University of Warwick
EC9A2 Advanced Macroeconomic Analysis
Problem Set #6: Income Inequality & Growth II
Jorge F. Chavez∗
December 3, 2012
Question 1
Consider the Galor-Zeira (1993) model. The population size of each generation is normalized
to 1, where Lt is the number of unskilled workers, Et is the number of skilled workers, and
Et + Lt = 1. The return to physical capital is R, the wage rate for unskilled workers is wu , the
wage rate of skilled workers is ws , and the cost of education is h.
(i) Individuals cannot borrow in order to invest in human capital: (θ = ∞)
(a) Find the dynamical system governing the evolution of transfers within a dynasty:
bit+1 = ϕ(bit ).
i
. Assume that we − wu > Rh so
Solution. Solving the individual’s UMP we know that bit+1 = βIt+1
that investing in education is profitable. Then:
{
bit+1
i
= βIt+1
=
[
(
)]
β we + R bit − h
ifbit ≥ h
[
]
β wu + Rbit
ifbit < h
( )
≡ φ bit
(1)
(2)
(b) Define sufficient conditions on ws and wu that assure that initial wealth distribution has
an effect on output in the long run. Show in a figure the dynamical system under these
conditions.
Solution. The situation we are looking for is depicted in figure 1. We need:
i. β(we − wu ) > βwu ⇔ we − wu > Rh, so that the intercept of the second piece is higher than
the intercept of the first piece of the φ(bit ) function. We had already stated this assumption.
ii. Rβ < 1 so that we make sure that both pieces will intersect the 45o degree line.
∗
e-mail:j.chavez-cotrado@warwick.ac.uk
1
EC9A2 (Fall 2012)
Problem Set # 6
iii. For a poverty trap we need the M kink in figure 1 to be below the 45o degree line. That is:
)
(
[ u
]
1
i
u
u
lim β w + Rbt = β [w + Rh] < h ⇒ w < h
−R
β
bit →h
iv. For the high-income steady state, we need the N kink in figure 1 to be above the 45o degree
line. That is , we need:
βwe > h ⇒ we >
h
β
Figure 1: Dynamic system bit+1 = φ(bit )
45o
bit+1
B
N
β (we − Rh)
M
βwu
0
A
bL
h
bH
bit
(c) Suppose now that in each period ε > 0 skilled individuals decide to leave their offspring
with no bequest: bit = 0. Under the restrictions on the parameters from part b, what will
characterize the long run income distribution in the economy?
Solution. Suppose assumptions in part (b) hold. This means that there is no social mobility (individuals from poor families cannot invest in education). Now each period there is an proportion
ϵ ∈ [0, 1] that exogenously decide to leave their offspring no bequests (bit = 0). Therefore:
Et+1 = (1 − ϵ) Et
This implies that limt→∞ Et+1 = 0. This is, in the long-run the whole population will be unskilled.
√
(d) Suppose now that wtu = A/ Lt . For a very small ϵ, find conditions on the parameters that
assure that the initial wealth distribution has an effect on the economy in the long-run.
i.e., conditions such that there are two locally stable steady states that the economy can
converge to. (Remember that the size of the working population is 1).
Solution. Recall that for the poverty trap we need wu < h(1/β − R), while for the high-income
steady state we need we > h/β. Only the first condition will be affected by the dynamics introduced
by ϵ and by the dependence of wu on Lt .
Jorge F. Chávez
2
EC9A2 (Fall 2012)
Problem Set # 6
• If Lt → 1 then wtu → A. In this case, to maintain the poverty trap we need A < h(1/β − R).
• For a Lt sufficiently small we could have
= wtu > h(1/β − R), so that the poverty trap will
(
)
disappear. Hence, there exists a cutoff L̂ such that √A = h β1 − R . Solving, we get:
√A
Lt
L̂
(
L̂ =
Aβ
we − Rh
)2
That is, as long as L0 ≥ L̂, we will have a poverty trap (provided that A < h(1/β − R)).
When L0 < L̂, there is no poverty trap and here Lt will converge to some value L̄ which implies
that the number of people that become skilled from t to t + 1 equals the number of people that
go from skilled to unskilled (because of the shock). The idea is that for a sufficiently small ϵ,
the condition that must guarantee this steady state is:
we − wu (L̄) = Rh
A
we − √
= Rh
L̄
Because otherwise, a fraction higher than ϵ will become skilled and the steady-state condition
will be violated. Then:
(
)2
A
L̄ =
we − Rh
(e) For each of the two steady states in part d, find (neglect ε in your calculations):
Solution.
