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How to reprice 15 products when price sensitivity differs by product

Course project, Michigan Ross MBA. Multinomial logit demand model and constrained price optimization in R. Not client or employer work.

R
R scriptDemand model and price optimization
SituationA retailer sells a line of 15 related products. Each product has its own cost and price.
ComplicationCustomers do not react to price in the same way for each product: features such as import origin change the sensitivity. One price rule for the full line leaves profit unused.
QuestionWhich price for each product gives the highest expected profit, within 20% of the price today?
AnswerRaise 12 prices and cut 3. Expected profit per customer rises 31%, with each price inside a ±20% limit.

Profit per customer is margin times the chance of a purchase, and the price moves the two in opposite directions

  • Expected profit per customer$1.08 today, $1.42 in the answer
    • Margin on each product
      • PriceThe lever: 15 prices

        Which prices change?

        12 up, 3 down. Eight go to the +20% limit.

      • Unit costFixed
    • Chance that the customer buys the product
      • Price sensitivityEstimated from 28,044 customer visits

        Does the sensitivity differ by product?

        Yes. Lower for imported and straight-shape products.

      • Price rangeThe limit: within 20% of the price today

        Does the limit decide the answer?

        Yes, for 8 products. The model asks for more, outside the range of the data.

      • Share of customers who buyThe limit: what the gain costs

        Does the share fall?

        Yes. From 10.5% to 7.2%.

Lever what the decision changes Limit what the answer must respect Fixed outside the decision

Expected profit per customer

+31%

$1.08 to $1.42, with each price inside a ±20% limit

Price changes

12 up, 3 down

8 products stop at the +20% limit

Share of customers who buy

10.5% to 7.2%

fewer sales at a higher margin: the cost of the gain

Twelve prices go up, eight of them to the limit, and three high-margin prices come down

ProductUnit costPrice nowNew priceChangeMargin nowNew margin
Product 1$35.00$36.04$43.25+20%*3%19%
Product 2$31.41$39.72$47.67+20%*21%34%
Product 3$41.45$47.19$56.63+20%*12%27%
Product 4$41.51$51.38$61.65+20%*19%33%
Product 5$81.97$118.20$141.84+20%*31%42%
Product 6$83.54$121.54$145.85+20%*31%43%
Product 7$51.04$86.86$71.00-18%41%28%
Product 8$56.55$84.08$74.90-11%33%24%
Product 9$78.16$99.79$119.74+20%*22%35%
Product 10$13.45$18.08$21.69+20%*26%38%
Product 11$82.27$102.80$108.96+6%20%24%
Product 12$84.31$87.36$93.39+7%3%10%
Product 13$83.84$85.10$92.93+9%1%10%
Product 14$98.18$119.01$112.53-5%18%13%
Product 15$102.72$134.21$141.76+6%23%28%

* The price stops at the +20% limit. Without the limit, the model asks for a larger increase, outside the range of prices in the data.

Customers are less sensitive to the price of imported and straight-shape products

Multinomial logit on 28,044 customer visits; 11% end with a purchase. A positive value makes the price effect weaker for products with that feature. "Clear" means clear at the 5% level.

The model

Demand

Each customer chooses one of 15 products or no purchase. The price effect is different for each product, through six product features.

Decision

The 15 prices, each within 20% of the current average price.

Objective

The highest expected profit per customer: margin times the probability of a purchase, summed over the products.

  • Eight products go to the +20% limit, so the limit decides most of the answer. A test in the market must come before larger increases.
  • The gain has a cost: the share of customers who buy falls from 10.5% to 7.2%. If customer count has value beyond this purchase, the increase must be smaller.
  • Products 7, 8 and 14 have high margins and price-sensitive customers. A lower price brings more than enough volume.
  • The demand model has no competitor prices. The result assumes that competitors do not react.