Inter-temporal Consumption Choice

1
Inter-temporal Consumption Choice
2
Economics 101

• In ECO100 we assume a unique happiness function
  for every individual (utility function). We call
  such function as the individuals subjective
  preference. Every individual is going to maximize
  his happiness subject to some constraints.
• Max utility subject to some constraints
• E.g., U(x, y) subject to PxX PyY W

3
Economics 101

• We can represent individuals preference, U(x,
  y), by indifference curves on the x-y diagram.

y
Represent higher levels of utility U0 lt U1
lt U2
U2
U1
U0
x
4
Economics 101

• The constraint of PxX PyY W can be shown as
  the budget line.

y
W/Py
Feasible Consumption Set
Budget line (Slope Px/Py)
x
W/Px
5
Economics 101

• Maximizing utility means picking the best
  feasible consumption point (C). The equilibrium
  condition is
• (slope of indifference curve) MRS Px/Py (Slope
  of budget line)
• Where MRS MUx/MUy

y
W/Py
C (with consumption of x and y)
U2
y
U1
U0
x
W/Px
x
6
Economics 101

• For inter-temporal consumption choice, we employ
  the same rationale. Now, x becomes current
  consumption (C0) and y becomes future consumption
  (C1). That means, our happiness depends on two
  things current and future consumption.

C1
U2
U1
U0
C0
7
Economics 101

• In order to have the indifference curves as
  described with nice convex-to-the-origin shape,
  we assume
• More is better than less.
• Diminishing marginal utility of consumption for a
  single period.

U(C0,C1)
MU gt 0 ? (?U / ?C0) gt 0 MU is diminishing i.e.,
(?2U / ?C2) lt 0
U U(C0,C1constant)
C0
8
Economics 101

• With the assumptions, we have the following
  diagram. The slope of the indifference curve
  represents the individuals subjective rate of
  time preference.
• We call the slope the marginal rate of
  substitution between current consumption and
  future consumption. The math expression is
• MRS MU(C0)/MU(C1) (?U / ?C0) / (?U / ?C1)

C1
U0
C0
9
The Constraint

• Recall that an individual maximizes his happiness
  subject to constraints. What are the constraints?
• It depends on the options available for the
  individual to allocate his wealth across
  different time periods.
• We study two options 1 Production opportunity
  and 2 participation of capital market.
• We assume the individual has endowment of Y0 and
  Y1 in the current and future periods
  respectively. So, we can plot the endowment point
  on the diagram.
• Constraint A With no wealth allocation across
  periods, his utility is U0.

C1
Y1
U0
C0
Y0
10
Production Opportunity

• Constraint B The individual can only invest in
  production opportunities to allocate wealth
  across periods
• Now, we introduce production opportunities that
  allow a unit of current savings/investment to be
  turned into more than one unit of future
  consumption.
• Assume the individual faces a schedule of
  productive investment opportunities. We line them
  up from the highest return to the lowest and plot
  them as follows
• Such decreasing marginal rate of return means
  diminishing marginal returns to investment
  because the more an individual invests, the lower
  the rate of return on the MARGINAL investment.

Marginal rate of return
Total investment
11
Production Opportunity

• Total investment in the current period is equal
  to current period endowment minus current
  consumption (i.e., Investment Y0-C0)
• With this in mind, we can plot the constraint on
  the C0-C1 space.
• We call this constraint the production
  opportunity set (POS).
• The slope of the POS is now called the Marginal
  Rate of Transformation (MRT) offered by the
  production/investment opportunity set.
• Investment (or dis-investment) means the
  individual can move its consumption point along
  POS.

C1
Y1
C0
Y0
12
Production Opportunity

• At the endowment point, the individual is not
  maximizing its utility subject to the constraint
  B. He can do better by investing more (i.e., move
  north-west along the POS) because at the
  endowment point, the return offered by investing
  is higher than his subject rate of time
  preference needed to make him feel indifferent.
  (For example, to sacrifice 1 unit of current
  consumption, he needs 1 0.2 units of future
  consumption to stay as happy. But if the return
  he can get is 30, that means investing can make
  him happier)
• The equilibrium is when he invests until the
  return offered by the marginal investment is just
  equal to its subjective rate of time preference.
  In math, we have (slope of POS) MRT MRS
  (slope of indifference curve)

C1
C1
U1
Y1
U0
C0
Y0
C0
13
Production Opportunity

• Messages
• This individual can achieve a higher utility
  (U1gtU0) by investing in production opportunities.
• His feasible consumption set expands with the
  introduction of production opportunities. With
  constraint A, he can only consume at the
  endowment point. With the introduction of
  production opportunity (a less restrictive
  constraint B), his feasible consumption set
  becomes all the points along the POS.
• This gives the rationale for inter-temporal
  consumption choice which also explains
  investment. If exposed to various investment
  opportunities, individuals want to take some of
  them in order to allocate wealth. Doing so would
  allow them to achieve higher utility level.

