
Inter-temporal Optimization of Consumption. Allocating wealth across time. Topics to be covered. (1) Individual preferences across time (2) Production opportunities (Real Investment) (2) Inter-temporal consumption With production opportunities only With capital markets only
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Allocating wealth across time
(2) Production opportunities (Real Investment)
U
MU
U = U(C)
MU > 0 (U / C) > 0
MU is diminishing
(2U / C2) < 0
MUC is high
1
MUC is low
MU
1
C
C
Single period utility functions
- exhibit diminishing utility (concave)
- As C , Utility , but each unit of
C increases U by less and less
When C is low, each unit of C has
high utility value
When C is high, each unit of C has
lower utility value
Convex Indifference Curves trace out combinations of C0 and C1for which the individual equally well-off
C1
Consumption
In period 1
C0 low, MU(C0) high
C1 high, MU(C1) low
1
C0 high, MU(C0) low
C1 low, MU(C1) high
1
U (Indifference Curve)
C0
Negative slope because as C0 falls, C1 must
increase to maintain the same level of utility
Consumption
In period 0
Represent higher levels
of utility
U0 < U1 < U2
C1
Consumption
In period 1
U2
U1
U0
C0
Consumption
In Period 0
C1
U0
C0
If C0 is to reduce by 1 unit:
(a) when C0 is high
less additional C1 is needed to make
this individual equally well off
(b) when C0 is low
more additional C1 is needed to make
this individual equally well off
How to show this graphically?
- Convex indifference curve
- the slope at a (ρa) < the slope at a (ρb)
C1
ρb
b
1
a
ρa
1
U0
C0
C1
C0
MRS = -
(slope of an indifference curve)
C1
C0
U = U0
Constraints: With production only
Marginal Rate of Return
(Future consumption – future endowment)
- production opportunities allow a unit of
current savings to be converted into output or
wealth in the future
ASSUMPTIONS
A1. Investments are arranged from the highest
rate of return to the lowest rate of return
A2. As total investment increases, the marginal rate
of return falls
A3. All investments are independent of one another
A4. All investments are perfectly divisible
A
B
Total investment
(current endowment – current consumption)
MRT = height of AB
= rate at which a dollar of consumption foregone today C0 is transformed
by productive investment into output tomorrow i.e. it is the MRT offered
by the production opportunity set
Marginal Rate of Return
(Future consumption – future endowment)
Production Opportunity Set (POS)
C1
A
Low rates of
return
High
High rates of
return
y1
endowment
point
Low
B
I0
I1
y0
C0
Total investment
(current endowment – current consumption)
I0
I1
Production Opportunity Set (POS)
C1
Given technology, the POS traces
out the maximum amount of C1 that
is feasible for any given amount of C0
Low rates of
return
High rates of
return
y1
endowment
point
How does individual maximize utility?
- make all investments in the production opportunity set that have rates of return higher than his subjective rate of time preference
y0
C0
I0
I1
PUTTING TOGETHER THE POS AND PREFERENCES
C1
In the absence of production
the individual would be forced
to consume at the endowment
point
y1
U0
The endowment
point
y0
C0
PUTTING TOGETHER THE POS AND PREFERENCES
C1
Small vertical arrow
- amount needed to be equally well
off in utility terms (ρ)
Large vertical arrow
- amount of return to deferring consum
today and undertaking investment
y1
0
1
U0
y0
C0
Individuals will make all investments in the production opportunity set
that have rates of return higher than his or her subjective rate of time preference
PUTTING TOGETHER THE POS AND PREFERENCES
This is the point where
MRT = MRS
(slope of = (slope of
POS) indifference
curve)
C1
*
0
y1
U1
The endowment
point
U0
I0
y0
C0
Message: Production opportunities allow individuals to achieve a higher
level of utility – i.e. Move to U1 > U0
from investing] individual to forego
current consumption]
=> Optimal action: Increase the amount investing until
Constraints: With Capital market only
Constraints: With Capital market only
The Budget Set (Capital Markets)
y0 = endowment in period 0
y1 = endowment in period 1
C1
This line (Capital market line)
traces out all affordable combinations
of C0 and C1, given the endowments y0 and y1
by borrowing and lending at market interest rate R
W1
By giving up,
or deferring one
unit of consumption
today, you get (1+R)
units tomorrow
The individual’s wealth:
(a) Measured at Period 0 (x-intercept):
W0 = y0 + y1 / (1+R)
(b) Measured at Period 1 (y-intercept):
W1 = y1 + y0 (1+R)
(1+R)
-1
y1
y0
W0
C0
Choosing the Optimal Consumption Path with Capital Markets
At point 0, the slope of the indifference curve
is less than the slope of the budget line
==> it is better-off to defer C0 today,
because the return (increase in C1) will be higher than needed to make him indifferent.
