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Applications of Functions inBusiness & Economics
Raymond Lapus
Mathematics Department
De La Salle University, Manila
10 February 2012
R. Lapus Applications of Functions in Business & Economics
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Preliminaries
Market
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Preliminaries
Market a group of buyers and sellers of a particular productor service
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Preliminaries
Market a group of buyers and sellers of a particular productor service
Competitive market
R. Lapus Applications of Functions in Business & Economics
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Preliminaries
Market a group of buyers and sellers of a particular productor service
Competitive market markets with many groups of buyersand sellers so that each has relatively small influence on price
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Preliminaries
Market a group of buyers and sellers of a particular productor service
Competitive market markets with many groups of buyersand sellers so that each has relatively small influence on price
supply and demand:
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Preliminaries
Market a group of buyers and sellers of a particular productor service
Competitive market markets with many groups of buyersand sellers so that each has relatively small influence on price
supply and demand: most useful model for competitive market
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Demand function
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Demand function
Demand refers to how much (quantity) a product/service is desired
by buyers.
R. Lapus Applications of Functions in Business & Economics
D d f i
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Demand function
Demand refers to how much (quantity) a product/service is desired
by buyers.The quantity demanded is the amount of a product so that peopleare willing to buy it at a certain price.
R. Lapus Applications of Functions in Business & Economics
D d f i
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Demand function
Demand refers to how much (quantity) a product/service is desired
by buyers.The quantity demanded is the amount of a product so that peopleare willing to buy it at a certain price.
Demand function
The function that illustrates the relationship between price p andquantity demanded x is referred as the demand function.
Notation:
p = f(x)
R. Lapus Applications of Functions in Business & Economics
D d f ti
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Demand function
Demand refers to how much (quantity) a product/service is desired
by buyers.The quantity demanded is the amount of a product so that peopleare willing to buy it at a certain price.
Demand function
The function that illustrates the relationship between price p andquantity demanded x is referred as the demand function.
Notation:
p = f(x)
The demand curve is the graph of a given demand function on thefirst quadrant of the Cartesian plane.
R. Lapus Applications of Functions in Business & Economics
L f D d
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Law of Demand
DefinitionIf all other factors remain equal,the higher the price of a good orservice, the lower the consumerdemand for the good or service,
and vice versa.
R. Lapus Applications of Functions in Business & Economics
L f D d
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Law of Demand
DefinitionIf all other factors remain equal,the higher the price of a good orservice, the lower the consumerdemand for the good or service,
and vice versa.
Consequence. The demand curve p = f(x) is monotonedecreasing. That is for any positive real numbers x1 and x2:
If x1 x2, then f(x1) f(x2).
R. Lapus Applications of Functions in Business & Economics
Some common demand curves: Lines
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Some common demand curves: Lines
The following curves can be a considered as demand function.
Type D1: Lines represented by
p = f(x) = mx + b
whose slope m 0 and p-intercept b> 0. Moreover, its(restricted) domain and range are:
Dom(f) = [0,
b/m]
Rng(f) = [0, b].
R. Lapus Applications of Functions in Business & Economics
Some common demand curves: Lines
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Some common demand curves: Lines
Example: The graph of p = f(x) = 0.5x + 20
R. Lapus Applications of Functions in Business & Economics
Some common demand curves: Semi parabolae
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Some common demand curves: Semi-parabolae
The following curves can be a considered as demand function.
Type D2A: Semi-parabolae opening to the left and whosevertex lies at V(h, 0) with h > 0. Its form is
p =
a(x h),where a < 0 and h > 0. Moreover, its (restricted) domain andrange are:
Dom(f) = [0, h]Rng(f) = [0,
ah].
R. Lapus Applications of Functions in Business & Economics
Some common demand curves: Semi parabolae
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Some common demand curves: Semi-parabolae
Example: The graph of p = f(x) =8(x 50)
R. Lapus Applications of Functions in Business & Economics
Common demand curves: semi-parabolae
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Common demand curves: semi-parabolae
The following curves can be a considered as demand function.
Type D2B: Semi-parabolae opening downwards and whosevertex lies at V(h, k). Its form is
p = a(x h)2
+ k,
such that a < 0, k> 0 andk/a < h 0. Moreover, its
(restricted) domain and range are:
Dom(f) = [0, h +k/a]
Rng(f) = [0, ah2 + k].
