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Econ 384 Intermediate Microeconomics II

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A.1 Compounding

A.2 Present Value

A.3 Present Value Decisions

A.4 Lifecycle Model

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.

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

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

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)

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

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:

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

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:

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}

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

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 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%,

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.

- 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

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)

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

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:

- 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

- 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

- 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

- 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)

- 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

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

- 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