Static Price Optimization

rm(list=ls(all=T))
options(digits=4, scipen=40)

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1. Logistic Distribution

1.1 Logistic vs Normal Distribution

\[f(x) = \frac{exp(-\frac{x-u}{s})}{s(1+exp(-\frac{x-u}{s}))^2} \quad ;\quad 1 - F(x) = \frac{1}{1+exp(\frac{x-u}{s})} \]

• mean = \(u\)
• scale = \(s\)
• variance = \(s^2 \pi^2 / 3\)

# parameters
u=0; sd=1; s=sqrt(sd*3/(pi^2))   

# PDF
par(cex=0.8)
curve(dlogis(x, u, s), u-6*s, u+6*s, col='cyan', lwd=2,
      main="Logistic Dist. (blue) vs Normal Dist. (red)\nPDF")
curve(dnorm(x,u,sd), col='red', lty=2, add=T)

# 1 - CDF
par(cex=0.8)
curve(1-plogis(x, u, s), u-6*s, u+6*s, col='cyan', lwd=2,
      main="Logistic Dist. (blue) vs Normal Dist. (red)\n1 - CDF")
curve(1-pnorm(x,u,sd), col='red', lty=2, add=T)

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2. Fitting the Price Response Function

# sample data
qty = c(461,493,469,339,60)
price = c(14,17,18,19,22)
plot(price, qty, pch=20, ylim=c(0,550), xlim=c(13,24))

We can observe: \(N \simeq 500\), \(u \simeq 20\), \(sd \simeq 6/6 = 1\)

mod = nls(qty ~ N/(1+exp((price - u)/s)), start=c(
  N=500, u=20, s=sqrt(1 * 3/(pi^2)))
  )
coef(mod)

       N        u        s 
490.6165  19.9344   0.8935 

N = coef(mod)[1]; u = coef(mod)[2]; s = coef(mod)[3]
par(cex=0.8)
plot(price, qty, pch=20, ylim=c(0,550), xlim=c(13,24), main=sprintf(
  "Logistic Price Response Function\nN=%.1f,  u=%.1f,  s=%.1f",
  N, u, s))
curve(N/(1+exp((x-u)/s)), col='orange', lwd=2, add=T)

---

3. Price Optimization

p = seq(13,24,0.1)
q = N * (1 - plogis(p, u, s))
r = p * q
i = which.max(r)
par(cex=0.8)
plot(p, p*q, type='l', main=sprintf(
  "Optimal Revenue $%.1f  at  $%.1f", r[i], p[i]))
abline(v=p[i], col='red')