R. Lapus Applications of Functions in Business & Economics
Common demand curves: semi-parabolae
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Common demand curves: semi parabolae
Example: The graph of p = f(x) = 0.01(x + 20)2 + 25
R. Lapus Applications of Functions in Business & Economics
Common demand curves: semi-parabolae
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Common demand curves: semi parabolae
Example: The graph of p = f(x) = 0.01(x + 20)2 + 25
Example: The graph of p = f(x) = 0.008x2 + 20
R. Lapus Applications of Functions in Business & Economics
Common demand curves: Semi-ellipses and semi-circles
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Common demand curves: Semi ellipses and semi circles
The following curves can be a considered as demand function.
Type D3: The right half of a semi-ellipse centered at C(0, 0)of the form:
p =b
aa2
x2,
where a > 0 and b> 0. Ifa = b, the resulting curve is asemi-circle. Moreover, its (restricted) domain and range are:
Dom(f) = [0, a]
Rng(f) = [0, b].
R. Lapus Applications of Functions in Business & Economics
Common demand curves: Semi-ellipses and semi-circles
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Common demand curves: Semi ellipses and semi circles
ExamplesThe graph of the following demand functions:
1 p = f(x) = 2520
400 x2 (left)
2 p = f(x) =
625 x2 (middle) and3 p = f(x) =
2530900 x2 (right)
R. Lapus Applications of Functions in Business & Economics
Common demand curves: Semi-hyperbolae
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S yp b
The following curves can be a considered as demand function.
Type D4A: The upper part of the left branch of asemi-hyperbolae centered at C(h, 0). Its form is given by
p=
b
a
(x
h)
2
a2
,
where h > a, a > 0 and b> 0. Moreover, its (restricted) domainand range are:
Dom(f) = [0, h a]Rng(f) = [0,
b
a
h2 a2].
R. Lapus Applications of Functions in Business & Economics
Common demand curves: Semi-hyperbolae
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yp
Example: The graph of p = f(x) = 13
(x 70)2 100
R. Lapus Applications of Functions in Business & Economics
Common demand curves: Semi-hyperbolae
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yp
The following curves can be a considered as demand function.
Type D4B: The left part of the lower branch of asemi-hyperbolae centered at C(0, k). Its form is given by
p = k
a
bb2 + x2,
where k> a, a > 0 and b> 0. Moreover, its (restricted) domainand range are:
Dom(f) = [0,b
a
a2 k2]Rng(f) = [0, k a].
R. Lapus Applications of Functions in Business & Economics
Common demand curves: Semi-hyperbolae
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yp
Example: The graph of p = f(x) = 22 2525 + x2
R. Lapus Applications of Functions in Business & Economics
Demand curve
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Given a demand curve represented by p = f(x):
Intercepts
R. Lapus Applications of Functions in Business & Economics
Demand curve
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Given a demand curve represented by p = f(x):
Intercepts
x-intercept of f(x) peak of the quantity demanded if thecommodity is free.
R. Lapus Applications of Functions in Business & Economics
Demand curve
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Given a demand curve represented by p = f(x):
Intercepts
x-intercept of f(x) peak of the quantity demanded if thecommodity is free.
p-intercept of f(x) maximum price.
R. Lapus Applications of Functions in Business & Economics
Illustration
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Consider the demand function of an apple x: f(x) = 53
x + 10,
where x is in dozens of boxes and p = f(x) is expressed in
PhP 1,000s.
0
1
2
3
4
5
6
7
8
9
10p
0 1 2 3 4 5 6x
p = 5
3x + 10
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Illustration
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Consider the demand function of an apple x: f(x) = 53
x + 10,
where x is in dozens of boxes and p = f(x) is expressed in
PhP 1,000s.
0
1
2
3
4
5
6
7
8
9
10p
0 1 2 3 4 5 6x
p = 5
3x + 10
The x-intercept of f(x) is at(6, 0).Interpretation: If the price
of an apple is free, themaximum quantitydemanded is six dozensboxes of apples.
R. Lapus Applications of Functions in Business & Economics
Illustration
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Consider the demand function of an apple x: f(x) = 53
x + 10,
where x is in dozens of boxes and p = f(x) is expressed in
PhP 1,000s.
