Time and the Discount Rate
• Project flows of costs and benefits are not
even over time.
Flow of Costs and Benefits
Consider 2 Investment Options
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
1
40 50
2
40 50
3
40 50
Sum 120 150
Net
10
10
10
30
Flow of Costs and Benefits
Net Benefits by Year
200
150
Net Benefits
100
Option 1
Option 2
50
0
1
2
-50
-100
-150
Year
3
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
Consumption T1
Intertemporal Consumption Choice
S1
x
x
S0
C1
C0
I0 (Intertemporal
Indifference Curve)
Consumption T0
Consumption T1
Intertemporal Consumption Choice
A Family of Indifference Curves
Consumption T0
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)
Consumption T1
Intertemporal Consumption Choice
“PPF” (Slope=-1)
S0 = C0
Se < Ce
Zero time preference
Positive time preference
450
C0
Ce
Consumption T0
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 = $112 in time 1
Interest rate=
(112/100-1) * 100
= 12%
• A unit-free measure, expressed as a ratio
Consumption T1
Interest Rate
150
112
100
i = 50%
i = 12%
i = 0%
100
Consumption T0
Time Preference of Consumers
• From consumers’ time preferences, derive
the supply of savings.
Determine amount of income and wealth that
households will save for the future at different
interest rates.
Consumption T1
Response to change in interest rate
S2
S1
i2
i1
i0
S0
Consumption T0
Response to change in interest rate
Interest
Rate
Ssavings
i2
x
x
i1
i0
x
S0
S1
S2
Savings ($)
Investment opportunities
• Investment opportunities generate the
demand for savings (capital)
– Intertemporal production possibility frontier
Production1
Intertemporal production possibility
frontier
I1
PPF
I0
P1
P0
Production0
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
Production1
Demand for Investment
i2
i2
I1
i1
I0
PPF
i0
Production0
Response to change in Interest Rate
Interest
Rate
i0
x
x
i1
i2
x
Demand for Invesment
I0
I1
I2
Investment Demand ($)
Capital Market
Supply
of
Savings
i
ieq
Demand
for
Investment
Qeq
Quantity ($)
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