7.1 Trade Discounts

A. Introduction to Mathematics of Merchandising

Imagine walking into a shopping mall during the holiday season. Every store displays an array of products, each tagged with a price, and every discount sign represents a carefully calculated decision. This is where the unseen world of merchandising mathematics plays a pivotal role.

In this chapter, we will explore the fundamental mathematical concepts essential to successful merchandising. We start by understanding the supply chain and the relationships between its members. Moving forward, we will delve into the calculations of trade discounts, both single and series, and understand the significance of cash discounts in financial transactions.

The art of pricing, through markup and markdown strategies, will be our next focus, uncovering how businesses balance profitability with market competitiveness. We will then study integrated real-world problems, providing a practical perspective on how mathematics is applied in various merchandising scenarios.

Merchandising is the strategic promotion and sale of products to consumers. It involves a range of activities including product selection, pricing, display, and advertising. The primary goal of merchandising is to attract customers and motivate purchases, thus driving sales and profitability for businesses. It is a key component in retail, but it also extends to various other sectors where goods are sold, be it physical products or online services.

The supply chain encompasses the entire process of producing and delivering a product or service, from the sourcing of raw materials to the delivery of the final product to the consumer. The supply chain is concerned with the flow of materials, information, and finances as they move from manufacturer to wholesaler to retailer to consumer.

Figure 7.1.1 Members of a typical supply chain

As shown in Figure 7.1.1, the typical members of a supply chain include:

• Manufacturers/Producers: They create the product, sourcing raw materials and turning them into finished goods.
• Distributors/Wholesalers: They buy in bulk from manufacturers and sell in smaller quantities to retailers or sometimes directly to consumers.
• Retailers: They sell products to the end consumer, either from physical stores or online platforms.
• Consumers: The end-users who purchase and use the products.

Some organizations may bypass one or more stages of the traditional supply chain. For instance, a manufacturer might also act as a retailer, selling directly to consumers, Apple is a notable example of this. Similarly, a wholesaler can sometimes also be a retailer, as is the case with Costco.

Each member of the supply chain adds costs to the product, from production, handling, and transportation to storage and marketing. As a product moves through the supply chain, these costs accumulate, and each member, except the consumer, adds a markup to cover their costs and earn a profit. The cumulative effect of these markups contributes to the final price paid by the consumer.

B. Trade Discounts

Manufacturers usually recommend the price a product should be sold to the consumer. This price is referred to as the Manufacturer’s Suggested Retail Price (MSRP), catalog price, or list price. It represents the intended final sale price before any discounts are applied. When manufacturers set the MSRP, they often take into account the potential discounts they will offer to supply chain members. When a manufacturer sells a product to the members of the supply chain, they deduct a certain amount from their list price, usually a percentage of the list price. This is known as a trade discount. They are often used in business-to-business transactions such as the name ‘trade’ discount. Trade discounts can serve other functions:

• Volume Incentives: Larger orders are often incentivized with bigger discounts, encouraging bulk purchases and ensuring more significant sales for manufacturers.
• Promoting New Products: Discounts can be used to encourage retailers to stock and promote new or less popular products.
• Maintaining Relationships: Offering discounts helps in building and maintaining long-term business relationships between various supply chain members.
• Competitive Strategy: In highly competitive markets, manufacturers might offer trade discounts to make their products more attractive to retailers compared to rival products.

Each member of the supply chain offers successive buyer trade discounts. the final price that a buyer pays for a product after all trade discounts have been applied is called the net price. If only one trade discount is offered to a buyer, it is called a single trade discount. If a buyer is qualified for more than one trade discount, the discount is referred to as a trade discount series.

C. Single Trade Discount

A single trade discount involves only one discount percentage being applied to the list price of a product. The amount of trade discount, denoted as ‘A’, is calculated by multiplying the discount rate, represented by ‘d’, by the list price, denoted as ‘L’.

 [asciimath]A=L*d[/asciimath]Formula 7.1a

The net price, denoted as ‘N’, is determined by subtracting the discount amount from the list price.

