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Unformatted text preview: Economics 3213 Answers to Problem Set 2: The Jungle Book
Prof. Xavier Sala-i-Martin
1. The Bare Necessities
i. The finding that the growth rate of real per capita GDP shows little relation to the initial level of real per capita GDP does not conflict with the neoclassical model. The Solow-Swan model predicts conditional, rather than absolute, convergence. Poor countries are probably very different from the rich ones in terms of their rates of saving s, population growth n, depreciation !, technology A, and even income share ". Hence they have different steady states. ii. Absolute convergence simply means that poor countries tend to grow faster than rich ones. If there is a negative relation between income per capita and growth rate, this is absolute convergence. Notice that if poor countries are growing faster, eventually they will catch up with the rich ones and then will all grow at the same rate. That's why it's called absolute convergence, or simply convergence. Conditional convergence means that poor countries grow faster than rich ones only if they have similar parameters (saving rate, population growth, technology, and depreciation). Notice that it implies that they have similar steady states. Hence poor and rich countries will converge only if they have the same steady states. Romer's AK model does not predict any kind of convergence. In the AK model, if two countries have the same parameters A, s, n, and !, they will be growing at the same rate forever. If one is richer than the other, the poor will never catch up. If, on the contrary, they have different parameters and hence different growth rates, the country with the higher growth rate will grow faster forever. The high-growth country will be eventually richer than the low-growth one, even if it was initially poor. Hence, there is no convergence. 2. Colonel Hathi's March
i. When people are poor (that is, when they have low k), they may be willing to sacrifice consumption in order to save, invest, and consume more in the future (like East Asian Tigers 40 years ago). When people are rich, they may prefer to consume now (like the U.S. now). If this is the case, the saving rate is higher for low k and lower for high k. In other words, s(k) is a decreasing function of k, where the first derivative is negative: s'(k) < 0. ii. A model with the AK technology and saving as a decreasing function of k will predict convergence. Since s(k) is a decreasing function of k, s(k)A will also be a decreasing function of k. If we assume that s(k) approaches 0 when k grows bigger and bigger, s(k)A must cross (n + !), which is still a constant. This will be a steady state. If a poor and a rich country have the same parameters, they will converge to the same steady state. Therefore, this is conditional convergence. , where s'(k) < 0. iii. The Solow-Swan model with a decreasing saving rate still predicts convergence. Now either , where s'(k) < 0, or , where s is a constant. The function s(k)A/k1-", where s(k) is decreasing in k, decreases faster than sA/k1-", where s is a constant. The two curves s(k)A/k1-" and sA/k1-" may intersect at the steady state, giving the same steady state whether the saving rate is decreasing or constant (Case 1 below). They may intersect at a point to the left of k*, giving a new lower steady state k** (Case 2), or to the right of k*, giving a higher steady state k*** (Case 3). They may not intersect at all, giving either a lower or a higher steady state (Cases 4 and 5). It all depends on the particular constant saving rate (e.g. s = 0.1 or 0.45) and the particular decreasing function of k (e.g. s (k) = 1/k or 2/(5k2)). If we are interested in comparing the speed of convergence, we must select two points that are located the same distance away from the old and the new steady state (k1 and k2 in the graphs). Since the saving curve is steeper when s(k) is decreasing in k, convergence from k2 to k** or k*** is always faster than from k1 to k*. Therefore, convergence is faster when saving is a decreasing function of k. 3. I Wanna Be Like You
i. Fertility is usually a decreasing function of k. In general, people in rich countries tend to have fewer children than in poor ones. As a country gets richer (that is, k grows), the wages of its residents also increase. Since raising children requires a lot of humans' time, especially women's time, having children represents a lot of forgone wages and hence a lot of forgone consumption. Having children becomes more expensive in terms of forgone income as a country becomes richer. In addition, poor countries typically do not have wellfunctioning social security systems and retirement benefits. Hence people in poor countries choose to have more children, so that at least one of them can take care of parents when parents become old or disabled. ii. Mortality is usually a decreasing function of k. In general, people in rich countries live longer than in poor ones. As a country gets richer, people are able to invest more in healthcare, thereby reducing the mortality rate. iii. If we define net migration as immigration (moving in) minus emigration (moving out), net migration is usually an increasing function of k. Poor countries tend to send migrants, while rich ones tend to receive them. iv. A model with the AK technology and a population growth rate as an increasing function of k will predict convergence. Since n(k) is now an increasing function of k, (n(k) + !) will also be an increasing function of k. If we assume that n(k) grows bigger and bigger when k grows bigger and bigger, (n(k) + !) and sA, which is still a constant, must cross. This will be a steady state. If a poor and a rich country have the same parameters, they will converge to the same steady state. Therefore, this is conditional convergence. , where n'(k) > 0. v. The Solow-Swan model with an increasing population growth predicts faster convergence than with a constant population growth rate. Now (n(k) + !), where n'(k) > 0, increases faster than (n + !), where n is a constant. ...
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This note was uploaded on 10/28/2008 for the course ECON W3214 taught by Professor Xavier during the Spring '06 term at Columbia.
- Spring '06