Budget Today or Tomorrow

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Budget Today or Tomorrow

• Here we study the properties of the budget line
  in the context of consumption over time.

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Budget Line
In the context of consumption today or
consumption tomorrow, the budget line is a bit
different than in the typical consumer behavior
model. One difference is we have consumption
today (C1) on the horizontal axis and consumption
tomorrow (C2) on the vertical axis. Another
difference is the presence of an endowment
point. It is assumed the consumer has an initial
set amount of possible consumption today and a
set amount of consumption next period and these
come from income each period, called M1 and M2.
The budget line must go through the endowment
point, but borrowing or lending (saving) can move
the consumer away from the endowment.
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Budget Line
C tomorrow C2
paid back interest Pay back interest
Endowment point (M1, M2)
C today C1
Borrow today
Lend today
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Budget Line
On the previous screen the endowment point shows
what the person can have in each period. If the
person borrows - takes more today than the
endowment - then next period 1 r must be paid
back for every 1 taken today. If the person lends
- gives some of todays consumption up - then
next period 1 r is received for every 1 given
up today. The slope of the budget is (1 r), or
1 r in absolute value.
5
Change in the interest rate
The original line has a lower interest rate
because when borrowing occurs less is given back
next period, or if lending occurs less is paid
back. So, the higher the interest rate the
steeper the curve through the endowment point.
C tom.
New
original
C today
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Example
The endowment point here means the income is
50000 today and 60000 tomorrow (Next year). Say
the interest rate is 20
C tomorrow
Vertical intercept.
paid back interest Pay back interest
Endowment point (50000, 60000)
C today
Borrow today
Lend today
Horizontal intercept
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Example continued
The vertical intercept means you consume nothing
today and everything next period. The 50000 this
period will earn interest and thus the total will
be 50000(1.2) 60000120000. The horizontal
intercept means you spent all you current income
and as much as you can borrow and use next years
income to pay it off. This would be 50000
(60000/1.2) 100000. This horizontal intercept
is the present value of lifetime income.
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Example continued
When the interest rate is 20 and the endowment
point is (50000, 60000) the budget line can be
thought of in math form in the following ways 1)
In terms of the future C2 1.2C1 120000, 2)
In terms of the present C1 (C2/1.2) 100000,
or 3) As we do in the graph C2 120000
1.2C1. The slope is -1.2 and means if you spend
1 today you give up the ability to consume 1.20
next year.
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Equation for budget line
When we have endowments M1 and M2 and interest
rate r the equation for C2 and C1 is C2 M2
M1(1 r) (1 r)C1. Thus, 1) If C1 0, C2
M2 M1(1 r). This is the vertical
intercept. 2) If C2 0, 0 M2 M1(1 r) (1
r)C1, or C1 M1 M21/(1 r). This is the
horizontal intercept and is called the present
value of lifetime income.
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Indifference Curves

• Here we study indifference curves in the context
  of consumption over time.

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Indifference Curves
Indifference curves in this context are basically
the same as we saw in the past. The curves slope
downward, do not cross, fill the graph (although
we do not always draw many in a graph), and are
convex (meaning they get flatter as you move down
the curve.)
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Present or Future Oriented
C tom.
C today
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Present or Future Oriented
On the previous screen we have a curve for two
separate people. Each one gives up a unit of C
today. The flat curve person does not need much C
tomorrow back in return for the C today given up.
This type of person is tomorrow oriented or
patient. The steep curve person needs more C
tomorrow (relative to flat curve) in return for
the C today given up. This type of person is
today oriented or impatient.
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Marginal rate of time preference (MRTP)
The absolute value of the slope of an
indifference curve at a point is called the MRTP.
The slope is change in C2 divided by the change
in C1. If the MRTP gt 1 the consumer has a
positive time preference meaning when giving up 1
unit of C1 more than 1 unit of C2 must be given
back to have the same utility. If the MRTP lt 1
the consumer has a negative time preference
meaning when giving up 1 unit of C1 less than 1
unit of C2 must be given back to have the same
utility. If the MRTP 1 1 the consumer has a
neutral time preference meaning when giving up 1
unit of C1 1 unit of C2 must be given back to
have the same utility.
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Equilibrium

• Given an interest rate, we see here the point
  consumers end up at in order to maximize their
  utility.