 [asciimath]N=L-A[/asciimath]Formula 7.2

By inserting the discount amount from Formula 7.1a into Formula 7.2, we establish a relationship between the net price and the list price, expressed as

 [asciimath]N = L - L*d[/asciimath]

 [asciimath]N= L(1 - d)[/asciimath]Formula 7.3a

The term [asciimath](1-d)[/asciimath], known as the net price factor (NPF), yields the net price when multiplied by the list price, similar to how the discount rate determines the discount amount. The discount rate and the net price factor are complementary to each other, and they are typically expressed as percentages. Formula 7.3a can also be expressed in terms of the net price factor.

 [asciimath]N=L*NPF[/asciimath]

where [asciimath]NPF=1-d[/asciimath].

Sometimes, we may know the discount amount and need to find the original list price or the discount rate. In these instances, we can adjust Formula 7.1a to isolate and solve for the unknown variable.

• If looking for the list price: [asciimath]L=A/d[/asciimath]   Formula 7.1b
• If looking for the trade discount rate: [asciimath]d=A/L[/asciimath] Formula 7.1c

When we know the net price and need to find the original list price or the discount rate, we can rearrange Formula 7.3a to isolate and solve for the unknown variable.

• If looking for the list price: [asciimath]L=N/(1-d)[/asciimath] Formula 7.3b
• If looking for the trade discount rate: [asciimath]d=1-N/L[/asciimath]    Formula 7.3c

D. Trade Discount Series

As previously mentioned, there are various reasons why trade discounts are offered. Occasionally, a buyer might qualify for multiple discounts, such as one for placing a large order and another for a marketing incentive. When a buyer is offered multiple discount rates, this set of discounts is known as a trade discount series. In such cases, When calculating the net price it is important to apply each discount rate in the series sequentially, rather than combining them into a single rate. For instance, if an item’s list price is $100 and the trade discount series offered is 10%, 20%, and 30%, the calculation should proceed in stages:

• Net price after the first discount rate [asciimath]= 100(1-10%) = $90[/asciimath]
• Net price after the second discount rate [asciimath]= 90(1-20%) = $72[/asciimath]
• Net price after the third discount rate [asciimath]= 72(1-30%) = $50.40[/asciimath]

Figure 7.1.2: Example of sequential application of discount series

To simplify the calculation of the final net price with multiple trade discounts, we can extend the Net Price Factor (NPF) method to a discount series. This approach consolidates all the calculations into one step. Here, the NPF for each discount rate is first multiplied together, and then this cumulative NPF is applied to the list price. This method is more efficient than applying each discount rate one at a time. For instance:

With a single discount rate, the net price formula is: [asciimath]N=L(1-d)[/asciimath]

For two discount rates, it becomes: [asciimath]N=L(1-d_1)(1-d_2)[/asciimath]

For three discount rates, the formula is: [asciimath]N=L(1-d_1)(1-d_2)(1-d_3)[/asciimath]

In general, if there are ‘n‘ discount rates, the net price is calculated using the formula:

 [asciimath]N=L(1-d_1)(1-d_2)...(1-d_n)[/asciimath]Formula 7.4

Where the product of the individual net price factor is the overall net price factor:

 [asciimath]NPF=underbrace((1-d_1))_(NPF_1)underbrace((1-d_2))_(NPF_2) ...underbrace((1-d_n))_(NPF_n)[/asciimath]

E. Single Equivalent Trade Discount for a Discount Series

When dealing with multiple discounts, it can be complex and time-consuming to understand the total impact. A single equivalent rate simplifies this by providing a clear, concise figure that represents the overall effect of all the discounts combined. This makes it easier to compare different discount strategies or options. The equivalent rate of discount for a discount series is a single percentage that represents the cumulative effect of multiple trade discounts applied in sequence. It is a way to summarize the overall impact of a discount series as if it were a single discount rate.

For a single trade discount, the discount rate [asciimath]d[/asciimath] is calculated as [asciimath]d=1-NPF[/asciimath]. Similarly, for a series of discounts, the single equivalent discount rate [asciimath]d_(eq)[/asciimath] is determined as one minus the overall net price factor. Therefore, you can obtain [asciimath]d_(eq)[/asciimath] by using the formula

[asciimath]d_(eq)=1-NPF[/asciimath]

 [asciimath]d_(eq)=1-[(1-d_1)(1-d_2)...(1-d_n) ][/asciimath]Formula 7.5

This single equivalent discount rate, when applied to the same list price, results in the same net price as that achieved through the series of discount rates.

Section 7.1 Exercises