Download 3.4 Marginal Functions in Economics

3.4
Marginal Functions in
Economics
Marginal Analysis
• Marginal analysis is the study of the rate of
change of economic quantities.
– An economist is not merely concerned with the value
of an economy's gross domestic product (GDP) at a
given time but is equally concerned with the rate at
which it is growing or declining.
– A manufacturer is not only interested in the total cost
of corresponding to a certain level of production of a
commodity, but also is interested in the rate of
change of the total cost with respect to the level of
production.
Supply
• In a competitive market, a relationship
exists between the unit price of a
commodity and the commodity’s
availability in the market.
• In general, an increase in the commodity’s
unit price induces the producer to increase
the supply of the commodity.
• The higher unit price, the more the
producer is willing to produce.
Supply Equation
• The equation that expresses the relation
between the unit price and the quantity
supplied is called a supply equation defined
by p = f ( x ) .
• In general, p = f ( x ) increases as x increases.
Demand
• In a free-market economy, consumer demand
for a particular commodity depends on the
commodity’s unit price.
• A demand equation p = f ( x ) expresses the
relationship between the unit price p and the
quantity demanded x.
• In general, p = f ( x ) decreases as x increases.
• The more you want to buy, the unit price should
be less.
Cost Functions
• The total cost is the cost of operating a
business. Usually includes fixed costs
and variable costs.
• The cost function C(x) is a function of the
total cost of operating a business.
• The actual cost incurred in producing an
additional unit of a certain commodity
given that a plant is already at a level of
operation is called the marginal cost.
Rate of Change of Cost Function
Suppose the total cost in dollars incurred each
week by Polaraire for manufacturing x
refrigerators is given by the total cost function
C ( x ) = 8000 + 200 x − 0.2 x 2
(0 ≤ x ≤ 400)
a. What is the actual cost incurred for
manufacturing the 251st refrigerator?
b. Find the rate of change of the total cost
function with respect to x when x = 250 .
a. the actual cost incurred for
manufacturing the 251st refrigerator is
C (251) − C (250)
[
]
− [8000 + 200(250) − 0.2(250) ]
= 8000 + 200(251) − 0.2(251)
2
2
= 45,599.8 − 45,500
= 99.80
b. The rate of change is given by the derivative
C ' ( x ) = 200 − 0.4 x
Thus, when the level of production is 250
refrigerators, the rate of change of the total
cost is
C ' (250) = 200 − 0.4(250) = 100
• Observe that we can rewrite
C (251) − C (250 )
C (251) − C (250) =
1
C (250 + 1) − C (250 ) C (250 + h ) − C (250)
=
=
1
h
• The definition of derivative tells us that
C (250 + h ) − C (250 )
C ' (250) = lim
h →0
h
• Thus, the derivative C' ( x ) is a good
approximation of the average rate of change of
the function C ( x ) .
Marginal Cost Function
• The marginal cost function is defined to be
the derivative of the corresponding total cost
function.
• If C ( x ) is the cost function, then C' ( x ) is its
marginal cost function.
• The adjective marginal is synonymous with
derivative of.
Revenue Functions
• A revenue function R(x) gives the revenue
realized by a company from the sale of x
units of a certain commodity.
• If the company charges p dollars per unit,
then R ( x ) = px .
• The demand function p = f ( x ) tells the
relationship between p and x. Thus,
R ( x ) = xf ( x )
Marginal Revenue Functions
• The marginal revenue function gives the
actual revenue realized from the sale of an
additional unit of the commodity given that
sales are already at a certain level.
• We define the marginal revenue function to
be R' ( x ) .
Profit Functions
• The profit function is given by P ( x ) = R ( x ) − C ( x )
where R and C are the revenue and cost
functions and x is the number of units of a
commodity produced and sold.
• The marginal profit function P' ( x ) measures
the rate of change of the profit function and
provides us with a good approximation of the
actual profit or loss realized from the sale of the
additional unit of the commodity.
Average Cost Function
• The average cost of producing units of the
commodity is obtained by dividing the total
production cost by the number of units
produced.
• The average cost function is denoted by C ( x )
and defined by
C (x )
x
• The marginal average cost function C' ( x )
measures the rate of change of the average
cost.
The weekly demand for the Pulser 25 color LED
television is
p = 600 − 0.05 x
(0 ≤ x ≤ 12,000)
where p denotes the wholesale unit price in
dollars and x denotes the quantity demanded.
The weekly total cost function associated with
manufacturing the Pulser 25 is given by
3
2
(
)
C x = 0.000002 x − 0.03x + 400 x + 80,000
where C(x) denotes the total cost incurred in
producing x sets.
a. Find the revenue function R and the profit
function P.
