i need solutions for first three examples, ASAP. for answer checking.

 

 

Math 1274 - Calculus II with Economic and
Business Applications
Pichmony Anhaouy Department of Mathematics & Statistics
Langara College
Chapter 6 : Applications of Integrals (II)
Part B - Continuous Income Streams and Consumers’ and Producers’
Surplus Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 1 Continuous Income Streams (PB, page 30)
(I) Total Income for a Continuous Income Stream:
Let’s consider the following example. Suppose that we have a trust
that pays us $2000 a year for 10 years.
What is the total amount we will receive from this trust by the end of
the 10th year?
The answer is simple. Since there are ten payments of $2, 000 each,
we will receive a total of 10 × $2, 000 = $20, 000.
Now, let’s look this problem from a calculus point of view. Let’s assume that the income stream is continuous at a rate of
$2, 000 per year. In the following figure, the area under the curve of f (t) = 2, 000 from
0 to t represents the income accumulated t years after the start.
Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 2 Rate (Dollars/Year) f(t) y = f(t) = 2000
2000 0 1 Time (Years) 1
2
1
4 Pichmony Anhaouy (panhaouy@langara.ca) 10 t Math 1274 Notes - Chapter 6, Part B Slide 3 For t = 1
1
year, the income would be (2, 000) = $500.
4
4
1
1
For t = year, the income would be (2, 000) = $1, 000.
2
2
For t = 1, the income would be 1(2, 000) = $2, 000. For t = 5.3 years, the income would be 5.3(2000) = $10, 600. It looks like that these numbers are the areas of rectangles with the
1 1
same length of 2000 and widths , , 1, and 5.3.
4 2
So, the total income over a ten year period, that is, the area
10(2, 000) = $20, 000.
But, this area is the same as the definite integral
10 f (t) dt = � �
0
$20, 000. 0 10 2, 000 dt = 2, 000t� Pichmony Anhaouy (panhaouy@langara.ca) 10
0 = 2, 000(10) − 2, 000(0) = Math 1274 Notes - Chapter 6, Part B Slide 4 In general, if the function f (t) is the rate of flow of a continuous
income stream, the total income produced (or total value) during the
period from t = a to t = b is given by
Total income = T V = � b a See the graph f (t) dt. y = f (t) y Total Income a
Pichmony Anhaouy (panhaouy@langara.ca) b t Math 1274 Notes - Chapter 6, Part B Slide 5 (II) Future Value of a Continuous Income Stream:
We have seen in previous calculus class, i.e., our Math 1174 here at
Langara, the following continuous compound interest formula
A = P ert ⇒ P = Ae−rt . P is the principal or present value, which is sometimes denoted by
PV ,
A is the amount or future value, which is sometimes denoted by F V ,
r is the annual rate of continuous compounding, which is expressed as
a decimal, and
t is the time, which is usually in years.
For example, if money is worth 10% compounded continuously, then
the future value of 2, 000 investment in 10 years is
A = 2, 000e0.10×10 = $5, 436.56.
Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 6 Now, we want to apply this future value concept to the income
produced by continuous income stream. Suppose that f (t) is the rate of flow of a continuous income stream
Suppose also that the income produced by this continuous income
stream is invested as soon as it is received at a rate r, compounded
continuously
We have already known how to compute the total income (or total
value) produced after T years, that is,
TV = � T 0 f (t) dt. But, how can we find the total of the income produced and the
interest earned by this income T V ?
Let’s look at the figure on the next page. Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 7 Money line: (For continuous income stream, take integral of each)
f (t)e rt f (t)er(T f (t) t 0
PV
(Now) t) T FV Flow of Money Chart In today’s dollars, the flow of a continuous income stream from t = 0
to t = T , is
F low = � T 0 f (t)e−rt dt. The future value of the stream at t = T is,
F V = erT � 0 T f (t)e−rt dt or F V = � 0 T f (t)er(T −t) dt. It does not have to start from t = 0 all the time. The time here is in
[0, T ]. In general, it could start from t = ti to t = tf . So, the time
interval is [ti , tf ]. Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 8 Example 1: A construction company is working on a project. The cost for
this project are: $1 million today, $1.5 million in two years time, and after
that, $0.75 million per year continuous over the next three years. Find the
present value of the cost if the interest is 3% compounded continuously. Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 9 Example 2: Suppose you win money for $1000 per week forever.
Assuming that the flow rate is f (t) = 52, 000 dollars per year and at the
interest rate of 4% compounded continuously. What would be the money
