Transcript V2007

Analyzing Accumulated Change
Integrals in Action
6.1 - Perpetual Accumulation and
Improper Integrals
Improper Integral
• Definite integrals have specific numbers for
both the upper limit and the lower limit.
• In this section, what happens to the
accumulation of change when one
or both of the limits of the integral
are infinite is considered.
• Improper integrals play a role in economics
and statistics as well as in other fields of study.
Improper Integral Evaluations
• An improper integral of the form
is evaluated by applying a limit:
• Replace  with a variable, N.
• Evaluate the limit of the integral as N
increases without bound, provided the limit
exists.
Evaluating Improper Integrals
• where F is an anti-derivative of f.
• Improper integrals show up in situations in
which quantities are being evaluated over an
indefinitely long interval.
Divergence and Convergence
• If the limit of an improper integral exists, the
improper integral converges.
• Sometimes the limit of an improper integral
does not exist. (The limit increases or
decreases without bound.) In this case,
improper integral diverges.
EXAMPLE
Example
EXAMPLE: 2, 6, 8, 12, 18, 20
Analyzing Accumulated Change
Integrals in Action
6.2 - Streams in Business and Biology
Income Stream and Flow Rates
• An income stream is a regular flow of money
that is generated by a business or an
investment.
• The rate of flow is a function R that varies
according to time t.
– Constant
%allocated + profit
– Linear
% allocated (profit ± increment * t)
– Exponential % allocated (1+ % increase)
– Exponential % allocated (1 - % decrease)
Future Value of a Continuous Income Stream
• Suppose that an income stream flows
continuously into an interest-bearing account at
the rate of R(t) dollars per year, where t is
measured in years and the account earns interest
at 100r% compounded continuously.
• The future value of the account at the end of T
years is
Flow Rates and Rates of Change
• The flow rate for a continuous income stream is not the
same thing as the rate of change of that stream. Using
the Fundamental Theorem of Calculus, the rate-ofchange function for future value is:
• The function F‘(t)= R(x) er(T-t) gives the rate of change
(after t years) of the future value (in T years) of an
income stream whose income is flowing continuously in
at a rate of per year. The rate-of-change function , rather
than the flow rate of the income stream, , is graphed
when illustrating future value as the area of a region
beneath a rate-of-change function.
Present Value of a Continuous Stream
• The present value of a continuous income stream
is the amount P that would need to be invested at
the present time so that it would grow to a
specified future value under stated conditions.
• Suppose that an income stream flows continuously
into an interest-bearing account at the rate of R(t)
dollars per year, where t is measured in years, and
that the account earns interest at the annual rate
of 100r% compounded continuously. The present
value of the account is
Present Value of a Continuous Income Stream
(when the future value is known)
Perpetual Income Streams
• When there is no specific end date to an income
stream, the stream is considered to flow in perpetuity.
• For example, when a company leases land rights from
its parent corporation to harvest bio-mass from that
land for the life of the company, the parent corporation
might, in turn, create a fund with the lease money that
allows it to purchase more land at some indefinite
time.
• Because there is no definite end date for the income
stream, it can be considered to flow in perpetuity.
• In the case of a perpetual income stream, the end time
T is considered to be infinite and the present value of
such a stream is calculated as the improper integral
Example : 2, 4, 8
Streams in Biology
• Biology and other fields involve situations
similar to income streams. An example of this
is the growth of populations of animals.
• Functions that model biological streams
where new individuals are added to the
population and the rate of survival of the
individuals is known are referred to as survival
and renewal functions.
Future Value of a Biological Stream
• The future value (in b years) of a biological
stream with initial population size P, survival
rate 100s% and renewal rate r(t), where t is
the number of years, is
+
b
EXAMPLE: 22
Analyzing Accumulated Change
Integrals in Action
6.3 - Calculus in Economics—Demand
and Elasticity
Demand function or Demand curve
• Understanding consumer demand is
important in economics, management, and
marketing.
• The amount of a good or service that
consumers buy can be considered as a
function of the price they have to pay.
• This function is known as a demand function
or demand curve.
Demand Functions
• Demand is affected by several factors, including
– utility (usefulness),
– necessity,
– the availability of substitutes,
– and buyer income.
However, when all other factors are held constant, the
quantity demanded can be considered as a function
of market price.
Demand Function
• A function giving the expected quantity of a
commodity purchased at a specified market price is
referred to as a demand function or demand
schedule.
• The demand function D relates the input variable p
(price per unit) with the output variable q = D(p)
(quantity).
• The graph of a demand function is referred to as a
demand curve.
The Law of Demand
• All other factors being constant, as the price of
a commodity increases, the market will react
by demanding less and, as the price of a
commodity decreases, the market will react by
demanding more.
“Why does my economics text show
demand as a function of quantity?”
• In economic theory, many functions (such as cost, revenue,
and profit) are graphed with quantity on the horizontal axis.
