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Future value and present value of an income stream
Suppose that we have a stream of income which is accrued continuously. (An ‘income stream’.)
Furthermore, we put this income into an account, and it then
accrues interest.
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Future value and present value of an income stream
Suppose that we have a stream of income which is accrued continuously. (An ‘income stream’.)
Furthermore, we put this income into an account, and it then
accrues interest.
We’d like to know the future value at some time T . (This is how
much the income plus interest is worth after time T .)
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Suppose that money is being transferred continuously into an
account over a time period 0 ≤ t ≤ T . Suppose that the rate
of this income is given by a function f (t), and suppose that the
account earns an interest rate of r (compounded continuously).
The future value of the income stream over the term T is:
FV = erT
Z T
0
f (t)e−rT dt =
Z T
0
f (t)er(T −t)dt
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The Present Value
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The Present Value
Suppose that we have a continuous stream of income with rate
f (t) and interest rate r, just like in the above situation.
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The Present Value
Suppose that we have a continuous stream of income with rate
f (t) and interest rate r, just like in the above situation.
The Present Value of this income stream is the amount of
money that would need to be placed into the account now (and
invested at an interest rate r compounded continuously) in order
to end up with the same amount of money at time T .
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The Present Value of an income stream that is deposited continuously at a rate f (t) into an account that earns interest at a
rate of r compounded continuously for a term of T years is:
PV =
Z T
0
f (t)e−rtdt.
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Examples
(1) Suppose that the continuous income stream with a rate
f (t) = 2000 is deposited into an account which earns interest
(compounded continuously) at a rate of 5% per year. What is
the future value of this income stream after 5 years?
What is the present value of this income for a term of 5 years?
(2) Suppose that the continuous income stream with a rate
f (t) = 1000e0.01t is invested in an account with interest compounded continuously at a rate of 7% per year. What is the
future value after 10 years, and the present value of this income
for a term of 10 years?
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Consumer willingness to spend
Suppose that we have a demand function
p = D(q)
So if q units are to be sold, the price must be p.
The total willingness to spend for q0 units is:
A(q0) =
Z q
0
0
D(q)dq.
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Consumer willingness to spend
Suppose that we have a demand function
p = D(q)
So if q units are to be sold, the price must be p.
The total willingness to spend for q0 units is:
A(q0) =
Z q
0
0
D(q)dq.
Think of it like this:
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Suppose that the demand function is:
p = D(q) = 100 − 2q.
We can write this as q = 50 − 2p (Remember: Everyone but an
economist thinks that quantity is a function of price, and not
the other way round...)
So, if the price is 100, then nobody will buy the function (since
this gives q = 0).
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Suppose that the demand function is:
p = D(q) = 100 − 2q.
We can write this as q = 50 − 2p (Remember: Everyone but an
economist thinks that quantity is a function of price, and not
the other way round...)
So, if the price is $100, then nobody will buy the function (since
this gives q = 0).
Now, if we reduce the price to $98, then we’ll get 1 person to
buy it. So we should (‘should’ according to an economist) charge
this person $98.
Now, reduce the price to $96, and you’ll pick up another customer (sucker).
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We continue in this manner.
If the price is $50, then there’s the extra money that the guy
who would have paid $98 was ‘willing’ to spend.
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We continue in this manner.
If the price is $50, then there’s the extra money that the guy
who would have paid $98 was ‘willing’ to spend.
So, the total willingness to spend for q0 units is the area under
the demand curve p = D(q) from q = 0 to q = q0.
[PICTURE]
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The consumers’ surplus is the total willingness to spend minus
the amount actually spent for q0 units (which is p0q0 where p0 =
D(q0)). So the consumers’ surplus at q0 units is
CS =
Z q
0
0
D(q)dq − p0q0,
where p0 = D(q0).
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On the flip side, at a particular level p0 = S(q0) of the supply
curve
p = S(q)
(which measures how much the produces want to charge to sell
q units), there are likely some producers who would have sold
there goods for less than p0. So there’s a producers’ surplus:
PS = p0q0 =
Z q
0
0
S(q)dq.
[PICTURE]
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Example
The boffins in R&D have determined the following data about
supply and demand curves for basketball shoes (where p is in
dollars and q is in tens of thousands of units bought/sold):
p = D(q) = 200 + 10q − q 2,
p = S(q) = 100 + 20q + q 2.
(1) What is the equilibrium price?
(2) What are the Consumers’ and Producers’ Surpluses at the
equilibrium price?
[Interesting that the boffins can find these functions, but not do
the calculus, isn’t it?]
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