EC9012 – EXAM 2020-21; Moavís section
Question 2 (25 marks)
Consider a ìGalor-Zeira economy: Individuals live for two periods in over_x005f lapping generations. Each individual has one parent and one o§spring. In the
rst period of their life, individuals are young.í 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 bi
t: 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.í The transfer bi
t is:
bi
t =
8
<
:
0 if It
i
(It
i ) if It
i >
;
where It
i 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: hi
t+1 = h(ei
t); where ei
t is the investment
in education of a young individual i in period t. The income of an adult is:
It
i = wh(ei
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: ei
t = bi
t + xt:
Suppose that hi
t+1 = h(ei
t)=1+ ei
t: That is, the level of human capital is
equal to one plus investment in education.
1. Find It
i
+1 as a function of bi
t (2 marks)
2. Find the function governing the evolution of transfers within each dynasty:
bi
t+1 = (bi
t) (3 marks)
3. For xt = 0 for all t; nd conditions on the parameters such that (bi
t) has
two steady states: a poverty trap, and a threshold above which income grows
endlessly. (3 marks)
4. Plot (bi
t) 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 bi
t 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 bi
t (holding constant
the average) on aggregate income in the short run: periods t + 1 and t + 2:
Explain brieáy.
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 bi
t on the rate
of economic growth in the long run Explain brieáy. (3 marks)
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)
9. Explain brieáy the reason behind the hope that microcredit will help the
poor escape poverty, and mention possible reason(s) behind the disappointing
outcomes of microcredit. (use no more than 110 words). (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) = Atkt
1=2
.
Productivity is a function of capital per worker. In particular: At = Bkt
; where
B > 0; and 2 (0; 1):The e§ect of kt on At is external to the rm,
and therefore
rms take At 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 = Bkt
(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)
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 brieáy.(4 marks)
6. Explain the e§ect of the depreciation rate on economic growth and on
welfare (utility). (5 marks)
Question 6 (25 marks)
Consider the following Principal-Agent problem:
3
Output produced by the agent (the farmer) can be either low or high: Y 2
fH; Lg; the agentís 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 agentís 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 doesnít 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 = .
The agent doesnít know the state of nature when deciding his e§ort level.
The agentís 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 agentís 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 ìpure 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 principalís 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 ìstick
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 principalís expected cost of employing an
agent that has the incentive to exert high e§ort, denoted Cs:(5 marks)
3. Find a condition on x that determines which contract, ìpure carrot or
ìstick and carrot, the principal will choose. (5 marks)
4. Explain in no more than a 120 words the two explanations discussed in the
lecture for granting property rights, and how they are related to information.
The two explanations are: (1) within the principal-agent model the explanation
for a ìpure carrot contract (no dismissal is granting the facto property rights
over land to farmers); (2) the Douglas Northís explanation based on the incentive
to invest. (10 marks)
Solution
Solution to Question 2
1. Find It
i
+1 as a function of bi
t (2 marks)
4
It
i
+1 = w
1 + bi
t + xt