0
1
2
3
4
5
6
7
8
9
10p
0 1 2 3 4 5 6x
p = 5
3x + 10
The x-intercept of f(x) is at(6, 0).Interpretation: If the price
of an apple is free, themaximum quantitydemanded is six dozensboxes of apples.
The p-intercept of f(x) is at(0, 10).Interpretation: Themaximum price of a dozenbox of apples is PhP 10,000.
R. Lapus Applications of Functions in Business & Economics
Supply function
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R. Lapus Applications of Functions in Business & Economics
Supply function
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Supply represents how much of a product the market can offer.
R. Lapus Applications of Functions in Business & Economics
Supply function
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Supply represents how much of a product the market can offer.
The quantity supplied is the amount of a certain good theproducers (e.g. sellers) are willing to supply at a certain price.
R. Lapus Applications of Functions in Business & Economics
Supply function
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Supply represents how much of a product the market can offer.
The quantity supplied is the amount of a certain good theproducers (e.g. sellers) are willing to supply at a certain price.
Supply function
The function that illustrates the relationship between price p andhow much of supplied good/service x to the market is referred asthe supply function.Notation:
p = g(x).
R. Lapus Applications of Functions in Business & Economics
Supply function
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Supply represents how much of a product the market can offer.
The quantity supplied is the amount of a certain good theproducers (e.g. sellers) are willing to supply at a certain price.
Supply function
The function that illustrates the relationship between price p andhow much of supplied good/service x to the market is referred asthe supply function.Notation:
p = g(x).
The supply curve is the graph of a given supply function on thefirst quadrant of the Cartesian plane.
R. Lapus Applications of Functions in Business & Economics
Law of Supply
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Definition
If all other factors being equal,the higher the price of a good orservice, the higher the quantityof goods or services offered by
suppliers, and vice versa.
R. Lapus Applications of Functions in Business & Economics
Law of Supply
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Definition
If all other factors being equal,the higher the price of a good orservice, the higher the quantityof goods or services offered by
suppliers, and vice versa.
Consequence: The supply curve is monotone increasing. That isfor any positive real numbers x1 and x2:
If x1 x2, then g(x1) g(x2).
R. Lapus Applications of Functions in Business & Economics
Some common supply curves: Lines
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The following curves can be a considered as supply function.
Type S1: Lines represented by
p = g(x) = mx + b
with slope m 0 and p-intercept b R.
R. Lapus Applications of Functions in Business & Economics
Some common supply curves: Lines
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Domain and range of a linear supply function
b> 0 b< 0
Dom(g) = [0,+) Dom(g) = [b/m,+)Rng(g) = [b,+) Rng(g) = [0,+)
R. Lapus Applications of Functions in Business & Economics
Some common supply curves: Lines
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Example: Graph of p = g(x) = 34
x + 9
R. Lapus Applications of Functions in Business & Economics
Some common supply curves: Lines
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Example: Graph of p = g(x) = 34
x + 9
Example: Graph of p = g(x) = 34
x 6
R. Lapus Applications of Functions in Business & Economics
Some common supply curves: Semi-parabolae
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The following curves can be a considered as supply function.
Type S2A: Semi-parabolae opening upwards whose vertexlocated at V(0, k) and k 0. Its form is
p = ax2
+ k,
where a > 0. Moreover, its (restricted) domain and range are:
Dom(g) = [0,+)Rng(g) = [k,+).
R. Lapus Applications of Functions in Business & Economics
Some common supply curves: Semi-parabolae
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Example: Graph of p = g(x) = 0.008x2
+ 10
R. Lapus Applications of Functions in Business & Economics
Some common supply curves: Semi-parabolae
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The following curves can be a considered as demand function.
Type S2B: Semi-parabolae opening to the right whose vertexlocated at V(h, 0) and k 0. Its form is
p = ax h,where a > 0. Moreover, its (restricted) domain and range are:
Dom(g) = [h,+)Rng(g) = [0,+).
R. Lapus Applications of Functions in Business & Economics
Some common supply curves: Semi-parabolae
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Example: Graph of p = g(x) = 3x 5
R. Lapus Applications of Functions in Business & Economics
Some common supply curves: Semi-hyperbolae
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The following curves can be a considered as demand function.