- Precisely, the amount that the capital market will reward this individual for deferring consumption is the distance (EB), which exceeds the amount the individual needs to be equally well off (DB).
The individual therefore moves to a higher level
of utility by lending money to the capital market
C1
*
E
D
0
U1
B
U0
C0
The individual continues moving along the capital market line to the point at which
the amount capital market rewards him is exactly what he needs to be equally well off
- at the tangency between the interest rate line and the indifference curve - point *
Consumption with Production and Capital Markets
In the absence of production and capital
markets, this individual can only consume at Pt 0.
Pt 1: Production alone
Pt C & P: Both Production and Capital Markets
C1
Capital Market
line
p
*
*
1
c
*
0
Uc&p
*
U1
U0
C0
At the optimum, there is a “separation” between the
Consumption and Production Decision (pt c & p)
The Inter-temporal Consumption Choice
Consumption
*
*
*
U’’’
U’
*
U’’
Production
0
U0
0
U0
U0
0
Both Capital Market & Production
Capital market line objectively determines production with the market interest rate,
And individual preference subjectively determines consumption with his rate of time preference
Production alone
Preferences determine
Production and
Consumption Point
(same point)
Capital Market Alone
There is no production,
and preference determine
consumption
[marginal rate of return on investing] = [the objective rate of interest]
[slope of POS] = [the objective rate of interest]
MRT = (1+r)
2. Choose the optimal Consumption pattern by borrowing or lending along the capital market line to equate your subjective rate of time preference with the market rate of interest
[slope of indifference curve] = [slope of market line]
MRS = (1+r)
This separation of decisions is known as the Fisher Separation Theorem
i.e the separation of the Consumption and Investment decisions
C1
Both individuals 1 and 2
make the same investment
decision: both produce at *,
and by using capital markets,
borrow or lend to achieve
their own optimum consumption point
That is, regardless of the shape
of their utility functions, they make
exactly the same production decision
In math:
MRS1 = MRS2 = -(1+R) = MRT
Individual 2
(stronger preference
for future consumption)
*
Individual 1
(stronger preference
for current consumption)
C0
ROLE OF CAPITAL MARKETS
INVESTMENT DECISION
marginal rate of return of investment = cost of capital (market interest rate)
Net Present Value Rule
At the optimal:
NPV of the least favourable project ~= zero
Fisher Separation Theorem
1. Production 2.Capital markets 3. Storage
____
1
(1+)
Consider the following utility function:
U= U(C0) + U(C1)
We now take a total derivative:
U'(C0)dC0 + [1/(1+ )]U'(C1)]dC1 = 0
Rearranging,
dC1/dC0 = -(1- ) [U'(C0)/ U'(C1)] slope of
indifference curve
- the slope of the indifference curve depends upon the relative marginal utilities as well as the subjective rate of time preference
As C0 MU
As C1MU
Slope of the indifference curve
along the 450 is -(1+ ) as
[U'(C0)/ U'(C1)] = 1
To the right of the 450 line, the slope
is less than 1 as
[U'(C0)/ U'(C1)] < 1
To the left of the 450 line, the slope
is greater than 1 as
[U'(C0)/ U'(C1)] > 1
C1
C0 =C1
1
U0
1
450
C0
dC1/dC0 = -(1- ) [U'(C0)/ U'(C1)]
slope of indifference curve
Therefore, even if > 0, the tradeoff between C0 and C1 can be < 1
if C0 is sufficiently high
U = log(C) + [1/(1+ )] log(C)
C1
C1*
U
C0
C0*
C0* = C1* = C* = 52.38