Problem Set 6 - income inequality and growth: random shooks, mobility
and aggregation
1. Consider the Galor-Zeira (1993) model. The population size of each
generation is normalized to 1 where  is the number of unskilled workers,
 is the number of skilled workers, and  +  = 1 The return to physical
capital is  the wage rate for unskilled workers is   the wage rate of
skilled workers is   and the cost of education is 
i. Individuals cannot borrow in order to invest in human capital:
( = ∞)
a. Find the dynamical system governing the evolution of transfers within a
dynasty:
+1 = ( )
b. Define sufficient conditions on  and  that assure that initial wealth
distribution has an effect on output in the long run. Show in a figure the
dynamical system under these conditions.
c. Suppose now that in each period   0 skilled individuals decide to
leave their offspring with no bequest:  = 0 Under the restrictions on
the parameters from part b, what will characterize the long run income
distribution in the economy?
√
d. Suppose now that  =   . For a very small , find conditions on
the parameters that assure that the initial wealth distribution has an effect
on the economy in the long-run. i.e., conditions such that there are two
locally stable steady states that the economy can converge to. (Remember
that the size of the working population is 1)
e. For each of the two steady states in part d, find (neglect  in your
calculations):
1. 
2. 
3. 
ii. There is a perfect loan market:
The young can lend and borrow from each other, repaying when old, and
the equilibrium interest rate between period  and  + 1 is +1  Hint: the
equilibrium interest rate that clears the loan market can not be less then
 − 1 (In this part  =  and  = 0)
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f. Find the dynamical system governing the evolution of output,   over
time, where  =   +   +  :
+1 = ( )
Present ( ) in a figure. (Note that you should distinguish between two
ranges of  :    and  ≥ )
g. What is the interest rate for loans between period  and  + 1, +1  for
   and for   ?
2. Consider the Maoz-Moav 1999 model, with the following production
function:
 =   +  
The cost of education,  , is uniformly distributed in the unit interval.
Assume that ( −  )  1 (this assumption assures that ̂  1  =  )
a. Find the dynamical system governing the evolution of   +1 = ( )
What is the steady state level of  ?
b. How is the slope and the intercept of the dynamical system, 0 ( ) and
(0) affected by a decline in  ? (assume 2   ) Explain the different
economic effects of  on investment decisions. hint: think of the effect of
changes in  on (1).
c. Suppose that a perfect loan market is introduced into the economy.
Individuals can borrow from other young individuals in order to finance
their education. Denote ̄ as the steady state education level. What are
the values of ̂ and ̂ ? What is the equilibrium interest rate as a function
of ̄ and wages?
d. Discuss the efficiency implications of the market for loans.
e. Find the dynamical system governing the evolution of  for the model
with a market for loans. (you may assume, for sake of simplifying your
analysis, that individuals do not consume in their first period, i.e.,  =
(1 − ) log +1 +  log +1 ).
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3. Consider an overlapping generations economy in which individuals consume and may invest in education in the first period of their lives. They work
and consume in the second period. The population size of each generation
is 1 Individuals receive in their first period a transfer from the government
 which they use for consumption and investment in education. The transfer to the young is financed by an income tax on the working population
(individuals in their second life period). The tax rate is  
Production is given by:
 =   +  
where      is the number of skilled workers in period  and  =
1 −  is the number of unskilled workers in  Therefore
 =  
Individuals’ preferences are represented by the utility function,
 = log  + log 
where  is consumption when young (first period) and  is consumption
when old (second period).
To become a skilled worker an individual has to purchase education when
young. The cost of education of individual ,  is indivisible. The cost of
education in the population is uniformly distributed in the unit interval.
There is no capital market in the economy. Individuals can not borrow or
lend. It is also impossible to store goods from one period to the next.
a. Find the dynamical system governing the evolution of   (Note that
your dynamical system should reflect the fact that  ≤ 1 for all  ).
b. Find a sufficient condition on  that assures that   1 for all 
c. Suppose now that there exists a perfect loan market. (The young can
lend and borrow from each other, repaying when old, and the equilibrium
interest rate between period  and  + 1 +1  assures that the demand for
loans is equal to the supply.). Find +1 as a function of      and +1
d. Find the dynamical system governing the evolution of  under the
assumption that a perfect loan market exists, and there is no consumption
in the first life period.
e. Explain why a negative effect of the tax on the incentive to purchase
education does not appear in both of the dynamical systems (with and
without loans).
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