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# 03 - Solutions to the Problems in the Textbook Conceptual...

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Solutions to the Problems in the Textbook Conceptual 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) = AN 1- θ K θ , where Y represents the level of output. (1 - θ ) and θ 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 that Y = F(K,N,H) = AN a K b H c 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 constant 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). 26

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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.
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