MSc Macro group B; EXAM 2020-21; Moavs section

Question 2 (25 marks)

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, individuals are young. They receive a transfer of in-

come from their parent and a transfer of income from the government. The

transfer from the parent to the o¤spring in period t 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 individuals are adults.The transfer bit is:

bit =

8<: 0 if I

i

t

(Iit ) if Iit >

;

where Iit is the income of individual i who is an adult in period t; and > 0..

That is, adults consume all their income if it is less than ; and transfer to their

o¤spring a fraction of income above : 2 (0; 1):

There is no taxation. Income from natural resources is used to
nance the

transfer xt:

The production of human capital is: hit+1 = h(e

i

t); where e

i

t is the investment

in education of a young individual i in period t. The income of an adult is:

Iit = wh(e

i

t): Where w is the wage rate per unit of human capital.

There is no physical capital in the economy and no credit markets: no bor-

rowing and no lending.

The young invest all their income in their education: eit = b

i

t + xt:

Suppose that hit+1 = h(e

i

t) = 1 + e

i

t: That is, the level of human capital is

equal to one plus investment in education.

1. Find Iit+1 as a function of b

i

t (2 marks)

2. Find the function governing the evolution of transfers within each dynasty:

bit+1 = (b

i

t) (3 marks)

3. For xt = 0 for all t;
nd conditions on the parameters such that (bit) has

two steady states: a poverty trap, and a threshold above which income grows

endlessly. (3 marks)

4. Plot (bit) under the conditions in part 3. (2 marks)

5. Under the conditions in part 3,
nd a necessary and su¢ cient condition

on xt that will eliminate the poverty trap. (3 marks)

Suppose now that xt = 0; and the conditions of part 3 hold. The initial

distribution of bit is uniform with an average above the threshold (above which

dynasties are in a growth path of income and below they converge to the poverty

trap), and the minimum is below this threshold.

6. What is the e¤ect of inequality in the distribution of bit (holding constant

the average) on aggregate income in the short run: periods t + 1 and t + 2:

Explain briey.

2

(Note that more inequality is a larger support for the distribution with no

change in the mean). (3 marks)

7. What is the e¤ect of inequality in the initial distribution of bit on the rate

of economic growth in the long run? Explain briey. (3 marks)

With more equality more individuals are above the threshold for a growth

path, thus aggregate growth in the long run will be higher despite a cost in the

short run.

Suppose now that individuals could also invest in physical capital with a

gross return of R > 0: The return to human capital is higher than the return to

physical capital: w > R:

Suppose further that wages are taxed at a rate to
nance the transfer to

the young xt:

8. Explain, within this model, the political economy mechanism for the e¤ect

of inequality on economic growth. (3 marks)

Suppose now that there is a perfect loan market. The young can borrow and

lend to each other and repay the loan when they are adults.

9. Will there be any investment in physical capital? And what will be the

equilibrium interest rate clearing the markets? (3 marks)

Question 5 (25 marks)

Consider the OLG model (Diamond 1965).

A generation of size L is born every period and lives for two periods. Individuals

supply labor inelastically, consume and save in their
rst life period (young),

and consume in the second (old).

The young save half of their income.

Output per worker as a function of capital per worker is: yt = f(kt) = Atk

1=2

t .

Productivity is a function of capital per worker. In particular: At = Bk

t ; where

B > 0; and 2 (0; 1):The e¤ect of kt on At is external to the
rm, and therefore

rms takeAt as given and factor prices are determined accordingly (Factor prices

are equal to their marginal product for a given At). The depreciation rate is

2 [0; 1]:

1. Find the wage rate wt as a function of kt for any given At. What is the

wage rate as a function of kt; when taking into account that At = Bk

t ? (4

marks)

2. Find the equation governing the evolution of kt over time: kt+1 = (kt):(4

marks)

3. Find a condition on such that the dynamical system is characterized

by a unique globally stable steady state (4 marks)

4. Find conditions on and B such that the economy is growing endlessly

in a constant growth rate. (4 marks)

