# Should you change a price?

Formulation of the problem

First, it is crucial to stress a very important yet often neglected principle: pricing and costing are two distinct and preferably separate issues. Unless a company has no competitor and can set its price (and even then), the price is not driven by what it costs to make the product. Price is driven by what customers are willing to pay for that product. In other words, the foundation of price is not the cost for the company, but the utility for the customer. The only situation where cost is a valid basis for pricing is when it is required in a governmental contract. This principle is very important because setting prices based on costs can trigger a death spiral.

A death spiral is the process by which a company goes out of business by calculating unit cost by allocating overhead to a diminished volume of business, raising prices to cover the resulting higher unit cost, further diminishing the volume of business.

The decision to change a price is therefore not based on the evolution of the unit cost, but solely on a trade-off between the volume impact and the price impact of the decision.

Classification based on relevance

In a pricing decision, only the impact on the contribution margin due to the change in volume and the change in price are relevant. To assess the price impact is straightforward once we know the changes in volume. As for the change in volume induced by the change in price, it requires making an assumption about the price elasticity of demand.

The price elasticity of demand ($$E_p^d$$) is a measure of the responsiveness of consumers to a change in a product’s price. It is calculated as the percentage of change in the demand divided by the percentage change in the price.

$E_p^d = \frac{\% \Delta Q}{\% \Delta P} = \frac{\frac{Q_A - Q_B}{Q_B}}{\frac{P_A - P_B}{P_B}} \\ \leftrightarrow \\ \% \Delta Q = E_p^d \times \% \Delta P = E_p^d \times \frac{P_A - P_B}{P_B}$

Where $$Q$$ refers to the volume (or demand), $$P$$ to the price, and the indices $$A$$ and $$B$$ indicate respectively after the change and before the change in price. Assuming a demand elasticity and using this formula, it is possible to infer the change in volume resulting from a change in price and then compute the impacts.

Net Economic Impact and indifference points

For this exercise, you can download here data and solutions if you did not already. Try to compute the net economic impact and the indifference point in volume change ($$\Delta Q$$) and in price change ($$\Delta P$$) assuming that you decrease the price of Cosmo by 0.4 euros (from 8 to 7.6) and that the price elasticity of demand is -3. The volume impact and price impact are represented in the following equation:

\begin{aligned} NEI & = + \Delta Q \times UCM_B && + (Q + \Delta Q) \times \Delta P \\ & = + (Q_A-Q_B) \times (P_B - V_{cB}) && + Q_A \times (P_A-P_B) \end{aligned}

Qualitative factors

Prices affect the perceived value of a product. Moreover they are easier to reduce than to increase. Therefore changes in prices should not be taken lightly because they may have a huge impact on product image and future sales.

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