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Introduction to Modern Economic Growth Exercise 10.6 Modify the Ben Porath model studied in Section 10.3 as follows First, assume that the horizon is finite Second, suppose that φ0 (0) < ∞ Finally, suppose that limx→h(0) φ0 (x) > Show that under these conditions the optimal path of human capital accumulation will involve an interval of full-time schooling with s (t) = 1, followed by another interval of on-the-job investment s (t) ∈ (0, 1), and finally an interval of no human capital investment, s (t) = How the earnings of the individual evolve over the life cycle? Exercise 10.7 Prove that as long as Y (t) = F (K (t) , H (t)) satisfies Assumptions and 2, the inequality in (10.29) holds Exercise 10.8 Show that equilibrium dynamics in Section 10.5 remain unchanged if δ < Exercise 10.9 Prove that the current-value Hamiltonian in (10.23) is jointly concave in (k (t) , h (t) , ik (t) , ih (t)) Exercise 10.10 Prove that (10.24) implies the existence of a relationship between physical and human capital of the form h = ξ (k), where ξ (·) is uniquely defined, strictly increasing and continuously differentiable Exercise 10.11 Prove 10.1 Show that the differential equation for consumption growth could have alternatively been written as c˙ (t) = [fh (k (t) , ξ (k (t))) − δh − ρ] c (t) εu (c (t)) Exercise 10.12 Consider the neoclassical growth model with physical and human capital discussed in Section 10.4 (1) Specify the consumer maximization problem in this economy (2) Define a competitive equilibrium (specifying firm optimization and market clearing conditions) (3) Characterize the competitive equilibrium and show that it coincides with the solution to the optimal growth problem Exercise 10.13 Introduce labor-augmenting technological progress at the rate g into the neoclassical growth model with physical and human capital discussed in Section 10.4 (1) Define a competitive equilibrium 503