R.E. Lucas, Jr, On the mechanics of economic development path. Then from(3), we have 8()/0(t)=-oK. Then from(4), we must have BA(t)N(t)bK()=p+aκ (6) That is, along the balanced path, the marginal product of capital must equal the constant value p+oK. With this Cobb-Douglas technology, the marginal product of capital is proportional to the average product, so that dividing(2) through by K(r)and applying(6)we obtain K()K(o) A(t)K()2-N() B B By definition of a balanced path, K(()/K(r)is constant so(7) implies that N(c(t)/K(r)is constant or, differentiating, that K(1 N(1) c(r) K(t)N(t)c(t)=k+λ (8) Thus per-capita consumption and per-capita capital grow at the common rate K. To solve for this common rate differentiate either(6)or(7)to obtain Then(7)may be solved to obtain the constant, balanced consumption-ci ratio N(rc(o)/K(ror, which is equivalent and slightly easier to interpret constant, balanced net savings rate s defined by B(K+入) (10) N(t)c(t)+R(t)p+oκ Hence along a balanced path, the rate of growth of per-capita magnitudes is simply proportional to the given rate of technical change, l, where the constant of proportionality is the inverse of labor's share, 1-B. The rate of time preference p and the degree of risk aversion o have no bearing on this long-run growth rate. Low time preference p and low risk aversion o induce a high savings rate s, and high savings is, in turn, associated with relatively high output levels on a balanced path. a thrifty society will, in the long run, b wealthier than an impatient one, but it will not grow faster In order that the balanced path characterized by(9)and(10) satisfy the transversality condition(5), it is necessary that p+ox >x+X[From(10),one sees that this is the same as requiring the savings rate to be less than capital's