Introduction to Modern Economic Growth where f (k) ≡ F (k, 1) is the standard per capita production function As usual, the wage rate is (9.4) w (t) = f (k (t)) − k (t) f (k (t)) 9.2.2 Consumption Decisions Let us start with the individual consumption decisions Savings by an individual of generation t, s (t), is determined as a solution to the following maximization problem max c1 (t),c2 (t+1),s(t) u (c1 (t)) + βu (c2 (t + 1)) subject to c1 (t) + s (t) ≤ w (t) and c2 (t + 1) ≤ R (t + 1) s (t) , where we are using the convention that old individuals rent their savings of time t as capital to firms at time t + 1, so they receive the gross rate of return R (t + 1) = + r (t + 1) (we use R instead of + r throughout to simplify notation) The second constraint incorporates the notion that individuals will only spend money on their own end of life consumption (since there is no altruism or bequest motive) There is no need to introduce the additional constraint that s (t) ≥ 0, since negative savings would violate the second-period budget constraint (given that c2 (t + 1) ≥ 0) Since the utility function u (·) is strictly increasing (Assumption 3), both con- straints will hold as equalities Therefore, the first-order condition for a maximum can be written in the familiar form of the consumption Euler equation (for the discrete time problem, recall Chapter 6), (9.5) u0 (c1 (t)) = βR (t + 1) u0 (c2 (t + 1)) Moreover, since the problem of each individual is strictly concave, this Euler equation is sufficient to characterize an optimal consumption path given market prices Solving this equations for consumption and thus for savings, we obtain the following implicit function that determines savings per person as (9.6) s (t) = s (w (t) , R (t + 1)) , 422