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### 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
- With both capital market and production opportunities
- (3) Fisher Separation Theorem
- (4) Role of Capital Markets
- (6) Unanimity Principle
- (7) Justifying the use of NPV

Inter-temporal Consumption Choice

- study consumption (C) and investment (I) decisions made by individuals and firms
- Simplified case: Robinson Crusoe’s case, 2-periods. He needs
- 1. to know his own subjective tradeoffs between C now (C0) and C next period (C1) (this information is embedded in the utility function)
- 2. to understand the feasible tradeoffs between present and future consumption that are technologically possible (this information is embedded in the production opportunity set (POS)
- the optimal consumption-investment decision establishes a subjective interest rate – the subjective rate of time preference ()
- This subjective interest rate represents the unique optimal rate of exchange between C now (C0) and C in the future (C1)
- Interest rates are the price of deferred consumption – by forgoing one dollar of consumption today, the market rewards with interest payments R

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

- The slope of the indifference curve
- measures the rate of trade-off between C0 and C1
- at the point of tangency
- This trade-off is called the marginal rate of
- substitution MRS between C0 and C1
- - this also reveals the individual’s subjective
- rate of time preference i (at each point on
- the indifference curve)
- - Precisely, slope = -

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

- Objective: maximizes utility subject to constraints.
- Constraints: only production is available for inter-temporal allocation of wealth
- A1. All investment outcomes are known with certainty
- A2. No transaction costs
- A3. No taxes
- A4. Decisions are in a two-period world (periods 0 and 1)
- y0 = is the individual’s endowment in period 0
- y1 = is the individual’s endowment in period 1

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

Benefits of Production Opportunities

- In the absence of investment opportunities, individuals are constrained to consume at his endowment point
- Investment opportunities expand the consumption set
- Individuals are therefore better off
- At point 0,
- MRT > MRS [the slope of POS] > [slope of utility function]
- [The return one obtains > [return required by the

from investing] individual to forego

current consumption]

=> Optimal action: Increase the amount investing until

- At point *, MRT=MRS, and I0 is the optimal level of investment

Constraints: With Capital market only

- Objective: maximizes utility subject to constraints.
- Constraints: only capital market is available for inter-temporal allocation of wealth
- A1. All investment outcomes are known with certainty
- A2. No transaction costs
- A3. No taxes
- A4. Decisions are in a two-period world (periods 0 and 1)
- y0 = is the individual’s endowment in period 0
- y1 = is the individual’s endowment in period 1

Constraints: With Capital market only

- Capital Market – individuals can borrow and lend, and thus facilitates the transfer of funds between lenders and borrowers
- With initial endowment of (y0 and y1) that has utility U0, individuals can reach any point along the capital market line through borrowing and lending at the market interest rate R.
- R = market rate of interest (reward to deferring consumption)
- I0 is the period 0 amount invested
- I1 is the period 1 value of this investment = I0(1+R)
- in the absence of capital markets, individuals would be constrained to consume at his endowment point (with U0)
- We will show how the ability to borrow and lend can improve an individual’s utility

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

The Consumption Decision

- Preferences (shape of utility curves) are unique to each individual
- Production opportunities are the same for everyone
- Capital markets (interest rates) are the same for everyone
- Preferences determine consumption
- Capital Market line (interest rate) determines production
- All individuals make the same production decision, independent of preferences

The Decision Making Process

- the decision-making process that takes place with production opportunities and capital market exchange opportunities occurs in two separate and distinct steps
- Choose the optimal production decision by taking on projects to the point where

[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

FISHER SEPARATION THEOREM

- Given perfect and complete capital markets (frictionless), the production decision is governed solely by an objective market criterion (represented by maximizing attained wealth) without regards to individual’s subjective preferences that enter into the consumption decision.
- This is extremely important for corporate finance
- regardless of the shape of individual investor’s indifference curve of a firm, every investor of that firm will direct the manager to the same production decision
- The investment decision is independent of individual preferences

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

The Separation of C and I decisions

- Both investors 1 and 2 will direct the manager of the firm to choose point (*)
- The investors simply takes the output of the firm and adapts it to their own subjective time preferences by borrowing and lending in the capital market
- The optimal production decision is separated from individuals utility preferences (thus individual investors are unanimous (Unanimity principle))