R ( x ) = px = (600 − 0.05 x )x
= 600 x − 0.05 x
2
P ( x ) = R ( x ) − C ( x ) = (600 x − 0.05 x
2
)
− (0.000002 x − 0.03x + 400 x + 80,000)
3
2
= −0.000002 x − 0.02 x + 200 x − 80,000
3
2
b. Find the marginal cost function, the marginal
revenue function, and the marginal profit
function.
C ' ( x ) = 0.000006 x − 0.06 x + 400
2
R' ( x ) = 600 − 0.1x
P' ( x ) = −0.000006 x − 0.04 x + 200
2
c. Compute C ' (2000) , R' (2000 ) , and P' (2000 )
and interpret your results.
•
•
•
Since C ' (2000 ) = 304 , the cost to manufacture
the 2001st LED TV is approximately 304.
Since R ' (2000 ) = 400 , the revenue increased
by manufacturing the 2001st LED TV is
approximately 400.
Since P ' (2000 ) = 96 , the profit for
manufacturing and selling the 2001st LED TV
is approximately 96.
Elasticity of Demand
• Question: when you produce more
commodities, do you actually get the more
revenue?
• It is convenient to write the demand function f in
the form x = f ( p ) ; that is, we will think of the
quantity demanded of a certain commodity as a
function of its unit price.
• Usually, when the unit price of a commodity
increases, the quantity demanded decreases
Suppose the unit price of a commodity is
increased by h dollars from p dollars to p+ h
dollars. Then the quantity demanded drops
from f (p) units to f(p + h) units. The
percentage change in the unit price is
h
(100)
p
and the corresponding percentage change in
the quantity demanded is
⎡ f ( p + h ) − f ( p )⎤
(100)
⎢
⎥
f ( p)
⎣
⎦
Percentage change in the quantity demanded
Percentage change in the unit price
⎡ f ( p + h ) − f ( p )⎤
(100)
⎢
⎥
f ( p)
⎣
⎦
=
h
(100)
p
p f ( p + h) − f ( p)
=
f ( p)
h
Elasticity of Demand
f ( p + h) − f ( p)
≈ f ' ( p)
h
• Since
previous ratio as
, we can write the
pf ' ( p )
E( p) = −
f ( p)
called the elasticity of demand at price p.
• We will see in section 4.1 that f ' ( p ) < 0
since f is decreasing. Because economists
would rather work with a positive value, we put
a negative sign.
Consider the demand equation
p = −0.02 x + 400
(0 ≤ x ≤ 20,000)
which describes the relationship between the
unit price in dollars and the quantity demanded
x of the Acrosonic model F loudspeaker
systems. Find the elasticity of demand E(p).
pf ' ( p )
p (− 50)
=
E( p) = −
f ( p ) − 50 p + 20,000
p
=
400 − p
• When
p = 100
1
, we have E (100) = . This result
3
tells us that when the unit price is set at $100
per speaker, an increase of 1% in the unit price
will cause a decrease of approximately 0.33% in
the quantity demanded.
• When p = 300 , we have E (300 ) = 3 . This result
tells us that when the unit price is set at $100
per speaker, an increase of 1% in the unit price
will cause a decrease of approximately 0.33% in
the quantity demanded.
Since the revenue is R ( p ) = px = pf ( p ) ,
the marginal revenue function is
R' ( p ) = f ( p ) + pf ' ( p )
⎡ pf ' ( p )⎤
= f ( p )⎢1 +
= f ( p )[1 − E ( p )]
⎥
f ( p) ⎦
⎣
1
• Since E (100 ) = < 1 , we have R ' (100 ) > 0 . If we
3
increase the unit price, the revenue increases.
• Since E (300) = 3 > 1 , we have R ' (300 ) < 0 . If we
decrease the unit price, the revenue decreases.
• If the demand is elastic at p [E(p)>1], then an
increase in the unit price will cause the revenue
to decrease, whereas a decrease in the unit
price will cause the revenue to increase.
• If the demand is inelastic at p [E(p)>1], then an
increase in the unit price will cause the revenue
to increase, whereas a decrease in the unit price
will cause the revenue to decrease.
• If the demand is unitary at p [E(p)>1], then an
increase in the unit price will cause the revenue
to stay about the same.
Consider the demand equation
p = −0.01x 2 − 0.2 x + 8 (0 ≤ x ≤ 20,000)
and the quantity demanded each week is 15. If
we increase the unit price a little bit, what will
happen to our revenue?
1.2
dx
dx
1 = −0.02 x
− 0.2 ⇒
=
dp
dp − 0.02 x
When x=15, p=2.75, f(p)=x=15, and f ' ( p ) =
(
2.75)(− 4 ) 11
pf ' ( p )
=
E( p) = −
=−
15
15
f ( p)
dx
= −4
dp