Type S3A: The upper section of the left branch of asemi-hyperbolae centered at C(0, 0) and whose vertex is atV(a, 0). This is defined by
p = ba
x2 a2,
where a > 0, b> 0. Moreover, its (restricted) domain and rangeare:
Dom(g) = [a,+)Rng(g) = [0,+).
R. Lapus Applications of Functions in Business & Economics
Some common supply curves: Semi-hyperbolae
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Example: Graph of p = g(x) =11
15x2
225
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Some common supply curves: Semi-hyperbolae
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The following curves can be a considered as demand function.
Type S3B: The left section of the upper branch of asemi-hyperbolae centered at C(0, 0) and whose vertex is atV(0, a). This is defined by
p =a
b
b2 + x2,
where a > 0 and b> 0. Moreover, its (restricted) domain andrange are:
Dom(g) = [0,+)Rng(g) = [a,+).
R. Lapus Applications of Functions in Business & Economics
Some common supply curves: Semi-hyperbolae
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Example: Graph of p = g(x) =10
17x2
+ 289
R. Lapus Applications of Functions in Business & Economics
Supply Curve
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Given a supply curve represented by p = g(x):
Intercepts
R. Lapus Applications of Functions in Business & Economics
Supply Curve
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Given a supply curve represented by p = g(x):
Intercepts
p-intercept of g(x) the lowest price at which thecommodity would be supplied
R. Lapus Applications of Functions in Business & Economics
Supply Curve
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Given a supply curve represented by p = g(x):
Intercepts
p-intercept of g(x) the lowest price at which thecommodity would be suppliedx-intercept of g(x) either the supplier produces theproduct for free or is unwilling to produce anything. Thelatter case is the point (0, 0).
R. Lapus Applications of Functions in Business & Economics
Supply curve: Illustration
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The supply function of an apple is defined by g(x) = x + 2, where
x is in dozens of boxes and p = g(x) is in PhP 10,000s.
0
1
2
3
4
5
6
7
8p
0 1 2 3 4 5 6x
p = x + 2
R. Lapus Applications of Functions in Business & Economics
Supply curve: Illustration
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The supply function of an apple is defined by g(x) = x + 2, where
x is in dozens of boxes and p = g(x) is in PhP 10,000s.
0
1
2
3
4
5
6
7
8p
0 1 2 3 4 5 6x
p = x + 2
The p-intercept of g(x) is at
(0, 2).Interpretation: PhP 20,000is the lowest price at whicha dozen box of apples will besupplied.
R. Lapus Applications of Functions in Business & Economics
Exercise
In this context let x be the quantity demanded/supplied and p be
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In this context, let x be the quantity demanded/supplied and p bethe price in hundred pesos. Perform the following tasks to thegiven equations.
Determine whether the following equations define the demandor supply for particular commodities by expressing p interms of x.
Compute and interpret the intercepts and sketch its graph
(Quadrant I only). (If you want see the graph viaGraphmatica, kindly replace use the variable y instead of p.)
1 p2 x2 = 25
2
x2
121 +p2
225 = 1
3 p+ 115
x 22 = 04 p = 0.1(x + 5)2 + 17.5
5 p 2.5x 1.5 = 06
p = (x + 2)2
7 (p+ 3)2 8x = 08
(p 1)225
x2
36= 1
R. Lapus Applications of Functions in Business & Economics
Market equilibrium
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Market equilibrium point (MEP)
The market equilibrium point exists in a market such that thequantity demanded by the consumers is equal to the quantitysupplied by the producers.
R. Lapus Applications of Functions in Business & Economics
Market equilibrium
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Market equilibrium point (MEP)
The market equilibrium point exists in a market such that thequantity demanded by the consumers is equal to the quantitysupplied by the producers.
Remark
MEP exists at point (xE, pE) whenever f(x) = g(x), andconversely.
R. Lapus Applications of Functions in Business & Economics
Market equilibrium
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Market equilibrium point (MEP)
The market equilibrium point exists in a market such that thequantity demanded by the consumers is equal to the quantitysupplied by the producers.
Remark
MEP exists at point (xE, pE) whenever f(x) = g(x), andconversely.
xE
is called market equilibrium quantity.
R. Lapus Applications of Functions in Business & Economics
Market equilibrium
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Market equilibrium point (MEP)
The market equilibrium point exists in a market such that thequantity demanded by the consumers is equal to the quantitysupplied by the producers.
