1.5 Quantitative Implications

We are often interested not just in a models qualitative implications, but in its quantitative predictions. If, for example, the impact of a moderate

growth path, actual investment equals break-even investment, (n+g + S)k*. Thus,

c* =f{k*)-{n + g + S)k*- (1.14)

k* is determined by s and the other parameters of the model, n, g, and 8; we can therefore write k* = k*{s, n, g, 8). Thus (1.14) implies

[nk4s,n,g,8)) -in.g. 5)]%. (1.15)

We know that the increase in s raises k*. Thus whether the increase raises or lowers consumption in the long run depends on whether f(k*)~ the marginal product of capital-is more or less than n + g + 8. Intuitively, when rises, investment (per unit of effective labor) must rise by n + g + 8 times the change in for the increase to be sustained. If /"(k*) is less than n+g + 8, then the additional output from the increased capital is not enough to maintain the capital stock at its higher level. In this case, consumption must fall to maintain the higher capital stock. If fik*) exceeds n + g + S,on the other hand, there is more than enough additional output to maintain at its higher level, and so consumption rises.

fik*) can be either smaller or larger than n+g + 8. This is shown in Figure 1.6. The figure shows not only in + g + 8)k and sfik), but also fik). On the balanced growth path, consumption equals output less break-even investment; thus is the distance between fik) and in + g + 8)k. In the top panel, fik*) is less than n+g + 8, and so an increase in the saving rate lowers consumption even when the economy has reached the new balanced growth path. In the middle panel, the reverse holds, and so an increase in s raises consumption in the long run.

Finally, in the bottom panel, fik*) just equals n + g + 6-that is, the fik) and in + g + 8)k lines are parallel at = k*. In this case, a marginal change in s has no effect on consumption in the long run, and consumption is at its maximum possible level among balanced growth paths. This value of * is known as the golden-rule level of the capital stock. We will discuss the golden-rule capital stock further in Chapter 2. Among the questions we will address are whether the golden-rule capital stock is in fact desirable and whether there are situations in which a decentraUzed economy with endogenous saving converges to that capital stock. Of course, in the Solow model, where saving is exogenous, there is no more reason to expect the capital stock on the balanced growth path to equal the golden-rule level than there is to expect it to equal any other possible value.

+ g + S)k sfik)

FIGURE 1.6 path

k*

Output, investment, and consumption on the balanced growth

increase in saving on growth remains large after several centuries, the result that the impact is temporary is of limited interest.

For most models, including this one, obtaining exact quantitative results requires specifying functional forms and values of the parameters; it often

ds {n + g + 8)-sfik*)

(1.20)

Two changes help in interpreting this expression. The first is to convert it to an elasticity by multiplying both sides by s / *. The second is to use the fact that sf(k*) = {n + g + 8)k* to substitute for s. Making these changes gives us

Tliis technique is known as implicit differentiation. Even though (1.17) does not explicitly give k* as a function of s, n, g, and S, it still determines how k* depends on those variables. We can therefore differentiate the equation with respect to s and solve for dk* Ids.

"We saw in the previous section that an increase in s raises k*.To check that this is also implied by equation (1.19), note that n -i- -i- is the slope of the break-even investment line and that sf(k*) is the slope of the actual investment line at k*. Since the break-even investment line is steeper than the actual investment line at k* (see Figure 1.2), it follows that the denominator of (1.19) is positive and thus that dk*Ids > 0.

also requires analyzing the model numerically. But in many cases, 11 is possible to learn a great deal by considering approximations around the long-run equilibrium. That is the approach we take here.

The Effect on Output in the Long Run

The long-run effect of a rise in saving on output is given by

ds as

where y* = f(k*) is the level of output per unit of effective labor on the balanced growth path. Thus to find ciy* /ds,we need to find dk* /ts.To do this, note that k* is defined by the condition that = 0; thus * satisfies

sf{k*(s, n,g, S)) = {n+g + S)k*(s, n, g, 8). (1.17)

Equation (1.17) holds for all values of s (and of n,g, and 8). Thus the derivatives of the two sides with respect to s are equal:

sf(k*) + f(k*) = (n+g+ 8), (1.18)

oS oS

where the arguments of k* are omitted for simpUcity. This can be rearranged to obtain

= fJJ (119)

ss (n+g + 8)-sf{k*y

Substituting (1.19) into (1.16) yields

dy* ¹*) *)