程序代写案例-EXAM 2020

MSc Macro group B; EXAM 2020-21; Moav’s section
Question 2 (25 marks)
Consider a “Galor-Zeira”economy: Individuals live for two periods in over-
lap
ping 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 brie‡y.
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 brie‡y. (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 brie‡y.(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 agent’s 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 agent’s 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 doesn’t 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 doesn’t know the state of nature when deciding his e¤ort level.
The agent’s 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 agent’s 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 principal’s 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 carrot”contract). 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 principal’s 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 carrot”or
“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 carrot”contract: 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 carrot”contract. Find the size of b (denoted br) that the principal
will include in the contract. Calculate the principal’s 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 carrot”and 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 brie‡y.
(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 isn’t 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 brie‡y. (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 brie‡y.(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 principal’s 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 carrot”contract). 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 principal’s 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 carrot”or
“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 carrot”contract: 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 carrot”contract. Find the size of b (denoted br) that the principal
will include in the contract. Calculate the principal’s 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 carrot”and 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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