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By Yuri Fonseca

## Demand models

In the previous post about pricing optimization (link here), we discussed a little about linear demand and how to estimate optimal prices in that case. In this post we are going to compare three different types of demand models for homogeneous products and how to find optimal prices for each one of them.

For the linear model, demand is given by:

$\displaystyle d(p) = \alpha p + \beta,$

where $\alpha$ is the slope of the curve and $\beta$ the intercept. For the linear model, the elasticity goes from zero to infinity. Another very common demand model is the constant-elasticity model, given by:

$\displaystyle \ln d(p) = \alpha \ln p + \beta,$
or

$\displaystyle d(p) = d_0 e^\beta p^\alpha = Cp^\alpha,$

where $\alpha$ is the elasticity of the demand and $C$ is a scale factor. A much more interesting demand curve is given by the logistic/sigmoide function:

$\displaystyle d(p) = C\frac{e^{\alpha p + \beta}}{1 + e^{\alpha p + \beta}} = \frac{C}{1+e^{-\alpha(p - p_0)}},$

where $C$ is a scale factor and $\alpha$ measures price sensitivity. We also can observe $p_0 = -\alpha/\beta$ as the inflection point of the demand.

Some books changes the signs of the coefficients using the assumption that $\alpha$ is a positive constant and using a minus sign in front of it. However, it does not change the estimation procedure or final result, it is just a matter of convenience. Here, we expect $\alpha$ to be negative in the three models.

In the Figure below we can check a comparison among the shapes of the demand models:

library(ggplot2)
library(reshape2)
library(magrittr)

linear = function(p, alpha, beta) alpha*p + beta
constant_elast = function(p, alpha, beta) exp(alpha*log(p)+beta)
logistic = function(p, c, alpha, p0) c/(1+exp(-alpha*(p-p0)))

p = seq(1, 100)
y1 = linear(p, -1, 100)
y2 = constant_elast(p, -.5, 4.5)
y3 = logistic(p, 100, -.2, 50)

df = data.frame('Prices' = p, 'Linear' = y1, 'Constant_elast' = y2, 'Logistic' = y3)
df.plot = melt(df, id = 'Prices') %>% set_colnames(c('Prices', 'Model', 'Demand'))

ggplot(df.plot) + aes(x = Prices, y = Demand) +
geom_line(color = 'blue', alpha = .6, lwd = 1) +
facet_grid(~Model)


Of course that in practice prices does not change between 1 and 100, but the idea is to show the main differences in the shape of the models.

All the models presented above have positive and negative points. Although local linear approximation may be reasonable for small changes in prices, sometimes this assumption is too strong and does not capture the correct sensitivity of bigger price changes. In the constant elasticity model, even though it is a non-linear relationship between demand and price, the constant elasticity assumption might be too restrictive. Moreover, it tends to over estimate the demand for lower and bigger prices. In a fist moment, I would venture to say that the logistic function is the most robust and realistic among the three types.

## Pricing with demand models

In a general setting, one have for the total profit function:

$\displaystyle L(p) = d(p)(p-c),$

where, $L$ gives the profit, $d$ is the demand function that depends of the price and $c$ is the marginal cost. Taking the derivative with respect to price we have:

$\displaystyle L'(p) = d'(p)(p - c) + d(p).$

Making $L'(p) = 0$ to calculate the optimum price (first order condition), we have:

$\displaystyle d'(p^\star)(p^\star - c) + d(p^\star) = 0$
$\displaystyle d'(p^\star)p^\star + d(p^\star) = d'(p^\star)c,$

which is the famous condition that in the optimal price, marginal cost equals marginal revenue. Next, let’s see how to calculate the optimum prices for each demand functions.

#### Linear model

For the linear model $d'(p) = \alpha$. Hence:

$\displaystyle d'(p^\star)p + d(p^\star) = d'(p^\star)c,$
$\displaystyle \alpha p^\star + \alpha p^\star + \beta = \alpha c,$
$\displaystyle p^\star = \frac{\alpha c - \beta}{2\alpha}.$

Example:

library(tidyverse)

# Synthetic data
p = seq(80,130)
d = linear(p, alpha = -1.5, beta = 200) + rnorm(sd = 5, length(p))
c = 75
profit = d*(p-c)

