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# Econ 384 Intermediate Microeconomics II - PowerPoint PPT Presentation

Econ 384 Intermediate Microeconomics II. Instructor: Lorne Priemaza Lorne.priemaza@ualberta.ca. A. Intertemporal Choice. A.1 Compounding A.2 Present Value A.3 Present Value Decisions A.4 Lifecycle Model. A.1 Compounding. If you invest an amount P for a return r, After one year:

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Econ 384 Intermediate Microeconomics II

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## Econ 384Intermediate Microeconomics II

Instructor: Lorne Priemaza

Lorne.priemaza@ualberta.ca

### A. Intertemporal Choice

A.1 Compounding

A.2 Present Value

A.3 Present Value Decisions

A.4 Lifecycle Model

### A.1 Compounding

If you invest an amount P for a return r,

After one year:

• You will make interest on the amount P

• Total amount in the bank = P(1+r) = P + Pr

After another year:

• You will make interest on the initial amount P

• You will make interest on last year’s interest Pr

• Total amount in the bank = P(1+r)2

This is COMPOUND INTEREST. Over time you make interest on the interest; the interest compounds.

### A.1 Compounding

Investment: \$100

Interest rate: 2%

Derived Formula:

S = P (1+r)t

S = value after t years

P = principle amount

r = interest rate

t = years

### A.1 Compounding Choice

Given two revenues or costs, choose the one with the greatest value after time t:

A: \$100 now B:\$115 in two years, r=6%

(find value after 2 years)

S = P (1+r)t

SA =\$100 (1.06)2 = \$112.36

SB =\$115

Choose option B

### A.1 Compounding Loss Choice

This calculation also works with losses, or a combination of gains or loses:

A: -\$100 now B: -\$120 in two years, r=6%

(find value after 2 years)

S = P (1+r)t

SA =-\$100 (1.06)2 = -\$112.36

SB =-\$120

Choose option A. (You could borrow \$100 now for one debt, then owe LESS in 2 years than waiting)

### A.2 Present Value

What is the present value of a given sum of money in the future?

By rearranging the Compound formula, we have:

PV = present value

S = future sum

r= interest rate

t = years

### A.2 Present Value Gain Example

What is the present value of earning \$5,000 in 5 years if r=8%?

Earning \$5,000 in five years is the same as earning \$3,403 now.

PV can also be calculated for future losses:

### A.2 Present Value Loss Example

You and your spouse just got pregnant, and will need to pay for university in 20 years. If university will cost \$30,000 in real terms in 20 years, how much should you invest now? (long term GIC’s pay 5%)

PV = S/[(1+r)t]

= -\$30,000/[(1.05)20]

= -\$11,307

### A.2 Present Value of a Stream of Gains or Loses

If an investment today yields future returns of St, where t is the year of the return, then the present value becomes:

If St is the same every year, a special ANNUITY formula can be used:

### A.2 Annuity Formula

PV = A[1-(1/{1+r})t] / [1- (1/{1+r})]

PV = A[1-xt] / [1-x] x=1/{1+r}

A = value of annual payment

r = annual interest rate

n = number of annual payments

Note: if specified that the first payment is delayed until the end of the first year, the formula becomes

PV = A[1-xt] / r x=1/{1+r}

### A.2 Annuity Comparison

Consider a payment of \$100 per year for 5 years, (7% interest)

PV= 100+100/1.07 + 100/1.072 + 100/1.073

+ 100/1.074

= 100 + 93.5 + 87.3 + 81.6 + 76.3 = \$438.7

Or

PV = A[1-(1/{1+r})t] / [1- (1/{1+r})]

PV = A[1-xt] / [1-x] x=1/{1+r}

PV = 100[1-(1/1.07)5]/[1-1/1.07] = \$438.72

### A.3 Present Value Decisions

When costs and benefits occur over time, decisions must be made by calculating the present value of each decision

-If an individual or firm is considering optionX with costs and benefits Ctx and Btx in year t, present value is calculated:

Where r is the interest rate or opportunity cost of funds.

