6 November 2000

Pricing Practices

Lecture Outline

• Markup Pricing
• Markup Pricing and Profit Maximization
• Price Discrimination
• Multiple Product and Joint Product Pricing
• Transfer Pricing: Pricing Across Divisions of a Firm

Lecture Notes:

Markup Pricing

Markup on cost: This form of markup pricing generates a profit margin (P-Cost, similar to profit contribution) expressed as a percentage of cost. This form of markup pricing is also referred to as cost-plus pricing:

Percentage Markup on cost = [(Price-Cost)/Cost]*100

Where 'Cost' refers to an average variable cost measure, plus an allocation of variable overhead cost (costs which vary with output but which cannot easily be directly attributable to a particular unit or units of ouput).

For example, suppose that AVC = $10, and that the firm allocates another $10 for variable overhead cost, to yield a unit cost of $20. If the firm routinely sets price equal to 150% of unit cost, then price = $1.5(20) = $30. Thus the

percentage markup on cost = [(30 - 20)/20]*100 = 50%

Thus, as shown above, if the firm has data on unit cost and their markup percentage, then:

markup price = (unit cost)*(1 + markup) = $20(1.5) = $30.

Alternatively, markup on price is a system in which firms calculate their markup as a percentage of price:

markup on price = (P - Cost)/P

One form of markup is not necessarily better or worse than the other; they simply offer different ways of expressing profit margins.

Optimal markup pricing should vary based on peak vs. off-peak capacity utilization periods. During peak periods, the firm should used fully allocated costs, including overhead costs. On the other hand, during off-peak periods there is excess capacity, and the firm must pay for the fixed cost of that capacity regardless of whether its utilized or not. Thus during off-peak periods it may make sense to charge prices that reflect a markup on incremental costs, which are primarily direct variable costs, rather than on fully allocated costs, in order to encourage sales (and profit contribution) during slack periods. For example, restaurants commonly charge lower prices on their lunch menu than on their dinner menu for similar quantities of food, which may reflect the fact that most sit-down restaurants have excess capacity during the day. Similarly, airlines offer lower rates during off-peak flying times, and movie theaters offer 'matinee' discount prices for movie screenings during the day.

On the demand side, we can argue that the key issue is to set markups based on 'price sensitivity' or elasticity of demand. If elasticity of demand is quite high, people perceive the product as having many substitutes, or people's incomes are relatively low and so the purchase is an important part of their budget. Either way, high elasticities indicate that a profit-maximizing firm should charge a low markup, while low elasticities (items with few substitutes or items that are an inconsequential element of a consumer's budget) indicate little price sensitivity, and a high markup will enhance profit. Recall from our previous chapter on product differentiation that high consumer travel cost (or strong brand loyalty) yield higher equilibrium prices in Hotelling-style models, which is consistent with the fact that high travel cost or strong brand loyalty means that the consumers have low demand elasticities for the particular products.

EXAMPLE: SEE TABLE 12.1 in HIRSCHEY/PAPPAS TEXT

Other examples: Senior citizen and children's discounts reflect the fact that these subsets of the population have less income and so are more price sensitive (still true for families with children, but much less true for seniors).

The fundamental idea is similar to that of congestion or peak-load pricing on toll bridges, toll highways, telephone long-distance service, and mainframe computer CPU time. Namely, when a facility is subject to congestion effects, some people are unable to get service at certain times because of binding capacity constraints. A profit-maximizing enterprise would like to make sure that the people who are unable to get service are on the lower end of the demand curve, with relatively low willingness-to-pay. In other words, congestion or peak-load pricing rises so that QS (at capacity) = QD. When this is seen as being unethical, such as during natural disasters, or is seen as being counter to maintaining customer 'goodwill', then alternative scarcity allocation schemes are needed, like reservation systems or determininations of 'need' or lotteries or queues (waiting lines).

Markup Pricing and Profit Maximization

P = MC*{1/(1+1/Ed)}

where Ed is the price elasticity of demand (without having taken absolute values), and is negative when the law of demand holds. Note that requiring a non-negative price is equivalent to requiring that the firm not sell to the inelastic portion of its demand curve (Ed less than 1 in absolute value), so that MR = MC at nonnegative values. So if the firm has an estimate of its marginal cost and price elasticity of demand at present output levels, the firm's optimal price can be derived rather easily by way of the formula given above.

