26CHAPTER 3GROWTH AND ACCUMULATIONSolutions to the Problems in the TextbookConceptual Problems:1. The production function provides a quantitative link between inputs and output. For example, the Cobb-Douglas production function mentioned in the text is of the form:Y = F(N,K) = AN1-K,where Y represents the level of output. (1 - ) and are weights equal to the shares of labor (N) and capital (K) in production, while A is often used as a measure for the level of technology. It can be easily shown that labor and capital each contribute to economic growth by an amount that is equal to their individual growth rates multiplied by their respective share in income. 2. The Solow model predicts convergence, that is, countries with the same production function, savings rate, and population growth will eventually reach the same level of income per capita. In other words, a poor country may eventually catch up to a richer one by saving at the same rate and making technological innovations. However, if these countries have different savings rates, they will reach different levels of income per capita, even though their long-term growth rates will be the same.3. A production function that omits the stock of natural resources cannot adequately predict the impact of a significant change in the existing stock of natural resources on the economic performance of a country. For example, the discovery of new oil reserves or an entirely new resource would have a significant effect on the level of output that could not be predicted by such a production function.4. Interpreting the Solow residual purely as technological progress would ignore, for example, the impact that human capital has on the level of output. In other words, this residual not only captures the effect of technological progress but also the effect of changes in human capital (H) on the growth rate of output. To eliminate this problem we can explicitly include human capital in the production function, such thatY = F(K,N,H) = ANaKbHc with a + b + c = 1. Then the growth rate of output can be calculated as Y/Y = A/A + a(N/N) + b(K/K) + c(H/H).5. The savings function sy = sf(k) assumes that a constant fraction of output is saved. The investment requirement, that is, the (n + d)k-line, represents the amount of investment needed to maintain a 27constant capital-labor ratio (k). A steady-state equilibrium is reached when saving is equal to the investment requirement, that is, when sy = (n + d)k. At this point the capital-labor ratio k = K/N is not changing, so capital (K), labor (N), and output (Y) all must be growing at the same rate, that is, the rate of population growth n = (N/N).6. In the long run, the rate of population growth n = (N/N) determines the growth rate of the steady-state output per capita. In the short run, however, the savings rate, technological progress, and the rate of depreciation can all affect the growth rate.7. Labor productivity is defined as Y/N, that is, the ratio of output (Y) to labor input (N). A surge in labor productivity therefore occurs if output grows at a faster rate than labor input. In the U.S. we have experienced such a surge in labor productivity since the mid-1990s due to the enormous growth in GDP. This surge can be explained from the introduction of new technologies and more efficient use of existing technologies. Many claim that the increased investment in and use of computer technology has stimulated economic growth. Furthermore, increased global competition has forced many firms to cut costs by reorganizing production and eliminating some jobs. Thus, with large increases in output and a slower rate of job creation we should expect labor productivity to increase. (One should also note that a higher-skilled labor force also can contribute to an increase in labor productivity, since the same number of workers can produce more output if workers are more highly skilled.) Technical Problems:1.a. According to Equation (2), the growth of output is equal to the growth in labor times the labor share plus the growth of capital times the capital share plus the rate of technical progress, that is,Y/Y = (1 - )(N/N) + (K/K) + A/A, where1 - is the share of labor (N) and is the share of capital (K). Thus if we assume that the rate of technological progress (A/A) is zero, then output grows at an annual rate of 3.6 percent, sinceY/Y = (0.6)(2%) + (0.4)(6%) + 0% = 1.2% + 2.4% = + 3.6%,1.b. The so-called Rule of 70 suggests that the length of time it takes for output to double can be calculated by dividing 70 by the growth rate of output. Since 70/3.6 = 19.44, it will take just under 20 years for output to double at an annual growth rate of 3.6%,1.c. Now that A/A = 2%, we can calculate economic growth as Y/Y = (0.6)(2%) + (0.4)(6%) + 2% = 1.2% + 2.4% + 2% = + 5.6%.Thus it will take 70/5.6 = 12.5 years for output to double at this new growth rate of 5.6%.282.a. According to Equation (2), the growth of o