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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) = ra (t) − c (t) + W (t) ,
and suppose that there are credit market imperfections so that a (t) ≥ 0.

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) , 1 − 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 > 0. Show that the optimal path of human capital investments
involves s (t) = 1 for some interval [0, T ] and then s (t) = s∗ for t ≥ T .
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