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Interest Rate Determination
Here we start with an example
and end with a theory of changes
in the interest rate.
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Suppose you have $1000 today and expect to receive another
$1000 one year from today. Your savings account pays an annual
interest rate of 25%, and your bank is willing to lend you money
at that same interest rate.
Let’s call c2 the amount you spend next year and c1 the amount
spend today. Let’s also call the endowment values today and next
year E1 and E2, respectively. r is the interest rate.
The budget line then has for
c2 = E2 + E1(1+r) – c1(1+r).
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 (1000/1.25) + 10002
= 1800.
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= E1, then c2 = E2, 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?
c2
(0, 2750)
Note the new intercepts and
endowment point.
(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. Your endowment next
year becomes 1500. How does
the budget shift?
c2
(0, 2750)
Note the new intercepts and
endowment point.
(0, 2250)
The budget shifts just like in the
previous example.
(1000, 1500)
(1400, 1000)
(1000, 1000)
(2200, 0)
(1800, 0)
c1
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If the interest rate rises to 50% the
budget rotates clockwise through
the budget. Note the new
intercepts.
c2
(0, 2500)
(0, 2250)
(1000, 1000)
(1666.67, 0) (1800, 0)
c1
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E. If you found the $400 today, or get a rise of $500 next year, either way your
budget ends up the same so you would consume the same amount today and
next year in either scenario.
f. On the next slide you see what it looks like at the original endowment of 1000
in each period and r=.25 when the person neither borrows nor lends.
g. Two slides from here you see what happens when the interest rate rise to
50%. Current spending is decreased, future spending is increased and the
person is better off. Remember the person started at the endowment point
here. The result may not hold for all types of starting points (like if the person
started out a borrower.)
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Here the endowment point yields
the most utility for the individual.
c2
(0, 2250)
(1000, 1000)
(1800, 0)
c1
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If the interest rate rises to 50% the
budget rotates clockwise through
the budget. The individual can
now get to a higher indifference
curve. The person will make c1
lower and c2 higher and be
happier.
c2
(0, 2500)
(0, 2250)
(1000, 1000)
(1666.67, 0) (1800, 0)
c1
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h. To show the income and
substitution effects you take the
new budget and shift it parallel so
it is tangent to the original
indifference curve.
c2
(0, 2500)
(0, 2250)
Here the real movement is from A
to C. But A to B is the sub. Effect.
C1 falls and c2 rises.
C
B
A
(1000, 1000)
From B to C is the income effect.
C1 rises and c2 could rise or fall,
here it looks like it rises.
Remember here when r rises c1
falls and c2 rises in total.
(1666.67, 0) (1800, 0)
c1
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Our next task is to use the framework presented so far, but in a
national economy sense to show under some conditions how the
interest rate can change. The model is really one of a closed
economy.
In the national economy closed to the rest of the world economy
c1 and c2 refer to production today and production next year.
We can only consume what we can produce.
Now, since in the economy every dollar borrowed is also every
dollar lent, the average or representative consumer is neither a
borrower or lender. In other words the endowment point would
be the optimal point. This point is shown in the graph on the
next slide. Remember that the interest rate is built into the slope
of the budget line. So we have an interest rate implicit in the
graph.
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Say we expect the future to get
brighter, meaning we expect to
make more in the future than
what we used to expect. The
budget shifts up and the
endowment point moves up the
straight line shown. Since the
representative consumer has to
be at the endowment we have
to end up at this point. But in
the graph you see that even the
representative consumer has
the urge to borrow. So, if there
is a relatively strong demand to
borrow for current consumption
the interest rate will rise to
chock off the excess demand
for borrowing. We see the final
point on the next slide.
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As the new budget rotates with
a higher interest rate we get a
tangency with the new
indifference curve.
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Now let’s consider a case where the
present gets bright, meaning we can
produce more today than we thought we
could. The budget line will shift right. We
see how on the next slide.
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The budget shifts right and the
endowment point moves over on
the straight line shown. Since the
representative consumer has to be
at the endowment we have to end
up at this point. But in the graph
you see that even the representative
consumer has the urge to lend. So,
if there is a relatively strong desire
to lend for more consumption next
year the interest rate will fall to stop
the urge to lend. We see the final
point on the next slide.
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