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					  Price elasticities
and linear regression
     A short review
             Price elasticities
• We are interested to measure the effect of
  price changes on quantity sold.
• The price elasticity can be computed as
  follows:
                    % Change in unit sales %Q Q P
 Price Elasticity                            
                     % Change in price      %P P Q

• An example:
  – Suppose we found out that if we raise price by
    5%, sales will drop by 10%
  – What is the demand elasticity with respect to
    price?
                   Solution
• Fill in the parts of the formula:
  – %ΔQ= - 0.1 (-10%)
  – %ΔP= 0.05 (5%)
• So the elasticity equals -0.1 / 0.05 = -2

• This elasticity is called point elasticity. Its
  value depends on the price-point where it
  is calculated
           Another example
• Suppose we observe the following data:
  – In period 1, sales equals 100 units, at a price
    of $ 8.00.
  – In period 2, sales equals 140 units, at a price
    of $ 6.00.
• What is the price elasticity?
                   Answer
• Compute the percentage changes:
  – Price was decreased from $8 to $6, which is
    25%
  – Sales increases from 100 to 140, which is
    40%.
• Fill this in in the formula:
  – The elasticity equals 0.4 / -0.25 = -1.6

• Next, we introduce arc-elasticity.
        Why arc-elasticity?
• Suppose price changes from $3.00 to
  $4.00
• This equals an increase of 33%.
• However, if we have a price changes from
  $4.00 to $3.00, we have a reduction of
  25%.
• Thus, there are two ways of viewing what
  happens between $3.00 and $4.00.
       Solution: arc-elasticity
• Use average price and quantity

                                Q / Q
       Price Arc - Elasticity 
                                P / P

• where the bar above the symbol means
  that we take the average of the observed
  values
     Example of arc-elasticity
• Company X has done the following
  marketing research for product Z
• When they price their product at $100, the
  demand equals 1000 units.
• If they price the product at $90, the
  demand equals 1200 units.
• What is the arc-elasticity for product Z?
                  Answer
• Use the formula:
  – The numerator equals
    (1200-1000) / ( average[1200,1000] )
  – The denominator equals
    (90-100) / ( average[90,100] )


• Combined, this yields an arc-elasticity of
  0.1818 / -0.1053 = -1.7273
     Arc versus point-elasticity
• We could also have computed the point
  elasticities for this product, for the two
  price points given
• Changing from $100 to $90, the point-
  elasticity would be -2
• Changing from $90 to $100, the point-
  elasticity would be -1.5
• As the two points get closer together, arc-
  elasticity approaches the point elasticities
    Regression

Use it to determine price
        sensitivity
                Regression
• Another way to obtain price sensitivity
  estimates is through regression
• Regression is a mathematical technique
  that we will use to relate sales to price
  levels
• This enables:
  – predictions of future sales levels
  – price optimization
   Simple linear demand model
                                         Input data for regression:
• A standard place to begin is:          p = price
                                         q = quantity
     qi = a + b pi + ei                  Results from the estimation:

• Elasticity is a function of b:         a = intercept
                                         b = slope

                  q p    p
   Elasticity         b
                  p q    q

• We can also use the model to predict profits
     Profit = q (p - c)
• Using calculus we can find the optimal price:
      profit                 bc  a  c a
              0         p 
                          *
                                       
        p                      2b     2 2b
                                            REMEMBER:
                                            Input data for regression:
                                            p = price

                   Example                  q = quantity
                                            Results from the estimation:
                                            a = intercept
                                            b = slope


• Suppose we apply regression to sales
  data, and find the following demand
  function:
  • q = a + b p = 100 – 4p
  • Unit cost equals $5
• What is the optimal price?
  • p* = [(-4 * 5) – 100] / [2*-4] = 15
• What is the price elasticity when p = 12?
  • When p = 12 then q = 52
  • So the elasticity = b * (p/q) = -0.92
           Is this really optimal??
• Use sensitivity analysis: using the regression result,
  compute q and profit for various levels of p. This gives the
  following table... and graph

       p       q   profit                        Profit for various prices
      $9      64    $256
                                       $450
     $10      60    $300
                                       $400
     $11      56    $336
                                       $350
     $12      52    $364               $300
     $13      48    $384      Profit
                                       $250
     $14      44    $396               $200
     $15      40    $400               $150
     $16      36    $396               $100
     $17      32    $384               $50
     $18      28    $364                 $0
     $19      24    $336                   $10    $12     $14               $16   $18   $20
                                                                price (p)
     $20      20    $300
            Log demand model
• If the relationship between price and sales is
  non-linear, it is better to use a log model:
                                        AGAIN:
     ln(qi )= a + b ln(pi ) + ei        Input data for regression:
• Elasticity is a function of b:        p = price
                                        q = quantity
                                        Results from the estimation:
                 q p
    Elasticity       b                a = intercept
                 p q                   b = slope


