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					Price Elasticity of Demand
        Overheads
How much would your roommate pay
   to watch a live fight?


How does Showtime decide how
much to charge for a live fight?
What about Hank and Son’s Concrete?


How much should they charge per square foot?

Can ISU raise parking revenue by raising parking fees?


Or will the increase in price drive demand
down so far that revenue falls?
All of these pricing issues revolve around the issue
of how responsive the quantity demanded is to price.



Elasticity is a measure of how responsive
one variable is to changes in another variable?
       The Law of Demand
The law of demand states that when
the price of a good rises,
and everything else remains the same,
the quantity of the good demanded will fall.


The real issue is how far it will fall.
The demand function is given by
       Q   D
                h(P, ZD )
       QD = quantity demanded
       P = price of the good
       ZD = other factors that affect demand
The inverse demand function is given by
               
        P h   1       D
                   (Q , ZD )

        P  g(Q , ZD )
                   D



To obtain the inverse demand function we
just solve the demand function for P
as a function of Q
Examples
QD = 20 - 2P
2P + QD = 20

2P = 20 - QD

P = 10 - 1/2 QD
Slope = - 1/2
Examples
QD = 60 - 3P
3P + QD = 60

3P = 60 - QD

P = 20 - 1/3 QD
Slope = - 1/3
One measure of responsiveness is slope
  For demand

           Q   D
                    h(P, ZD )

The slope of a demand curve is given by the
change in Q divided by the change in P

                ΔQ       D
         slope
                 ΔP
For inverse demand

          P  g(Q , ZD )
                       D



The slope of an inverse demand curve is given by
the change in P divided by the change in Q

                 ΔP
         slope
                ΔQ D
Examples
QD = 60 - 3P

Slope = - 3

P = 20 - 1/3 QD
Slope = - 1/3
Examples
QD = 20 - 2P

Slope = - 2

P = 10 - 1/2 QD
Slope = - 1/2
We can also find slope from tabular data
         Q    P
         0    10
  Q    2    9     P
         4    8
         6    7
         8    6
         10   5

  slope ΔQ D    
                 2   2
          ΔP    1
Demand for Handballs
                       Q    P
                       0    10
                       1    9.5
                       2    9
                       3    8.5
                       4    8
                       5    7.5
                       6    7
                       7    6.5
                       8    6
                       9    5.5
                       10   5
                       11   4.5
                       12   4
                       13   3.5
                       14   3
                       15   2.5
                       16   2
                       17   1.5
                       18   1
                       19   0.5
                       20   0
Q    P
0    10                         Demand for Handballs
1    9.5
2    9
                   11
3    8.5


           Price
4    8             10
5    7.5            9
6    7
7    6.5            8
8    6              7
9    5.5
                    6
10   5
                    5
11
12
     4.5
     4
                                                                    P
                    4
13   3.5
14   3              3
15   2.5            2
16   2
                    1
17   1.5
18   1              0
19   0.5                0   2    4   6   8   10 12 14 16 18 20 22
20   0
                                                         Quantity
       Q          P
       0          10                          Demand for Handballs
       1          9.5

Q           
       2          9
       3 P
                                11
                  8.5




                        Price
       4          8             10

                                                   Q = 2 - 4 = -2
       5          7.5            9
       6          7
       7          6.5            8

                                                           P = 9 - 8 = 1
       8          6              7
       9          5.5
                                 6
       10         5
       11         4.5            5
       12         4              4
       13         3.5
       14         3              3
                                         slope     ΔP  1
       15         2.5
       16         2
                                 2
                                                   ΔQ D   2
                                 1
       17         1.5
       18         1              0
       19         0.5                0    2    4   6   8    10 12 14 16 18 20 22
       20         0
                                                                        Quantity
Problems with slope as a measure of responsiveness