ROLE OF CAPITAL MARKETS

- capital markets allow the efficient transfer of funds from lenders to borrowers
- Individuals who have insufficient wealth to take advantage of all their investment opportunities that yield rates of return higher than the market rate are able to borrow funds and invest more than they would without capital markets i.e., funds can be efficiently allocated from individuals with few production opportunities and great wealth to individuals with many opportunities and insufficient wealth
- All borrowers and lenders are better off than in the absence of capital markets

INVESTMENT DECISION

- The objective of the firm is to maximize the wealth of its shareholders.
- This is the same as [maximizing the present value of the shareholders lifetime Consumption] = [maximizing the price per share per stock]

How to max shareholder’s 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 net present value rule:

Net Present Value Rule

- Calculate the NPV for all available (independent) projects. Those with positive NPV are taken.

At the optimal:

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.

Fisher Separation Theorem

- The separation principle implies that the maximization of the shareholder’s 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 individual’s production optimum is independent of his resources and tastes
- If asked to vote on their preferred production decisions at a shareholder’s 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

- Principals = Shareholders, Agent = Manager of the firm.
- Ownership ≠ control, so there is no reason to believe that managers will always act in the best interest of the shareholders (Consider all the accounting scandals)
- If the shareholders can costlessly monitor management decisions, they can be sure that management really does make every decision in a way that maximizes their wealth
- In reality, owners must incur non-trivial monitoring costs in order to keep the manager in line => owners (shareholders) face a trade-off between monitoring costs and forms of compensation that will cause the agent to always act in the owners interest
- Fisher separation holds only with strong assumptions: (I.e frictionless markets)
- I.e can costlessly monitor managers
- decisions always made to maximize value of the firm

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

Inter-temporal Choice

- an endowment, y, can be allocated Inter-temporally in three ways

1. Production 2.Capital markets 3. Storage

- in order to move resources from period 0 to period 1 using
- 1. Production, we can move north-west in the graph at the rate of (slope of the POS)
- 2. Capital markets, we can move north-west or south-east along the capital market line at the rate of (1+ R) where R = market interest rate
- 3. Storage, we can move north-west at the rate of 1. This requires additional assumptions of non-perishable goods and no storage costs which is not required for 1 and 2 above.
- Under what conditions is the individual made better off from an Inter-temporal reallocation?
- To answer that question we need to consider the slope of the indifference curve relative to the rate of transformation in each of 1 to 3 above. General rule: If rate of time preference < rate of transformation, then the individual is better off transforming using either of the 3 ways.

An exercise for self-study

____

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

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 individual’s wealth is given by the equation
- W = y0 + [1/(1+R)]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/(1+R)]y1
- Derive the optimal consumption paths, assuming
- W=100, R=10%, =10%
- W=100, R=5%, =10%
- W=100, R=10%, =5%
- We are ignoring production opportunities in this example

The Optimization Problem

- Set up the constrained optimization problem
- L = log(C0) + [1/(1+ )] log(C1) + [ W – C0 – C1/(1+R)]
- The first order conditions
- L/C0 = (1/C0) - = 0 = 1 / C0
- L/C1 = (1/(1+ )) (1/C1) - /(1+R) = 0 = [(1+R)/(1+)] (1/ C1)]
- [1 / C0] = [(1+R)/(1+)] (1/ C1)]
- C0* = [(1+ )/(1+ R)] C1*
- If = R C0* = C1*
- If > R C0* > C1*
- If < R C0* < C1*

C1

C1*

U

C0

C0*

Optimal Consumption Paths

- Solve for the following three cases
- a) W=100, R=10%, =10%
- C0 = (1.10/1.10)C1 W = C0 + C1 /(1+R) = 100

C0* = C1* = C* = 52.38

- b)W=100, R=5%, =10%
- c)=100, R=10%, =5%

Storage

- For an individual to benefit from storage (assuming non-perishable goods and no storage costs,
- [slope of the indifference curve] < 1 (in absolute value)
- dC1/dC0 = -(1- ) [U'(C0)/ U'(C1)]
- and assuming log utility:
- dC1/dC0 = [-(1- )](1/C0) / (1/C1) = [-(1- )]C1/C0
- In other words, simple storage makes an individual better off if
- (1+ ) C1/C0 < 1 or (1+ ) < C0/C1
- Again, this assumes log utility i.e. storage may be beneficial if C0 is high relative to C1

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