There is a simple inverse relation between the optimal markup and the price sensitivity of demand. The optimal markup is large when the underlying price elasticity of demand is low; the optimal markup is small when the underlying price elasticity of demand is high.

Optimal Markup on Cost

Recall from Chapter 4 that there is a direct relation among marginal revenue, price elasticity of demand, and the profit-maximizing price for a product. This relation was expressed as

To maximize profit, a firm must operate at the activity level at which marginal revenue equals marginal cost. Because marginal revenue always equals the right side of Equation, at the profit-maximizing output level, it follows that MR = MC and

AND

Equation 12.8 provides a formula for the profit-maximizing price for any product in terms of its price elasticity of demand. The equation states that the profit-maximizing price is found by multiplying marginal cost by the term

To derive the optimal markup-on-cost formula, recall from Equation that the price established by a cost-plus method equals cost multiplied by the expression (1 + Markup on Cost). Equation implies that marginal cost is the appropriate cost basis for cost-plus pricing and that

By dividing each side of this expression by MC and subtracting 1 yields the expression

After simplifying, the optimal markup on cost, or profit-maximizing markup-on-cost, formula can be written

The optimal markup-on-cost formula can be illustrated through use of a simple example. Consider the case of a leading catalog retailer of casual clothing and sporting equipment that wishes to offer a basic two-strap design of Birkenstock leather sandals for easy on-and-off casual wear. Assume the catalog retailer pays a wholesale price of $25 per pair for Birkenstock sandals and markets them at a regular catalog price of $75 per pair. This typical $50 profit margin implies a standard markup on cost of 200 percent because

In a preseason sale, the catalog retailer offered a discounted “early bird” price of $70 on Birkenstock sandals and noted a moderate increase in weekly sales from 275 to 305 pairs per week. This $5 discount from the regular price of $75 represents a modest 6.7 percent markdown. Using the arc price elasticity formula, the implied arc price elasticity of demand on Birkenstock sandals is

In the absence of additional evidence, this arc price elasticity of demand EP = –1.5 is the best available estimate of the current point price elasticity of demand. Using Equation, the $75 regular catalog price reflects an optimal markup on cost of 200 percent because

Optimal Markup on Price

Just as there is a simple inverse relation between a product’s price sensitivity and the optimal markup on cost, so too is there a simple inverse relation between price sensitivity and the optimal markup on price. The profit-maximizing markup on price is easily determined using relations derived previously. Dividing each side of Equation by P yields the expression

Subtracting 1 from each side of this equation and simplifying gives

Then, multiplying each side of this expression by –1 yields

Notice that the left side of Equation is an expression for markup on price. Thus, the optimal markup-on-price formula is

The optimal markup-on-price formula can be illustrated by continuing with the previous example of a catalog retailer and its optimal pricing policy for Birkenstock leather sandals. As you may recall from that example, the catalog retailer pays a wholesale price of $25 per pair for Birkenstock sandals, markets them at a regular catalog price of $75 per pair, and the arc price elasticity of demand EP = –1.5 is the best available estimate of the current point price elasticity of demand. This typical $50 profit margin implies a standard markup on price of 66.7 percent because

If it can again be assumed that the arc price elasticity of demand EP = –1.5 is the best available estimate of the current point price elasticity of demand, the $75 regular catalog price reflects an optimal markup on price because

Table shows the optimal markup on marginal cost and on price for products with varying price elasticities of demand. As the table indicates, the more elastic the demand for a product, the more price sensitive it is and the smaller the optimal margin. Products with relatively less elastic demand have higher optimal markups. In the retail grocery example, a very low markup is consistent with a high price elasticity of demand for milk. Demand for fruits and vegetables during their peak seasons is considerably less price sensitive, and correspondingly higher markups reflect this lower price elasticity of demand.

Another Optimal Markup Example

The use of the optimal markup formulas can be further illustrated by considering the case of Betty’s Boutique, a small specialty retailer located in a suburban shopping mall. In setting its

Optimal Markup on Marginal Cost and Price at Various Price Elasticity Levels

initial $36 price for a new spring line of blouses, Betty’s added a 50 percent markup on cost. Costs were estimated at $24 each: the $12 purchase price of each blouse, plus $6 in allocated variable overhead costs, plus an allocated fixed overhead charge of $6. Customer response was so strong that when Betty’s raised prices from $36 to $39 per blouse, sales fell only from 54 to 46 blouses per week. Was Betty’s initial $36 price optimal? Is the new $39 price suboptimal? If so, what is the optimal price? At first blush, Betty’s pricing policy seems clearly inappropriate. It is always improper to consider allocated fixed costs in setting prices for any good or service; only marginal or incremental costs should be included. However, by adjusting the amount of markup on cost or markup on price employed, Betty’s can implicitly compensate for the inappropriate use of fully allocated costs. It is necessary to carefully analyze both the cost categories included and the markup percentages chosen before judging a given pricing practice.

To determine Betty’s optimal markup, it is necessary to calculate an estimate of the point price elasticity of demand and relevant marginal cost, and then apply the optimal markup formula. Betty’s standard cost per blouse includes the $12 purchase cost, plus $6 allocated variable costs, plus $6 fixed overhead charges. However, for pricing purposes, only the $12 purchase cost plus the allocated variable overhead charge of $6 are relevant. Thus, the relevant marginal cost for pricing purposes is $18 per blouse. The allocated fixed overhead charge of $6 is irrelevant for pricing purposes because fixed overhead costs are unaffected by blouse sales.

The $3 price increase to $39 represents a moderate 7.7 percent rise in price. Using the arc price elasticity formula, the implied arc price elasticity of demand for Betty’s blouses is

If it can be assumed that this arc price elasticity of demand _P = –2 is the best available estimate of the current point price elasticity of demand, the $36 price reflects an optimal markup of 100 percent on relevant marginal costs of $18 because

Similarly, the $36 price reflects an optimal markup on price because

Betty’s actual markup on relevant marginal costs per blouse is an optimal 100 percent, because

Similarly, Betty’s markup on price is an optimal 50 percent, because

Therefore, Betty’s initial $36 price on blouses is optimal, and the subsequent $3 price increase should be rescinded. This simple example teaches an important lesson. Despite the improper consideration of fixed overhead costs and a markup that might at first appear unsuitable, Betty’s pricing policy is entirely consistent with profit-maximizing behavior because the end result is an efficient pricing policy. Given the prevalence of markup pricing in everyday business practice, it is important that these pricing practices be carefully analyzed before they are judged suboptimal.

The widespread use of markup pricing methods among highly successful firms suggests that the method is typically employed in ways that are consistent with profit maximization. Far from being a naive rule of thumb, markup pricing practices allow firms to arrive at optimal prices in an efficient manner.