Inter-temporal Optimization of Consumption

Allocating wealth across time

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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

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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

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CONSUMER THEORYU

C C

MU

MU

U = U(C)

MU > 0 (U / C) > 0

MU is diminishing(2U / C2) < 0 1

1

MUC is low

MUC is high

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

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INDIFFERENCE CURVES

C1

C0

ConsumptionIn period 1

Consumption In period 0

C0 low, MU(C0) high

C1 high, MU(C1) low

C0 high, MU(C0) low

C1 low, MU(C1) high

1

1 U (Indifference Curve)

Convex Indifference Curves trace out combinations of C0 and C1 for which the individual equally well-off

Negative slope because as C0 falls, C1 must increase to maintain the same level of utility

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INDIFFERENCE CURVES

C1

C0

ConsumptionIn period 1

Consumption In Period 0

U0

U1

U2

Represent higher levelsof utility U0 < U1 < U2

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What is the slope?

C1

C0

U0

-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 = -

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MRS is not constant!

C1

C0

1

1 U0

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 lowmore 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)

ρaa

b

MRS = -C1

C0C1

C0 U = U0

(slope of an indifference curve)

ρb

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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

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INVESTMENT OPPORTUNITIES

Total investment (current endowment – current consumption)

Marginal Rate of Return(Future consumption – future endowment)

B

A

- production opportunities allow a unit of current savings to be converted into output or wealth in the future

ASSUMPTIONSA1. Investments are arranged from the highest rate of return to the lowest rate of returnA2. As total investment increases, the marginal rate of return fallsA3. All investments are independent of one anotherA4. All investments are perfectly divisible

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

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INVESTMENT OPPORTUNITIES

Total investment(current endowment – current consumption)

Marginal Rate of Return(Future consumption – future endowment)

B

A

C1

C0y0

y1

endowmentpoint

High rates of return

Low rates of return

Production Opportunity Set (POS)

I0 I1

Low

High

I0

I1

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INVESTMENT OPPORTUNITIES

C1

C0y0

y1

endowmentpoint

High rates of return

Low rates of return

Production Opportunity Set (POS)

I0

I1

Given technology, the POS tracesout the maximum amount of C1 thatis feasible for any given amount of C0

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

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C1

C0y0

y1

The endowmentpoint

U0

In the absence of productionthe individual would be forcedto consume at the endowmentpoint

PUTTING TOGETHER THE POS AND PREFERENCES

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C1

C0

U0

PUTTING TOGETHER THE POS AND PREFERENCES

y0

y1

1

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

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

0

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C1

C0y0

y1The endowmentpoint U0

U1

PUTTING TOGETHER THE POS AND PREFERENCES

This is the point where MRT = MRS

(slope of = (slope ofPOS) indifference curve)

Message: Production opportunities allow individuals to achieve a higherlevel of utility – i.e. Move to U1 > U0

0

*

I0

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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

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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

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• 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

Constraints: With Capital market only

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The Budget Set (Capital Markets)y0 = endowment in period 0y1 = endowment in period 1

y0

W1

C0

C1

y1

W0

This line (Capital market line)traces out all affordable combinationsof C0 and C1, given the endowments y0 and y1

by borrowing and lending at market interest rate R

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

(1+R)

By giving up,or deferring oneunit of consumptiontoday, you get (1+R)units tomorrow

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Choosing the Optimal Consumption Path with Capital Markets

C1

C0

U0

U10B

D

E

At point 0, the slope of the indifference curveis 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 levelof utility by lending money to the capital market

*

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 *

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C1

C0

U0

Consumption with Production and Capital Markets

0

In the absence of production and capitalmarkets, this individual can only consume at Pt 0.Pt 1: Production alonePt C & P: Both Production and Capital Markets

U1

Capital Marketline

At the optimum, there is a “separation” between the Consumption and Production Decision (pt c & p)

Uc&p

*

*

c

p

1*

*

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The Inter-temporal Consumption Choice

0

*

0

*

0

*

U0

U’

U0 U0

U’’U’’’

Production alone

Preferences determineProduction and Consumption Point(same point)

Capital Market Alone

There is no production, and preference determineconsumption

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

*

Consumption

Production

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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

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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

1. 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 Theoremi.e the separation of the Consumption and Investment decisions

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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

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C0

C1

*

Both individuals 1 and 2make the same investmentdecision: both produce at *,and by using capital markets,borrow or lend to achieve their own optimum consumption point

That is, regardless of the shapeof their utility functions, they makeexactly the same production decision

In math: MRS1 = MRS2 = -(1+R) = MRT

Individual 1(stronger preference for current consumption)

Individual 2(stronger preference for future consumption)

Consider two individuals

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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 marketsINVESTMENT 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]

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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.

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Again and again,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

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Principal-Agent Problem• 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

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TOPICS for self-interest1. Different techniques for selecting investment projects

a) Payback methodb) Accounting rate of returnc) Net present valued) Internal rate of return

2. Measuring shareholder wealth (you need to know how to compute NPV)

3. Economic profits vs Accounting profits

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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.

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An exercise for self-studyConsider 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

____1(1+)

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An Exercise

450

U0

C0 = C1

C0

C1

1

1

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 slopeis less than 1 as

[U'(C0)/ U'(C1)] < 1

To the left of the 450 line, the slopeis greater than 1 as

[U'(C0)/ U'(C1)] > 1dC1/dC0 = -(1- ) [U'(C0)/ U'(C1)] slope of indifference curve

Therefore, even if > 0, the tradeoff between C0 and C1 can be < 1if C0 is sufficiently high

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

• We are ignoring production opportunities in this example

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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

*

C0

C1

U

C0*

C1*

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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%

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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