Under the restrictions on A and we the economy could converge to two steady states: if L0 ≤ L̂ it
will converge to L̄ if L0 > L̂ it will converge to L = 1. Then:
If L0 ≤ L̂
If L0 > L̂
L:
1
L̄
wu :
A
we − Rh
b:
βA
1−βR
β(we −Rh)
1−βR
(ii) There is a perfect loan market: The young can lend and borrow from each other,
repaying when old, and the equilibrium interest rate between period t and t + 1, is rt+1 . Hint:
the equilibrium interest rate that clears the loan market can not be less then R − 1. (In this part
wtu = wu and ε = 0).
(f) Find the dynamical system governing the evolution of output, Yt , over time, where Yt =
we Et + wu Ut + RKt :
Yt+1 = ψ(Yt ).
Present ψ(Yt ) in a figure. (Note that you should distinguish between two ranges of Yt :
βYt < h, and βYt ≥ h).
Jorge F. Chávez
3
EC9A2 (Fall 2012)
Problem Set # 6
Solution. We need to distinguish between two cases:
• Case 1: βYt < h
In this case, the economy as a whole will no have enough resources to finance education to every
individual. Because we are working under the assumption of perfect markets, there will be no
investing in physical capital and so K = 01 . For this we need 1 + rt+1 > R, otherwise some
individuals willing to borrow money to invest in human capital will be left aside. In equilibrium
a fraction βYt /h will invest in human capital while the rest 1 − βYt /h will remain unskilled:
(
)
(
)
βYt
βYt
Yt+1 = wu 1 −
+ ws
h
h
Appendix A1 derives this result from first principles.
• Case 2: βYt ≥ h
In this case, the economy as a whole is capable of financing everyone’s need of credit for investing
in human capital and what is more, there is room for a surplus of (aggregate) bequests, which
means that there is room for investing in physical capital. The fact that all individuals are now
skilled (because they are not credit constrained) implies that:
Yt+1 = ws + R (βYt − h)
Once again, this outcome can also be derived from first principles. See Appendix A2.
In sum:
Yt+1

(
)
(
)
 wu 1 − βYt + ws βYt
h
h
=

s
w + R (βY − h)
t
ifβYt < h
ifβYt ≥ h
(g) What is the interest rate for loans between period t and t + 1, rt+1 , for βYt < h and for
βYt > h?
Solution. If βYt < h we need to impose conditions that assure that individuals will find it more
attractive to invest in human capital for individuals to be indifferent between investing in human
capital and lending their surplus to others. Also, to make sure that there is no one invests in physical
capital, the return R must be lower than 1 + rt+1 , otherwise there markets will not clear as some
individuals will prefer to invest in K than to lend to other individuals who demand loans to finance
their education. That is:
(1 + rt+1 ) h = ws − wu > Rh
Finally, if βYt ≥ h then we will need (1 + rt+1 ) = R, otherwise individuals will not invest in physical
capital (if (1 + rt+1 ) > R and loan markets will not clear if (1 + rt+1 ) < R.
1
This means that we need to assume a production function in which capital is not essential for production.
Jorge F. Chávez
4
EC9A2 (Fall 2012)
Problem Set # 6
Question 2
Consider the Maoz-Moav 1999 model, with the following production function:
Yt = we Et + wu Lt
The cost of education, hi , is uniformly distributed in the unit interval. Assume that β(we −wu ) <
1 (this assumption assures that ĥi < 1, i = u, e)
(a) Find the dynamical system governing the evolution of Et , Et+1 = ϕ(Et ). What is the steady
state level of Et ?
Solution. Solving the UMP problem and taking into account the discrete nature of the investment
decision in education, we can find a cutoff level of hit , such that individuals will be indifferent between
investing or not in education:
[ e
u ]
wt+1 − wt+1
ĥit = bit
e
wt+1
s
u
= ws . Also recall that homoethetic preferences imply thtat bit+1 =
= wu and wt+1
Here wt+1
i
i
βIt+1 = βwt+1 . [Hence, ]an individual from an educated family will invest in education if and only if
hit ≤ ĥe = βwe
we −wu
we
capital if and only if
Et+1
hit
. Similarly, an individual from an uneducated family will invest in human
[ e u]
w −w
. Note that both cutoffs are the same for all individuals.
