1 EC9012
Exam & solution 2019-20 Moav's section
Question 2 (25 marks)
Aggregate output in the economy in period t is:
Yt = F (Kt; Lt) =
8<: AK
1=2
t L
1=2 if Kt=Lt 4
A (Lt +Kt=4) if Kt=Lt > 4
:
1. Find output per worker as a function of capital per worker: yt = f(kt):
(2 marks)
2. Find the marginal product of capital f 0(kt):(2 marks)
3. Find the wage rate wt:(2 marks)
Consider the Solow model with no population growth, full depreciation, and
a saving rate of 1/2: n = 0, = 1; s = 1=2:
4. Find the function governing the evolution of capital per worker over time:
kt+1 = (kt): (2 marks)
5. Find 0:(2 marks)
6. Find the smallest A for which the model is characterized by endless
growth. (3 marks)
Suppose now that A is a function of kt: In particular:
At = A(kt) =
8<: 2 if kt 4
6 if kt > 4
:
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7. Find the function kt+1 = (kt):(3 marks)
8. Find all stable steady states. (3 marks)
9. Plot a …gure of (kt) and the 45 degree line. Clearly mark the steady
states and the threshold kt = 4: (3 marks)
Consider now the OLG model. Output per worker is f(kt) = Ak
1=2
t and that
the utility function is: ut = c
y
t + c
o
t+1=2
10. Find the saving of the young in t; syt ; as a function of wt and kt+1:(3
marks)
Question 5 (25 marks)
Consider the endogenous R&D growth model
The …nal good produced by each worker in the …nal good sector is yt = At;
where, At = it: it is new knowledge (inventions) purchased by the worker. Note
that unlike the model in the lecture notes, the producers of the …nal good can
only use new inventions, and not the old stock of knowledge: At1:
In each period a population of size N joins the economy. Individuals are
active one period in which they work in the …nal good sector or in the R&D
sector. The number of workers in the R&D sector is H: The number of workers
in the …nal good sector is L: L+H = N .
The number of non-rival inventions produced in t is:
it = At1H1=2:
Inventions are made at the beginning of the period and sold to producers of the
…nal good. The income of each R&D worker is the number of innovations per
R&D worker, it=H; multiplied by the price of an innovation and the number
of …nal good producers who each purchase all the innovations. The price of an
innovation is 2 (0; 1]:
1. Find the income IHt of each R&D worker and the income of each …nal
good worker ILt for any given H: (5 marks)
2. Find the equilibrium allocation H. (5 marks)
3. Find the equilibrium growth rate. Is there a “scale e¤ect”? Explain the
economics of negative growth under the speci…c assumptions of the model in
this question. (5 marks)
4. Find the allocation H that maximizes output in any given one period. (5
marks)
5. An optimal policy that maximizes income over more than one period
would be to set at:
a. = 1=3
b. > 1=3
c. = 1=2
d. > 1=2
e. = 1
Choose the correct answer and explain brie‡y. (5 marks)
Question 6 (25 marks)
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Consider a “Galor-Zeira”economy: Individuals live for two periods in over-
lapping generations. Each individual has one parent and one o¤spring. In the
…rst period of their life (young), individuals receive a transfer of income from
their parent and a transfer of income from the government. The transfer in
period t from the parent to the o¤spring in dynasty i is bit: The transfer from
the government is xt (all the young in t receive the same xt). In the second
period of life (adults), individuals allocate their income as following: a fraction
is transferred to their o¤spring and a fraction 1 is consumed. There is no
physical capital in the economy and individuals cannot borrow or lend.
Young individuals, if they have the required resources, could invest in school-
ing. The cost of schooling is h units of income. Thus, a young individual i in t
could invest in schooling only if bit + xt h:
In adulthood the wage rate of an educated worker is we and the wage rate
of an uneducated workers is wu:
The government taxes labour income (which is the only income individuals
have). In particular, a fraction of all labour income is taxed, and therefore,
the after tax income is (1 )we for an educated worker and (1 )wu for an
uneducated worker.
All the tax revenue is transferred to the young. Each young individual
receives xt units of income, equal to the average tax revenue per worker: xt =
(Etw
e + (1 Et)wu) ; where Et is the fraction of educated workers in period
t.
Assume that wu < h, we > h; and we > wu + h:
1. Find the lowest xt that permits o¤spring of an uneducated parent to
investment in human capital. (4 marks)
2. Find the required tax rate as a function of Et that would permit each
young individual in the economy to invest in schooling. That is, …nd the lowest
permitting education to all. Denote this tax rate by min: How is that tax rate
a¤ected by Et (does it increase or decrease with Et)? Explain in one sentence
the e¤ect of Et on the required tax rate. (4 marks)
Suppose now that the young consume any income that isn’t invested in
schooling. The utility function of individual i who is young in t is: uit = c
i
t +
cit+1 + b
i
t+1: That is, individuals are indi¤erent between consuming one unit of
income when young (in t) and using one unit of income when old (in t+ 1) for
consumption or transfer to their o¤spring.
3. Find a condition on wages, the cost of schooling, and the tax rate, such
that an individual who can invest in education will choose to do so. Find the
highest tax rate possible that will not prevent investment in schooling. Denote
this tax rate max: (4 marks)
Suppose now that the tax rate is su¢ ciently high so that the o¤spring of the
uneducated receive a transfer from the government that is su¢ ciently high for
investment in education if they prefer to do so and at the same time, the tax
rate is below the highest possible that will not prevent investment in schooling.
