3. l[Homer 1.?) Find the elaaticitj.r of output per unit of effective labor on the balanced growth path, p\

uantltatlve mp 'cations We are usually interested not just in a model's qualitative implications, but in its quantitative predictions. If, for example, the impact of a moderate 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 also requires analyzing the model numerically. But in many cases, it is possi- ble 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 (1.21) as as where y* = is the level of output per unit of effective labor on the balanced growth path. Thus to find y*/s, we need to find ak*/s. To do this, note that k* is defined by the condition that k = O. Thus k* satisfies (1.22) Chapter 1 THE SOLOW GROWTH MODEL Equation (1.22) holds for all values of s (and of n, g, and 6). Thus the deriva- tives of the two sides with respect to s are equal:14 SC ) as (1.23) where the arguments of k* are omitted for simplicity. This can be rearranged to obtain'S as Substituting (1.24) into (1.21) yields (1.24) (1.25) as (n+g+) Two changes help in interpreting this expression. The first is to convert it to an elasticity by multiplying both sides by s/ y*. The second is to use the fact that = (n + g + )k* to substitute for s. Making these changes gives us