Extra Practice Problems # 4
Jorge F. Chavez∗
December 3, 2012
Question 1: Human Capital in the Solow Model
Assume that the production function is given by:
Yt = Ktα Htλ (At Lt )1−α−λ
where Yt is aggregate output, Kt is physical capital, Ht is human capital, At is the level of
technology, and Lt is labor. Assume that α > 0, λ > 0 and α + λ < 1. The rates of growth of
Lt and At are n and g, respectively. Output can be used on a one-for-one basis for consumption
or investment in either type of capital. Both types of capital depreciate at the rate δ. Assume
that gross investment in physical capital and gross investment in human capital are fractions
sK and sH of output, respectively.
(a) Obtain the laws of motion of stock of physical capital per effective worker kt = Kt /(At Lt )
and the stock of human capital per effective worker ht = Ht /(At Lt ).
(b) What are the steady-state values of physical capital, human capital and output, all in per
unit of effective labor?
(c) What is the growth rate of output per capita in steady state?
i. If we think of all countries as being at their steady-state, can this model explain why
income per capita grows at different rates across countries?
ii. What if countries are at various distances from their steady-state?
(d) This augmented Solow model can be tested empirically with cross-country data if we assume
that all countries are in their steady-state.
i. Derive a log-linear regression equation for output per worker that you could estimate
with a simple regression, assuming that you have measures of siK , siH , δ i , ni for each
country i and that g and A0 are known and constant across countries.
ii. Give 1 or 2 examples of some problems that might arise in estimating such equations
∗
e-mail:j.chavez-cotrado@warwick.ac.uk
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Question 2: A variation of the Galor-Zeira model
Consider the following variation in the nature of the credit market in the Galor-Zeira model: a
borrower has the option of “taking the money and run” (i.e., not repay a loan at all) but incurs
a fixed cost of d from this which does not vary with loan size. The credit market is competitive,
and the cost of lending is constant equal to r.
Each individual lives for two periods, and leaves a bequest to a single heir. In the first period,
an individual either works as an unskilled worker or decides to acquire human capital at a fixed
schooling cost of h. In the second period, the individual works as a skilled worker if human
capital was acquired previously, and as an unskilled worker otherwise. Labor supply is inelastic,
and there is no consumption in the first period. Second period wealth between consumption and
bequest. The wage rate for unskilled and skilled labor are also constant: wn and ws respectively,
where ws > wn .
(a) Describe how the possibility of defaulting a loan affects the way credit market work.
(b) Provide a condition for human capital investments to be undertaken at all. In what follows,
assume this condition holds.
(c) Given the distribution of inherited wealth for any given generation, characterize the nature
of human capital investment decisions in the population.
(d) Describe the resulting dynamics of the distribution of inherited wealth. Provide sufficient
conditions for the emergence of wealth inequality in the long run.
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