Example 2
• Group R Group P
• Time 0 5,000 4,000
• Time T 10,000 2,000
• Conclusion 1 the rate of growth of real GDP for
  group R is positive (100 between time periods 0
  and T) the rate of growth for group P is
  negative (-50 between time periods 0 and T).
  There is a lack of Beta-convergence in this
  example.
• Conclusion 2 the distance in income per head
  increases from 1,000 as of time 0 to 8,000 as
  of time T (and the standard deviation increases
  from 707 to 5,657 between time periods 0 and
  T). There is a lack of Sigma convergence in this
  example.

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Corollary from examples 1 and 2

• It is not possible for the income gap between
  groups R and P to narrow down if the initially
  poor, P, does not grow faster than the initially
  rich, R.
• In other words, Beta-convergence is a necessary
  condition for sigma convergence.

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Three numerical examples (3)

• Example 3
• Group R Group P
• Time 0 10,000 5,000
• Time T 5,000 10,000
• Conclusion 1 the rate of growth of real GDP per
  head for group P is positive (actually, 100)
  the rate of growth for group R is negative
  (actually, -50). Therefore, we have
  Beta-convergence in this example.
• Conclusion 2 the distance has not changed it
  was 5,000 as of time 0 and it still is 5,000 as
  of time T (the standard deviation stayed the same
  at 3,536 between time periods 0 and T). There
  is not sigma convergence

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Corollary from example 3

• Beta-convergence is not a sufficient condition
  for sigma-convergence.
• In other words, P growing faster than R is not
  enough to guarantee a fall in the standard
  deviation of GDP per head in the cross-section.

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What are the implications of the standard growth
model?

• There is confusion around this issue.
• Some authors claim that the neoclassical model of
  growth implies absolute Beta-convergence. That
  claim is incorrect.
• BUT
• The Solow-Swan model only predicts conditional
  Beta-convergence. Only if the parameters
  characterizing the steady states of R and P are
  the same both groups will share the same steady
  state. In that case, and only in that case, the
  initially poor, P, will grow faster than the
  initially rich, given diminishing returns to
  capital accumulation.

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What the data say

• There is evidence of conditional Beta-
  convergence within homogeneous regions. The speed
  of convergence has been estimated to be quite low
  (the gap narrows down at a rate of 2-3 per year,
  the so-called Iron law of convergence)
• There is also evidence of absolute or
  unconditional Beta-divergence at the world level
  at different horizons (1830-present
  1950-present).
• Therefore, there is no evidence of
  Sigma-convergence at the world level.