Problem Set 4
FE411 Spring 2007
Rahman
INSTRUCTIONS: Use your own paper to answer the questions. Please turn in your problem sets with your name clearly marked on the front page and all pages stapled together. You are encouraged to work together, but you must hand in your own work. You must show your work for credit and answer in complete sentences when appropriate (such as when the question asks you to “describe” or “explain”).
Chapter 7:
4) We assume α = 0.33. Then, in order to calculate the ratio of factor accumulation relative to the U.S. for each country, we utilize the following equation:
Factor Accumulation = kαh1-α
where the ratio of physical capital relative to the U.S., k, and the ratio of human capital relative to the U.S., h, are given for each country. In order to calculate the ratio of productivity relative to the U.S. for each country, we utilize the following equation:
Productivity = y / (kαh1-α),
where the denominator is simply the previous ratio of factor accumulation and y is the ratio of output per worker relative to the U.S. for each country. The results of each of these calculations for each country are listed below:
/ ACountry / Factor Accumulation / Productivity
Sweden / 0.9 / 0.74
Mauritius / 0.51 / 0.97
Jordan / 0.45 / 0.49
The table shows that Sweden has the highest level of factor accumulation relative to the U.S. among all the countries. In fact, the level of factor accumulation in Sweden is nearly that of the U.S. This allows us to conclude that differences in factor accumulation cannot be responsible for the difference in output per worker. Instead, the relative level of productivity, which is not much different from the relative output level (0.67) plays the largest role in explaining income relative to the U.S. By contrast, the relative level of productivity in Mauritius is by far the largest among all countries, and this level of productivity is nearly that of the U.S. level of productivity. So productivity cannot explain the difference in output per capita. Rather, it is the relative level of factor accumulation that is largely responsible for the differences in output per worker relative to the U.S.
5) We assume α = 0.33.
/ (%) / /Country / Growth Rate of Factors / Growth Rate of Productivity / Fraction of Growth due to Factor Growth / Fraction of Growth due to Productivity Growth
Argentina / 1.02 / 0.15 / 0.87 / 0.13
Austria / 1.52 / 1.54 / 0.5 / 0.5
Chile / 0.98 / 1.02 / 0.49 / 0.51
By the table, we see that Argentina’s factor growth contributes most to overall growth, and Chile’s productivity growth contributes most of overall growth.
6) The effect of using data on school days increases the level of education for richer countries and lessens the level of education for poorer countries. As a result, there will be a stronger correlation between levels of human capital, as measured by the days of schooling, and output per worker. The stronger correlation ensures a greater emphasis on the factors of production by magnifying the variable h. The role of productivity in explaining variations in output per worker among countries will diminish.
Chapter 8:
5) In diagrams (not drawn), it can be shown that the increase in the fraction of labor devoted to R&D in Country 1 will create a drop in the level of output per worker but an increase in the growth rate of productivity as well as output per worker. Country 1 behaves in accordance with the one-country model. However, the speed of growth in productivity in Country 1 raises the steady state A1/ A2 ratio. Consequently, μc, the cost of copying in Country 2, falls. The fall in this cost will raise productivity in Country 2, so the growth rate of output per worker in Country 2 will also rise. There will be no jump in output for country 2 (up or down), but the fall in the cost of copying will place Country 2 on a permanently higher growth path. In the long run, the growth rates of output for both Country 1 and Country 2 will be equal, with Country 1 still at the higher level, and the gap in income between the two will be smaller than before.
7) a. In the steady state, the growth of A1 equals the growth of A2. Therefore:
Rearranging and solving for μc, we get,
Setting the above steady state condition equation to the specified cost-of-copying function, we get
Rearranging, we find the solution to be:
Without the exponent, the ratio of technology in Country 1 to Country 2 would be determined proportionally by the ratio of the fraction of the labor force employed in R&D. However because we assume 0<β<1, the ratios will not be proportional. As the value of β falls to zero, the proportional difference in the level of technology between the two countries grows extremely large, and as the value of β rises to one, the proportional difference in the level of technology between the two countries matches the proportional difference in the fraction of worker devoted to R&D.
b. If we assume β = ½, μi = 10, γA,1 = 0.2, and γA,2 = 0.1, we can solve the previous equation to get:
That is, the steady-state ratio of technology in Country 1 to technology in Country 2 is 4.
Chapter 9:
1) The annual growth rate of productivity is given by the following equation:
We are given a value of 1/3 for β and 0 for the growth of y, leaving the growth of population as the only unknown. To solve for it, we use the standard growth equation with the initial population as 4 million and the final population after 10,000 years as 170 million. The equation is:
Now we substitute to find our growth rate of productivity over this period.
That is, the growth rate of productivity over this period was roughly 0.0125% per year.
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