A Marketing Approach to Price Determination

Model Code

library(pricesensitivitymeter)
library(ggplot2)
library(knitr)

set.seed(38)

# standard van Westendorp Price Sensitivity Meter Analysis
# input directly via vectors

tch <- round(rnorm(n = 500, mean = 25, sd = 5), digits = 1)
ch <- round(rnorm(n = 500, mean = 35, sd = 10), digits = 1)
ex <- round(rnorm(n = 500, mean = 45, sd = 10), digits = 1)
tex <- round(rnorm(n = 500, mean = 60, sd = 5), digits = 1)

output_psm_demo1 <- psm_analysis(toocheap = tch,
  cheap = ch,
  expensive = ex,
  tooexpensive = tex)

# additional analysis with Newton Miller Smith Extension
# input via data.frame

pint_ch <- sample(x = c(1:5), size = length(tex),
  replace = TRUE, prob = c(0.1, 0.1, 0.2, 0.3, 0.3))

pint_ex <- sample(x = c(1:5), size = length(tex),
  replace = TRUE, prob = c(0.3, 0.3, 0.2, 0.1, 0.1))

data_psm_demo <- data.frame(tch, ch, ex, tex, pint_ch, pint_ex)

output_psm_demo2 <- psm_analysis(toocheap = "tch",
  cheap = "ch",
  expensive = "ex",
  tooexpensive = "tex",
  pi_cheap = "pint_ch",
  pi_expensive = "pint_ex",
  data = data_psm_demo,
  intersection_method = "min",
  interpolation_steps=.01)

# Exctract the cdf functions for plotting
figureoutput <- output_psm_demo2$data_vanwestendorp


# Hack package function to extract intersection points of too cheap/expensive and too expensive/cheap
identify_intersection <- function(data, var1, var2, method) {
  first_intersection_pos <- which(data[, var1] >= data[, var2])[1]

  if(is.na(first_intersection_pos)) { # if no intersection: return NA
    return(NA)
  } else { # otherwise, run the actual function
  all_intersections_pos <- which(data[, var1] == data[first_intersection_pos, var1] &
                                   data[, var2] == data[first_intersection_pos, var2])

  all_intersections_prices <- data[all_intersections_pos, "price"]

  switch(method,
         min = {min(all_intersections_prices)},
         max = {max(all_intersections_prices)},
         mean = {mean(all_intersections_prices)},
         median = {median(all_intersections_prices)})
  }
}

pricerange_lower2 <- identify_intersection(data = figureoutput,
                                            var1 = "ecdf_expensive",
                                            var2 = "ecdf_toocheap",
                                            method = "min")


identify_intersection2 <- function(data, var1, var2, method) {
  first_intersection_pos <- which(data[, var1] <= data[, var2])[1]

  if(is.na(first_intersection_pos)) { # if no intersection: return NA
    return(NA)
  } else { # otherwise, run the actual function
  all_intersections_pos <- which(data[, var1] == data[first_intersection_pos, var1] &
                                   data[, var2] == data[first_intersection_pos, var2])

  all_intersections_prices <- data[all_intersections_pos, "price"]

  switch(method,
         min = {min(all_intersections_prices)},
         max = {max(all_intersections_prices)},
         mean = {mean(all_intersections_prices)},
         median = {median(all_intersections_prices)})
  }
}
pricerange_upper2 <- identify_intersection2(data = figureoutput,
                                            var1 = "ecdf_cheap",
                                            var2 = "ecdf_tooexpensive",
                                            method = "min")

optimalpp <- output_psm_demo2$opp
indifferencepp <- output_psm_demo2$idp

# Build plot using ggplot with the cdfs and the intersection points
psm_plot <- ggplot(figureoutput, aes(x=price, y=ecdf_toocheap, color="Too Cheap")) +
            geom_line() +
            geom_line(aes(x=price, y=ecdf_cheap, color="Cheap")) +
            geom_line(aes(x=price, y=ecdf_expensive, color="Expensive")) +
            geom_line(aes(x=price, y=ecdf_tooexpensive, color="Too Expensive")) +
            ylab("Cumulative Distribution") +
            geom_vline(xintercept=output_psm_demo2$opp, line=1, linetype="dashed") +
            geom_vline(xintercept=output_psm_demo2$idp, line=1, linetype="dotdash") +
            geom_vline(xintercept=pricerange_lower2, line=1, linetype="dotted") +
            geom_vline(xintercept=pricerange_upper2, line=1, linetype="dotted") +
            theme_minimal() + theme(panel.grid.minor = element_blank(), panel.grid.major = element_blank()) +
            ggtitle("Van Westendrop Price Sensitivity Example") +
            scale_color_manual(name = "", values = c("Too Cheap" = "navyblue", "Cheap" = "blue", 
                                                            "Expensive" = "green", "Too Expensive"="darkgreen"))

Sample Survey Data

For this example, the 500 survey responses are randomly generated. To get a sense for the data, the below table shows the survey responses across the four key variables: tch, ch, ex, tex. Note, given this data was randomly generated from a distribution, like any analysis, nonsensical responses are automatically dropped from inclusion (i.e. tch < ch < ex < tex).

library(kableExtra)
library(knitr)

displaydata <- data_psm_demo[,1:4]

kable(head(displaydata, n=10)) %>%
  kable_styling(bootstrap_options = "striped", full_width = F)

Cumulative Distribution Calculation

Using the survey responses, each variable is rank ordered from smallest to largest and a cumulative distribution is calculated for all responses. The cumulative distribution for each variable is then plotted against the price points. The below data snapshot demonstrates this idea.

library(kableExtra)
library(knitr)

displaydata2 <- figureoutput[,1:5]

kable(head(displaydata2, n=15)) %>%
  kable_styling(bootstrap_options = "striped", full_width = F)

Graphical Analysis & Interpretation

A quick look at the below figure helps to clarify the value of this analysis. In its simplest form, there four key points of note as stated by Allestee in the r pricesensitivitymeter documentation:

1. Point of Marginal Cheapness (MGP): Below this price point, there are more respondents that consider the price as “too cheap” than respondents who consider it as “expensive” (intersection of “too cheap” and “expensive”). This is interpreted as the lower limit of the range of acceptable prices.

2. Point of Marginal Expensiveness (MEP). Above this price point, there are more respondent that consider the price as “too expensive” than there are respondents who consider it as “cheap” (intersection of “cheap” and “too expensive”). This is interpreted as the upper limit of the range of acceptable prices.

3. Indifference Price Point (IDP): The same number of respondents perceives the price as “cheap” and “expensive” (intersection of “cheap” and “expensive”). In van Westendorp’s interpretation, this is either the median price paid in the market or the price of an important market-leader.

4. Optimal Price Point (OPP): The same number of respondents perceives the product as “too cheap” and “too expensive” (intersection of “too cheap” and “too expensive”). van Westendorp argues that this is the value for which the respondents’ resistance against the price is particularly low.

psm_plot

Van Westendorp Price Sensitivity Meter Analysis

Based on this analysis, this test product should be priced between $33.8-$51.3, has an optimal price point of $37.7, and an indifference price point of $41.8.