1 Teaching Convergence: what should undergrads know? Federico Guerrero Department of Economics University of Nevada

2 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?

3 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.

4 Three numerical examples (1) Example 1: Group RGroup 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

5 Three numerical examples (2) Example 2: Group RGroup 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.

6 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.

7 Three numerical examples (3) Example 3: Group RGroup 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

8 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.

9 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.

10 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; present). Therefore, there is no evidence of Sigma- convergence at the world level.