• The demand curve is likewise drawn with quantity on the
horizontal axis.
• Reading the economics text carefully can reveal whether
price or quantity is considered to be the input variable.
• If quantity is considered to be the input variable, the
economics text is dealing with the inverse demand function.
If price is considered to be the input variable, the economics
text is simply drawing the demand function on a reversed
set of axes to keep quantity on the horizontal axis.
Consumer Expenditure
• Consumer expenditure is the price per unit of a
commodity times the quantity purchased by
consumers.
• Assuming that demand is satisfied, the quantity
purchased is the same as the quantity in
demand.
• This expenditure is represented graphically as
the area of a rectangle under the demand curve.
• For a commodity with demand function D and
price per unit p, when market price is fixed at
p0, the quantity demanded is written
q0 = D(p0) and
Consumer expenditure = p0 . q0
Consumer Surplus
• Consumer surplus is the amount that
consumers are willing and able to spend but
do not actually spend because the market
price was fixed at p0. This is the amount that
consumers have in excess from not having to
spend as much as they were willing and able.
Consumer Willingness and Ability to Spend
• When consumers demand a certain quantity
of a commodity, they are willing and able to
pay more (as an aggregate) than they actually
spend at the correlating market price.
• Consumer willingness and ability to spend is
the total amount of money consumers have
available and are willing to pay to obtain a
certain quantity of a commodity.
Consumer Willingness and Ability to Spend
• For a continuous demand function D, the
amount that consumers are willing and able to
spend for a certain quantity of a commodity is
given by
Consumer
Expenditure
Consumer Surplus
Consumer Willingness and Ability to Spend
• where p0 is the market price at which q0 units
are in demand, and pmax is the price above
which consumers will purchase none of the
commodity.
• (If the demand function approaches but does
not cross the input axis, the integral is
improper with an upper limit of .)
Example: 12, 14, 24
Price Elasticity of Demand
• Elasticity is a measure of the responsiveness of a function’s
output to a change in its input variable.
• Because the measures of the quantities of output and input
are normally very different, elasticity uses a ratio of
percentage rates of change to compare relative changes.
• For a commodity with differentiable demand function D
and price per unit p, the price elasticity of demand is
• Demand is elastic when |ƞ| > 1 and inelastic when |ƞ| < 1 .
Demand is at unit elasticity when |ƞ| = 1 .
• Price elasticity of demand is normally negative
Example: 28
Analyzing Accumulated Change
Integrals in Action
6.4 - Calculus in Economics—Supply
and Equilibrium
Calculus in Economics—
Supply and Equilibrium
• Supply Function
– A function giving the expected quantity of a
commodity supplied at a specified market price is
referred to as a supply function or supply
schedule.
– The supply function S relates the input variable
p(price per unit) with the output variable q = S(p)
(quantity).
– The graph of a supply function is referred to as a
supply curve.
Shutdown
• The shutdown price, ps , is the lowest market
price that producers are willing and able to
accept to supply any quantity of a certain
commodity.
• The shutdown point is the point (ps, S(ps))on
the supply curve that marks the conditions
(market price and quantity)under which the
production of a commodity will shut down.
Producer Revenue
• Producer revenue is the price per unit of a
commodity times the quantity supplied to the
market (assuming the entire quantity is also
purchased at market price).
• Producer revenue is represented graphically as
the area of a rectangle with one corner at the
origin (0, 0) and the opposite corner on the
supply curve at point (p0, q0) where p0 is the
market price.
• Producer revenue = p0 . q0
Producer Willingness and Ability to
Receive
• For a continuous supply function S, the minimum
amount producers are willing and able to receive
for a certain quantity of a commodity is given by
• where p0 is the market price at which q0 units are
supplied, and ps is the shutdown price.(If there is
no shutdown price, ps =0.)
Producer Surplus
• The amount that producers receive in excess
of the minimum amount that they require to
supply a certain quantity of a commodity is
known as producer surplus.
• Producer surplus benefits society as a whole
because producers are able to use surplus to
develop better or different products, invest in
other economic sectors, or pass it on to their
employees as money or added benefits.
Producer Surplus
• Producer surplus is calculated as producer
revenue minus producer willingness and
ability to receive:
Producer
revenue
Producer surplus =
Producer willingness to pay
Equilibrium and Social Gain
• The market price and quantity (p*, q*) at which
supply equals demand is called the equilibrium
point.
• At the equilibrium price p*, the quantity
demanded by consumers coincides with the
quantity supplied by producers. This quantity is
q*.
• Society benefits when consumers and/or
producers have surplus funds.
total social gain
• When the market price of a product is the
equilibrium price for that product, the total
benefit to society is the consumers’ surplus
plus the producer surplus. This amount is
known as the total social gain.
EXAMPLE: 10, 16, 22, 24