3

5. Plot (kt) for > 1=2 and a 45 degree line. Does the saving rate has a

negative e¤ect on output and growth? (Growth is the rate of change in kt for

any kt): Explain briey.(4 marks)

Suppose now that the utility function of the young in period t is ut =

ct + ct+1; = 0; and < 1=2:

6. Find the equation governing the evolution of kt over time: kt+1 = (kt):

(Hint: Remember that A(kt) is an externality when calculating the return to

capital Rt+1). (5 marks)

Question 6 (25 marks)

Consider the following Principal-Agent problem:

Output produced by the agent (the farmer) can be either low or high: Y 2

fH;Lg; the agents e¤ort can also be either low or high. The state of nature can

be either good or bad. Output is a function of e¤ort and the state of nature.

In particular, output is high if and only if the state of nature is good and the

agent exerts high e¤ort.

The agents cost of high e¤ort is : The cost of low e¤ort is zero. The

probability of a good state of nature is p. The principal doesnt observe the

state of nature or the e¤ort of the agent. The economy exists for two periods.

The principal designs a contract to maximize her expected income. The

contract includes a bonus payment b if output is high and could include dismissal

as punishment if output is low. The cost to the principal of dismissing the agent

is x. The value for the agent of maintaining the job is V > 0: (You do not need

to calculate V ; take it as a given parameter). V < =p:

The contract must include a minimum wage w > 0. w. w 6= .

The agent doesnt know the state of nature when deciding his e¤ort level.

The agents utility is equal to expected income net of the cost of e¤ort and the

expected value of not being dismissed (keeping the land). The agents outside

option is a zero utility. That is, if the gent is dismissed, he will have a zero

utility in the next period instead of V:

There is no signal on the state of nature.

1. Suppose punishment is not part of the contract. (It is a pure carrot

contract). Find the incentive compatibility constraint of the agent. Find the

size of b (denoted bc) that the principal will include in the contract. Calculate

the principals expected cost of employing an agent that has the incentive to

exert high e¤ort. Denote the cost: Cc. (5 marks)

2. Suppose punishment (dismissal) is included in the contract (It is a stick

and carrotcontract). The agent is punished if output is low. Find the incentive

compatibility constraint. Find the size of b (denoted bs) that the principal will

include in the contract. Calculate the principals expected cost of employing an

agent that has the incentive to exert high e¤ort, denoted Cs:(5 marks)

4

3. Find a condition on x that determines which contract, pure carrotor

stick and carrot,the principal will choose. (5 marks)

Suppose now that with probability q the true level of e¤ort of the agent is

revealed to the principal. (The principal randomly chooses a fraction q of all the

agents and inspects their e¤ort. The agents do not know if they are inspected

or not. They just know that with probability q their e¤ort level is revealed).

Consider the following revised stick and carrotcontract: the principal will

dismiss an agent if and only if output is low and it was revealed that the agent

did not exert high e¤ort..That is, an agent that was not inspected will not be

dismissed. (Think carefully about the probability that an agent that exerts high

e¤ort will be dismissed).

4. Find the incentive compatibility constraint of the agent for the revised

stick and carrotcontract. Find the size of b (denoted br) that the principal

will include in the contract. Calculate the principals expected cost of employing

an agent that has the incentive to exert high e¤ort, Cr. (4 marks)

5. Is there a combination of q > 0 and x > 0 such that the principal will

choose pure carrotand not the revised stick and carrot? Explain (3 marks)

6. Suppose now that x is su¢ ciently small such that the principal prefers

the (non revised) stick and carrot contract over the pure carrot contract.

(as calculated in part 3 above). Find a condition on q that determines which

contract the principal will choose: stick and carrot or revised stick and

carrot. What is the e¤ect of transparency (high q) on property rights (the

probability farmers are not dismissed) (3 marks)

Solution

Solution to Question 2

1. Find Iit+1 as a function of b

i

t (2 marks)

Iit+1 = w

1 + bit + xt

2. Find the function governing the evolution of transfers within each dynasty:

bit+1 = (b

i

t) (3 marks)

bit =

8<: 0 if w

1 + bit + xt

$ bit =w 1 xt

(w

1 + bit + xt

) if w 1 + bit + xt > $ bit > =w 1 xt ;