≤ ĥu = βwu
we
( )
( )
= Et F ĥe + Lt F ĥu
(
)
(
)
= Et Pr hit ≤ ĥe + (1 − Et ) Pr hit ≤ ĥu
Given uniform distribution:
[ e
]
[ e
]
u
u
e w −w
u w −w
Et+1 = Et βw
+ (1 − Et ) βw
we
we
[(
)
]
[
]
wu
wu
e
u
u
= Et β 1 − e (w − w ) +βw 1 − e
w
w
|
{z
}
| {z }
A
B
Hence in steady state:
Ē =
B
1−A
(b) How is the slope and the intercept of the dynamical system, ϕ′ (Et ) and ϕ(0), affected by a
decline in wu ? (assume 2wu < we ). Explain the different economic effects of wu on investment
decisions. hint: think of the effect of changes in wu on ϕ(1).
Solution. Recall:
Jorge F. Chávez
5
EC9A2 (Fall 2012)
Problem Set # 6
Effect on the slope:
2β
∂A
= e (we − wu ) (−1) < 0
∂wu
w
Effect on the intercept:
]
[
∂B
2wu
=
β
1
−
> 0Because 2wu < we
∂wu
we
Now, regarding the effect of an increase in wu on investment decisions, we can see how such
change affects the cutoffs for educated and for uneducated individuals. It is easy to see that
∂ ĥe
∂wu < 0 (this is a substitution effect: now the option of becoming unskilled is relatively
u
∂ ĥ
more attractive), and because of the assumption that 2wu < we , ∂w
u > 0 (this is an income
effect: income for unskilled is increased and therefore more people will be able to afford the
cost of education).
That is, the gap between the cutoffs is reduced with an increase in wu as shown in figure 2.
To see which effect dominates, we can follow the hint and see the effect on ϕ(1) = β(we −wu )
(that is, the number of educated people tomorrow when everyone is educated today. It is
easy to see that ∂ϕ(1)
∂wu = −β < 0, which suggest that the substitution effect dominates.
Figure 2: The gap effect of wu ↑
Gap 2
Gap 1
0
ĥu1
ĥu2
ĥe2
ĥe1
As wu ↑ these area additional
As wu ↑ these area additional
indiv. from uneducated families
indiv. from educated families
that are able to invest in
education
that will stop investing in
education
1
(c) Suppose that a perfect loan market is introduced into the economy. Individuals can borrow
from other young individuals in order to finance their education. Denote Ē as the steady
state education level. What are the values of ĥu and ĥe ? What is the equilibrium interest
rate as a function of Ē and wages?
Solution. Allowing for borrowing, the individual’s UMP is now:
{
}
max
log cit + (1 − β) log cit+1 + β log bit+1
{cit ,cit+1 ,bit+1 ,Bti }
s.t.
cit + δ i hit ≤ bit + Bti
i
cit+1 + bit+1 ≤ wt+1
− Rt+1 Bti
cit , cit+1 , bit+1 ≥ 0
Bti ≥ −H
Jorge F. Chávez
6
EC9A2 (Fall 2012)
Problem Set # 6
where Bti is the borrowing (Bti < 0) or lending (Bti > 0) decision and H is an (for now) exogenous
borrowing limit.2
As usual we can solve the problem backwards. At time t + 1 solves, given an optimal choice for time
t decision variables:
{
( i )
( i
)
}
i
i
v wt+1
≡
max
(1
−
β)
log
w
−
R
t+1 Bt − bt+1 + β log bt+1
t+1
i
i
i
0≤bt+1 ≤wt+1 −Rt+1 Bt
Solving for interior solution3 :
[ i
]
− Rt+1 Bti
bit+1 = β wt+1
Hence:
( i )
v wt+1
( i
[ i
])
( [ i
])
= (1 − β) log wt+1
− Rt+1 Bt − β wt+1
− Rt+1 Bti + β log β wt+1
− Rt+1 Bti
)
( i
= log wt+1
− Rt+1 Bti + M
where M = (1 + β) log(1 + β) + β log(β) is a constant term.