We can also use the optimal pricing formula to derive the optimal, profit-maximizing markup rule. If we use a 'markup on cost' rule, recall that the markup price established by way of a cost-plus rule is established as follows:

markup price = (unit cost)*(1 + markup)

If we use marginal cost to be our cost factor for 'unit cost', we can re-write this optimal markup price equation as:

markup price = MC*(1 + markup)

If we set this markup price equal to the optimal, profit-maximizing price from the elasticity formula, we get:

MC*[1 + markup] = MC*{1/(1 + 1/Ed)}

we can then cancel the MC terms and express the optimal, profit maximizing markup as:

optimal markup = {1/(1 + 1/Ed)} - 1

With a bit of algebraic manipulation (creating common denominators), we get:

optimal, profit-maximizing markup on cost = -1/(1 + Ed)

In percentage terms, this optimal markup is:

100*(-1/(1 + Ed))

Note that the above rule is subject to a non-negativity constraint on price, which in turn requires that Ed must be elastic (i.e., Ed be no smaller than 1 in absolute value). The profit-maximizing markup is inversely related to the price elasticity of demand. When demand is only somewhat elastic (e.g., Ed = -1.1), the markup is 10, or 1000%, while when demand is much more elastic (e.g., Ed = -3), then the markup is .5, or 50%.

Price Discrimination

• A good made in Arcata is priced at $10 in Arcata and $12 in Portland, when the price difference reflects shipping costs
• Last-minute airline reservations are more expensive than ones made several weeks in advance, when the price difference reflects the cost that last-minute reservations place on airlines that are not given the time to select the optimal sized aircraft for a given flight)
• Charging less for child-size portions, when the price difference reflects the reduced cost of producing a smaller unit size
• Charging higher prices for delivered goods than when the customer picks up, when the price difference reflects the delivery cost

• There must be consumers (or market segments) with different price elasticities of demand
• The firm must be able to identify these consumers (or market segments), either directly or indirectly through their revealed preference
• The firm must be able to prevent arbitrage (resales from high elasticity consumers or market segments to low elasticity consumers or market segments)

Degrees of PD:

• First-degree PD: The firm charges each consumer exactly their willingness-to-pay. Under first-degree price discrimination, the firm gets all the gains from trade. Tyranny. Requires that the firm have a great deal of market power and a great deal of information on seller valuations.
• Second-degree PD: Charging different prices for different size of purchase quantities. High markups on small orders, and smaller markups on larger orders. Note that second-degree PD only occurs when the markup exceeds any handling cost difference for small vs. large orders (i.e., for large orders, the unit handling cost is lower than for small orders). Often times we see second-degree PD that creates a wholesale-retail price difference, or a retail rate-commercial (contractor) rate difference.
• Third-degree PD: Perhaps the most common form of PD. The firm separates its overall demand into segments that are distinguished by their elasticity characteristics, and charges different prices to different segments. A prime example are senior/student discounts at movie theaters, on airline flights, restaurants, and for newspaper and magazine subscriptions (there are many other examples). Another is that under traditional (non-HMO) health care, hospitals used to charge people with insurance higher rates than those without, because those who pay full price will be much more price sensitive than those whose insurance companies pay.

Real-World Example of Third-degree PD:

As reported in the 9 May 1996 issue of the New York Times, a Federal judge in Chicago gave preliminary approval to an amended $351 million settlement of a lawsuit over pricing by some of the nation's biggest pharmaceutical companies. The original suit was brought in 1994 by nearly 40,000 retail pharmacies, against several major drug manufacturers. The pharmacies, which as a group represent approximately 45 percent of the retail drug business, argued that they were being unfairly charged higher wholesale prices for drugs than HMOs and other managed care businesses, which together represent about 55 percent of the retail drug market. Part of the agreement required that the drug companies not deny discounts to retail pharmacies simply because they were not HMOs or managed care customers.

Now do we think that this example of PD is an example of drug companies having wholesale market power, and that retail pharmacies have lower price elasticity of demand than HMOs? Or do we think that the big HMOs have some monopsony power and so can negotiate better prices? The latter case has an interesting antitrust history. In the 1930s, the rise of supermarkets like A&P led to these big firms having the capacity to negotiate more favorable wholesale prices than the "mom and pop" stores. These "mom and pop" stores collectively had political clout, however, and the result was that many states enacted "fair trade" laws that allowed companies to set minimum (wholesale) prices, thus abrogating the big chains' wholesale market power. The Miller Tydings Act of 1937 provided Federal support for these state "fair trade" laws, and remained in place until the passage of the Consumer Goods Pricing Act of 1975.

