Introduction to Modern Economic Growth where c0 refers to next period’s consumption Using this relationship, the consumption Euler equation becomes u0 (c) = β (1 + r) u0 (c0 ) (6.30) This form of the consumption Euler equation is more familiar and requires the marginal utility of consumption today to be equal to the marginal utility of consumption tomorrow multiplied by the product of the discount factor and the gross rate of return Since we have assumed that β and (1 + r) are constant, the relationship between today’s and tomorrow’s consumption never changes In particular, since u (·) is assumed to be continuously differentiable and strictly concave, u0 (·) always exists and is strictly decreasing Therefore, the intertemporal consumption maximization problem implies the following simple rule: (6.31) if r = β − c = c0 and consumption is constant over time if r > β − c < c0 and consumption increases over time if r < β − c > c0 and consumption decreases over time The remarkable feature is that these statements have been made without any reference to the initial level of asset holdings a (0) and the sequence of labor income {w (t)}∞ t=0 It turns out that these only determine the initial level of consumption The “slope” of the optimal consumption path is independent of the wealth of the individual Exercise 6.10 asks you to determine the level of initial consumption using the transversality condition and the intertemporal budget constraint, and also contains a further discussion of the effect of changes in the sequence of labor income {w (t)}∞ t=0 on the optimal consumption path 6.5.2 Dynamic Programming Versus the Sequence Problem To get more insights into dynamic programming, let us return to the sequence problem Also, let us suppose that x is one dimensional and that there is a finite horizon T Then the problem becomes max {x(t+1)}T t=0 T X β t U(x (t) , x (t + 1)) t=0 subject to x (t + 1) ≥ with x (0) as given Moreover, let U(x (T ) , x (T + 1)) be the last period’s utility, with x (T + 1) as the state variable left after the last period (this utility could be thought of as the “salvage value” for example) 289