San Francisco State University ECON 560 Summer Midterm Exam 2. Monday, July hour 15 minutes

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1 San Francisco State University Michael Bar ECON 560 Summer 2018 Midterm Exam 2 Monday, July 30 1 hour 15 minutes Name: Instructions 1. This is closed book, closed notes exam. 2. No calculators or electronic devices of any kind are allowed. 3. Show all the calculations, and explain your steps. 4. If you need more space, use the back of the page. 5. Fully label all graphs. 6. Use a ruler to draw neat graphs. Good Luck

2 1. (40 points). Consider the Malthusian model discussed in class, and described as follows. Consumers: Like to consume food (YY tt ). Each consumer supplies 1 unit of labor. Producers: Produce food using land and labor. Output of food at time tt is given by YY tt = AA tt ΛΛ θθ 1 θθ, 0 < θθ < 1, where AA tt is productivity level at time tt, ΛΛ is (fixed) land, and is the number of workers, which is also the size of the population. Population: evolves according to +1 = gg(yy tt ), where gg(yy tt ) is the growth rate of population as a function of output per capita yy tt = YY tt /. It is assumed that there is some subsistence level of consumption per capita yy such that yy tt > yy gg(yy tt ) > 1, yy tt < yy gg(yy tt ) < 1 and yy tt = yy gg(yy tt ) = 1. a. (5 points). Derive the equation of output per capita (yy tt ) and the law of motion of output per capita (yy tt+1 as a function of yy tt ) for this model. yy tt = YY tt Output per capita: = AA ttλλ θθ 1 θθ = AA tt Λ θθ Law of motion of output per capita: One way is to multiply and divide yy tt+1 by AA tt, and identify yy tt on the right hand side: yy tt+1 = AA tt+1 ΛΛ θθ θθ ΛΛ = (AA +1 /AA tt )AA tt = AA tt+1/aa tt tt+1 gg(yy tt ) gg(yy tt ) θθ yy tt Another way is to divide yy tt+1 by yy tt and rearrange: yy AA tt+1 ΛΛ θθ AA tt+1 tt+1 LL = tt+1 AA yy tt AA tt Λ = tt θθ 1 θθ 1 θθ +1 = AA tt+1 AA tt 1 θθ 1 θθ gg(yy tt ) = AA tt+1 AA tt gg(yy tt ) θθ yy tt+1 = AA tt+1/aa tt gg(yy tt ) θθ yy tt 1

3 b. (5 points). Suppose that Chinese economy, prior to 19 th century, behaved like the Malthusian model described in this section. The productivity level in China is fixed at AA tt = 10 tt, and the population growth function is gg(yy tt ) = yy tt aa, aa = 5 The land is Λ = 100 and land share is θθ = 1. Solve for the steady state level of output 3 per capita (yy ) and the steady state population level (LL ). Steady state output per capita gg(yy ) = 1 yy aa = 1 yy = aa = 5 Steady state population yy = AA Λ LL θθ 1 yy AA θθ Λ = LL LL = Λ AA yy 1 θθ = = = 800 2

4 c. (10 points). Suppose that at some time ττ the productivity in China increased by 50% (i.e. AA ττ = AA ττ+1 = = 15), and stays at this new level forever (once-and-for-all increase in productivity). Assuming that prior to the change, the Chinese economy was at the original steady state; calculate the immediate impact of the productivity change on the Chinese economy. That is, find the output per capita yy ττ, the net population growth rate gg(yy ττ ) 1, and the population size in the following period, i.e. LL ττ+1. Output per capita at time ττ: yy ττ = AA ττ Λ θθ = LL ττ = = 7.5 Alternatively, using the law of motion of output per capita: yy ττ = AA ττ/aa ττ 1 gg(yy ττ 1 ) θθ yy ττ 1 = 15/10 5 = Population growth rate at time ττ: gg(yy ττ ) = yy ττ aa = = 1.5 The net growth rate of population is: gg(yy ττ ) 1 = 0.5 = 50% Population size at time ττ + 1: LL ττ+1 = gg(yy ττ )LL ττ = = 1,200 3

5 d. (5 points). Given the change in the last section, solve for the long-run (steady state) level of output per capita (yy ) and the steady state population level (LL ). Steady state output per capita gg(yy ) = 1 yy aa = 1 yy = aa = 5 Observe, that there is no changes compared to section a. Steady state population LL = Λ AA 1 3 θθ 15 yy = = = 2,700 4

