Introduction to Modern Economic Growth in Theorem 7.2 to a simple economic problem More interesting economic examples are provided later in the chapter and in the exercises Example 7.1 Consider a relatively common application of the techniques developed so far, which is the problem of utility-maximizing choice of consumption plan by an individual that lives between dates and (perhaps the most common application of these techniques is a physical one, that of finding the shortest curve between two points in the plane, see Exercise 7.4) The individual has an instantaneous utility function u (c) and discounts the future exponentially at the rate ρ > We assume that u : [0, 1] → R is a strictly increasing, continuously differentiable and strictly concave function The individual starts with a level of assets equal to a (0) > 0, earns an interest rate r on his asset holdings and also has a constant flow of labor earnings equal to w Let us also suppose that the individual can never have negative asset position, so that a (t) ≥ for all t Therefore, the problem of the individual can be written as max1 [c(t),a(t)]t=0 Z exp (−ρt) u (c (t)) dt subject to a˙ (t) = r [a (t) + w − c (t)] and a (t) ≥ 0, with an initial value of a (0) > In this problem, consumption is the control variable, while the asset holdings of the individual are the state variable To be able to apply Theorem 7.2, we need a terminal condition for a (t), i.e., some value a1 such that a (1) = a1 The economics of the problem makes it clear that the individual would not like to have any positive level of assets at the end of his planning horizon (since he could consume all of these at date t = or slightly before, and u (·) is strictly increasing) Therefore, we must have a (1) = With this observation, Theorem 7.2 provides the following the necessary conditions for an interior continuous solution: there exists a continuously differentiable costate variable λ (t) such that the optimal path of consumption and asset holdings, (ˆ c (t) , aˆ (t)), satisfy a consumption Euler equation similar to equation (6.29) may have an empty interior, making it impossible that an interior solution exists See, for example, Exercise 7.15 321