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8 The Impact on Output The increase in s shifts the actual investment line upward, and so k* rises. This is shown in Figure 1.4. does not immediately jump to the new value of *, however. Initially, is equal to the old value of *. At this level, actual investment now exceeds breakeven investmentmore resources are being devoted to investment than are needed to hold constantand so is positive. Thus begins to rise. It continues to rise until it reaches the new \alue of fc*, at which point it remains constant. The behavior of output per worker, Y/L, is something we are likely to be particularly interested in. Y/L equals Af(k). When is constant, Y/L grows at rate g, the growth rate of A. When is increasing, Y/L grows both because A is increasing and because is increasing. Thus its growth rate exceeds g. When reaches the new value of k*, however, again only the growth of A contributes to the growth of Y/L, and so the growth rate of Y/L returns to g. Thus a permanent increase in the saving rate produces a variable of the model is growing at a constant rate. On the balanced growth path, the growth rate of output per worker is determined solely by the rate of technological progress. The balanced growth path of the Solow model fits several of the major styUzed facts about growth described by Kaldor (1961). In most of the major industriaUzed countries over the past century, it is a reasonable first approximation to say that the growth rates of labor, capital, and output are each roughly constant. The growth rates of output and capital are about equal (so that the capitaloutput ratio is approximately constant) and are larger than the growth rate of labor (so that output per worker and capital per worker are rising). The balanced growth path of the Solow model has these properties. 1.4 The Impact of a Change in the Saving Rate The parameter of the Solow model that policy is most likely to affect is the saving rate. The division of the governments purchases between consumption and investment goods, the division of its revenues between taxes and borrowing, and its tax treatments of saving and investment are all likely to affect the fraction of output that is invested. Thus it is natural to investigate the effects of a change in the saving rate. For concreteness, we will consider a Solow economy that is on a balanced growth path, and suppose that there is a permanent increase in s. In addition to demonstrating the models implications concerning the role of saving, this experiment will illustrate the models properties when the economy is not on a balanced growth path.
(n + g + S)k NtVvAk) ) FIGURE 1.4 The effects of an increase in the saving rate on investment temporary increase in the growth rate of output per worker: is rising for a time, but eventually it increases to the point where the additional saving is devoted entirely to maintaining the higher level of k. These results are summarized in Figure 1.5. fo denotes the time of the increase in the saving rate. By assumption, s jumps at time to and remains constant thereafter, rises gradually from the old value of k* to the new value. The growth rate of output per worker, which is initially g, jumps upward at to and then gradually retums to its initial level. Thus output per worker begins to rise above the path it was on and gradually settles into a higher path parallel to the first." In sum, a change in the saving rate has a level effect but not a growth effect: it changes the economys balanced growth path, and thus the level of output per worker at any point in time, but it does not affect the growth rate of output per worker on the balanced growth path. Indeed, in the Solow model only changes in the rate of technological progress have growth effects; all other changes have only level effects. The Impact on Consumption If we were to introduce households into the model, their welfare would depend not on output but on consumption: investment is simply an input into production in the future. Thus for many purposes we are likely to be more interested tn the behavior of consumption than in the behavior of output. "The reason that Figure 1.5 shows the log of output per worker rather than its level is that when a variable is growing at a constant rate, a graph of the log of the variable as a function of time is a straight line. That is, the growth rate of a variable is the derivative with respect to time of the log of the variable: d \n(X)/dt = (l/X)dX/dt = X/X.
1.4 The Impact of a Change in the Saving Rate 17 Growth rate of Y/L InY/L to t FIGURE 1.5 The effects of an Increase in the saving rate Consumption per unit of effective labor equals output per unit of effective labor, f(k), times the fraction of that output that is consumed, I  s. Thus, since s changes discontinuously at fo and does not, initially consumption per unit of effective labor jumps downward. Consumption then rises gradually as rises and s remains at its higher level. This is shown in the last panel of Figure 1.5. Whether consumption eventually exceeds its level before the rise in s is not immediately clear. Let * denote consumption per unit of effective labor on the balanced growth path, c* equals output per unit of effective labor, fik*), minus investment per unit of effective labor, sf(k*). On the balanced
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