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Econ 384 Intermediate Microeconomics II. Instructor: Lorne Priemaza [email protected] 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

Econ 384Intermediate Microeconomics II

Instructor: Lorne Priemaza

[email protected]

A intertemporal choice
A. Intertemporal Choice

A.1 Compounding

A.2 Present Value

A.3 Present Value Decisions

A.4 Lifecycle Model

A 1 compounding
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 compounding1
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
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
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
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
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
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
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
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
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


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
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.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 example1
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
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
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,


If you only consumed when young:

c1=i1+i2 /(1+r)

Lifecycle budget constraint
Lifecycle Budget Constraint

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

Often point E is referred to as the endowment point.


Old Consumption





i1+i2 /(1+r)

Young Consumption

A 4 lifecycle budget constraint1
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:

A 4 lifetime utility
A.4 Lifetime Utility

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


  • 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
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


  • 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.










Marginal rate of intertemporal substitution mris
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
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


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










A 4 maximization conclusion
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


Utility is always

maximized at the

tangent to the

indifference curve




A conclusion
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