Introduction to Modern Economic Growth The steady-state (stationary) solution of this optimal control problem involves µ˙ (t) = and h˙ (t) = 0, and thus implies that (10.16) x∗ = φ0−1 (r + ν + δ h ) , where φ0−1 (·) is the inverse function of φ0 (·) (which exists and is strictly decreasing since φ (·) is strictly concave) This equation shows that x∗ ≡ s∗ h∗ will be higher when the interest rate is low, when the life expectancy of the individual is high, and when the rate of depreciation of human capital is low To determine s∗ and h∗ separately, we set h˙ (t) = in the human capital accumulation equation (10.13), which gives φ (x∗ ) δ ¢ ¡h 0−1 φ φ (r + ν + δ h ) = δh h∗ = (10.17) Since φ0−1 (·) is strictly decreasing and φ (·) is strictly increasing, this equation implies that the steady-state solution for the human capital stock is uniquely determined and is decreasing in r, ν and δ h More interesting than the stationary (steady-state) solution to the optimization problem is the time path of human capital investments in this model To derive this, differentiate (10.14) with respect to time to obtain µ˙ (t) x˙ (t) = εφ0 (x) , µ (t) x (t) where xφ00 (x) >0 εφ (x) = − φ (x) is the elasticity of the function φ0 (·) and is positive since φ0 (·) is strictly decreasing (thus φ00 (·) < 0) Combining this equation with (10.15), we obtain (10.18) x˙ (t) = (r + ν + δ h − φ0 (x (t))) x (t) εφ (x (t)) Figure 10.1 plots (10.13) and (10.18) in the h-x space The upward-sloping curve corresponds to the locus for h˙ (t) = 0, while (10.18) can only be zero at x∗ , thus the locus for x˙ (t) = corresponds to the horizontal line in the figure The arrows of motion are also plotted in this phase diagram and make it clear that the steady-state 471