Remark
MEP exists at point (xE, pE) whenever f(x) = g(x), andconversely.
xE
is called market equilibrium quantity.pE is called market equilibrium price.
R. Lapus Applications of Functions in Business & Economics
Illustration 1
Consider the demand function p = f (x) = 20 4x and the supply
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Consider the demand function p = f(x) = 20 4x and the supplyfunction p = g(x) = 0.25x2, where p is price in PhP 100s and x is
in units.
R. Lapus Applications of Functions in Business & Economics
Illustration 1
Consider the demand function p = f (x) = 20 4x and the supply
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Consider the demand function p = f(x) = 20 4x and the supplyfunction p = g(x) = 0.25x2, where p is price in PhP 100s and x is
in units.Graph.
R. Lapus Applications of Functions in Business & Economics
Illustration 1
Consider the demand function p = f (x) = 20 4x and the supply
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Consider the demand function p = f(x) = 20 4x and the supplyfunction p = g(x) = 0.25x2, where p is price in PhP 100s and x is
in units.Graph.
The MEP is obtained by finding xE in the equation f(x) = g(x)and then looking for pE by evaluating either f or g at xE.
R. Lapus Applications of Functions in Business & Economics
Illustration 1 (contd)Solution.
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f(x) = g(x)
20 4x = 0.25x2
0 = 0.25x2 + 4x 20
= x = 4
42 4(0.25)(20)2(0.25)
x = 4360.5
x = 20 (rejected) or x = 4 (accepted).= xE = 4.
R. Lapus Applications of Functions in Business & Economics
Illustration 1 (contd)Solution.
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f(x) = g(x)
20 4x = 0.25x2
0 = 0.25x2 + 4x 20
= x = 4
42 4(0.25)(20)2(0.25)
x = 4360.5
x = 20 (rejected) or x = 4 (accepted).= xE = 4.
To get pE, we have
pE = g(xE) = g(4) = 0.25(4)2 = 0.25(16) = 4.
R. Lapus Applications of Functions in Business & Economics
Illustration 1 (contd)Solution.
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f(x) = g(x)
20 4x = 0.25x2
0 = 0.25x2 + 4x 20
= x = 4
42 4(0.25)(20)2(0.25)
x = 4360.5
x = 20 (rejected) or x = 4 (accepted).= xE = 4.
To get pE, we have
pE = g(xE) = g(4) = 0.25(4)2 = 0.25(16) = 4.
Hence, the market equilibrium quantity is 4 units and the
market equilibrium price is PhP 400.
R. Lapus Applications of Functions in Business & Economics
Illustration 2The supply function of a commodity is p = g(x) = 0.4x + 8 andit di d d f ti i f ( ) 17 0 2 H
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its corresponding demand function is p = f(x) = 17 0.2x. Here,x is in units and p is in USD 1,000s. The MEP can be obtained
by setting f(x) = g(x) and then solving the resulting equation.That is,
f(x) = g(x)
0.4x + 8 = 17
0.2x
0.4x + 0.2x = 17 80.6x = 9
= xE = 15.
R. Lapus Applications of Functions in Business & Economics
Illustration 2The supply function of a commodity is p = g(x) = 0.4x + 8 andits s di d d f ti is f ( ) 17 0 2 H
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its corresponding demand function is p = f(x) = 17 0.2x. Here,x is in units and p is in USD 1,000s. The MEP can be obtained
by setting f(x) = g(x) and then solving the resulting equation.That is,
f(x) = g(x)
0.4x + 8 = 17
0.2x
0.4x + 0.2x = 17 80.6x = 9
= xE = 15.
Solving pE, we obtain,pE = f(15) = 0.4(15) + 8 = 6 + 8 = 14.
Hence, the MEP is 15 units of commodity and price
amounting to USD 14,000.
R. Lapus Applications of Functions in Business & Economics
Exercise
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We make the following assumptions about supply and demand.The supplier will produce 1,000 units when the selling price isPhP 20 per unit and will produce 1,500 units if the price isPhP 25 per unit.
Consumers will demand 1,500 units when the selling price isPhP 20 per unit but that the demand will decrease by 10% ifthe price increases by 5%.
Both supply and demand functions are assumed to be linear.
