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15 resources) are available in finite supply must eventually bring growth to a halt. This problem asks you to address this idea in the context of the Solow model. Let the production function be = K(AL)lRl, where R is the amount of land. Assume a > 0, /3 > 0, and a + p < 1. The factors of production evolve according to = sY  SK, A = gA, L = nL, and R = 0. ia) Does this economy have a unique and stable balanced growth path? That is, does the economy converge to a situation in which each of Y, K, I, A, and R are growing at constant (but not necessarily equal) rates? If so, what are those growth rates? If not, why not? (b) In light of your answer, does the fact that the stock of land is constant imply that permanent growth is not possible? Explain intuitively. 1.11. Embodied technological progress. (This follows Solow, 1960, and Sato, 1966.) One view of technological progress is that the productivity of capital goods built at t depends on the state of technology at t and is unaffected by subsequent technological progress. This is known as embodied technological progress (technological progress must be "embodied" in new capital before it can raise output). This problem asks you to investigate its effects. (a) As a preliminary, let us modify the basic Solow model to make technological progress capitalaugmenting rather than laboraugmenting. So that a balanced growth path exists, assume that the production function is CobbDouglas: ( ) = " Assume that A grows at rate : Ait) = M(f). Show that the economy converges to a balanced growth path, and find the growth rates of Y and on the balanced growth path. (Hint: show that we can write YUAiL) as a function of X/( "*1), where = a/(l  a). Then analyze the dynamics of /iAT).) ib) Now consider embodied technological progress. Specifically, let the production function be y(f} = Jit)Lit), where Jit) is the effective capital stock. The dynamics of J(r) are given by j(r) = sAit)Yit)  dJit). The presence of the Ait) term in this expression means that the productivity of investment at f depends on the technology at f. Show that the economy converges to a balanced growth path. What are the growth rates of and J on the balanced growth path? (Hint: let Jit) = Jit)/Ait). Then use the same approach as in (a), focusing on 7/( *1) instead of/( *!).) ic) What is the elasticit) of output on the balanced growth path with respect to s7 (d) In the vicinity of the balanced growth path, how rapidly does the economy converge to the balanced growth path? ie) Compare your results for (c) and (d) with the corresponding results in the text for the basic Solow model. 1.12. Consider a Solow economy on its balanced growth path. Suppose the growthaccounting techniques described in Section 1.7 are applied to this economy.
(a) What fraction of growth in output per worlcer does growth accounting attribute to growth in capital per worlcer? What fraction does it attribute to technological progress? (b) How can you reconcile your results in (a) with the fact that the Solow model implies that the growth rate of output per worker on the balanced growth path is determined solely by the rate of technological progress? L 13. ia) In the model of convergence and measurement error in equations (1.33)(1.34), suppose the true value of b is 1. Does a regression of ln(F/N)iq7q  ln(y/N)i87o 0 a constaut and ln(y/N)i87C) yield a biased estimate of b? Explam. ib) Suppose there is measurement error in measured 1979 income per capita but not in 1870 income per capita. Does a regression of ln(F/N)i979 ln(y/JV)]87o on a constant and In(y/N)i87c yield a biased estimate of b? Explain.
Chapter 2 BEHIND THE SOLOW MODEL: INFINITEHORIZON AND OVERLAPPINCCENERATIONS MODELS This chapter investigates two models that resemble the Solow model but in which the dynamics of economic aggregates are determined by decisions at the microeconomic level. Both models continue to treat the growth rates of labor and knowledge as exogenous. But the models derive the evolution of the capital stock from the interaction of maximizing households and firms in competitive markets. As a result, the saving rate is no longer exogenous, and it need not be constant. The first model is conceptually the simplest. Competitive firms rent capital and hire labor to produce and sell output, and a fixed number of infinitelylived households supply labor, hold capital, consume, and save. This model, which was developed by Ramsey (1928), Cass (1965), and Koopmans (1965), avoids all market imperfections and all issues raised by heterogeneous households and links among generations. It therefore provides a natural benchmark case. The second model is the overlappinggenerations model developed by Diamond (1965). The key difference between the Diamond model and the RamseyCassKoopmans model is that the Diamond model assumes that there is continual entry of new households into the economy. As we will see, this seemingly small change has important consequences.
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