Time and the Discount Rate

• Project flows of costs and benefits are not even over time.

Flow of Costs and Benefits

Option1

Year Cost Benefit Net

1 100 0 -100

2 10 0 -10

3 10 150 140

Sum 120 150 30

Option 2

Year Cost Benefit Net

1 40 50 10

2 40 50 10

3 40 50 10

Sum 120 150 30

Consider 2 Investment Options

Flow of Costs and BenefitsNet Benefits by Year

-150

-100

-50

0

50

100

150

200

1 2 3

Year

Ne

t B

en

efi

ts Option 1 Option 2

Flow of Costs and Benefits

• These two options have very different flows of costs and benefits over time.

• Is there any difference between these two options?

• Which is preferred?

Flow of Costs and Benefits

• In order to address this question, need to understand how the future is valued relative to the present:– Intertemporal time preferences– The interest rate

Time Preference of Consumers

• Consider consumers have a stock of wealth today, and must choose to consume between today and tomorrow.

• Intertemporal indifference map– Time indifference– Intertemporal indifference curve

Intertemporal Consumption Choice

Consumption T0

Con

sum

ptio

n T

1

C0

S0

S1

I0 (Intertemporal Indifference Curve)

C1

x

x

Intertemporal Consumption Choice

Consumption T0

Con

sum

ptio

n T

1A Family of Indifference Curves

Time Preference of Consumers

• Expect consumers to have “positive time preference” :– prefer consumption today rather than in the future

• Why?– There is a chance that will not be able to consume in

the future • (in the long run we are all dead)

– Expectation that income will be higher in the future (economy will grow)

• (Goods and services will be more abundant in the future as a result of economic growth)

Intertemporal Consumption Choice

Consumption T0

Con

sum

ptio

n T

1

Slope=-1

C0

S0 = C0

Ce

Se < Ce

The price of current consumption

• Income or wealth that is consumed today is not available to be saved for consumption in the future.

• What is the price of consuming today?

• Interest rate: amount that savings today can be increased in the future, through investment.

Interest Rate

• The price (or opportunity cost) of consuming a dollar of income or wealth today rather than saving for consumption in the future

• Expressed as a percentage increase:– (K1/K0)-1} * 100

K0 money invested in time 0

K1 money obtained from investment in time 1

Interest Rate

• Example:

• Invest $100 in time 0

• Return = $112 in time 1

• Interest rate=(112/100-1) * 100

= 12%

• A unit-free measure, expressed as a ratio

Interest Rate

Consumption T0

Con

sum

ptio

n T

1

100

100

i = 0%

112

150

i = 12%

i = 50%

Time Preference of Consumers

• From consumers’ time preferences, derive the supply of savings.

Determine amount of income and wealth that households will save for the future at different interest rates.

Response to change in interest rate

Consumption T0

Con

sum

ptio

n T

1

S0

S1

S2

i0

i1

i2

Response to change in interest rate

i0

Interest Rate

Savings ($)S0

x

i1

S1

i2

S2

x

x

Ssavings

Investment opportunities

• Investment opportunities generate the demand for savings (capital)– Intertemporal production possibility frontier

Intertemporal production possibility frontier

Production0

Pro

duct

ion 1

P0P1

I1

I0

PPF

Demand for Investment

• Rank potential investments according to the rate of return, from highest to lowest.

• This will give the derived demand schedule for savings

Demand for Investment

Production0

Pro

duct

ion 1

PPF

i0

i1

i2

I0

I1

i2

Response to change in Interest Rate

Interest Rate

Investment Demand ($)

i0

I0

x

i1

I1

i2

I2

x

x

Demand for Invesment

Capital Market

i Supplyof

Savings

Demandfor

Investment

Quantity ($)

ieq

Qeq

Interest rate

• In market equilibrium

• interest rate (r) = MRS = MRT

• In reality: transactions costs associated with matching up suppliers and demanders of savings:– Financial intermediation (banks)– Spread between savings and borrowing rate– Risk premia