First period:
max
{
Bti ∈[−H,+∞)
)
( i )}
(
log bit + Bti − δ i hit + v wt+1
Solving for interior solution4 :
[
]
i
− Rt+1 bit − δ i hit
wt+1
i
Bt =
2Rt+1
i
∈ {we , wu }:
Now we use wt+1
Bte
=
Btu
=
[
]
we − Rt+1 bit − hit
2Rt+1
wu − Rt+1 bit
2Rt+1
Individuals will choose to invest in human capital if and only if:
(
)
(
)
log bit + Bte − 1 × hit + log (we − Rt+1 Bte ) > log bit + Btu − 0 × hit + log (wu − Rt+1 Btu ) (3)
After some algebra (see Appendix A3), condition (3) becomes:
(
(
))
(
)
(
)
log we + Rt+1 bit − hit > log wu + Rt+1 bit
⇔ we + Rt+1 bit − hit > wu + Rt+1 bit (4)
Hence, there exists a cutoff level ĥit such that the individual will invest in education if an only if
2
3
4
This is to prevent Ponzi games. The idea is that the agent can borrow only up to his ability to repay. Perhaps
next period you will be dealing with such constraint. Now is a mere formality here.
Is easy to see that this is the only plausible solution
Here we have only 1 corner. I think the way to rule out the corner would be to show that hitting it (borrowing
−H would imply no consumption.
Jorge F. Chávez
7
EC9A2 (Fall 2012)
Problem Set # 6
hit < ĥit . We can get ĥit from:
(
)
we + Rt+1 bit − ĥit = wu + Rt+1 bit
Then:
ĥit =
we − wu
= ĥe = ĥu
Rt+1
Then the law of motion of Et+1 is:
Et+1 = Et ĥt + (1 − Et ) ĥt = ĥt
Finally, in steady-state:
Ē =¯
ˆh =
we − wu
R̄
⇒
R̄ =
we − wu
Ē
(d) Discuss the efficiency implications of the market for loans.
Solution. The introduction of a “perfect loan market” implies that the cutoff for investing in education is now independent of the individual’s dynastic origin (i.e. their parent’s level of education and
therefore their level of bequest). The individual’s investment decision is now a function of only its
idiosyncratic realization of the cost of education hit . This implies that more individuals are allowed
to acquire education and become skilled, relative to the case in which there were two cutoffs.
The efficiency gain comes from the fact that, by introducing a single cutoff, perfect loan markets
imply that now some “rich” individuals with relatively high educational cost—who before would have
invested in education in the case with no perfect loans market—decide not to invest in human capital
and lend their bequests instead to individuals that are unable to finance the cost of education. That
is there is a reallocation of resources towards a more efficient usage.
(e) Find the dynamical system governing the evolution of Et for the model with a market
for loans. (you may assume, for sake of simplifying your analysis, that individuals do not
consume in their first period, i.e., ut = (1 − β) log ct+1 + β log bt+1 ).
Solution. If consumption when young ctt does not enter the utility function, and assuming homothetic
i
preferences, individuals will bequeath a constant share of their income: bit+1 = βIt+1
. Then, the
aggregate amount the economy invests in education is at time t is the sum of all bequests in the
economy:
∫ 1
∫ 1
i
Iti = BYt
βIt = β
0
0
On the other hand, the aggregate cost of investment will be the sum of all costs effectively paid by
individuals who invest in education, that is by individuals with hit < ĥt which in equilibrium must
be equal to BYt :
∫
ĥt
BYt =
0
Jorge F. Chávez
ĥ
( )
1 ( )2 t
1 ( )2
ĥt
hit f hit di = hit =
2
2
0
8
EC9A2 (Fall 2012)
Problem Set # 6
Then, using ĥt = Et+1 and Yt = we Et + wu (1 − Et ) we get:
2βYt = (Et+1 )
2
⇔
1/2
Et+1 = 2β[we Et + wu (1 − Et )]
Finally because Et+1 cannot be greater than 1:
{
}
1/2
Et+1 = min 1, 2β[we Et + wu (1 − Et )]
Jorge F. Chávez
9
EC9A2 (Fall 2012)
Problem Set # 6
Question 3
Consider an overlapping generations economy in which individuals consume and may invest in
education in the first period of their lives. They work and consume in the second period. The
population size of each generation is 1. Individuals receive in their first period a transfer from
the government bt which they use for consumption and investment in education. The transfer to
the young is financed by an income tax on the working population (individuals in their second
life period). The tax rate is τ.