 Slope depends on the units of measurement

 The same slope can be associated with
 very different percentage changes
Examples
QD = 200 - 2P
2P + QD = 200
2P = 200 - QD
P = 100 - 1/2 QD
slope  ΔP  1
       ΔQ D   2
                                      Q    P
Consider data on racquets             0    100
                                      1    99.5
Let P change from 95 to 96            2    99
                                      3    98.5
                                      4    98
 P = 96 - 95 = 1                     5    97.5
                                      6    97
 Q = 8 - 10 = -2                     7    96.5

                             Q   
                                      8
                                      9
                                      10
                                           96
                                           95.5
                                           95
                                                   P
                                      11   94.5
                                      12   94
                                      13   93.5
                                      14   93
A $1.00 price change when P = $95.00 is tiny
   Graphically for racquets
                                 Demand for Racquets
                   102

           Price   100
                    98
                    96

Slope = - 1/2       94
                    92
                    90
                    88
                         0   2   4   6   8   10   12   14     16   18

      Large % change in Q                                   Quantity

                                         Small % change in P
Graphically for hand balls
                                        Demand for Handballs
                           11




                   Price
                           10


P=7-6=1
                            9
                            8
                            7
                            6

 Q = 6 - 8 = -2            5
                                                                            P
                            4
                            3
Slope = - 1/2               2
                            1
                            0
                                0   2    4   6   8   10 12 14 16 18 20 22
                                                                 Quantity


 Large % change in Q                             Large % change in P
So slope is not such a good measure
of responsiveness

Instead of slope we use percentage changes

The ratio of the percentage change in one variable
to the percentage change in another variable
is called elasticity
The Own Price Elasticity of Demand
is given by         ΔQ   D

                 D           D
      %ΔQ                 Q
 εD      
      %ΔP                 ΔP
                           P
 There are a number of ways to compute
           percentage changes
Initial point method for computing
The Own Price Elasticity of Demand
Price Elasticity of Demand
   (Initial Point Method)             P     Q

                                                 Q
                                      6     8
              ΔQ D           P       5.5   9
                                      5     10
      %ΔQ D    QD
 εD                                 4.5   11
      %ΔP      ΔP                     4     12
               P
       (8 10)
  εD 
           8
                     (8  10)    6
        (6 5)           8     (6  5)
           6
          2      6    
                       12
                          1.5
          8       1   8
Final point method for computing
The Own Price Elasticity of Demand
Price Elasticity of Demand
   (Final Point Method)               P     Q

                                                 Q
                                      6     8
             ΔQ D            P       5.5   9
                                      5     10
     %ΔQ D    QD
εD                                  4.5   11
     %ΔP      ΔP                      4     12
              P
       (8    10)
  εD 
           10
                      (8  10)    5
        (6   5)          10    (6  5)
            5
          2        5    
                         10
                            1.0
          10        1   10
The answer is very different
depending on the choice of the
base point

So we usually use
The midpoint method for computing
The Own Price Elasticity of Demand
Elasticity of Demand Using the Mid-Point
                  ΔQ D
     εD  %ΔQ D  Q D
          %ΔP      ΔP
                    P
    ΔQ D  Q1  0 or Q0  1
               Q         Q
     For QD we use the midpoint of the Q’s
                    1
           Q   D
                    (Q1  Q0 )
                    2
Similarly for prices

  ΔP  P1  0 or P0  1
           P         P

  For P we use the midpoint of the P’s

         1
      P  ( P1  P0 )
         2
           ΔQ D
        D     D
εD  %ΔQ  Q
     %ΔP    ΔP
             P
      (Q1  0 )
           Q
      1                   (Q1  0 )
                               Q
        ( Q1  Q0 )
      2                   (Q1  Q0 )
ε                    
       (P1  0 )
              P           (P1  0 )
                               P
      1
        ( P1  P0 )       ( P1  P0 )
      2
Price Elasticity of Demand
  (Mid-Point Method)                Q     P
              ΔQ D                  8     6
                                    9     5.5
      %ΔQ D   QD                    10    5
 εD                               11    4.5
      %ΔP      ΔP
                                    12    4
               P
       (Q1 Q0 )
       (Q1  Q0 )        (Q1  0 ) (P1 P0 )
                              Q
                    
