Introduction to Modern Economic Growth To make the problem interesting, we also assume that a (0) < ∞ and P∞ t=0 (1 + r)−t w (t) < ∞, so that the individual has finite wealth, and thus can achieve only finite value (utility) Let us now write the recursive formulation of the individual’s maximization problem The state variable is a (t), and consumption can be expressed as c (t) = a (t) + w (t) − (1 + r)−1 a (t + 1) With standard arguments and denoting the current value of the state variable by a and its future value by a0 , the recursive form of this dynamic optimization problem can be written as ê â Ă ¢ u a + w − (1 + r)−1 a0 + βV (a0 ) V (a) = max a ≥0 Clearly u (·) is strictly increasing in a, continuously differentiable and strictly concave in both a and a0 Moreover, since u (·) is continuously differentiable and the individual’s wealth is finite, V (a (0)) is also finite Thus all of the results from our analysis above, in particular Theorems 6.1-6.6, apply and imply that V (a) is differentiable and a continuous solution a0 = π (a) exists Moreover, we can use the Euler equation (6.19), or its more specific form (6.22) for one-dimensional problems, can be used to characterize this solution In particular, we have ¡ ¢ (6.29) u0 a + w − (1 + r)−1 a0 = u0 (c) = β (1 + r) V (a0 ) This important equation is often referred to as the “consumption Euler” equation It states that the marginal utility of current consumption must be equal to the marginal increase in the continuation value multiplied by the product of the discount factor, β, and the gross rate of return to savings, R It captures the essential economic intuition of dynamic programming approach, which reduces the complex infinite-dimensional optimization problem to one of comparing today to “tomorrow” Naturally, the only difficulty here is that tomorrow itself will involve a complicated maximization problem and hence tomorrow’s value function and its derivative are endogenous But here the envelope condition, (6.23), again comes to our rescue and gives us V (a0 ) = u0 (c0 ) , 288