3. For xt = 0 for all t;
nd conditions on the parameters such that (bit) has

two steady states: a poverty trap, and a threshold above which income grows

endlessly. (3 marks)

> w

and

w > 1

5

4. Plot (bit) under the conditions in part 3. (2 marks)

Zero up to =w 1 > 0 and linear at a slope w > 1 from =w 1

5. Under the conditions in part 3,
nd a necessary and su¢ cient condition

on xt that will eliminate the poverty trap. (3 marks)

w (1 + xt) > ! xt > =w 1

Suppose now that xt = 0; and the conditions of part 3 hold. The initial

distribution of bit is uniform with an average above the threshold (above which

dynasties are in a growth path of income and below they converge to the poverty

trap), and the minimum is below this threshold.

6. What is the e¤ect of inequality in the distribution of bit (holding constant

the average) on aggregate income in the short run: periods t + 1 and t + 2:

Explain briey.

(Note that more inequality is a larger support for the distribution with no

change in the mean). (3 marks)

Inequality in b has no e¤ect on income in t+1: All the transfers are invested

in human capital and the return to human capital is constant. It could have a

positive e¤ect on income in t+2 as inequality increases aggregate transfers and

thereby investment in human capital. Inequality reduces the consumption of in-

dividuals that leave a zero bequest. Inequality has only a positive e¤ect because

investment in human capital isnt subject to diminishing marginal product.

7. What is the e¤ect of inequality in the initial distribution of bit on the rate

of economic growth in the long run? Explain briey. (3 marks)

With more equality more individuals are above the threshold for a growth

path, thus aggregate growth in the long run will be higher despite a cost in the

short run.

Suppose now that individuals could also invest in physical capital with a

gross return of R > 0: The return to human capital is higher than the return to

physical capital: w > R:

Suppose further that wages are taxed at a rate to
nance the transfer to

the young xt:

8. Explain, within this model, the political economy mechanism for the e¤ect

of inequality on economic growth. (3 marks)

If the tax rate is too high such that (1)w < R; there will be no investment

in human capital, and hence lower output and growth. This could happen if

the current generation might prefer to heavily tax the wage income of the adult

workers to obtain a higher x even if it will imply a lower return on their own

investment

Suppose now that there is a perfect loan market. The young can borrow and

lend to each other and repay the loan when they are adults.

6

9. Will there be any investment in physical capital? And what will be the

equilibrium interest rate clearing the markets? (3 marks)

no and 1 + r = w

Solution to question 5

1. Find the wage rate wt as a function of kt for any given At. What is the

wage rate as a function of kt; when taking into account that At = Bk

t ? (4

marks)

wt = Atk

1=2

t =2 = Bk

+1=2

t =2

2. Find the equation governing the evolution of kt over time: kt+1 = (kt):(4

marks)

kt+1 = (kt) = wt=2 = Bk

+1=2

t =4

3. Find a condition on such that the dynamical system is characterized

by a unique globally stable steady state (4 marks)

< 1=2

4. Find conditions on and B such that the economy is growing endlessly

in a constant growth rate. (4 marks)

= 1=2 and B > 4:

5. Plot (kt) for > 1=2 and a 45 degree line. Does the saving rate has a

negative e¤ect on output and growth? (Growth is the rate of change in kt for

any kt): Explain briey.(4 marks)

The curve is increasing and concave, crossing the 45 degree line once from

below. This is an unstable steady state. Zero is a locally stable steady state.

Above the unstable steady state the economy is converging to in
nity. An

increase in the saving rate of the young will shift the function up, and the

threshold steady state to the left. The result will be a higher growth rate for

any kt > 0:

Suppose now that the utility function of the young in period t is ut =

ct + ct+1; = 0; and < 1=2:

6. Find the equation governing the evolution of kt over time: kt+1 = (kt):

(Hint: Remember that A(kt) is an externality when calculating the return to

capital Rt+1). (5 marks)

Rt = Atk

1=2

t =2 = Bk

1=2

t since < 1=2; Rt is decreasing in kt: Therefore

there is a unique k such that Rt is larger than 1 below k and smaller above.

