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20 2.4 Welfare 51      \ t    FIGURE 2.5 The saddle path 
possible to make anyone better off without making someone else worse off. Since the conditions of the First Welfare theorem hold in our model, the equilibrium must be Paretoefficient. And since all households have the same utility, this means that the decentraUzed equilibrium produces the highest possible utiUty among allocations that treat all households in the same way. To see this more clearly, consider the problem facing a social planner who can dictate the division of output between consumption and investment at each date and who wants to maximize the lifetime utility of a representative household. This problem is identical to that of an individual household except that, rather than taking the paths of w and r as given, the planner takes into account the fact that these are determined by the path of k, which is in turn determined by (2.23). The intuitive argument involving consumption at consecutive moments used to derive (2.19) or (2.22) applies to the social planner as well: reducing by at time f and investing the proceeds allows the planner to increase at time f i by ( ( )) ( +0) , lo j (j) jog je path chosen by the planner must satisfy (2.22). And since equation (2.23) giving the "Note that this change does affect r and w over the (brief) interval from f to fiAt. r falls by fik) times the change in k, while w rises by "(k)k times the change in k. But the effect of these changes on total income (per tmit of effective labor), which is given by the change in w plus times the change in r, is zero. That is, since capital is paid its marginal product, total payments to labor and to previously existing capital remain equal to the previous level of output (again per unit of effective labor). This is just a specific instance of the general result that the pecuniary externalitiesextemalittes operating through pricesbalance in the aggregate under competition.
"A formal solution to the planners problem involves the use of the calculus of variations. For a formal statement and solution of the problem, see Blanchard and Fischer (1989, pp. 3843). For an introduction to the calculus of variations, see Section 8.2, Kamien and Schwartz (1991), or Dixit (1990, Chapter 10). > evolution of reflects technology, not preferences, the social planner must obey it as weU. Finally, as with the households optimization problem, paths that require that the capital stock becomes negative can be ruled out on the groimds that this is not feasible, and paths that cause consumption to approach zero can be ruled out on the grounds that they do not maximize households utility. In short, the solution to the social planners problem is for the initial value of to be given by the value on the saddle path, and for and to then move along the saddle path. That is, the competitive equilibrium maximizes the welfare of the representative household.! 2.5 The Balanced Growth Path Properties of the Balanced Growth Path The behavior of the economy once it has converged to Point E is identical to that of the Solow economy on the balanced growth path. Capital, output, and consumption per imit of effective labor are constant. Since and are constant, the saving rate, (y  c)/y, is also constant. The total capital stock, total output, and total consumption grow at rate n+g. And capital per worker, output per worker, and consumption per worker grow at rate g. Thus the central implications of the Solow model concerning the driving forces of economic growth do not hinge on its assumption of a constant saving rate. Even when saving is endogenous, growth in the effectiveness of labor remains the only possible source of persistent growth in output per worker. And since the production function is the same as in the Solow model, one can repeat the calculations of Chapter 1 demonstrating that significant differences in output per worker can arise from differences in capital per worker only if the differences in capital per worker, and in rates of return to capital, are enormous. The Balanced Growth Path and the GoldenRule Level of Capital The only notable difference between the balanced growth paths of the Solow and RamseyCassKoopmans models is that a balanced growth path with a capital stock above the goldenrule level is not possible in the RamseyCassKoopmans model. In the Solow model, a sufficiently high saving rate causes the economy to reach a balanced growth path with the property that there
Qualitative Effects Since the evolution of is determined by technology rather than preferences, p enters the equation for but not the one for k. Thus only the = 0 are feasible alternatives that involve higher consumption at every moment. In the RamseyCassKoopmans model, in contrast, saving is derived from the behavior of households whose utility depends on their constunption, and there are no externalities. As a result, it cannot be an equilibrium for the economy to follow a path where higher consumption can be attained in every period; if the economy were on such a path, households would reduce their saving and take advantage of this opportunity. This can be seen in the phase diagram. Consider again Figure 2.5. If the initial capital stock exceeds the goldenrule level (that is, if k(0) is greater than the associated with the peak of the = 0 locus), initial constunption is above the level needed to keep constant; thus is negative, gradually approaches *, which is below the goldenrule level. Finally, the fact that k* is less than the goldenrule capital stock implies that the economy does not converge to the balanced growth path that yields the maximum sustainable level of c. The intuition for this result is clearest in the case of g equal to zero, so that there is no longrim growth of consumption and output per worker. In this case, k* is defined by f(.k*) = p (see [2.22]) and is defined by [( ) = n, and our assumption that p  n  {1  e)g > 0 simplifies to p > n. Since k* is less than kcu, an increase in saving starting at = k* would cause constunption per worker to eventually rise above its previous level and remain there (see Figure 1.5). But, because households value present consumption more than future consumption, the benefit of the eventual permanent increase in constunption is bounded. At some pointspecifically, when exceeds k*the tradeoff between the temporary shortterm sacrifice and the permanent longterm gain is sufficiently unfavorable that accepting it reduces rather than raises lifetime utility. Thus converges to a value below the goldenrule level. Because k* is the optimal level of for the economy to converge to, it is known as the modified goldenrule capital stock. 2.6 The Effects of a Fall in the Discount Rate Consider a RamseyCassKoopmans economy that is on its balanced growth path, and suppose that there is a fall in p, the discount rate. Since p is the parameter governing households preferences between current and future constunption, this change is the closest analogue in this model to a rise in the saving rate in the Solow model.
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