Economic growth and economic development 70

Introduction to Modern Economic Growth output δk(t) f(k(t)) f(k*) consumption sf(k(t)) sf(k*) investment k* k(t) Figure 2.4 Investment and consumption in the steady-state equilibrium the capital-labor ratio k∗ ∈ (0, ∞) is given by (2.17), per capita output is given by (2.18) y ∗ = f (k∗ ) and per capita consumption is given by (2.19) c∗ = (1 − s) f (k∗ ) Proof The preceding argument establishes that (2.17) any k∗ that satisfies (2.16) is a steady state To establish existence, note that from Assumption (and from L’Hopital’s rule), limk→0 f (k) /k = ∞ and limk→∞ f (k) /k = Moreover, f (k) /k is continuous from Assumption 1, so by the intermediate value theorem (see Mathematical Appendix) there exists k∗ such that (2.17) is satisfied To see uniqueness, differentiate f (k) /k with respect to k, which gives w ∂ [f (k) /k] f (k) k − f (k) = = − < 0, (2.20) ∂k k k where the last equality uses (2.14) Since f (k) /k is everywhere (strictly) decreasing, there can only exist a unique value k∗ that satisfies (2.17) 56