       (P1 P0 )         (P1  0 ) (Q1 Q0 )
                              P
       (P1  P0 )      (8  10) (6  5)
                     
                        (6  5) (8  10)
                         2)
                       ( (11)      22    11
                                       
                        (1) (18)    18     9
Classification of the elasticity of demand
   Inelastic demand
   When the numerical value of the elasticity of demand
   is between 0 and -1.0, we say that demand is inelastic.

          %ΔQ D
         
          %ΔP 
                <
                1
               

           %ΔQ D          <     %ΔP
Classification of the elasticity of demand
   Elastic demand
   When the numerical value of the elasticity of demand
   is less than -1.0, we say that demand is elastic.

           %ΔQ D
          
           %ΔP 
                 >
                 1
                

           %ΔQ D          >     %ΔP
Classification of the elasticity of demand
 Unitary elastic demand
 When the numerical value of the elasticity of demand
 is equal to -1.0, we say that demand is unitary elastic.


           %ΔQ D
                
                 1
           %ΔP 
                
           %ΔQ D               %ΔP
Classification of the elasticity of demand

Perfectly elastic - D = -     horizontal

Perfectly inelastic - D = 0      vertical
Elasticity of demand with linear demand
 Consider a linear inverse demand function
      P A  D
            BQ

 The slope is (-B) for all values of P and Q
 For example,
     P  12 0.5Q D
 The slope is -0.5 = - 1/2
                                                               P      Q
                     Demand for Diskettes                      12     0
                                                               11.5   1
        13                                                     11     2
Price




        12
        11                                                     10.5   3
        10                                                     10     4
         9                                                     9.5    5
         8
         7                                                     9      6
         6                                                     8.5    7
         5

                                                        P             Q
         4                                                     8      8
         3                                                     7.5    9
         2
         1                                                     7      10
         0                                                     6.5    11
             0   2   4   6   8   10 12 14 16 18 20 22          6      12
                                             Quantity          5.5    13

                                                        P                 Q
                                                               5      14
                 ΔQ D                                          4.5    15
                       2.0                                   4      16
                  ΔP                                           3.5    17
                                                               3      18
The slope is constant but the       P
                                    12
                                           Q
                                           0
elasticity of demand will vary      11.5
                                    11
                                           1
                                           2
               ΔQ D                 10.5   3
                                    10     4
       %ΔQ D   QD
  εD                              9.5    5
       %ΔP      ΔP                  9      6
                                    8.5    7
                P
                             P                 Q
                                    8      8

        ΔQ (P1  P0 )              7.5    9
                                    7      10
         ΔP (Q1  Q0 )              6.5    11
                                    6      12
     (8 10) (8  7)               5.5    13
       (8 7) (8  10)              5      14
                                    4.5    15
                                    4      16
        2)
      ( (15)     30   5
                                 3.5    17
       (1) (18)   18    3           3      18
The slope is constant but the       P
                                    12
                                           Q
                                           0
elasticity of demand will vary      11.5
                                    11
                                           1
                                           2
               ΔQ D                 10.5   3
                                    10     4
       %ΔQ D   QD                   9.5    5
  εD                              9      6
       %ΔP      ΔP                  8.5    7
                P                   8      8

      ΔQ (P1  P0 )                7.5    9
                                    7      10
       ΔP (Q1  Q0 )                6.5    11
                                    6      12
     (14 16) (5  4)              5.5    13
       (5  ) (14  16)
                             P 
           4
                                                 Q
                                    5      14
                                    4.5    15
                                    4      16
        2)
      ( (9 )     18   3
                                 3.5    17
       (1) (30)   30    5           3      18
  The slope is constant but the
  elasticity of demand will vary

A linear demand curve becomes more inelastic
as we lower price and increase quantity
                                   P smaller