C1
C1
U1
Y1
U0
C0
Y0
C0
14
Capital market

• Now, instead of one individual, lets assume
  there are many individuals in the economy. Some
  are lenders, while others are borrowers. Among
  them, there are opportunities to borrow and lend
  at the market-determined interest rate (r).
• Constraint C No production opportunity. But
  individuals can lend/borrow at r.
• We can then graph the borrowing and lending
  opportunities along the capital market line.
• Now, we introduce the concept of wealth. Wealth
  of an individual is the present value of his
  current and future endowment. Thus,
• W0 Y0 Y1/(1r)

C1
Y0(1r) Y1
Capital market line with Slope -(1r)
Y1
C0
Y0
W0
15
Capital market

• The feasible consumption set is now all the
  points along the capital market line.
• Moving North-west along the capital market line,
  the individual can achieve a higher utility
  (U2gtU0)
• This individual is now lending (Y0-C0) amount of
  money, and will get back ((1r)(Y0-C0)) in the
  next period so that he can consume a total of
  C1 Y1(1r)(Y0-C0).
• Equilibrium condition is
• MRS (1r)

C1
Y0(1r) Y1
C1
Y1
U2
U0
C0
Y0
W0
C0
16
Production and capital market

• Constraint D Individuals can now borrow/lend at
  r invest in production opportunities.
• With only production opportunity, the individual
  achieve U1 only. But if capital market is
  introduced, he can actually do better.
• At point , the individual can borrow more money
  at rate r from the capital market, and be able to
  invest more and get return higher than r. Until
  in equilibrium, he reaches , where the return
  on the marginal investment is equal to the market
  interest rate r. At this point, his wealth is
  maximized.
• Now his wealth is W0 P0 P1/(1r) which is
  larger than W0.
• With his wealth maximized, he chooses (C0, C1)
  to consume and yield him U3.

C1
P1
(C0, C1)
U3
Y1
U1
U0
C0
Y0
P0
W0
17
Fisher Separation Theorem

• Message
• The decisions of production and consumption
  involve 2 distinct steps.
• 1st step Choosing production point by moving
  along POS and produce at the point where MRT
  (1 r), this means return on the marginal
  investment is just equal to market interest rate.
• 2nd step Choosing consumption point by moving
  along the capital market line and consume at the
  point where MRS (1 r), this means subjective
  rate of time preference is equal to market
  interest rate.
• We call this the FISHER SEPARATION THEOREM. The
  important point is the production point is
  governed solely by objective criteria, namely,
  the set of opportunities available and the market
  interest rate. This is independent of
  individuals subjective rate of time preferences.

C1
P1
(C0, C1)
U3
Y1
U1
U0
C0
Y0
P0
W0
18
Fisher Separation Theorem

• Implication 1
• As the graph below shows, with two different
  individuals that differs only in their subjective
  preferences, given the same opportunity set, both
  of them would choose the exact same point of
  production regardless of the difference of their
  preferences.
• Exercise Read the formal definition of Fisher
  Separation Theorem on page 10. What are the
  required assumptions and why?

C1
Individual 2
P1
Individual 1
Y1
C0
Y0
P0
W0
19
Fisher Separation Theorem

• Implication 2
• The role of capital market the market channel
  funds from the lenders to the borrowers. Setting
  demand supply, we have a market-determined r.
  Given the individuals exposure to his own
  production opportunities, he may decide whether
  to lend or borrow money. By allowing lending and
  borrowing, those who need money can get financed,
  while those have excess fund will be able to lend
  out and earn interest. Everyone is made better
  off.
• In short, as shown below, capital market is
  important because everyone can be happier with
  it.

C1
Individual 2
P1
Individual 1
Y1
C0
Y0
P0
W0
20
Fisher Separation Theorem

• Implication 3
• Consider the following two investors investing
  all their money on the stocks of a single firm.
  Their well-being is thus tied to the well-being
  of the firm. Consider the firm is making decision
  of what to produce.
• Fisher Separation Theorem implies even the two
  investors differ in their subjective perception
  of how to consume between now and future, they
  both has one unified objective, i.e, to maximize
  their current wealth.
• Doing so means the firm can maximize its value.
  This is the same as investing until the return on
  the marginal investment is just equal to the cost
  of capital, i.e, the market interest rate.

