Economic growth and economic development 337

Introduction to Modern Economic Growth the discrete time intertemporal consumer optimization problem in Example 6.5, in particular, equation (6.31) The only variable to determine in order to completely characterize the consumption profile is the initial value of the costate variable This comes from the budget constraint of the individual together with the observation that the individual will run down all his assets by the end of his planning horizon, thus a (1) = Now using the consumption rule, we have ê â a (t) = R a (t) + w − u0−1 [Rλ (0) exp ((ρ − R) t)] The initial value of the costate variable, λ (0), then has to be chosen such that a (1) = You are asked to complete the details of this step in Exercise 7.6 Example 7.1 applied the results of Theorem 7.2 It may at first appear that Theorem 7.1 is more convenient to use than Theorem 7.2, since it would enable us to directly formulate the problem as one of dynamic optimization rather than first argue about what the terminal value of the state variable, a (1), should be (based on economic reasoning as we did in Example 7.1) However, as the continuation of the previous example illustrates, this is not necessarily the case: Example 7.1 (continued) Let us try to apply Theorem 7.1 to the economic environment in Example 7.1 The first-order necessary conditions still give λ (t) = λ (0) exp (−Rt) However, since λ (1) = 0, this is only possible if λ (t) = for all t ∈ [0, 1] But then the Euler equation c (t)) = λ (t) R, exp (−ρt) u0 (ˆ which still applies from the necessary conditions, cannot be satisfied, since u0 > by assumption This implies that when the terminal value of the assets, a (1), is a choice variable, there exists no solution (at least no solution with an interior continuous control) How is this possible? The answer is that Theorem 7.1 cannot be applied to this problem, because there is an additional constraint that a (t) ≥ We would need to consider a version of Theorem 7.1 with inequality constraints The necessary conditions with inequality 323