Problem Set 7 - inequality and growth - the political economy approach
1. Consider an economy in which one third of the population has an income
(before tax) of  one third has an income of  and one third has an income
of  where      A tax rate  ∈ (0 1) is proposed. A fraction
 ∈ (0 1] of the tax revenue is equally distributed among all individuals,
where 1 −  is the distortion cost of the tax.
a. Find a condition (defined over    and ) that will assure a majority
support for the proposed tax.
b. What is the interpretation of the condition for  = 1?
2. Suppose ability of individual  is   and income  is given by
 =  + 
where    0  is uniformly distributed in the unit interval.
a. Find the cumulative distribution of wealth,  ()
b. Find the density function  ()
c. Calculate the Gini coefficient. (Gini: twice the surface between the diagonal and the Lorenz curve. Lorenz curve: the share of income of a fraction
of the population plotted on the fraction of population ranked according to
income). Show that inequality is increasing in 
d. A fraction  of wealth is taxed and equally redistributed. Suppose that
the tax is distortionary - a fraction   1 of the tax is redistributed. Find
the income level ̂ that is the threshold for supporting the policy. Show that
the majority of the population would oppose the tax:  (̂)  12 What
property of the income distribution is the reason for this result?
e. Plot  (̂) against  for   1 and for   1
f. Suppose that the larger is the support for the policy, i.e., the larger is
 (̂) the tax rate  is larger. What is the effect of income inequality on the
economy if   1 and if   1?
3. Consider an economy that exists for two periods. In the first period
individuals’ wealth is invested in physical and human capital. In the second
period production takes place, production factors are paid their marginal
product, and members of the economy consume all their income. The economy consists of  capitalists who each owns a wealth of  units in the first
period, and workers, whose number is normalized to one. Workers’ wealth
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in the first period is zero. Workers learn in the first period, and supply one
unit of time to the labor force in the second period. A fraction  of capitalists’ wealth is taxed in order to finance public schooling for the workers.
Hence the investment in education per worker is  =   The resulting
units of labor are () =  ;   1 Capitalists do not go to public schools
and do not work. Output is given by  =   1−  The stock of capital in
the economy is equal to the savings of the capitalists, i.e., their wealth net
of tax payment:  = (1 −  )
a. Find the tax rate that maximizes output
b. Find the optimal tax rate from the viewpoint of the workers
c. Find the optimal tax rate from the viewpoint of the capitalists
d. Explain your findings. In particular, without any further calculations,
explain which parts of your conclusions depend on the specific human capital production function and on the specific output production function,
and what is the importance of the assumption that capitalists do not get
educated and do not work.
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