Inter-temporal Consumption Choice

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The Basic Economics

• In ECO100 we assume a unique happiness function for every individual (utility function). We call such function as the individual’s 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

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The Basic Economics

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

x

y

U0

U1

U2

Represent higher levels of utility U0 < U1 < U2

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The Basic Economics

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

x

y

Budget line (Slope = Px/Py)

W/Py

W/Px

Feasible Consumption Set

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The Basic Economics• 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

x

y

U0

U1

U2

W/Py

W/Px

C* (with consumption of x* and y*)

x*

y*

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The Basic Economics• 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.

C0

C1

U0

U1

U2

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The Basic Economics• 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)

C0

U = U(C0,C1=constant)

MU > 0 (U / C0) > 0

MU is diminishingi.e., (2U / C2) < 0

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The Basic Economics• With the assumptions, we have the following diagram. The slope of the

indifference curve represents the individual’s 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)

C0

C1

U0

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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.

C0

C1

U0

Y0

Y1

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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.

Total investment

Marginal rate of return

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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.

C0

C1

Y0

Y1

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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)

C0

C1

Y0

Y1

U0

U1

C*0

C*1

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Production OpportunityMessages:

– This individual can achieve a higher utility (U1>U0) 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.

C0

C1

Y0

Y1

U0

U1

C*0

C*1

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Capital market• Now, instead of one individual, let’s 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/(1+r)

C0

C1

Y0

Y1

W0

Capital market line with Slope = -(1+r)

Y0(1+r) + Y1

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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 (U2>U0)• This individual is now lending (Y0-C*0) amount of money, and will get back

((1+r)(Y0-C*0)) in the next period so that he can consume a total of C*1= Y1+(1+r)(Y0-C*0).

• Equilibrium condition is:MRS = (1+r)

C0

C1

Y0

Y1

W0

Y0(1+r) + Y1

U0

U2

C*0

C*1

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Production and capital marketConstraint 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 W*0 = P*0 + P*1/(1+r) which is larger than W0.• With his wealth maximized, he chooses (C*0, C*1) to consume and yield him U3.

C0

C1

Y0

Y1

U0

U3

(C*0, C*1)P*1

P*0

U1

W*0

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Fisher Separation TheoremMessage:• 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 individual’s subjective rate of time preferences.

C0

C1

Y0

Y1

U0

U3

(C*0, C*1)P*1

P*0

U1

W*0

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Fisher Separation TheoremImplication 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?

C0

C1

Y0

Y1

P*1

P*0 W*0

Individual 1

Individual 2

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Fisher Separation TheoremImplication 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 individual’s 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.

C0

C1

Y0

Y1

P*1

P*0 W*0

Individual 1

Individual 2

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Fisher Separation TheoremImplication 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.

C0

C1

Y0

Y1

P*1

P*0 W*0

Individual 1

Individual 2

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Fisher Separation TheoremImplication 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

C0

C1

Y0

Y1

P*1

P*0 W*0

Individual 1

Individual 2

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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 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.

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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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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

4. Agency problem

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An exercise for self-studyConsider 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

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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%