Practice Problem, Third-degree PD:

Suppose that a major college has its football team in a major Bowl. Suppose further that the college controls ticket sales as a monopolist, can prevent arbitrage with campus cops (!), and has a very good estimate of each market segment's demand. Suppose that alumni ticket inverse demand for this event is estimated to be:

Pa = $205 - $.006Q

While the inverse demand for tickets by students is estimated to be:

Ps = $105 - $.002Q

Suppose that the marginal cost of seating at the college sporting event is given by:

MC = $5

Then if the college is a monopoly supplier of access to the sporting event (i.e., there are no tall adjacent buildings where professors can sell window views...), the college will maximize its profit by selling tickets at different prices to these two different market segments:

ALUMNI:

MR = MC: 5 = 205 - $.012Q; Q = 16,667 alumni tickets

Pa = 205 - .006*(16,667) = $105

STUDENTS:

MR = MC: 5 = 105 - .004Q; Q = 25,000 student tickets

Ps = 105 - .002*(25,000) = $55

IN-CLASS EXERCISE:

• Calculate the college's profit from the above 3rd degree PD scheme
• Now suppose instead that the college charges a uniform monopoly price to the combined student/alumni demand.
  • Re-write each market segment demand in terms of Q = a - bP
  • Sum individual demands (Qtot = Qs + Qa)
  • Re-write this overall market demand back in inverse form: P = x - yQ
  • Calculate MRtot, set it equal to MC, find the optimal Q and P
  • Calculate profit when the college charges a single, uniform monopoly price for tickets
• Compare profits under 3rd degree PD with the uniform pricing scheme.

Multiple Product and Joint Product Pricing

Consider the most simple illustration of demand interactivity -- a two-good product line of a monopolist with constant marginal costs. The monopolists problem is to maximize the joint profit from the two product lines:

profit, good A: (Pa - Ca)Da(Pa, Pb)

profit, good B: (Pb - Cb)Db(Pa, Pb)

To choose the profit-maximizing Pa and Pb, the monopolist sets the derivatives of profits with respect to each price equal to zero (note that I'm using 'd' to refer to a partial derivative):

d(total profit)/dPa = d(profit, good a)/dPa + d(profit, good b)/dPa = 0

d(total profit)/dPb = d(profit, good a)/dPb + d(profit, good b)/dPb = 0

Example: Consider a camera shop that sells both cameras and film. Using the analysis given above, and recognizing that cameras and film are complementary goods, it is entirely possible that the optimal set of prices is such that the shop charges a price below cost for cameras if doing so sufficiently increases demand for film, allowing the increase in profit from film to offset the decline in profit from cameras.

In addition to demand interrelationships, firms can also have production interrelationships. One example are production complementarities. For example, a forest products mill produces waste wood that can easily be burned as a biomass fuel to generate electricity. A professor can produce a textbook as a byproduct of the teaching process. These are often times referred to as joint products.

EXAMPLE: Suppose that a doughnut shop sells both doughnuts as well as the doughnut 'centers', and that these products are (of course) produced in fixed proportions of 1:1.

Suppose that estimated demand for doughnuts (per dozen) is:

Pd = $5 - $.015Qd

Implying that MRd = $5 - $0.03Qd

The estimated demand for doughnut centers (per dozen) is:

Pc = $2 - $0.01Qc

Implying that MRc = $2 - $0.02Qc

Suppose that the total cost of producing doughnuts is given by:

TC = $150 + $1Q + 0.01Q²

Implying that MC = 1 + 0.02Q.

To determine joint marginal revenue, we must first construct joint total revenue:

TR = PdQd + PcQc = (5 - .015Qd)Qd + (2 - 0.01Qc)Qc

Note that Qd = Qc = Q because of fixed proportions. Thus joint total revenue is:

TR = 7Q - 0.025Q²

So joint marginal revenue is:

MR = 7 - 0.05Q

Optimal output is given by the following condition:

MR=MC: 7 - .05Q = 1 + .02Q

Implying that Q = 114.29 (dozen) doughnuts (and centers) will maximize profit from joint production. At Q = 114 dozen, the market-clearing price per dozen for doughnuts is:

Pd = $5 - $.015*114 = $3.29

While the market-clearing price per dozen of doughnut centers is:

Pc = $2 - $0.01*114 = $0.86

Transfer Pricing: Pricing Across Divisions of a Firm

When there is no external market for the product/input, as when McDonalds corporation 'sells' worker training services to its corporate-owned McDonalds outlets, then the optimal transfer price should be the marginal cost of the service. Why? Because if price is below marginal cost, excess training will be consumed, while if price if above marginal cost, then less than the optimal level of training will be consumed (recall that P = MC is required for optimal resource allocation).

The transfer pricing problem is a bit more complicated when there is an external market for the product/input being transferred across profit centers. The typical case is one in which a vertically integrated firm can produce an input for internal use, or for sale in an imperfectly competitive external market. In this case the optimal transfer price must 'clear the market' at an output level (Q) where MC = joint MR from both internal and external sales.