if you could take it all today? Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 10 Example 3: You want to have $200, 000 when you retire. You open an
account that pays 8% interest compounded continuously. Each year you
put $10, 000 into it.
a) How many more years that you have to work before you can retire? b) However, after you retired, you convert this $200, 000 into an
retirement income fund (RIF) which also pays the same interest. Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 11 b) (Continues) You want to be able to withdraw continuously each year
from this saving for the next 20 years after retirement. How much will
be each withdraw ? Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 12 Example 4: Suppose that the rate of change (flow rate) of a continuous
income stream produced is given by f (t) = 5, 000e0.04t .
a) Find the total income (TV) during the first 5 years.
TV = � 0 5 f (t) dt = � 5 0 5, 000e0.04t dt = 125, 000e0.04t � = $27, 675.
5
0 b) Find the future value of this income stream at 12%, compounded
continuously for 5 years.
F V = erT � 0 T f (t)e−rt dt = e0.12×5 � 5, 000e0.04t ⋅ e−0.12t dt = 5
1
e−0.08t � = $37, 545.
0
−0.08
0
c) How much is the total interest earned by this income stream during the
five year period?
$37, 545 − $27, 675 = $9, 870. 5, 000e0.6 � 5 e−0.08t dt = 5, 000e0.6 0 5 Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 13 Consumers’ and Producers’ Surplus (PB, page 34)
Let p = D(x) be the price-demand equation for a product, where x is
the number of items sold at a price p each.
Suppose that p0 is the current price and x0 is the number of units
that can be sold at this price.
If the price is higher than p0 , then the demand x is less than x0 .
The consumers’ surplus is the total savings to consumers who are
willing to pay more than p0 for the product, but still able to buy the
product at that price.
It can be shown that the consumers’ surplus, CS, at the price level of
p0 is defined by
CS = � 0 x0 �D(x) − p0 � dx = � 0 x0 D(x) dx − p0 x0 , which is the area between p = p0 and p = D(x) from x = 0 to x = x0 . Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 14 p p p = S(x) CS
p0 p0 PS
p = D(x)
x x0 Pichmony Anhaouy (panhaouy@langara.ca) x0 Math 1274 Notes - Chapter 6, Part B x Slide 15 Similarly, if p = S(x) is the price-supply equation for a product, p0 is
the current price, and x0 is the current supply level, then the total
money gained to producers who are willing to supply the products at
a lower price than p0 but are still able to supply the products at this
price is called producers’ surplus, P S, which is defined by
PS = � 0 x0 �p0 − S(x)� dx = p0 x0 − � 0 x0 S(x) dx, which is the area between p = p0 and p = S(x) from x = 0 to x = x0 . In a free market, the price of a product is determined by the
relationship between supply and demand. The point (x0 , p0 ) is the
intersection point of p = D(x) and p = S(x). It is called the
equilibrium point. The price p0 is called the equilibrium price and x0
is called the equilibrium quantity.
If a price ps stabilizes at the equilibrium price p¯, then it is the price
level that will determine both the consumers’ surplus and the
producers’ surplus.
Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 16 The total of the consumers’ and producers’ surplus is called the total
gains (from trade). From the graph in the figure below, we can see
that this total gains is the area between the two curves p = D(x) and
p = S(x).
T otal Gains = CS + P S = � x0 0 p �D(x) − S(x)� dx. p = S(x) CS
p0 PS
p = D(x)
x0
Pichmony Anhaouy (panhaouy@langara.ca) x Math 1274 Notes - Chapter 6, Part B Slide 17 Example 1: Find the consumers’ surplus at a price level of $8 for the
price-demand equation p = D(x) = 20 − 0.05x. Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 18 Example 2: Find the producers’ surplus at a price level of $20 for the
price-demand equation p = S(x) = 2 + 0.0002x2 . Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 19 Example 3: Suppose that the price-demand equation and the price-supply
equation are given by
p = D(x) = 20 − 0.05x, and p = S(x) = 2 + 0.0002x2 . Find the consumers’ surplus and the producers’ surplus at the equilibrium
price level. Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 20 Example 4: Suppose that the price-demand equation and the price-supply
equation are given by p = D(x) and p = S(x). The equilibrium price and
quantity are (x0 , p0 ) = (200, 100). Also, suppose that the amount of the
producers’ surplus is $12, 067, and
� 0 200 �D(x) − S(x)� dx = 28, 067 (Area between the curves). a) What is the amount of the consumers surplus? Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 21 Example 4 (Continues)
b) What does � 0 200 D(x) dx evaluate to? Pichmony Anhaouy (panhaouy@langara.ca) Math 1274 Notes - Chapter 6, Part B Slide 22