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30

"These issues are briefly discussed further in Section 6.14.

Finally, Panel (d) shows a case in which kf+i is not uniquely determined by kf: when kf is between ka and , there are three possible values of kf+i. This can happen if saving is a decreasing function of the interest rate. When saving is decreasing in r, saving is high if individuals expect a high value of kf+] and therefore expect r to be low, and is low when individuals expect a low value of kf+7. If saving is sufficiently responsive to r, and if r is sufficiently responsive to k, there can be more than one value of kf+i that is consistent with a given kf. Thus the path of the economy is indeterminate: equation (2.60) (or [2.66]) does not fully determine how evolves over time given its initial value. This raises the possibility that self-fulfilling prophecies and sunspots can affect the behavior of the economy and that the economy can exhibit fluctuations even though there are no exogenous disturbances. Depending on precisely what is assumed, various dynamics are possible.-*

Thus assuming that there are overlapping generations rather than infinitely-lived households has potentially important implications for the dynamics of the economy: for example, sustained growth may not be possible, or it may depend on initial conditions.

At the same time, the model does no better than the Solow and Ramsey models at answering our basic questions about growth. Because of the Inada conditions, kf+i must be less than kf for kf sufficiently large. Specifically, since the saving of the young cannot exceed the economys total output, kf+] must be less than or equal to f{kt)/l(l -t- n)(l -t- g)]. And because the marginal product of capital approaches zero as becomes large, this must eventually be less than kf. The fact that kf+i is eventually less than kf implies that unbounded growth of is not possible. Thus, once again, growth in the effectiveness of labor is the only potential source of long-run growth in output per worker. Because of the possibility of multiple k*s, the model does imply that otherwise identical economies can converge to different balanced growth paths simply because of differences in their initial conditions. But, as in the Solow and Ramsey models, we can account for quantitatively large differences in output per worker in this way only by positing immense differences in capital per worker and in rates of return.

2.13 The Possibility of Dynamic Inefficiency

The one major difference between the balanced growth paths of the Diamond and Ramsey-Cass-Koopmans models involves welfare. We saw that the equilibrium of the Ramsey-Cass-Koopmans model maximizes the welfare of the representative household. In the Diamond model, individuals born at different times attain different levels of utihty, and so the appropriate



way to evaluate social welfare is not clear. If we specify welfare as some weighted sum of the utilities of different generations, there is no reason to expect the decentralized equilibrium to maximize welfare, since the weights we assign to the different generations are arbitrary.

A minimal criterion for efficiency, however, is that the equilibrium be Pareto-efficient. It turns out that the equilibrium of the Diamond model need not satisfy even this standard. In particular, the capital stock on the balanced growth path of the Diamond model may exceed the golden-rule level, so that a permanent increase in consumption is possible.

To see this possibility as simply as possible, assume that utility is logarithmic, production is Cobb-Douglas, and g is zero. Equation (2.63) (together with the definition of D in [2.61]) implies that in this case the value of on the balanced growth path is

k* =

-ll/d-a)

1+ 2+

(1 - a)

(2.67)

Thus the marginal product of capital on the balanced growth path, ak*" , is

f(k*) =

(l + n)(2+p).

(2.68)

The golden-rule capital stock is defined by fkcR) n. f{k*) can be either more or less than [( ). In particular, for a sufficiently small, f(k*) is less than f ( )-the capital stock on the balanced growth path exceeds the golden-rule level.

To see why it is inefficient for k* to exceed kcR, imagine introducing a social planner into a Diamond economy that is on its balanced growth path with k* > kcR.lf the planner does nothing to alter k, the amount of output per worker available each period for consumption is output, f{k*), minus the new investment needed to maintain at k*, nk*. This is shown by the crosses in Figure 2.14. Suppose instead, however, that in some period, period to. the planner allocates more resources to consumption and fewer to saving than usual, so that capital per worker the next period is kcR, and that thereafter he or she maintains at kcR Under this plan, the resources per worker available for consumption in period to are f(k*)+{k- kcR)- nkcR. In each subsequent period, the output per worker available for consumption is fikcR)- nkcR. Since kcR maximizes f{k) - nk, f ( )- nkcR exceeds f{k*)-nk*. And since k* is greater than kcR, f{k*) + {k* - kcR) - nkcR is even larger than fikcR) ~ nkcR. The path of total consumption under this policy is shown by the circles in Figure 2.14. As the figure shows, this policy makes more resources available for consumption in every period than the policy of maintaining at k*. Given this, it must be possible for the planner to allocate consumption between the young and the old each period to make every generation better off.



©

©©©©©©©

X X X X X X X

«0 f

X maintaining at k" > .;

© reducing to kR in period to

FIGURE 2.14 How reducing to the golden-rule level affects the path of consumption per worker

Thus the equilibrium of the Diamond model can be Pareto-inefficient. This may seem puzzling: given that markets are competitive and there are no externalities, how can the usual result that equilibria are Pareto-efficient fail? The reason is that the standard result assumes not only competition and an absence of externalities, but also a finite number of agents. Specifically, the possibility of inefficiency in the Diamond model stems from the fact that the infinity of generations gives the planner a means of providing for the consumption of the old that is not available to the market. If individuals in the market economy want to consume in old age, their only choice is to hold capital, even if its rate of return is low. The planner, however, need not have the consumption of the old determined by the capital stock and its rate of return. Instead, he or she can divide the resources available for consumption between the young and old in any manner. The planner can take, for example, one unit of labor income from each young person and transfer it to the old; since there are I -t- n young people for each old person, this increases the consumption of each old person by 1 -i- n units. The planner can prevent this change from making anyone worse off by requiring the next generation of young to do the same thing in the following period, and then continuing this process every period. If the marginal product of capital is less than n~that is, if the capital stock exceeds the golden-rule level-this way of transferring resources between youth and old age is more efficient than saving, and so the planner can improve on the decentralized allocation.

Because this type of inefficiency differs from conventional sources of inefficiency, and because it stems from the intertemporal structure of the economy, it is known as dynamic inefficiency.

Problem 2.19 investigates the sources of dynamic inefficiency further.



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