Economic Growth-Miderm 1, fall 2011

Transcription

1 Economic Growth-Miderm 1, fall 2011 David Glancy September 23rd 1 Problem 1 The US population is slightly under 350 billion, and the world population is slightly over 7 billion, so 5% is a reasonable approximation for the percentage of the population in the US. 2 Problem a In the solow model, new investment is the exogenous investment rate (γ) times output per worker (y = Ak 1 3 ) minus depreciation per worker, which is proportional to existing capital per worker (δk). Thus we get the following dynamics for capital per worker. k = γak 1 3 δk In the steady state, this will be equal to zero: k ss = ( γa δ ) 3 2 Using that each country has the same A and delta, the ratio is: you. k A ss k B ss = ( γa 3 γ ) B 2 = = 8 8 If you find an error or typo please send an to david glancy@brown.edu. Thank 1

2 3 Problem a False, population growth but no technological progress means that output per capita is constant in the steady state. However, as population is increasing, we need total output to increase at the rate n to keep per capita income constant. 3.2 b True, it is fixed at f(k ss ) where k ss solves k = γf(k) δk = c True, it grows at the rate n so that per worker output is constant 3.4 d True, that the level is constant means a constant growth rate of 0. 4 Problem a Note that: growth = y t+1 y t 1 = Akα t+1 1. As α doesn t vary, it will be Akt α the country with the higher value of +1 which grows faster. Dividing the normal capital accumulation function by yields: +1 = γk α 1 t (n + δ) Since the two countries have the same ouput percapita and production functions, capital is the same in both countries so we can write that: ( +1 ) A ( +1 ) B = (γ A γ B )kt α 1 (n A n B ) As the two countries have the same steady state: 0 = γ A k α 1 ss (n A + δ) = γ B k α 1 ss (n B + δ) 2

3 0 = (γ A γ B )k α 1 ss (n A n B ) Finally, we are given that < k ss. Note that (γ A γ B ) > 0 and α 1 < 0 making (γ A γ B )kt α 1 (n A n B ) a deacreasing function of which takes a value of zero at k ss and thus a value greater than 0 below that level, allowing us to conclude that growth is higher in A. 5 Problem 5 In the malthusian model, there is a relationship between output perworker and population growth: ˆL = f(y), f > 0 and a negative relationship between population and output per capita due to some diminishing fixed resource, typically land. y = g(l), g < a New seed raises productivity and shifts up g(l). Thus y goes up immediately, but the steady state income per capita f 1 (0) is unchanged. Income goes up immediately because L is unchaged in levels but each worker is more productive, and steady state total income rises because L ss rises while y ss stays the same. 5.2 b New land does the same as (a). There is an instant increase in productivity raising output perworker, then population growth returns us to the old income per capita level. 5.3 c People wanting more children shifts up f. There are the same number of equally productive workers, so income per capita and total income aren t immediately effected. However, the steady state level of income falls as people have more children, while total income rises because there are more workers. 3

4 5.4 d Half the population dying means there are fewer workers but more land per worker, so per capital income rises while total income falls. The population growth takes us to the old steady state income per capita and total income as none of the curves shifted. 6 Problem a TFR is the average number of kids a woman would have if she has the current age specific rates and lives to the end of her reproductive life, thus is is the sum of the rates for the year. It is maximized in year 2013 at b Cohort specific TFR comes from adding along the diagonals, we see that those born in 2012 have a TFR of c Age Women born in 2011 decide to only give birth in year 3 with certainty instead of half giving birth in year 2 and half in 3. Consequently the TFR drops in 2013 because only the 2010 cohort is giving birth, but the TFR stays at 1 for every cohort. 7 Problem 7 We want to determine the growth rate of population if population doubles every 25 years (NRR=2 and generation length =25 years). Recall that the 4

5 rule of 70 approximates: doubling time= 70 = g = 70 g 25 3% per year. which is a bit under 5