Introduction to Modern Economic Growth Proposition 2.8 Suppose Assumptions and hold and f (k) = af˜ (k) Denote the steady-state equilibrium level of the capital-labor ratio by k∗ (a, s, δ, n) and the steady-state level of output by y ∗ (a, s, δ, n) when the underlying parameters are given by a, s and δ Then we have ∂k∗ (a, s, δ, n) ∂k∗ (a, s, δ, n) ∂k∗ (a, s, δ, n) ∂k∗ (a, s, δ, n) > 0, > 0, < and 0, > 0, < 0and < ∂a ∂s ∂δ ∂n The new result relative to the earlier comparative static proposition is that now a higher population growth rate, n, also reduces the capital-labor ratio and output per capita The reason for this is simple: a higher population growth rate means there is more labor to use the existing amount of capital, which only accumulates slowly, and consequently the equilibrium capital-labor ratio ends up lower This result implies that countries with higher population growth rates will have lower incomes per person (or per worker) 2.5 Transitional Dynamics in the Continuous Time Solow Model The analysis of transitional dynamics and stability with continuous time yields similar results to those in Section 2.3, but in many ways simpler To this in detail, we need to remember the equivalents of the above theorems for differential equations Other useful results on differential equations are provided in the Mathematical Appendix Theorem 2.4 Consider the following linear differential equation system (2.34) x˙ (t) = Ax (t) + b with initial value x (0), where x (t) ∈ Rn for all t, A is an n × n matrix and b is a n × column vector Let x∗ be the steady state of the system given by Ax∗ + b = Suppose that all of the eigenvalues of A have negative real parts Then the steady state of the differential equation (2.34) x∗ is globally asymptotically stable, in the sense that starting from any x (0) ∈ Rn , x (t) → x∗ Proof See Luenberger (1979, Chapter 2) or Simon and Blume (1994, Chapter Ô 25) 71