# Fit of the demand model
model1 = lm(d~p)
profit.fitted = model1$fitted.values*(p - c) # Pricing Optimization alpha = model1$coefficients[2]
beta = model1$coefficients[1] p.max.profit = (alpha*c - beta)/(2*alpha) # Plots df.linear = data.frame('Prices' = p, 'Demand' = d, 'Profit.fitted' = profit.fitted, 'Profit' = profit) ggplot(select(df.linear, Prices, Demand)) + aes(x = Prices, y = Demand) + geom_point() + geom_smooth(method = lm)  ggplot(select(df.linear, Prices, Profit)) + aes(x = Prices, y = Profit) + geom_point() + geom_vline(xintercept = p.max.profit, lty = 2) + geom_line(data = df.linear, aes(x = Prices, y = Profit.fitted), color = 'blue')  #### Constant elasticity model For the constant elasticity model, since $\lim_{\Delta \rightarrow 0}\frac{\Delta D}{\Delta p} = d'(p)$, we have that: $\displaystyle \epsilon = \frac{\%D}{\%p} = \frac{p\Delta D}{D\Delta p} = -\frac{d'(p)p}{D}.$ Therefore, $\displaystyle d'(p^\star)p^\star + d(p^\star) = d'(p^\star)c,$ $\displaystyle \frac{d'(p^\star)p^\star}{d(p^\star)} + 1 = \frac{d'(p^\star)c}{d(p^\star)},$ $\displaystyle -\epsilon + 1 = \epsilon \frac{c}{p^\star},$ $\displaystyle p^\star = \frac{\epsilon c}{1-\epsilon} = \frac{c}{1-1/\epsilon}.$ Moreover, knowing that $\frac{\%D}{\%p} \sim \frac{\Delta \ln D}{\Delta \ln p}$ and using the constant elasticity model, we have that: $\displaystyle \epsilon \sim \lim_{\Delta \rightarrow0} \frac{\Delta \ln D}{\Delta \ln P} = \frac{d\ln D}{d\ln p} = \alpha.$ Thus, we can calculate the optimum profit price for the constant elasticity model as: $\displaystyle p^\star = \frac{c}{1 - \frac{1}{|\alpha|}}$ It is interesting to note that one needs Example: # Synthetic data p = seq(80,130) d = constant_elast(p, alpha = -3, beta = 15)*exp(rnorm(sd = .15, length(p))) c = 75 profit = d*(p-c) # Fitting of demand model model2 = lm(log(d)~log(p)) profit.fitted = exp(model2$fitted.values)*(p - c)

# pricing optimization
alpha = model2$coefficients[2] p.max.profit = c/(1-1/abs(alpha)) # Plots df.const_elast = data.frame('Prices' = p, 'Demand' = d, 'Profit.fitted' = profit.fitted, 'Profit' = profit) ggplot(select(df.const_elast, Prices, Demand)) + aes(x = log(Prices), y = log(Demand)) + geom_point() + geom_smooth(method = lm)  ggplot(select(df.const_elast, Prices, Profit)) + aes(x = Prices, y = Profit) + geom_point() + geom_vline(xintercept = p.max.profit, lty = 2) + geom_line(data = df.const_elast, aes(x = Prices, y = Profit.fitted), color = 'blue')  #### Logistic model For the logistic function, one can check that $d'(p) = \alpha d(p)(1-d(p)/C)$. Thus: $\displaystyle d'(p^\star)(p^\star - c) + d(p^\star) = 0,$ $\displaystyle \alpha d(p^\star)(1-d(p^\star)/C)(p^\star-c) + d(p^\star) = 0,$ $\displaystyle \alpha(1-d(p^\star)/C)(p^\star-c) + 1 = 0,$ $\displaystyle \frac{\alpha e^{-\alpha(p^\star - p_0)}(p^\star - c) + 1+ e^{-\alpha(p^\star - p_0)}}{1+ e^{-\alpha(p^\star - p_0)}} = 0,$ $\displaystyle \alpha(p^\star-c)+1]e^{-\alpha(p^\star - p_0)} + 1 = 0.$ Since the last equation above does not have an analytical solution (at least we couldn’t solve it), one can easily find the result with a newton-step algorithm or minimization problem. We will use the second approach with the following formulation: $\displaystyle \min_{p \in \mathbb{R}} \big{(}[\alpha(p-c)+1]e^{-\alpha(p - p_0)} + 1\big{)}^2$ Example: # Objective functions for optimization demand_objective = function(par, p, d) sum((d - logistic(p, par[1], par[2], par[3]))^2) price_objective = function(p, alpha, c, p0) (exp(-alpha*(p-p0))*(alpha*(p-c)+1) + 1)^2 # A cleaner alternative for pricing optimization is to min: price_objective2 = function(p, c, alpha, C, p0) -logistic(p, C, alpha, p0)*(p-c) # synthetic data p = seq(80,130) c = 75 d = logistic(p, 120, -.15, 115) + rnorm(sd = 10, length(p)) profit = d*(p-c) # Demand fitting, we can't use lm anymore par.start = c(max(d), 0, mean(d)) # initial guess demand_fit = optim(par = par.start, fn = demand_objective, method = 'BFGS', p = p, d = d) par = demand_fit$par # estimated parameters for demand function
demand.fitted = logistic(p, c = par[1], alpha = par[2], p0 = par[3])
profit.fitted = demand.fitted*(p - c)

# Pricing Optimization, we don't have a closed expression anymore
price_fit = optim(mean(p), price_objective, method = 'BFGS',
alpha = par[2], c = c, p0 = par[3])

# or

price_fit2 = optim(mean(p), price_objective2, method = 'BFGS',
c = c, C = par[1], alpha = par[2], p0 = par[3])

# both results are almost identical
p.max.profit = price_fit\$par

# Graphics
df.logistic = data.frame('Prices' = p, 'Demand' = d, 'Demand.fitted' = demand.fitted,
'Profit.fitted' = profit.fitted, 'Profit' = profit)

ggplot(select(df.logistic, Prices, Demand)) + aes(x = Prices, y = Demand) +
geom_point() +
geom_line(data = df.logistic, aes(x = Prices, y = Demand.fitted), color = 'blue')


ggplot(select(df.logistic, Prices, Profit)) + aes(x = Prices, y = Profit) +
geom_point() + geom_vline(xintercept = p.max.profit, lty = 2) +
geom_line(data = df.logistic, aes(x = Prices, y = Profit.fitted), color = 'blue')


I hope you liked the examples. In the next post we will discuss about choice models, which are demand models when products are heterogeneous. Goodbye and good luck!

## References

Phillips, Robert Lewis. Pricing and revenue optimization. Stanford University Press, 2005.