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Thread: Price Elasticity based on log regression

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    Price Elasticity based on log regression




    Hello genius people!!

    1) If i understand correctly, In the regression equation below,

    WS= White shoes
    BS=Black shoes


    Sales of WS = β1 (Price of WS) - β2(price of BS) + constant

    Then, can I say ....β1 = price elasticity of WS and
    β2 = price elasticity of BS

    Or

    log (sales of WS) = β1*log (Price of WS) - β2* log (price of BS) + constant

    is it now β1 = price elasticity of WS and β2 = price elasticity of BS ?

    Why is log applied to calculate price elasticity?

    2) If WS has two promotion
    a) Price discount of 15%
    b) Buy one pair and get another pair free
    Then how to incorporate it in the regression equation (OLS), considering both promotions appear only for two month separately I mean how the data should be set up? Should I have separate variables for these two promotions? and should i have 0 for rest of the months when there is no promotion?

    Sales of WS = β1 (Price of WS) - β2(Discount 15%) β3 (buy one get one) β4(price of BS) + Constant

    Would the above equation be correct and would respective β values would be elasticity of each variable??
    Thanks

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    Re: Price Elasticity based on log regression

    Let some other folks reply before you take this and run with it, but my thoughts:

    I'm assuming you expect the coefficient of BS to be negative if you are using:
    Sales of WS = β1 (Price of WS) - β2(price of BS) + constant as the model.

    That is, you need to make sure that as price of BS is rising that sales of WS is also increasing. Since the two are substitutes, a rise in the price of one, all else held constant, should create a drop in demand for that product and substitution towards the other.

    The log is being use because the demand curve is likely curvilinear. This transforms the data to something more linear to properly meet regression linear requirements.

    I haven't given it great thought, but I'd call B1 the price elasticity of demand for WS and B2 the cross price elasticity of demand.

    Keep in mind, however, that elasticity changes at different points along the demand curve. Unless the demand curves are iso-elastic (unlikely), the model may not be all that good for inference too far beyond the mean, though this is just conjecture on my part. Let someone else verify this assumption... and all my others for that matter. Taking the log may ameliorate this issue, though I'm not sure and its too far past my bedtime to think about it.

    I'll let you do your own homework..... Look at dummy variables and consider if they may be useful.

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    Re: Price Elasticity based on log regression

    Well, β1 is the price elasticity of WS sure and I got confirmation on that and thank for expressing the same view, however still not sure about β2 interpretation. going by your logic,
    β1 can be interpreted as - Price Elasticity of WS and Cross Price Elasticity of BS and similarly
    β2 can be interpreted as - Price Elasticity of BS and Cross Price Elasticity of WS ??

    I am pretty sure about the logic of the elasticity function (increase and decrease in Price and impact on sales), i am struggling in terms of how to use these details in the equation and interpretation of the results.

    You are correct, the elasticity will be different at each price point on demand curve, if i am using a time series data then the elasticity is different at different periods (Would it be correct if i take the average of elasticity of a product for 12 months and refer it to as the price elasticity of that product for year1 ?)

    Thanks for explaining the log part, i will look for more information on the same to explain the same phenomenon in a simple way (common man English, if i may say) but thanks it did help me.

    again, my concern here is how to introduce, the promotion details (I mean the interpretation of the second equation) can β2 be explained as promotion price elasticity?

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    Re: Price Elasticity based on log regression

    First,

    The log form is the only correct specification of this problem since the definition of elasticity is %change in demand / %change in price. Using the log model also takes care of the problem of elasticity being different for various levels of X.

    Your final equation would be correct if Y, X1 and X4 were logs. B2 and B3 are defined as the marginal effects of the promotions. They are not elasticities.

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    Re: Price Elasticity based on log regression

    Hi Timmy, you have addressed the very important aspect of the log function........thank you very much indeed.
    So, if i understand correctly, on a time series data when i get the final β values for the second equation (with log), then i can straight away say it is the price elasticity of that particular product/brand! - Thanks, this is very important.

    The only clarification or the interpretation I want to understand is, is it correct to say,

    β1 can be interpreted as - Price Elasticity of WS and Cross Price Elasticity of BS and similarly
    β2 can be interpreted as - Price Elasticity of BS and Cross Price Elasticity of WS ??

    For your last comment, can you explain the logic? The reason i ask is, in the last equation the X2 is 5% discount on regular price so the input data is still the price
    (regular price - discount of 5%) they why not call it promotion price elasticity (considering that i'll apply log to this variable also) ? and by the way in the input data file the variable X2 will have readings for lets say 6 months out of 24 months should then i have zero in all other period (months) or will it be good to have a dummy variable as gentleman has explained above?
    Last edited by PKA111; 01-29-2013 at 01:19 PM.