### A.3 PV Decisions Example

• A firm can:

• Invest \$5,000 today for a \$8,000 payout in year 4.

• Invest \$1000 a year for four years, with a \$2,500 payout in year 2 and 4

• If r=4%,

### A.3 PV Decisions Example

2) Invest \$1000 a year for four years, with a \$2,500 payout in year 2 and 4

If r=4%,

Option 1 is best.

### A.4 Lifecycle Model

• Alternately, often an individual needs to decide WHEN to consume over a lifetime

• To examine this, one can sue a LIFECYCLE MODEL*:

*Note: There are alternate terms for the Lifecycle Model and the curves and calculations seen in this section

### A.4 Lifecycle Budget Constraint

Assume 2 time periods (1=young and 2=old), each with income and consumption (c1, c2, i1, i2) and interest rate r for borrowing or lending between ages

If you only consumed when old,

c2=i2+(1+r)i1

If you only consumed when young:

c1=i1+i2 /(1+r)

### Lifecycle Budget Constraint

The slope of this constraint is (1+r).

Often point E is referred to as the endowment point.

i2+(1+r)i1

Old Consumption

i2

E

O

i1

i1+i2 /(1+r)

Young Consumption

### A.4 Lifecycle Budget Constraint

Assuming a constant r, the lifecycle budget constraint is:

Note that if there is no borrowing or lending, consumption is at E where c1=i1, therefore:

• In the lifecycle model, an individual’s lifetime utility is a function of the consumption in each time period:

U=f(c1,c2)

• If the consumer assumptions of consumer theory hold across time (completeness, transitivity, non-satiation) , this produces well-behaved intertemporal indifference curves:

### A.4 Intertemporal Indifference Curves

• Each INDIFFERENCE CURVE plots all the goods combinations that yield the same utility; that a person is indifferent between

• These indifference curves have similar properties to typical consumer indifference curves (completeness, transitivity, negative slope, thin curves)

Intertemporal Indifference Curves

c2

• Consider the utility function U=(c1c2)1/2.

• Each indifference curve below shows all the baskets of a given utility level. Consumers are indifferent between intertemporal baskets along the same curve.

2

U=2

1

U=√2

0

c1

1

2

4

### Marginal Rate of Intertemporal Substitution (MRIS)

• Utility is constant along the intertemporal indifference curve

• An individual is willing to SUBSTITUTE one period’s consumption for another, yet keep lifetime utility even

• ie) In the above example, if someone starts with consumption of 2 each time period, they’d be willing to give up 1 consumption in the future to gain 3 consumption now

• Obviously this is unlikely to be possible

### A.4 MRIS

• The marginal rate of substitution (MRIS) is the gain (loss) in future consumption needed to offset the loss (gain) in current consumption

• The MRS is equal to the SLOPE of the indifference curve (slope of the tangent to the indifference curve)

### A.4 Maximizing the Lifecycle Model

• Maximize lifetime utility (which depends on c1 and c2) by choosing c1 and c2 ….

• Subject to the intertemporal budget constraint

• In the simple case, people spend everything, so the constraint is an equality

• This occurs where the MRIS is equal to the slope of the intertemporal indifference curve:

Maximizing Intertemporal Utility

c2

Point A: affordable, doesn’t maximize utility

Point B: unaffordable

Point C: affordable (with income left over) but doesn’t maximize utility

Point D: affordable, maximizes utility

IBL

D

B

C

IIC2

A

IIC1

c1

0

### A.4 Maximization Conclusion

Lifetime utility is maximized at 817,316 when \$797,619 is consumed when young and \$837,500 is consumed when old.

*Always include a conclusion

Maximizing IntertemporalUtility

c2

Utility is always

maximized at the

tangent to the

indifference curve

U=817,316

c1

0

### A. Conclusion

• Streams of intertemporal costs and benefits can be compared by comparing present values

• To examine consumption timing, one can use the LIFECYCLE MODEL:

• An intertemporal budget line has a slope of (1+r)

• The slope of the intertemporal indifference curve is the Marginal Rate of Intertemporal Substitution (MRIS)

• Equating these allows us to Maximize