C1
Individual 2
P1
Individual 1
Y1
C0
Y0
P0
W0
21
Fisher Separation Theorem

• Implication 3
• MRT (1 r) is the point where both of the two
  individuals would agree for the firm to produce.
• This is exactly the famous project selection
  rule, the positive Net Present Value rule. The
  firm value is maximized by taking all projects
  that have positive NPV.
• NPV -initial investment present value of
  future payout discounted by cost of capital.
• Cost of capital r

C1
Individual 2
P1
Individual 1
Y1
C0
Y0
P0
W0
22
How to max shareholders wealth?

• We again uses Fisher Separation Theorem
• Given perfect and complete capital markets, the
  owners of the firm (shareholders) will
  unanimously support the acceptance of all
  projects until the least favourable project has
  return the same as the cost of capital.
• In the presence of capital markets, the cost of
  capital is the market interest rate.
• The project selection rule, i.e., equate
• marginal rate of return of investment cost of
  capital (market interest rate)
• Is exactly the same as the positive net present
  value rule
• Net Present Value Rule
• Calculate the NPV for all available (independent)
  projects. Those with positive NPV are taken.
• At the optimum
• NPV of the least favourable project zero
• This is a rule of selecting projects of a firm
  that no matter how individual investors of that
  firm differ in their own opinion (preferences),
  such rule is still what they are willing to
  direct the manager to follow.

23
Again and again, Fisher Separation Theorem

• The separation principle implies that the
  maximization of the shareholders wealth is
  identical to maximizing the present value of
  lifetime consumption
• Since borrowing and lending take place at the
  same rate of interest, then the individuals
  production optimum is independent of his
  resources and tastes
• If asked to vote on their preferred production
  decisions at a shareholders meeting, different
  shareholders will be unanimous in their decision
• ? unanimity principle
• Managers of the firm, as agents for shareholders,
  need not worry about making decisions that
  reconcile differences in opinion among
  shareholders i.e there is unanimity
• The rule is therefore
• take projects until the marginal rate of return
  equals the market interest rate taking all
  projects with ve NPV

24
TOPICS for self-interest

• Different techniques for selecting investment
  projects
• Payback method
• Accounting rate of return
• Net present value
• Internal rate of return
• Measuring shareholder wealth (you need to know
  how to compute NPV)
• Economic profits vs Accounting profits
• Agency problem

25
An exercise for self-study

• Consider the following utility function
• U U(C0) 1/(1 ?)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 ?

26
An Exercise
As C0 ? MU ? As C1 ? MU ? 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) lt 1 To the left of the 450 line, the
slope is greater than 1 as U'(C0)/ U'(C1) gt 1
C1
C0 C1
1
U0
1
450
C0
dC1/dC0 -(1- ?) U'(C0)/ U'(C1) slope of
indifference curve
Therefore, even if ? gt 0, the tradeoff between C0
and C1 can be lt 1 if C0 is sufficiently high
27
Another Numerical Example

• Assume individuals can borrow and lend, but no
  production
• Suppose that the utility function for consumption
  is
• U log(C) 1/(1 ?) log(C)
• The individuals wealth is given by the equation
• W y0 1/(1R)y1
• where R is the rate of interest and
• ? is the subjective rate of time preference
• If an individual is to maximize utility, then we
  know that the present value of consumption must
  equal wealth W y0 1/(1R)y1
• Derive the optimal consumption paths, assuming
• W100, R10, ? 10
• W100, R5, ? 10
• W100, R10, ? 5
• We are ignoring production opportunities in this
  example

28
The Optimization Problem

• Set up the constrained optimization problem
• L log(C0) 1/(1 ?) log(C1) ? W C0
  C1/(1R)
• The first order conditions
• ?L/?C0 (1/C0) - ? 0 ? ? 1 / C0
• ?L/?C1 (1/(1 ?)) (1/C1) - ?/(1R) 0 ? ?
  (1R)/(1?) (1/ C1)
• 1 / C0 (1R)/(1?) (1/ C1)
• C0 (1 ?)/(1 R) C1
• If ? R ? C0 C1
• If ? gt R ? C0 gt C1
• If ? lt R ? C0 lt C1

C1
C1
U
C0
C0
29
Optimal Consumption Paths

• Solve for the following three cases
• a) W100, R10, ? 10
• C0 (1.10/1.10)C1 ? W C0 C1 /(1R) 100
• ?C0 C1
  C 52.38
• b)W100, R5, ? 10
• c)100, R10, ? 5