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4. Find Et+1 for any Et: (4 marks)
5. Find Et+1 for any Et; if the tax rate is zero: = 0: (4 marks)
6. Find an implicit condition on Et such that a tax policy, ; designed
to increase education exists. Explain how a poverty trap could exist in the
economy, despite a government that chooses the optimal tax rate to reduce
poverty. In a case of such a poverty trap, suggest an alternative policy to
increase education. (5 marks)
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Solution to Question 2
Aggregate output in the economy in period t is:
Yt = F (Kt; Lt) =
8<: AK
1=2
t L
1=2 if Kt=Lt 4
A (Lt +Kt=4) if Kt=Lt > 4
:
1. Find output per worker as a function of capital per worker: yt = f(kt):
(2 marks)
yt = f(kt) =
8<: Ak
1=2
t if kt 4
A (1 + kt=4) if kt > 4
:
2. Find the marginal product of capital f 0(kt):(2 marks)
f 0(kt) =
8<: Ak
1=2
t =2 if kt 4
A=4 if kt > 4
:
3. Find the wage rate wt:(2 marks)
wt =
8<: Ak
1=2
t =2 if kt 4
A if kt > 4
:
Consider the Solow model with no population growth, full depreciation, and
a saving rate of 1/2: n = 0, = 1; s = 1=2:
4. Find the function governing the evolution of capital per worker over time:
kt+1 = (kt):
kt+1 = (kt) =
8<: Ak
1=2
t =2 if kt 4
A (1 + kt=4) =2 = A (1=2 + kt=8) if kt > 4
:
5. Find 0
0(kt) =
8<: Ak
1=2
t =4 if kt 4
A=8 if kt > 4
:
6. Find the smallest A for which the model is characterized by endless
growth:
A = 8
Suppose now that A is a function of kt: In particular:
At = A(kt) =
8<: 2 if kt 4
6 if kt > 4
:
5
7. Find the function kt+1 = (kt)
kt+1 = (kt) =
8<: k
1=2
t if kt 4
3 + 3kt=4 if kt > 4
:
8. Find all the stable steady states
A (1 + k=4) =2 = k ! k = 4A8A ! for A = 6; k = 12
Ak1=2=2 = k ! k = A2=4! for A = 2; k = 1
9. Plot a …gure of (kt) and the 45 degree line. Clearly mark the steady
states and the threshold kt = 4:
Figure: increasing and concave from zero to 2 at k = 4; a jump at k = 4 to
6: Linear and increasing for k > 4. steady states at 1 and 12.
Consider now the OLG model. Output per worker is f(kt) = Ak
1=2
t and that
the utility function is: ut = c
y
t + c
o
t+1=2
10. …nd the saving of the young in t as a function of wt and kt+1:
st =
8<: wt if Rt > 2$ kt+1 < A
2=16
[0; wt] if Rt = 2$ kt+1 = A2=16
0 if Rt < 2$ kt+1 > A2=16
:
Solution to Question 5
1. IHt = At1H
1=2L = At1H1=2 (N H) ; ILt = (1 )At1H1=2
2. IHt = I
L
t ! At1H1=2 (N H) = (1 )At1H1=2 !
(N H) = (1 )H ! N = H
3. Yt = LAt
At = At1H1=2 ! growth: At=At1 1 = H1=2 1 = 1=2N1=2 1:
There is a scale e¤ect: N has a positive e¤ect on growth. Growth can be
negative (if 1=2N1=2 < 1): The reason is that old innovations can not be used
in the production process.
4. Find the allocation H that maximizes output in any given one period.
Yt = At1H1=2 (N H) ; dAt1H
1=2(NH)
dH =
1
2
p
H
At1 (N 3H) = 0;H =
N=3
5. The answer is > 1=3: = 1=3maximizes output in one period. However,
since there is a positive externality from one period to the next, there are gains
from higher investment in t even on the account of output in t for the sake
of output in t + 1 and later periods. = 1=2 or > 1=2 is just noise. = 1
maximizes the growth rate of knowledge At+1=At but output is constant at zero.
Solution Question 6
1. (1 )wu + x = h! x = h (1 )wu
2. (Etwe + (1 Et)wu) = h (1 )wu
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(Etw
e + (1 Et)wu) + wu wu = h
(Etw
e + (1 Et)wu wu) = h wu
min =
h wu
Etwe + (1 Et)wu wu =
h wu
Et (we wu) + (1 )wu
The required tax rate is decreasing in E since we > wu: The reason is that
with more educated workers total income is higher and thus a lower tax rate is
su¢ cient for the required tax revenue.
3. condition for investment:
(1 )(we wu) > h
highest tax rate
max = 1 h
we wu =
we wu h
we wu
4. Et+1 = 1 for any Et:
5. Et+1 = Et
6. there exist a policy to reduce poverty if: min < max ! hw
u
Et(wewu)+(1)wu <
wewuh
wewu
With a low Et it could be the case that min > max and there is no feasible
that would lead to an increase in E: An alternative policy would be to choose
a tax rate that is < max and allocate the transfer x only to a su¢ ciently
small fraction of the population such that it is su¢ cient to allow the o¤spring
of the uneducated to invest in education.
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