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    Re: Price Elasticity based on log regression

    Consider this - the definition of elasticity of f(x) with repect to x

    El_x f(x) \equiv \frac{x}{f(x)} \frac{d f(x)}{dx}

    and considers this

    log(y) = a + b \cdot log (x) + c \cdot log (z)

    now you want to interpret b so you notice that \frac{d log(y)}{d log(x)} = b then you say to yourselve thats fine but what is: \frac{d log(y)}{d log(x)}? And then you use the chainrule:


    \frac{d log f(x)}{ dx} = \frac{d log f(x)}{ d log (x)}   \cdot \frac{d log x}{dx} and
    \frac{1}{f(x)} \cdot \frac{df(x)}{dx} = \frac{d log f(x)}{ d log (x)}   \cdot \frac{1}{x}
    \frac{x}{f(x)} \cdot \frac{df(x)}{dx} = \frac{d log f(x)}{ d log (x)}

    and hence according to the definition of the elasticity:


    El_x f(x) = \frac{x}{f(x)} \cdot \frac{df(x)}{dx} = \frac{d log f(x)}{ d log (x)}


    So using log-log will always give you an elasticity. But price elasticity and cross price elasticity are certain types of elasticities:


    Using the following regression:

    log(Y) = \beta_0 + \beta_1 log(X_1)  + \beta_2 log(X_2)  + \epsilon

    and letting Y be sales - or demand - of a cetrain product and X_1 be the price of the same product \beta_1 is the elasticity of demand for product A with respect to change in price of A hence it is the price elastictity according to the definition:

    Price elasticity of demand is a measure used in economics to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price.

    and similarly \beta_2 is the elasticity of demand for product A with respect to change in price of product B hence it is the crossprice elastictity according to the definition:

    In economics, the cross elasticity of demand or cross-price elasticity of demand measures the responsiveness of the demand for a good to a change in the price of another good.



    Also when you use the log-model you are assuming constant elasticity. If you want to test for non-constant elasticity use quadratic term and test their significance:

    log(Y) = \beta_0 + \beta_1 log(X_1) + \beta_2 [log(X_1)]^2 + \beta_3 log(X_2) + \beta_4 [log(X_2)]^2 + \epsilon
    Last edited by JesperHP; 01-31-2013 at 09:27 AM.

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    Re: Price Elasticity based on log regression

    Hi there Jasper! thanks for your valuable input! As point of further discussion. What would be your opinion on using actual price point as the input or creating an index of price and use that as the input.
    I prefer (based on my understanding/learning) actual price point is a far better approach, than using Price Index.

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    Re: Price Elasticity based on log regression

    If you're concerned with price elasticity, then you'll want to use the actual price. An index isn't a real value. It's basically a made up value. So what does a "percentage change in ..." mean with an index? It means whatever you want it to mean! You made the index. There's a pretty simple reason why we use logs to estimate price elasticity in regression models: the log-change is an approximation for a percentage change. Thus, on the usual interpretation of a regression model ("a one unit change in ...") with a log variable will mean exactly what we want to know: "a percentage change in ..."

    Note, when calculating elasticity, you don't need a log model. You can use a log-linear or linear-log model, too. The only difference is you'll need to use the mean of the response or independent variable as the numerator or denominator (respectively? May be the other way around) to estimate the elasticity. I have some documents at work on this that I found when I needed a refresher, and the slides were pretty easy to digest with good examples. I'll post them later this week.

    To elaborate on the log-change, consider the percentage change between two points A and B. How do we calculate it?

    \frac{B - A}{B}

    Of course, this assumes a change from A to B. If it were the other way it would be

    \frac{A - B}{A}

    If we don't know or don't presume a directional change, we're simply going to take the midpoint percentage change.

    \frac{|A - B|}{0.5\times (A + B)}

    It is this midpoint percentage change that the log-difference approximates. Give it a try. It's accurate to a few decimal places.
    You should definitely use jQuery. It's really great and does all things.

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    Re: Price Elasticity based on log regression

    Quote Originally Posted by PKA111 View Post
    Hi there Jasper! thanks for your valuable input! As point of further discussion. What would be your opinion on using actual price point as the input or creating an index of price and use that as the input.
    I prefer (based on my understanding/learning) actual price point is a far better approach, than using Price Index.
    Lets start with the regression

    \log y = \beta_0 + \beta_1 \log x_1 + \epsilon

    I'm not entirely sure I interpret you're question corerctly but if you use an indexation of X_1 let's define this:

    x_{index} \equiv \frac{x_1}{A} where A is some constant - possibly a value of X_1 at some timepoint. Then I think this is what happens in the regression:

    \log y = \beta_0 + \beta_1  \log (x_1/A)+ \epsilon

    \log y = \beta_0 -  \beta_1 log A  + \beta_1  \log (x_1)+ \epsilon

    \log y = \gamma_0 + \beta_1  \log (x_1)+ \epsilon

    where \gamma_0  \equiv \beta_0 -  \beta_1 log A

    Hence you will still estimate \beta_1 which is the elasticity you wanted - I assume - but your constantterm will change... but you can try this out in practice by running the regression with indexed variable and without indexing X_1. Remember though the above deduction was based on the properties of the log-function and are hence only usable when using logged variable. You would still have to log the indexed variable as is done in the regression eqaution.
    Last edited by JesperHP; 02-02-2013 at 11:42 PM.

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    Re: Price Elasticity based on log regression


    Hi Bryan,
    Thanks for your explanation, the first paragraph makes a lot of sense. In the second paragraph I get where you are coming from and taking cue from your comment i'll try it out first to see the difference. However, I saw this video on Youtube (below is the link) where it is explained that a log-log model is more appropriate (my interpretation)to calculate the price elasticity (although the author explain log, log-lin, lin-log, log-log model). Will be great to see the example slides you talked about.

    http://www.youtube.com/watch?v=rN8Ll...B6330BD7BB8147

    Also, It is my personal experience and I have noticed that lots of corporate companies in west cost (US of course) use price index to calculate price elasticity, similarly lot companies in Europe and in Asia Pacific prefer to use actual price point (not i cannot say there is a correlation between the geography and price models, it is just an observation)

    As mentioned by Jesper above, it will be a good idea to work out both options and see what difference it makes.

    Jasper, thanks again for your valid comments!

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