Chapter 8 Logarithmic Transformations

Exercise 8.1 This is a theoretical modeling exercise, relating to demand modeling. Suppose we have daily recorded data for the variables

  • \(y\) = number of airline passengers leaving from Molde airport with destination Oslo.
  • \(x\) = average price of tickets used that day.
  • \(w\) = day of week, 1 = Monday, \(\ldots\), 7 = Sunday.

in all models, just write the relation you suggest between variables, ignore random terms. a. Write the general constant elasticity model for the price - demand relation in this case. b. Write a model that would allow statistical testing and estimation of day-to-day demand variations. It is assumed that demand is stable Tuesday to Thursday, peaking on Friday, Sunday and Monday, and low on Saturday. Your model should capture at least these effects. Write also the logarithmic transformed model. Make sure it is linear. * One could suspect that most travelers will try to avoid going on Saturday. Perhaps mainly very price-sensitive customers will go on Saturday if they find cheap tickets. Price sensitivity is to a large extent measured by the price elasticity. If you go back to your original model from a., it assumes price elasticity is the same on every weekday. Try to modify the model to allow estimating a change in price elasticity on Saturdays. Can this model be linearized by logarithmic transformation? Does it seem necessary to calculate new variables to estimate the possible change in elasticity?

Exercise 8.2 Find the file meat_brands.sav in Canvas. The data is described as follows. A supermarket sells two brands of ground meat, A and B. Brand A is cheaper, and brand B is considered a quality brand. Imagine the supermarket have fixed prices for each week, then possibly change it between weeks. One can imagine at least three types of buyers of the product:

  • Highly price-conscious ones, who care mainly about the price of the products.
  • “Opportunists,” who will buy the quality brand unless price difference is large.
  • Quality seekers (or people with too much money) who always buy brand B (almost) regardless of prices.

Weekly sale volumes in kg, of each brand are recorded along with prices for each brand. Let \(y_A, y_B\) be the sale volumes for brand A, B, while \(x_A, x_B\) are prices in the same week. The classical demand models state \[ y_A = c_A x_A^{\alpha_A} x_B^{\alpha_B}, \quad y_B = c_B x_A^{\beta_A} x_B^{\beta_B}\;. \]

  1. In the model for \(y_A\), if any of the parameters \(\alpha_A, \alpha_B\) are different from 0, what sign would you expect on each? What would it mean if \(\alpha_B = 0\) in the model?
  2. Theory tells us that the relation between logarithms of the variables is linear. Use linear regressions in R to estimate parameters \(\alpha_A\) etc. for the two models. Take out eventual nonsignificant variables and rerun regressions if necessary.
  3. Discuss the differences you read from the estimated elasticities. In what way does the price of brand A (B) affect the sale of brand A? In what way does the price of brand A (B) affect the sale of brand B? Could some of the effects be explained by the possibly different customer groups as mentioned above?
  4. Difficult: Discuss whether (and how) advertisement and short-term-low-price campaigns e.g. for meat A could invalidate the models we have used here. This means, do you see possible market effects we can not estimate with our simple models? Does it matter that almost all households in the town has a deep-freezer? Could some modifications of the models help?