8.1 Not this year

8.2

  1. The expression should be decreasing as a function of \(x_A\) and increasing as a function of \(x_B\), so we should have \(\alpha_A < 0\),\(\alpha_B > 0\). If \(\alpha_B = 0\), the term relating to brand B becomes \(x_B^0 = 1\), so the price of brand B does not affect the sale of brand A if this is the case.

It is always a good idea to have a look at the data:

meatdata <- read.csv("M:/Undervisning/Undervisningh21/Data/meat_brands.csv")
#make a matrix plot of all variables
plot(meatdata)

summary(meatdata)
##      saleA            saleB           priceA          priceB     
##  Min.   : 525.0   Min.   : 90.0   Min.   : 89.0   Min.   :119.0  
##  1st Qu.: 779.2   1st Qu.:114.2   1st Qu.:110.2   1st Qu.:136.0  
##  Median : 845.0   Median :125.5   Median :121.0   Median :139.0  
##  Mean   : 883.1   Mean   :130.2   Mean   :117.5   Mean   :139.7  
##  3rd Qu.: 966.0   3rd Qu.:146.5   3rd Qu.:124.0   3rd Qu.:145.0  
##  Max.   :1614.0   Max.   :179.0   Max.   :135.0   Max.   :159.0

We see larger sales of brand A in general, and somewhat higher prices in general for brand B. From the various scatterplot only the sale vs price relation for brand A is very clear. No other effects appear visually as very strong. (That does not mean there are no other effects, as we will see.) There are no “dangerous” outlier data points, so it looks good for a regression analysis.

  1. Ok, doing a few initial experiments show that while saleA is affected by both prices, the saleB is not significantly affected by priceA, so we can consider the following three models.
library(stargazer)


#brand A sales full model
regA <- lm(log(saleA) ~ log(priceA) + log(priceB), data = meatdata)

#same for B
regB <- lm(log(saleB) ~ log(priceA) + log(priceB), data = meatdata)

#reduced model for saleB
regB2 <- update(regB, . ~ . - log(priceA))
regB2 <- stargazer(regA, regB, regB2,
          type = "text", 
          keep.stat=c("n", "rsq"),
          model.numbers=FALSE,
          column.labels = c("A", "B", "B2"))
## 
## ==========================================
##                   Dependent variable:     
##              -----------------------------
##              log(saleA)     log(saleB)    
##                   A         B        B2   
## ------------------------------------------
## log(priceA)   -2.077***   0.179           
##                (0.164)   (0.234)          
##                                           
## log(priceB)    0.589**   -0.861** -0.823**
##                (0.238)   (0.339)  (0.334) 
##                                           
## Constant      13.744***  8.254*** 8.921***
##                (1.315)   (1.874)  (1.652) 
##                                           
## ------------------------------------------
## Observations     50         50       50   
## R2              0.773     0.123    0.112  
## ==========================================
## Note:          *p<0.1; **p<0.05; ***p<0.01
  1. We can read somehting like this:

This can be interpreted as follows. Typical buyers of brand A are quite price sensitive, if the price goes up, they tend to stop buying, and they do not generally switch to brand B (because price A is largely unrelated to sale B).

Typical buyers of brand B are much less price sensitive. If the price B goes up, some sale of B is lost, but at the same time we see sale A going up, so we can interpret this as customers swithcing to brand A when brand B is getting more expensive.

We can note in connection with this, that in practice, we often think about say 10 or 20% price jumps, and in this case we just multiply up the effect, so for example a 10% higher price A would lead to about 21% lost sales for A. Or we can think oppositely: A 10% drop in price B would increase the sale of B by about 8%.

  1. Briefly: Widely advertised low price campaigns for meat A could have at least two effects:

  1. Customers who usually shops elsewhere are attracted by the offer, so the size of the market temporarily changes. The constant \(c_A\) in the model describes a fixed market size where prices are known to all. We may need to model changes in \(c_A\) that may occur in campaign weeks.

  2. Since people have freezers, many would buy lots of meat in campaign weeks and store at home. That would possibly “dry-out” the market in subsequent weeks. Again, the number of potential buyers (size of market) of meat A may vary from week to week. The supermarket should be prepared to order less meat in weeks following a substantial low price campaign.

Without going into any detail, clearly modifications to such models are possible. Regarding point 1, dummy variables indicating extra advertising/campaign weeks could be included in a sensible way. For point 2, the time order of weeks seems to matter. One could have dummies indicating one, two, three weeks after campaigns. It would be possible to model changes in market size with this.