εD  %ΔQ D  ΔQ (P1  P0 )
     %ΔP     ΔP (Q1  Q0 )             Q larger


  The elasticity gets closer to zero
The slope is constant but the
elasticity of demand will vary
Q    P    Elasticity   Expenditure
0    12          0
2    11   -23.0000     22
4    10   -7.0000      40
6    9    -3.8000      54
8    8    -2.4286      64
10   7    -1.6667      70
12   6    -1.1818      72
14   5    -0.8462      70
16   4    -0.6000      64
18   3    -0.4118      54
20   2    -0.2632      40
22   1    -0.1429      22
24   0    -0.0435      0
The slope is constant but the
elasticity of demand will vary
Q    P    Elasticity   Expenditure
0    12          0
2    11   -23.0000     22
4    10   -7.0000      40
6    9    -3.8000      54
8    8    -2.4286      64
10   7    -1.6667      70
12   6    -1.1818      72
14   5    -0.8462      70
16   4    -0.6000      64
18   3    -0.4118      54
20   2    -0.2632      40
22   1    -0.1429      22
24   0    -0.0435      0
 Note
We do not say that demand is elastic
or inelastic …..

We say that demand is elastic or
inelastic at a given point
Example
              ΔQ    D

          D       D
      %ΔQ      Q
εD         
      %ΔP      ΔP
                P
      ΔQ (P1  P0 )
    
      ΔP (Q1  Q0 )

          Constant with linear demand
The Own Price Elasticity of Demand
and Total Expenditure on an Item

How do changes in an items price affect
expenditure on the item?

If I lower the price of a product, will the increased
sales make up for the lower price per unit?
  Expenditure for the consumer
  is equal to revenue for the firm

Revenue = R = price x quantity = PQ

Expenditure = E = price x quantity = PQ
Modeling changes in price and quantity

 P = change in price

 Q = change in quantity

   The Law of Demand says that
   as P increases Q will decrease

    P         Q
So
P = initial price
P = change in price
P + P = final price

Q = initial quantity
Q = change in quantity

Q + Q = final quantity
So
Initial Revenue = PQ
P + P = final price
Q + Q = final quantity
Final Revenue = (P + P) (Q + Q)

              = P Q + P Q + P Q + P Q
Now find the change in revenue
 R = final revenue - initial revenue
     = P Q + P Q + P Q + P Q - P Q
     = P Q + P Q + P Q

 %R = R / R = R / P Q
   ΔR   ΔP Q  P ΔQ  ΔP ΔQ
      
   PQ            PQ
We can rewrite this expression as follows

   ΔR   ΔP Q   P ΔQ   ΔP ΔQ
                  
   PQ    PQ     PQ      PQ
        ΔP    ΔQ    ΔP ΔQ
  %ΔR           
         P     Q     PQ
  %ΔR %ΔP %ΔQ
Classification of the elasticity of demand
  Inelastic demand
        D
  %ΔQ
 
  %ΔP 
             <
            1     %ΔQ D       <   %ΔP
           
            +      -
 %ΔR    %ΔP %ΔQ
 % Q and % P are of opposite sign so
 %R has the same sign as %P
Classification of the elasticity of demand
   Inelastic demand
         D
  %ΔQ
 
  %ΔP 
         <
        1   %ΔQ D             <   %ΔP
       
       -    +
 %ΔR %ΔP %ΔQ
 % Q and % P are of opposite sign so
 %R has the same sign as %P
   Lower price  lower revenue
   Higher price  higher revenue
Classification of the elasticity of demand
  Elastic demand
  %ΔQ D
 
  %ΔP 
            >
           1    %ΔQ D         >   %ΔP
          
            +    -
 %ΔR     %ΔP %ΔQ
 % Q and % P are of opposite sign so
 %R has the opposite sign as %P
  Higher price  lower revenue
   Lower price  higher revenue
Classification of the elasticity of demand
  Unitary elastic demand
  %ΔQ D
           