6 e. (15 points). Draw the time paths of output per capita, population growth rate and total population in China, that illustrate the economy before, during and after the change in productivity. Use the values found in previous sections to fully label the diagrams. yy tt gg(yy tt ) ττ time ττ time 2, ττ time 5

7 2. (5 points). Based on our analysis of the Malthusian model, circle the correct statement. A once-and-for-all increase in productivity leads to a. Permanent increase in standard of living and population. b. Temporary increase in standard of living and population. c. Permanent increase in standard of living and temporary increase in population. d. Temporary increase in standard of living and permanent increase in population. e. None of the above. 3. (5 points). Demographic Transition is (circle the correct answer): a. Transition from low standard of living to high standard of living. b. Transition from low interest rates to high interest rates. c. Transition from low mortality and fertility rates to high mortality and fertility rates. d. Transition from high mortality and fertility rates to low mortality and fertility rates. e. None of the above. 6

8 4. (10 points) The following table shows data for a planet of Aissur, populated by species that live for a maximum of five years. In addition, all the people are female, who are nonetheless able to reproduce. Age (from last Birthday) Population in 2017 Age specific fertility rates Probability of surviving to next age Population in ( ) = = = = = 80 Total Calculate the population at each age and the total population in

9 5. (10 points). The next table presents key demographic data for Japan, during the years Period Life Expectancy Total Fertility Rate Net Reproduction Rate Based on the above table, the Net Reproduction Rate declined during , despite the sizable increase in life expectancy. Explain how this is possible. Your explanation must utilize the formula of Net Reproduction Rate and make reference to the given table. The Net Reproduction Rate formula is: NNNNNN = 1 2 ππ ii FF ii ii=0 On the one hand, the increase in Life Expectancy means that more women survive through their childbearing years, which is reflected in the increased probabilities of being alive at age ii (ππ ii ). On the other hand, we see in the table that the Total Fertility Rate declined significantly, which is reflected in declining age specific fertility rates (FF ii ). The decline in fertility must have bigger impact than decline in mortality, for the Net Reproduction Rate to decrease. In the above NRR formula, we indicate the increase in ππ ii with one arrow, and the decrease in fertility with two arrows, to stress that the decline in fertility dominated. Indeed, the decline in fertility was dramatic, from about 3 children per woman to about 1.48 children per woman (about half). 8

10 6. (20 points). Assume that the aggregate output is produced according to YY tt = AA tt KK tt θθ (h tt ) 1 θθ, 0 < θθ < 1, where YY tt is the total real GDP, AA tt is the Total Factor Productivity, KK tt is the total physical capital, is the number of workers, and h tt is human capital per worker. a. The next table presents data on two countries. yy ii yy jj αα ii αα jj AA ii AA jj h 1 θθ ii h jj kk θθ ii kk jj 35 1? Based on the above table, if the only difference between the two countries was productivity, what would be the ratio of country ii to country jj GDP per capita? Using the cross-country accounting formula, gives: yy ii yy jj = αα ii αα jj AA ii AA jj h ii h jj 35 = 1 AA ii AA jj AA ii AA jj = 7 1 θθ kk θθ ii kk jj b. The next table shows how the average wage increases in years of education in a sample of countries. Years of schooling ,10, Marginal return Based on the above table, how would you estimate the human capital per worker in a country where the average education per worker is 13.7 years? Assume that human capital of workers with no schooling is h 0, and that human capital of workers with education level ee is proportional to the relative wage of workers with this education level. You only need to write the formula that you would use if you had a calculator. h(13.7) = h

11 7. (10 points). Assume that the aggregate output is produced according to YY tt = AA tt KK tt θθ (h tt ) 1 θθ, 0 < θθ < 1. We derived the approximate growth accounting formula for output per capita: yy NN αα + AA + θθkk + (1 θθ)h a. Assume that the capital share in both countries is 0.35, calculate the growth rate in productivity in both countries. Country yy NN αα kk h AA China 5% 0% 2% 2% 5% 0 [0.35 2% %] = 3% Japan 1% 0% 1% 1% 1% 0 [0.35 1% %] = 0% The growth in productivity was calculated using: AA = yy αα θθkk + (1 θθ)h b. Based on the above accounting, is the growth in standard of living in both countries sustainable? Briefly explain. In China, the growth in standard of living in fueled by growth in productivity, while in Japan productivity is stagnant. Based on the Solow growth model, without growth in productivity, sustained growth in standard of living is impossible. Therefore, we conclude that growth in standard of living in China is sustainable, but in Japan it is not (if the current stagnation in productivity persists). 10