Production is given by:
Yt = we Et + wu Ut
where we > wu , Et is the number of skilled workers in period t, and Ut = 1 − Et is the number
of unskilled workers in t. Therefore
bt = τ Yt
Individuals’ preferences are represented by the utility function,
u = log cy + log co
where cy is consumption when young (first period) and co is consumption when old (second
period).
To become a skilled worker an individual has to purchase education when young. The cost of
education of individual i, hi is indivisible. The cost of education in the population is uniformly
distributed in the unit interval.
There is no capital market in the economy. Individuals can not borrow or lend. It is also
impossible to store goods from one period to the next.
(a) Find the dynamical system governing the evolution of Et . (Note that your dynamical system
should reflect the fact that Et ≤ 1 for all t ).
Solution. The UMP is:
{
}
i,t
max
log ci,t
t + log ct+1
2
{ctt ,ctt+1 }∈R+
s.t.
i i,t
ci,t
t + δ ht ≤ bt
i,t
i,t
ct+1 ≤ (1 − τ ) It+1
Note that because population is normalized to 1, all individuals receive the same level of transfers bt
and:
∫ 1
bt di = bt = βYt
0
The UMP can be rewritten as discrete variable optimization problem:
{ (
)
(
)}
i,t
i i,t
max
log
b
−
δ
h
+
log
[1
−
τ
]
I
t
t
t+1
i
δ ∈{0,1}
Jorge F. Chávez
10
EC9A2 (Fall 2012)
Problem Set # 6
We then need to compare the level of utility that the individual with realization of education cost
hit would ge if she invests in education with the level of utility if she does not invest:
(
)
log bt − hi,t
+ log ([1 − τ ] we ) > log (bt ) + log ([1 − τ ] wu )
t
)
(
> wu bt . Hence, the individual will invest in
It is easy to see that this reduces to we bt − hi,t
t
education if and only if it gets hit below a cutoff value ĥt :
[ e
]
w − wu
hi,t
<
b
≡ ĥt
t
t
we
Then, using the assumption that hit is distributed uniform with support [0, 1]:
Et+1 = ĥt Et + ĥt (1 − Et ) = ĥt
which implies:
[
Et+1
= τ Yt
we − wu
we
]
(5)
[
= τ (we Et + wu (1 − Et ))
we − wu
we
]
(6)
where I am using the fact that bt = βYt : and Yt = we Et + wu (1 − Et )
Finally, after some rearrangements:
[
]
2
wu e
τ (we − wu )
Et+1 =
E
+
τ
(w − wu ) ≡ φ(Et )
t
we
we
(b) Find a sufficient condition on τ that assures that Et < 1 for all t.
Solution. Note that the slope and intercept of the linear function φ(Et ) are both positive. To ensure
that we always keep Et ≤ 1 we need to make sure that φ(1) ≤ 1. Note that having a slope less that
1 is not sufficient for Et ≤ 1 as is illustrated by the red phase curve in figure 3 (for Et close to 1, we
see that Et+1 > 1). Then we need:
[
2
τ (we − wu )
we
]
<1 ⇔
ϕ (1) = τ (we − wu ) < 1 ⇔
τ<
we
2
(we − wu )
1
τ<
e
(w − wu )
(c) Suppose now that there exists a perfect loan market. (The young can lend and borrow from
each other, repaying when old, and the equilibrium interest rate between period t and t + 1,
Rt+1 , assures that the demand for loans is equal to the supply.). Find Rt+1 as a function of
we , wu , τ and Et+1.
