Chapter Ten
Intertemporal Choice
What Are We Doing in this Chapter?
We are applying our method of consumer choices to consumer choices over time
Intertemporal Choice
Persons often receive income in “lumps”; . monthly salary.
How is a lump of income spread over the following month (saving now for consumption later)?
Or how is consumption financed by borrowing now against income to be received at the end of the month?
Present and Future Values
Begin with some simple financial arithmetic.
Take just two periods; 1 and 2.
Let r denote the interest rate per period.
Future Value
Given an interest rate r the future value one period from now of $1 is
Given an interest rate r the future value one period from now of $m is
Present Value
Suppose you can pay now to obtain $1 at the start of next period.
What is the most you should pay?
$1?
No. If you kept your $1 now and saved it then at the start of next period you would have $(1+r) > $1, so paying $1 now for $1 next period is a bad deal.
Present Value
Q: How much money would have to be saved now, in the present, to obtain $1 at the start of the next period?
A: $m saved now becomes $m(1+r) at the start of next period, so we want the value of m for which m(1+r) = 1 That is, m = 1/(1+r), the present-value of $1 obtained at the start of next period.
Present Value
The present value of $1 available at the start of the next period is
And the present value of $m available at the start of the next period is
The Intertemporal Choice Problem
Let m1 and m2 be incomes received in periods 1 and 2.
Let c1 and c2 be consumptions in periods 1 and 2.
Let p1 and p2 be the prices of consumption in periods 1 and 2.
The Intertemporal Choice Problem
The intertemporal choice problem: Given incomes m1 and m2, and given consumption prices p1 and p2, what is the most preferred intertemporal consumption bundle (c1, c2)?
For an answer we need to know:
the intertemporal budget constraint
intertemporal consumption preferences.
Intertemporal Choice
If c1 units are consumed in period 1 then the consumer spends p1c1 in period 1, leaving m1 - p1c1 saved for period 1. Available income in period 2 will then be so
Intertemporal Choice
rearranged is
This is the “future-valued” form of the budget constraint since all terms are expressed in period 2 values. Equivalent to it is the “present-valued” form
where all terms are expressed in period 1 values.
The Intertemporal Budget Constraint
c1
c2
m2/p2
m1/p1
0
0
Saving
Borrowing
Slope =
Price Inflation
Define the inflation rate by p where
For example, p = means 20% inflation, and p = means 100% inflation.
Price Inflation
We lose nothing by setting p1=1 so that p2 = 1+ p .
Then we can rewrite the budget constraint as
Price Inflation
rearranges to
so the slope of the intertemporal budget constraint is
Price Inflation
When there was no price inflation (p1=p2=1) the slope of the budget constraint was -(1+r).
Now, with price inflation, the slope of the budget constraint is -(1+r)/(1+ p). This can be written as r is known as the real interest rate.
Real Interest Rate
gives
For low inflation rates (p » 0), r » r - p . For higher inflation rates this approximation becomes poor.
Comparative Statics
The slope of the budget constraint is
The constraint becomes flatter if the interest rate r falls or the inflation rate p rises (both decrease the real rate of interest).
Comparative Statics
c1
c2
m2/p2
m1/p1
0
0
slope =
Comparative Statics
c1
c2
m2/p2
m1/p1
0
0
slope =
Comparative Statics
c1
c2
m2/p2
m1/p1
0
0
slope =
The consumer saves.
Comparative Statics
c1
c2
m2/p2
m1/p1
0
0
slope =
The consumer saves. An increase in the inflation rate or a decrease in the interest rate “flattens” the budget constraint.
Comparative Statics
c1
c2
m2/p2
m1/p1
0
0
slope =
If the consumer saves then saving and welfare are reduced by a lower interest rate or a higher inflation rate.
Comparative Statics
c1
c2
m2/p2
m1/p1
0
0
slope =
Comparative Statics
c1
c2
m2/p2
m1/p1
0
0
slope =
Comparative Statics
c1
c2
m2/p2
m1/p1
0
0
slope =
The consumer borrows.
Comparative Statics
c1
c2
m2/p2
m1/p1
0
0
slope =
The consumer borrows. A fall in the inflation rate or a rise in the interest rate “flattens” the budget constraint.
Comparative Statics
c1
c2
m2/p2
m1/p1
0
0
slope =
If the consumer borrows then borrowing and welfare are increased by a lower interest rate or a higher inflation rate.
Summary
We treat intertemporal choices as regular consumer choice problems by letting the prices be
P1 and P2/(1+r)
Then everything studied before follows.
