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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

:

1

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 briey. (5 marks)

Question 6 (25 marks)

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Consider a Galor-Zeiraeconomy: 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 isnt 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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