           1        %ΔQ D         %ΔP
  %ΔP 
          
            +        -
 %ΔR    %ΔP %ΔQ
 % Q and % P are of opposite sign so their
 effects will cancel out and %R = 0.
       Tabular data                     Elastic
            Q    P    Elasticity   Revenue
Price falls 0    12          0
            2    11   -23.0000     22
            4    10   -7.0000      40
            6    9    -3.8000      54
            8    8    -2.4286      64
            10   7    -1.6667      70   Inelastic
            12   6    -1.1818      72
Price falls 14   5    -0.8462      70
            16   4    -0.6000      64
            18   3    -0.4118      54
            20   2    -0.2632      40     Revenue falls
            22   1    -0.1429      22
            24   0    -0.0435      0
Graphical analysis
                                                          Q   P    Elasticity Revenue
                 Demand for Diskettes
                                                    0         12             0
                                             Demand 2         11    -23.0000 22
        13                                          4         10    -7.0000 40
Price




        12                                   P0, Q0 6         9     -3.8000 54
        11                                          8         8     -2.4286 64
        10                                   P1, Q1 10        7     -1.6667 70
         9       B                                  12        6     -1.1818 72
         8
         7                                          14        5     -0.8462 70
         6                                          16        4     -0.6000 64
         5
                                                    18        3     -0.4118 54
         4
         3
                 C         A                        20        2     -0.2632 40
         2                                          22        1     -0.1429 22
         1                                          24        0     -0.0435 0
         0
             0    2    4   6   8   10 12 14 16 18 20 22
                                               Quantity
                      Lose B, gain A, revenue rises
  Graphical analysis
                                                          Q   P    Elasticity Revenue
                 Demand for Diskettes
                                                    0         12             0
                                             Demand 2         11    -23.0000 22
        13                                          4         10    -7.0000 40
Price




        12                                   P0, Q0 6         9     -3.8000 54
        11                                          8         8     -2.4286 64
        10                                   P1, Q1 10
         9                                                    7     -1.6667 70
         8                                          12        6     -1.1818 72
         7                                          14        5     -0.8462 70
         6                                          16        4     -0.6000 64
         5
                                                    18        3     -0.4118 54
         4
         3
         2
                           A                        20
                                                    22
                                                              2
                                                              1
                                                                    -0.2632 40
                                                                    -0.1429 22
         1                                    B     24        0     -0.0435 0
         0
             0    2    4   6   8   10 12 14 16 18 20 22
                                               Quantity
                      Lose A, gain B, revenue falls
Factors affecting the elasticity of demand

 Availability of substitutes


 Importance of item in the buyer’s budget
Availability of substitutes
  The easier it is to substitute for a good,
   the more elastic the demand

   With many substitutes, individuals will
   move away from a good whose price increases
Examples of goods with “easy “substitution
   Gasoline at different stores
   Soft drinks
   Detergent
   Airline tickets

   Local telephone service
Narrow definition of product
  The more narrowly we define an item,
  the more elastic the demand

  With a narrow definition, there will lots of
  substitutes
Examples of narrowly defined goods
   Lemon-lime drinks
   Corn at a specific farmer’s market
   Vanilla ice cream
   Food

   Transportation
“Necessities” tend to have inelastic demand



   Necessities tend to have few substitutes
Examples of necessities
   Salt
   Insulin
   Food
   Trips to Hawaii

   Sailboats
Demand is more elastic in the long-run


There is more time to adjust in the long run
Examples of short and long run elasticity
   Postal rates

   Gasoline

   Sweeteners
Factors affecting the elasticity of demand
Importance of item in the buyer’s budget

   The more of their total budget consumers
   spend on an item,
   the more elastic the demand for the good

   The elasticity is larger because the item has
   a large budget impact
“Big ticket” items and elasticity
   Housing
   Big summer vacations
   Table salt
   College tuition
The End

				
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