Jorge F. Chávez
11
EC9A2 (Fall 2012)
Problem Set # 6
Figure 3: Dynamic system Et+1 = φ(Et )
45o
Et+1
1.2
1.0
0.8
0.6
0.4
τ
wu
we
A
e
(w − wu )
0.2
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4 Et
Solution. The problem is now:
{
}
max
log ci,t
+ log ci,t
t
t+1
2
{ctt ,ctt+1 ,Bti,t }∈R+
s.t.
i i
i
ci,t
t + δ ht ≤ bt + Bt
i,t
i
ci,t
t+1 + Rt+1 Bt ≤ (1 − τ ) It+1
cit , cit+1 ≥ 0
Bti ≥ −H
We can put everything in terms of the borrowing decision Bti,t and to solve the problem taking as
given the investment in education decision:
{ (
)
(
)}
i,t
i,t
i,t
i i
max
log
b
+
B
−
δ
h
+
log
(1
−
τ
)
I
−
R
B
t
t+1 t
t
t
t+1
i,t
i,t
0≤Bt ≤bt −δ i ht
Solving for interior solution we find:
(
)
i,t
bt − δ i hit Rt+1 − It+1
(1 − τ )
i,t
Bt =
2Rt+1
As usual the individual decides whether to invest in education if and only if:
(
)
(
)
i,t
i,t
i,t
i
log bt + Bt,δ
+ log (1 − τ ) It+1
− Rt+1 Bt,δ
i =1 − ht
i =1
(
)
(
)
i,t
i,t
i,t
i
> log bt + Bt,δ
+ log (1 − τ ) It+1
− Rt+1 Bt,δ
i =0 − ht
i =0
Jorge F. Chávez
12
EC9A2 (Fall 2012)
Problem Set # 6
where:
i,t
Bt,δ
i =1
=
i,t
Bt,δ
i =0
=
(
)
bt − hit Rt+1 − we (1 − τ )
2Rt+1
bt Rt+1 − wu (1 − τ )
2Rt+1
This can be solved in exactly the same fashion as in question 2, part (c) (see appendix). The end
result is the cutoff ĥt :
hit <
(we − wu ) (1 − τ )
≡ ĥt
Rt+1
Finally, because ĥt = Et+1 :
(we − wu ) (1 − τ )
Et+1
Rt+1 =
(7)
(d) Find the dynamical system governing the evolution of Et under the assumption that a perfect
loan market exists, and there is no consumption in the first life period.
Solution. If there is no consumption in the first period, then all tax revenues will be allocated to
finance the accumulation of human capital (supply of funds). As in question 2, part (c), this amount
must equal the demand for investment:
∫
ĥt
τ Yt =
0
ĥ
( )
1 ( )2 t
1
1 ( )2
2
hit f hit di = hit =
ĥt = (Et+1 )
2
2
2
0
Then, using Yt = we Et + wu (1 − Et ):
2
(Et+1 ) = 2τ [we Et + wu (1 − Et )]
which implies:
Et+1 = (2τ [we Et + wu (1 − Et )])
1/2
Once again, Et+1 cannot be greater than 1, so:
{
}
1/2
Et+1 = min 1, (2τ [we Et + wu (1 − Et )])
Finally, plugging the above condition into (7), the expression we wanted to find is:
Rt+1 =
(we − wu ) (1 − τ )
1/2
(2τ [we Et + wu (1 − Et )])
(e) Explain why a negative effect of the tax on the incentive to purchase education does not
appear in both of the dynamical systems (with and without loans).
Jorge F. Chávez
13
EC9A2 (Fall 2012)
Problem Set # 6
Solution. Without loans the dynamical system is:
[
]
2
τ (we − wu )
wu
Et+1 =
Et + τ e (we − wu ) ≡ φ(Et )
e
w
w
While with loans, the dynamical system is:
{
}
1/2
Et+1 = min 1, (2τ [we Et + wu (1 − Et )])
Jorge F. Chávez
14
EC9A2 (Fall 2012)
Problem Set # 6
Appendix
A1 Aggregation when βYt < h
i , thus:
UMP implies that bit+1 = βIt+1
bit+1





[
]
β wu + (1 + rt+1 ) bit
bit < b̂
[ s
(
)] i [ )
i