Rt = 1! Bk1=2t = 1! kt = (1=B)

1

1=2 = k

And

st =

8<: wt if kt+1 <

k

[wt; 0] if kt+1 = k

0 if kt+1 > k

7

kt+1 =

(

Bk

+1=2

t =2 if kt < k^

(1=B)

1

1=2 if kt k^

where k^ is given by Bk^+1=2=2 = (1=B)

1

1=2

Bk^+1=2=2 = (1=B)

1

1=2

k^+1=2 =

2

B+1=2

k^ =

2

1

+1=2

B

Solution to Question 6

1. Suppose punishment is not part of the contract. (It is a pure carrot

contract). Find the incentive compatibility constraint of the agent. Find the

size of b (denoted bc) that the principal will include in the contract. Calculate

the principals expected cost of employing an agent that has the incentive to

exert high e¤ort. Denote the cost: Cc. (5 marks)

Incentive compatibility constraint :

w + pbc w ! bc = =p

Cost of incentivizing the agent: Cc = w + pbc = w +

2. Suppose punishment (dismissal) is included in the contract (It is a stick

and carrotcontract). The agent is punished if output is low. Find the incentive

compatibility constraint. Find the size of b (denoted bs) that the principal will

include in the contract. Calculate the principals expected cost of employing an

agent that has the incentive to exert high e¤ort, denoted Cs:(5 marks)

Incentive compatibility constraint :

w + pbs + pV w ! bs = =p V

Cost of incentivizing the agent: Cs = w+pbs+(1p)x = w+
pV+(1 p)x

3. Find a condition on x that determines which contract, pure carrotor

stick and carrot,the principal will choose. (5 marks)

choose stick and carrot if Cc Cs $

w + w + pV + (1 p)x

pV

1 p x

Punishment is included in the contract if the cost to the principal is su¢ ciently

low: x pV1p

8

Suppose now that with probability q the true level of e¤ort of the agent is

revealed to the principal. (The principal randomly chooses a fraction q of all the

agents and inspects their e¤ort. The agents do not know if they are inspected

or not. They just know that with probability q their e¤ort level is revealed).

Consider the following revised stick and carrotcontract: the principal will

dismiss an agent if and only if output is low and it was revealed that the agent

did not exert high e¤ort..That is, an agent that was not inspected will not be

dismissed. (Think carefully about the probability that an agent that exerts high

e¤ort will be dismissed).

4. Find the incentive compatibility constraint of the agent for the revised

stick and carrotcontract. Find the size of b (denoted br) that the principal

will include in the contract. Calculate the principals expected cost of employing

an agent that has the incentive to exert high e¤ort, Cr. (4 marks)

Incentive compatibility constraint (an agent exerting e¤ort is never dis-

missed, a shirking agent will be dismissed with probability q):

w + pbr + V w + (1 q)V

! br = =p qV=p

Cost of incentivizing the agent under the revised stick and carrot contract:

Cr = w + pbr = w + qV

(in equilibrium, under the revised stick and carrot, there is no punishment:

agents exert high e¤ort and therefore are never revealed as shirking)

5. Is there a combination of q > 0 and x > 0 such that the principal will

choose pure carrotand not the revised stick and carrot? Explain (3 marks)

No. Cc = w + > Cr = w + qV

The inspection allows a threat of punishment that works as an incentive for

the agent, allowing a smaller bonus, but comes at no cost for the principal.

6. Suppose now that x is su¢ ciently small such that the principal prefers

the (non revised) stick and carrot contract over the pure carrot contract.

(as calculated in part 3 above). Find a condition on q that determines which

contract the principal will choose: stick and carrot or revised stick and

carrot. What is the e¤ect of transparency (high q) on property rights (the

probability farmers are not dismissed) (3 marks)

choose revised S&C if Cs Cr $

w + pV + (1 p)x w + qV

q p (1 p)x

V

q^

We assume (part 3 ) that pV1p x! q^ 0:

9

Hence: if q > q^ high transparency regime - the principal will choose

the revised S&C contract and the agents are never dismissed. So transparency

supports property rights.

10

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