Introduction to Modern Economic Growth the notation k for capital per worker in the next section From the definition of κ, the law of motion of effective capital-labor ratios can be written as R1 bi (t − 1) di K (t) = 0R (10.30) κ (t) ≡ H (t) hi (t) di Factor prices are then given by the usual competitive pricing formulae: (10.31) R (t) = f (κ (t)) and w (t) = f (κ (t)) − κ (t) f (κ (t)) , with the only noteworthy feature that w (t) is now wage per unit of human capital, in a way consistent with (10.28) An equilibrium in this overlapping generations economy is a sequence of bequest o n and consumption levels for each individual, [hi (t)]i∈[0,1] , [ci (t)]i∈[0,1] , [bi (t)]i∈[0,1] ∞ t=0 that solve (10.27) subject to (10.28) a sequence of effective capital-labor ratios, {κ (t)}∞ t=0 , given by (10.30) with some initial distribution of bequests [bi (0)]i∈[0,1] , and sequences of factor prices, {w (t) , R (t)}∞ t=0 , that satisfy (10.31) The characterization of an equilibrium is simplified by the fact that the solution to the maximization problem of (10.27) subject to (10.28) involves (10.32) ci (t) = ηmi (t) and bi (t) = (1 − η) mi (t) , and substituting these into (10.27), we obtain the indirect utility function (see Exercise 10.16): (10.33) mi (t) ả hi (t) , a which the individual maximizes by choosing hi (t) and recognizing that mi (t) = w (t) hi (t) + R (t) bi (t − 1) The first-order condition of this maximization gives the human capital investment of individual i at time t as: ả hi (t) , (10.34) aw (t) = γ a or inverting this relationship, defining γ 0−1 (·) as the inverse function of γ (·) (which is strictly increasing) and using (10.31), we obtain (10.35) hi (t) = h (t) ≡ aγ 0−1 [a (f (κ (t)) − κ (t) f (κ (t)))] An important implication of this equation is that the human capital investment of each individual is identical, and only depends on the effective of capital-labor ratio 483 ,