08 applications of functions (1)

7/27/2019 08 Applications of Functions (1)

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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


Competitive market

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Preliminaries


Competitive market markets with many groups of buyersand sellers so that each has relatively small influence on price

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Preliminaries



supply and demand:

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Preliminaries



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.


D d f i
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Demand function


by buyers.The quantity demanded is the amount of a product so that peopleare willing to buy it at a certain price.


D d f i


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Demand function



Demand function

The function that illustrates the relationship between price p andquantity demanded x is referred as the demand function.

Notation:

p = f(x)


D d f ti
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Demand function



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.


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.


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).


Some common demand curves: Lines
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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].


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Example: The graph of p = f(x) = 0.5x + 20


Some common demand curves: Semi parabolae
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Some common demand curves: Semi-parabolae


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].


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)


Common demand curves: semi-parabolae


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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].


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Common demand curves: semi parabolae

Example: The graph of p = f(x) = 0.01(x + 20)2 + 25


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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


Common demand curves: Semi-ellipses and semi-circles
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Common demand curves: Semi ellipses and semi circles


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].


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)


Common demand curves: Semi-hyperbolae
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S yp b


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].


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yp

Example: The graph of p = f(x) = 13

(x 70)2 100




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yp


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].


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yp

Example: The graph of p = f(x) = 22 2525 + x2


Demand curve
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Given a demand curve represented by p = f(x):

Intercepts


Demand curve
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Intercepts

x-intercept of f(x) peak of the quantity demanded if thecommodity is free.


Demand curve
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Intercepts

x-intercept of f(x) peak of the quantity demanded if thecommodity is free.

p-intercept of f(x) maximum price.


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


Illustration
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x + 10,


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.


Illustration
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x + 10,


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.


Supply function
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Supply function
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Supply represents how much of a product the market can offer.


Supply function


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The quantity supplied is the amount of a certain good theproducers (e.g. sellers) are willing to supply at a certain price.


Supply function


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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).


Supply function
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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.


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.


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).


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.


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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,+)


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Example: Graph of p = g(x) = 34

x + 9


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x + 9


x 6


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,+).


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Example: Graph of p = g(x) = 0.008x2

+ 10


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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,+).


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Example: Graph of p = g(x) = 3x 5


Some common supply curves: Semi-hyperbolae
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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,+).


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Example: Graph of p = g(x) =11

15x2

225


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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,+).


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Example: Graph of p = g(x) =10

17x2

+ 289


Supply Curve
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Given a supply curve represented by p = g(x):

Intercepts


Supply Curve
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Intercepts

p-intercept of g(x) the lowest price at which thecommodity would be supplied


Supply Curve
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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).


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


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.


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


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.


Market equilibrium
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Remark

MEP exists at point (xE, pE) whenever f(x) = g(x), andconversely.


Market equilibrium
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Remark


xE

is called market equilibrium quantity.


Market equilibrium
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Remark


xE

is called market equilibrium quantity.pE is called market equilibrium price.


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.


Illustration 1

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in units.Graph.


Illustration 1

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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.


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.


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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


To get pE, we have

pE = g(xE) = g(4) = 0.25(4)2 = 0.25(16) = 4.


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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


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.


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.


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.


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.


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.


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Definition


The surplus of a commodity occurs when g(x) > f(x) forp pE.


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Definition


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).


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Definition



S+ = g(x) f(x).

The shortage or scarcity of a commodity occurs whenf(x) > g(x) for 0 p pE.


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Definition



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).


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.


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.


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.


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.

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08 applications of functions (1)

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