Exercise 2.14. Consider the discrete-time Solow growth model with constant population growth at the rate n, no technological change, constant depreciation rate of δ and a constant saving rate s. Assume that the per capita production function is given by the following continuous but non-neoclassical function: f (k) = Ak +

b, where A, b > 0. (1) Explain why this production function is non-neoclassical (i.e., why does it violate Assumptions 1 and 2 above?). (2) Show that if A − n − δ = 1, then for any k (0) 6= b/2, the economy settles into an asymptotic cycle and continuously fluctuates between k (0) and b − k (0). (3) Now consider a more general continuous production function f (k) that does not satisfy Assumptions 1 and 2, such that there exist k1, k2 ∈ R+ with k1 6= k2 and k2 = f (k1) − (n + δ) k1 k1 = f (k2) − (n + δ) k2. Show that when such (k1, k2) exist, there may also exist a stable steady state. (4) Prove that such cycles are not possible in the continuous-time Solow growth model for any (possibly non-neoclassical) continuous production function f (k). [Hint: consider the equivalent of Figure 2.9 above]. (5) What does the result in parts 2 and 3 imply for the approximations of discrete time by continuous time suggested in Section 2.4? (6) In light of your answer to part 6, what do you think of the cycles in parts 2 and 3?