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Introduction to Modern Economic Growth 10.11 Exercises Exercise 10.1 Formulate, state and prove the Separation Theorem, Theorem 10.1, in an economy in discrete time Exercise 10.2 (1) Consider the environment discussed in Section 10.1 Write the flow budget constraint of the individual as a˙ (t) = (t) − c (t) + W (t) , and suppose that there are credit market imperfections so that a (t) ≥ Construct an example in which Theorem 10.1 does not apply Can you generalize this to the case in which the individual can save at the rate r, but can only borrow at the rate r0 > r? (2) Now modify the environment so that the instantaneous utility function of the individual is u (c (t) , − l (t)) , where l (t) denotes total hours of work, labor supply at the market is equal to l (t) − s (t), so that the individual has a non-trivial leisure choice Construct an example in which Theorem 10.1 does not apply Exercise 10.3 Derive equation (10.9) from (10.8) Exercise 10.4 Consider the model presented in Section 10.2 and suppose that the discount rate r varies across individuals (for example, because of credit market imperfections) Show that individuals facing a higher r would choose lower levels of schooling What would happen if you estimate the wage regression similar to (10.12) in a world in which the source of difference in schooling is differences in discount rates across individuals? Exercise 10.5 Consider the following variant of the Ben Porath model, where the human capital accumulation equation is given by h˙ (t) = s (t) φ (h (t)) − δ h h (t) , where φ is strictly increasing, continuously differentiable and strictly concave, with s (t) ∈ [0, 1] Assume that individuals are potentially infinitely lived and face a Pois- son death rate of ν > Show that the optimal path of human capital investments involves s (t) = for some interval [0, T ] and then s (t) = s∗ for t ≥ T 502