UNIVERSITY OF WARWICK
Department of Economics
Course EC9A2
End of Term Makeup Test
Time Allowed: Two and a half hours
Answer ALL 3 questions
Question 1
Output in period t is a function of physical capital K t and human capital H t :
Yt
F (K t , H t )
A[ K t1 / 2
H t1 / 2 ]2 .
Human capital is measured in units of labour.
a. Show that this production function is subject to constant returns to scale.
(4 marks)
b. Find output per unit of human capital as a function of the physical/human
capital ratio: f (k t )
Yt / H t , where k t
Kt / Ht .
(4 marks)
c. Find the marginal product of physical capital
Rt
as a function of
kt
(4 marks)
d. Find
Rt
for k t
and for k t
0
(4 marks)
e. Find the marginal product of human capital
wt as a function of k t
(4 marks)
f. Each worker has ht units of human capital; ht
H t / Lt where Lt is the
number of workers. Find output per worker Yt / Lt as a function of k t and ht .
(4 marks)
Suppose now that the saving rate is
s
(saving in each period t is
sYt ) , the size of
population is constant, there is full depreciation of physical capital and the stock
of human capital is constant. There is no investment in human capital and
EC9A2
therefore the stock of physical capital in each period is equal to saving in the
preceding period and ht h is constant over time.
g. Find the dynamical system governing the evolution of capital per unit of
(k t ) .
human capital over time: k t 1
(4 marks)
Suppose now that savings are allocated equally between investment in human
capital and investment in physical capital: K t 1 H t 1 sYt / 2
h. Find the dynamical system governing the evolution of output Yt
1
(Yt ) .
(4 marks)
i.
Find a condition on A that assures growth in the steady state.
(4 marks)
(Total 36 Marks)
Question 2
Consider an economy in which income, denoted x, is uniformly distributed across
all individuals between 1 and 1+2b:
x ~ U [1, 1 2b]
b 0
The government taxes income at a rate and equally distributes the tax revenue
among all individuals, net of an administrative cost, which is a fraction 1
of the
tax revenue. That is, only a fraction of the tax revenue is redistributed.
a. Find average income (before tax) in the economy, denoted x . Present the
income after redistribution (disposable income) of each individual, ~
x i , as a
function of pre-tax income,
and average income, x .
x i , the tax rate,
, the distortion parameter,
,
(5 marks)
b. Find the income level, x̂ above which individuals prefer that there is no tax
and below they prefer the tax exists.
(6 marks)
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EC9A2
c. Find the highest level of inequality, measured by b, such that no one
supports that tax. Denote this level of b as
b̂ .
(6 marks)
d. Find the fraction of individuals who support the tax as a function of the
parameter b for b bˆ .
What is the effect of b on the fraction who support the tax if =1? Explain
your result in two sentences. Your explanation should include the average
income and the median income.
(6 marks)
e. Find the effect of inequality, as captured by b, on taxation and aggregate
welfare.
(6 marks)
(Total 30 Marks)
Question 3
Consider an economy in which individuals live for two periods in overlapping
generations. In the first period of their life individuals receive a transfer of bti units
from their parent and allocate it optimally between investment in human capital
and physical capital. In the second period of life individuals allocate their income
1 of income above some
from labour and capital as following: a fraction
0 is transferred to their offspring. They consume the rest of their
threshold
(1 )(Income ) if Income
income: Consumption
otherwise they
consume their entire income. Each individual has one parent and one offspring.
Labour income of individual i, who is working in period t+1, is 1
A ln(1
eti ) . The
return to capital is R. individuals cannot borrow for the sake of investment in
education.
a. Find a necessary condition on A for investment in education to be positive.
(7 marks)
b. Find the optimal unconstrained level of investment in human capital as a
function of the return to physical capital.
(7 marks)
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EC9A2
c. Find the investment in human capital as a function of the transfer an individual
receives and the return on physical capital.
(7 marks)
Suppose now that R=0. That is, there is no investment in physical capital.
d. Find the dynamical system governing the evolution of bequests in a dynasty:
bti 1
(bti ) .
(7 marks)
e. Find necessary conditions such that the initial distribution of wealth has an
impact on the long run steady state of the economy.
(6 marks)
(Total 34 marks)
END
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