Introduction to Modern Economic Growth Show that this is only possible if x00 (t) = 0, so that the shortest path between two points is a straight-line Exercise 7.5 Prove Theorem 7.2, in particular, paying attention to constructing feasible variations that ensure x (t1 , ε) = x1 for all ε in some neighborhood of What happens if there are no such feasible variations? Exercise 7.6 (1) Provide an expression for the initial level of consumption c (0) as a function of a (0), w, r and β in Example 7.1 (2) What is the effect of an increase in a (0) on the initial level of consumption c (0)? What is the effect on the consumption path? (3) How would the consumption path change if instead of a constant level of labor earnings, w, the individual faced a time-varying labor income profile given by [w (t)]1t=0 ? Explain the reasoning for the answer in detail Exercise 7.7 Prove Theorem 7.4 Exercise 7.8 * Prove a version of Theorem 7.5 corresponding to Theorem 7.2 [Hint: instead of λ (t1 ) = 0, the proof should exploit the fact that x (1) = xˆ (1) = x1 ] Exercise 7.9 * Prove that in the finite-horizon problem of maximizing (7.1) or (7.11) subject to (7.2) and (7.3), fx (t, xˆ (t) , yˆ (t) , λ (t)) > for all t ∈ [0, t1 ] implies that λ (t) > for all t ∈ [0, t1 ] Exercise 7.10 * Prove Theorem 7.6 Exercise 7.11 Prove Theorem 7.11 Exercise 7.12 Provide a proof of Theorem 7.15 Exercise 7.13 Prove that in the discounted infinite-horizon optimal control problem considered in Theorem 7.14 conditions (7.51)-(7.53) are necessary Exercise 7.14 Consider a finite horizon continuous time maximization problem, where the objective function is W (x (t) , y (t)) = Z t1 f (t, x (t) , y (t)) dt with x (0) = x0 and t1 < ∞, and the constraint equation is x˙ (t) = g (t, x (t) , y (t)) Imagine that t1 is also a choice variable 365