Calculate the MEP. Graph the supply and demand curves together
with the MEP.
R. Lapus Applications of Functions in Business & Economics
Surplus and scarcity
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Definition
Let f(x) be the demand function, g(x) be the supply function and(xE, pE) be the MEP.
R. Lapus Applications of Functions in Business & Economics
Surplus and scarcity
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Definition
Let f(x) be the demand function, g(x) be the supply function and(xE, pE) be the MEP.
The surplus of a commodity occurs when g(x) > f(x) forp pE.
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Surplus and scarcity
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Definition
Let f(x) be the demand function, g(x) be the supply function and(xE, pE) be the MEP.
The surplus of a commodity occurs when g(x) > f(x) forp pE. The amount of surplus is obtained as:
S+ = g(x) f(x).
R. Lapus Applications of Functions in Business & Economics
Surplus and scarcity
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Definition
Let f(x) be the demand function, g(x) be the supply function and(xE, pE) be the MEP.
The surplus of a commodity occurs when g(x) > f(x) forp pE. The amount of surplus is obtained as:
S+ = g(x) f(x).
The shortage or scarcity of a commodity occurs whenf(x) > g(x) for 0 p pE.
R. Lapus Applications of Functions in Business & Economics
Surplus and scarcity
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Definition
Let f(x) be the demand function, g(x) be the supply function and(xE, pE) be the MEP.
The surplus of a commodity occurs when g(x) > f(x) forp pE. The amount of surplus is obtained as:
S+ = g(x) f(x).
The shortage or scarcity of a commodity occurs whenf(x) > g(x) for 0 p pE. The amount of shortage is
obtained as: S
= f(x) g(x).
R. Lapus Applications of Functions in Business & Economics
Illustration 4
Consider the demand function p = f(x) = 20 4x and the supplyf ti ( ) 0 25 2 h i i i PhP 100 d i
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function p = g(x) = 0.25x2, where p is price in PhP 100s and x isin units.
R. Lapus Applications of Functions in Business & Economics
Illustration 4
Consider the demand function p = f(x) = 20 4x and the supplyfunction p g ( ) 0 25 2 where p is price in PhP 100s and is
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function p = g(x) = 0.25x2, where p is price in PhP 100 s and x isin units.
From Illustration 1, the MEP is (4, 4). There is a surplus whenp = 4.5 since f(4.5) = 2 is smaller than g(4.5) = 5.0625. Hence
S+ = g(4.5)
f(4.5) = 3.0625.
Translating this value back to the problem, we get PhP 306.25 asthe amount of surplus.
R. Lapus Applications of Functions in Business & Economics
Illustration 4
Consider the demand function p = f(x) = 20 4x and the supplyfunction p = g (x) = 0 25x 2 where p is price in PhP 100s and x is
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function p = g(x) = 0.25x2, where p is price in PhP 100 s and x isin units.
From Illustration 1, the MEP is (4, 4). There is a surplus whenp = 4.5 since f(4.5) = 2 is smaller than g(4.5) = 5.0625. Hence
S+ = g(4.5)
f(4.5) = 3.0625.
Translating this value back to the problem, we get PhP 306.25 asthe amount of surplus.
When p = 3.5, we get f(3.5) = 6 and g(3.5) = 3.0625. Hence,
there is a shortage of
S
= g(3.5) f(3.5) = 2.9375.
This amounts to PhP 293.75.
R. Lapus Applications of Functions in Business & Economics
Exercise
1 The market equilibrium point of a certain product occurshe 2 300 its a e od ced a d sold at PhP 250 e it
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when 2,300 units are produced and sold at PhP 250 per unit.The lowest price at which the producer is willing to supply theproduct is PhP 100 while the highest price at which theconsumer will buy it is PhP 400. Find the demand and supplyequations assuming they are both linear.
2 Given the equations 30x + 4p+ p2 = 1796 and
4p+ p2
30x 4 = 0, where x represents quantity inhundreds of units and p represents the unit price in tens ofpesos.
(a) Identify which equation represents the demand curve andwhich one represents the supply curve.
(b) Interpret the intercepts of the given equations.(c) Determine the market equilibrium price and the marketequilibrium quantity.
(d) Sketch the curves on the same coordinate axes and indicatethe market equilibrium point.
R. Lapus Applications of Functions in Business & Economics
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