• We can also use the model to predict profits
     Profit = q (p - c)
• Using calculus we can find the optimal price:
       profit                     c
               0         p 
                            *

         p                      1
                                    1
                                    b
     Relationship with cost-plus
• Note that this optimal price
                             c
                     p 
                      *
                               1
                            1
                               b
  resembles the cost-plus markup rule:
     p = c / (1 - margin)
• In cost-based pricing, this margin is often determined
  beforehand, which is generally not a good pricing
  approach.
• However, using regression, it is the customer’s price
  sensitivity which enters the cost-plus markup rule, by
  setting margin = -1/b
• This margin maximizes profit
                          Example
• Suppose we find that there is a non-linear
  relationship between sales and price…
  – Then we estimate the log-demand model.
• Suppose we find the following parameters:
  – ln(q) = 1 - 2 ln(p)
• What is the optimal price, using unit cost=5?
  • Answer: p* = 5 / [1+ 1/-2 ] = 10
• What is the price elasticity when p = 12?
  • Answer: The elasticity is equal to b, so it’s equal to -2
                When to use what?
•   Price elasticity alone can only be used to determine an optimal price if
    we are dealing with a constant elasticity across prices (for example:
    log-demand model). Each price-quantity point on the function has the
    same elasticity, but there is only one price that maximizes profit.
•   For linear price-quantity schedules (for example: linear regression), we
    saw that we need both the slope and intercept to determine optimal
    price. Such demand functions have an infinite number of point
    elasticities (it is different for each price), but only one point maximizes
    profits.
•   Using the slope from a finite interval as an approximation of point
    elasticity will yield a different estimate, depending on whether the
    movement is "up" or "down". Arc elasticity averages these
    movements to produce a single value for elasticity. However, arc
    elasticity, nor point elasticity alone, based on these intervals, can be
    used to find optimal prices.
•   When in doubt, use a table of values for prices, quantity, and profits to
    confirm optimal price calculations. These tables have the added value
    of providing insight into the sensitivity of prices, quantities, and profits.
General tip for completing this quiz..
• It makes sense to create a spreadsheet
  with important formulas, while making the
  quiz.
• You can find the important formulas in this
  PPT
• This will save you a lot of time !
        Excel and regression
• Excel contains a Regression Add-In
  – (See Tools | Data Analysis | select regression)
• If Data Analysis doesn’t show up:
  – you may need to add an add-in, through
    Tools | Add-ins. Choose Analysis ToolPak

• On the next slides, we have a step-by-step
  description of how to perform regression in
  Excel
    How to perform regression in Excel
•       Save the data to an Excel-file
        (there’s a link under the table)
•       Open the Excel-file
•       The regression add-in accepts
        only variables in columns, so
        you may need to transpose the
        data:
    •     Copy the data
    •     Select a destination cell (e.g. A7)
    •     Select PasteSpecial and check
          ‘transpose’
    •     Click okay
•       Run the regression Add-in (see
        next slide)
        Run the regression add-in
• Select Tools | Data Analysis | Regression

• Set Input Y Range:
   – click next to the
     textbox
   – Select the range
• Do same for X
  range
• Check Labels if
  your ranges
  contain the variable
  labels
                              Results
  Regressing sales on price will produce the following result.
    The most important numbers are in bold
Interpretation of output:
1. The R-Square describes how
    well the model fits the observed
    data. Closer to 1 means better
    fit.
2. Intercept displays the value for
    a in the regression equation
    qi = a + b pi + ei
3. Price displays the value for b
4. The standard errors tells us how
   certain we are about these
   values. The lower the standard
   error, relative to the coefficient
   value, the more certain we are.
          Log-demand model
• If you need to use regression for the log-
  demand model
    ln(qi )= a + b ln(pi ) + ei
• then
  – You will need to ln-transform the data first
  – Use extra columns for these data (e.g. 2
    slides back: columns D and E)
  – In these cells, use the formula LN(…)
  – Apply regression to these cells, in stead of the
    original ones.
                Finally…
• You can use the output to compute
  optimal prices and predict sales.
• Use the formulas in this PPT

• ... and make that Excel-sheet with useful
  formulas, and use it for all your future
  pricing questions!