Measuring Marketing Effectiveness: Cobb-Douglas Production Functions


Introduction, Data, and Program

Measuring the effectiveness of a marketing channel is difficult due to the large amount of variables and other confounding factors. The field of Marketing Mix Modelling was first developed by econometricians to accurately estimate the impact of marketing on consumer packaged goods, since manufacturers of those goods had access to good data on sales and marketing support.

This post is going to use concepts from microeconomics and econometrics to understand the effectiveness of Television (TV), Newspaper, and Radio on the sales of a good. These data come from the the textbook “An Introduction to Statistical Learning with Applications in R”.  I have provided these data along with the R program used to derive the marketing estimates derived in this post, please see the links below:


Market Mix Modelling R Program


Marketing Production Function

Production functions are used in economics to model the relationship between inputs and outputs.  Production functions are very flexible and have been used in various branches of economics.  Agricultural economists use production function to model how different inputs effect crop yields, educational production functions have been used to model how different classroom inputs effect children’s learning, and macroeconomists have used production functions to understand how labor and capital inputs effect the total national output. I’m going to use a production function to model how different marketing inputs effect sales, per the following equation:


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The majority of inputs that go into production experience diminishing marginal returns, therefore I take the multiplicative form of the production function and take natural logarithms to both sides of the equation.  This is the famous translog equation. The translog equation has the nice property of converting the multiplicative form of the production function into a linear model that can be estimated using Ordinary Least Squares (OLS Regression). Another nice property of the translog equation is that the coefficient (betas)  or a regression analysis can be interpreted as elasticities.  Elasticities are measures of % change in the outcome variable (sales) as a result of a % change in one of the input marketing variables.  The stand alone variable ‘alpha’ captures all non-marketing variables that effect sales, this is called the baseline in Marketing Mix Modelling (MMM).  In this post I will not use other explanatory variables (store traffic, seasonality, other promotions, etc.) to keep things simple, but a robust analysis of the effectiveness of marketing should include additional variables to control for these factors.


Statistical Estimates


The simple set of scatter plots show that television appears to have the strongest impact on sales.  Radio has a modest effect on sales, but newspaper appears to be weakly correlated with sales. The data also support the notion of marginal diminishing returns, which further motivates the logarithmic transformation of the production function.


Nespaper Radio Tv


Scatter plots can only reveal so much to do a proper analysis economists use econometric estimates of the translog function described above.  This ensures that we are controlling for other factors when measuring the impact of each of the marketing variables, and is shown below:

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The results shows that for every 1% increase in TV advertising you’d expect to get a .34% increase in sales.  A 1% increase in the Newspaper budget only increases sales by .01%, the smallest of all the elasticity estimates.  A 1% increase in the Radio budget accounts for a modest .17 increase in sales.

How much should the company spend on each form of advertising? The data in this example doesn’t show how many impressions or people where reached with the money spent.  In order to provide a proper optimal allocation one would need to know the cost-per-impression, but assuming the cost per impression (CPM) are constant one can simply take the ratio of each elasticity relative to the sum of all elasticities to come up with the optimal marketing mix.


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A following post will do a proper optimization using Lagragian Optimization of the production function, which will take into account the total cost of advertising on each channel.