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A borrower
Notice at the endowment the consumers
indifference curve goes through the budget
steeper then the budget- they are willing to pay
back more than they have to, so they borrow
today and become happier than at the endowment.
C tom.
Endowment point
C today
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A lender
Notice at the endowment the consumers
indifference curve goes through the budget
flatter than the budget - They get more in the
future than they require to have the same utility
so they lend today and are happier doing so.
C tom.
Endowment point
C today
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Note
Both the lender and the borrower at the point of
equilibrium have the MRTP 1the interest rate
and this is greater than 1. This means that both
borrowers and lenders have positive time
preferences in equilibrium. Again, this means
when giving up 1 unit of C1 more than 1 unit of
C2 must be given back to have the same
utility. Also note that with a given interest
rate some people are lenders and some borrowers
based on their preferences. Later on we show how
folks might change from being a lender to a
borrower, and vice versa, depending on changes in
the interest rate.
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Changes in Equilibrium

• Here we study how the consumer position changes
  given changes in the interest rate.

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Change in the interest rate
C tom.
l
n
Endowment point
m
C today
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Change in the interest rate
We have seen in the past that as the interest
rate falls the budget line becomes flatter. At
the highest interest rate in the example on the
previous screen, we see the individual go to
point l (and is actually a lender.) This point
has a certain amount of C today involved (as well
as a certain of C tom.) As the interest rate
falls the consumer moves to point m and then
point n. So the amount of current consumption
rises as the interest rate falls. The point here
is that the demand for current consumption is a
function of the interest rate. In fact, we say
as the interest rate falls the quantity demanded
for current consumption rises.
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Change in the interest rate
This is the demand for current consumption curve
and is derived from the graph two slides before
this one.
r
l
m
n
C today
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Permanent income hypothesis
Lets consider an example where your income will
1000 in each of two years and the interest rate
will be 25. C2 M2 M1(1r) (1r)C1. a.
Suppose that you save all of your money to spend
next year. How much will you be able to spend
next year? This is the same as asking on the
budget what is C2 when C1 0? C2 would be 1000
1000(1.25) 2250. How much will you be able
to spend today is like what is C1 if C2 0. C1
would be 1000 (1000/1.25) 1800.
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b. Suppose you borrow 800 and spend 1800
today. How much will you be able to spend next
year? If C1 1800, C2 1000 1000(1.25)
1800(1.25) 0. c. The graph is on the next
slide with C1 on the horizontal and C2 on the
vertical axis. Note the vertical intercept is
(0, 2250), the horizontal intercept is (1800, 0)
and the endowment point is (1000, 1000) The slope
(2250-0)/(0-1800) -1.25, so the slope shows
that the price of spending 1 today means you can
not spend 1.25 next year. Note if C1 M1, then
C2 M2, and vice versa. This means the person
can have their endowment point and neither borrow
or lend.
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c2
(0, 2250)
(1000, 1000)
(1800, 0)
c1
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Say you find 400 in your desk drawer. Your
endowment today becomes 1400. How does the
budget shift? Note the new intercepts and
endowment point.
c2
(0, 2750)
(0, 2250)
(1400, 1000)
(1000, 1000)
(2200, 0)
(1800, 0)
c1
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Say you will get 500 more in pay next year but
this year you only have 1000. Your endowment
next year becomes 1500. How does the budget
shift? Note the new intercepts and endowment
point. The budget shifts just like in the
previous example.
c2
(0, 2750)
(0, 2250)
(1000, 1500)
(1400, 1000)
(1000, 1000)
(2200, 0)
(1800, 0)
c1
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Now say the person has endowment 1000 and 1000
initially. They consume at point A. If C1 and C2
are normal goods, then if the endowment in period
1 rises to 400, or the endowment in period 2
rises to 500, the individual will end up here.
c2
(0, 2750)
(0, 2250)
A
(1400, 1000)
(1000, 1000)
(2200, 0)
(1800, 0)
c1
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In this example we see if income in period 1 goes
up 40 consumption in period 1 is not likely to
go up 40. Some of the increase is spread out
into the next year. Similarly, if income next
period goes up 50 (Say you expect to graduate
and make more money) your consumption in period 2
is not likely to go up 50. Permanent income is
the present value of our lifetime income and we
saw this has the horizontal intercept. Given our
preferences, permanent income is what determines
our consumption pattern. Another way to say this
is that our consumption pattern over time is
influenced not only by the income in he period in
which we consume, but by the income in every
period.