=
β
w
−
(1
+
r
)
h
−
b
bt ∈ b̂, h
t+1
t



[
(
)]

β ws + (1 + rt+1 ) bit − h
bit ≥ h
≡ φ(bit )
Because population is normalized to 1, we can assume that there is a continuum of individuals
along the [0, 1] range, with mass equal 1. Furthermore, we can assume
[ )that in principle there
i
i
is a measure N1 of individuals with bt < b̂, a measure N2 with bt ∈ b̂, h and a measure N3 of
rich individuals, with bit ≥ h. Note that:
∫
∫
1
1
i
βIt+1
di
bit+1 di =
0
0
[∫
∫
i
It+1
di
= β
+
i∈N1
]
∫
i
It+1
di
i
It+1
di
+
i∈N2
i∈N3
= βYt+1
Then:
∫
[
βYt+1 =
u
β w + (1 +
rt+1 ) bit
]
∫
i∈N1
∫
[
(
)]
β ws − (1 + rt+1 ) h − bit
di +
i∈N2
[
(
)]
β ws + (1 + rt+1 ) bit − h
+
i∈N3
[
=
]
∫
u
bit di
w N1 + (1 + rt+1 )
[
]
∫
+ w N2 − (1 + rt+1 ) hN2 + (1 + rt+1 )
s
i∈N1
bit di
i∈N2
[
]
∫
+ w N3 − (1 + rt+1 ) hN3 + (1 + rt+1 )
s
bit di
i∈N3

{∫
3
∑
u
s
= w N1 + w (N2 + N3 ) + (1 + rt+1 ) 
j=1
Jorge F. Chávez
}
bit di

− (N2 + N3 ) h
i∈Nj
15
EC9A2 (Fall 2012)
Problem Set # 6
where the term in brackets in the last equality will be zero as loan markets clear:
∫
∫
∫
i
i
βIt di +
βIt di +
βIti di − (N2 + N3 ) h = 0
i∈N1
i∈N2
i∈N3
where I am using the fact that bit = βIti . Finally:
[∫
]
∫
∫
i
i
i
β
It di +
It di +
It di − (N2 + N3 ) h = βYt − (1 − N1 ) h = 0
i∈N1
i∈N2
i∈N3
where I am using N1 = 1 − N2 − N3 . Then 1 − N1 =
(
)
(
)
βYt
u
s βYt
Yt+1 = w 1 −
+w
h
h
βYt
h
and finally,
A2 Aggregation when βYt ≥ h
In this case, all individuals will bequeath following the third piece of the φ(bit ) function. Hence,
aggregating:
∫ 1
∫ 1
[
(
)]
i
β
It+1 di =
β ws + (1 + rt+1 ) bit − h di
0
0
Using the fact that
∫1
0
adi = a
(∫
s
∫1
0
)
1
bit di
Yt+1 = w + R
di = a × 1 = a for some constant a ∈ R
−h
0
= ws + R (βYt − h)
A3 Derivation for condition 4 in question 2, part (c)
We had:
(
)
(
)
log bit + Bte − 1 × hit + log (we − Rt+1 Bte ) > log bit + Btu − 0 × hit + log (wu − Rt+1 Btu )
Replacing Bte and Btu :
Bte
=
Btu =
Jorge F. Chávez
]
[
we − Rt+1 bit − hit
2Rt+1
wu − Rt+1 bit
2Rt+1
16
EC9A2 (Fall 2012)
Problem Set # 6
(
)
(
[
])
we −Rt+1 [bit −hit ]
we −Rt+1 [bit −hit ]
i
i
e
log bt +
− ht + log w − Rt+1
2Rt+1
2Rt+1
(
)
(
[ u
])
u −R
i
i
w
b
w
−R
b
t+1 t
t+1 t
+ log wu − Rt+1
> log bit + 2Rt+1
2Rt+1
Rearranging a little bit we get:
(
log
(
(
))
(
))
( u
)
( u
)
we + Rt+1 bit − hit
we + Rt+1 bit − hit
w + Rt+1 bit
w + Rt+1 bit
+log
> log
+log
2Rt+1
2
2Rt+1
2
(8)
Note that we can play a little bit with the properties of logarithms to get:
(
log
)
(
(
)
(
))
we + Rt+1 bit − hit
we + Rt+1 bit − hit
Rt+1
= log
+ log Rt+1
2Rt+1
2Rt+1
)
( u
( u
)
w + Rt+1 bit
w + Rt+1 bit
= log
log
Rt+1
+ log Rt+1
2Rt+1
2Rt+1
which are the last term of the LHS and the last term of the RHS of inequality (8). Then we end
up having the following condition:
[

)
([
( i
) ]2
]
i
e
u+R
i 2
w + Rt+1 bt − ht
w
b
t+!
t
log 
Rt+1
Rt+1  > log
2Rt+1
2Rt+1
Jorge F. Chávez
17