By de Croix D.l., Michel P.
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Savings are positive and smaller than ﬁrst-period income. This implies 0< w s(w, f (k)) < . k k For a ﬁxed w > 0 the limit of w/ k when k → ∞ is 0. This implies20 lim k→+∞ s(w, f (k)) = 0. k As a consequence, for (k, w) = k 1 + n − s(w, f (k)) , k we have lim k→+∞ (k, w) = 1 + n > 0. k This implies that (k, w) is positive for large values of k. We now study the sign of (k, w) when k goes to 0. The decreasing function f (k) admits a limit when k goes to 0. We distinguish two cases according to whether this limit is finite (case 1) or infinite (case 2): r Case 1: lim k→0 f (k) = f (0) is ﬁnite.
Under the assumption H3 this is equivalent to If g ( k) ¯ < 1. If g ( k) ¯ > 1, k¯ is unstable. The condition sw ω < 1 + n − sR f , or to m ( k) ¯ ¯ g ( k) > 1 is equivalent to m ( k) > 1. 10 (Stability of monotonic dynamics) Assume H1, H2, and H3, and consider a steady state k¯ > 0. ¯ ≥ 0 (monotonic dynamics with myopic foresight), k¯ In the case where m ( k) is respectively stable, unstable, or non-hyperbolic for the two dynamics when m (k) is respectively <1, >1, or = 1. ¯ < 0, k¯ is stable for the rational dynamics, but it may be In the case where m ( k) stable (m (k) > −1) or unstable (m (k) < −1) for the myopic dynamics.
In this case, the savings function s(w, f (0)) is well deﬁned and is positive. Then we have lim k→0 (k, w) = lim [(1 + n)k − s(w, f (k))] = −s(w, f (0)) < 0. k→0 r Case 2: lim k→0 f (k) = +∞. The return on savings becomes infinite as k approaches 0. 21 This implies that lim k→0 (k, w) = lim [k(1 + n) − s(w, f (k))] < 0. k→0 r Sub-case 2: lim k→0 s(w, f (k)) = 0. This is the case when savings go to zero as the interest rate goes to inﬁnity. This property of the savings function implies that the second-period consumption d = f (k)s(w, f (k)) tends to +∞.
A theory of economic growth by de Croix D.l., Michel P.