Pricing under Monopoly Conditions

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Pricing under Monopoly Conditions

Accounting and Management Information System 3300

The Ohio State University

David E. Wallin

Version: Autumn 2014
While true monopolies are difficult to find, consideration of monopoly pricing allows a useful exploration of a number of accounting issues. We shall explore a number of issues relative to pricing in a monopoly.

A First Example

The subdivision of Acorn Acres (AA) is a new, upscale, 100-home development in the town of Croniesville. Residents of AA have complained to the town council recently about the times of the day lawns were mowed and had expressed a safety concern regarding lawn mowing companies operating in the subdivision. The town council subsequently passed legislation that limited mowing hours and permitted only one firm, Acorn Acres Lawn Service (AALS), to sell mowing services in AA.

Roscoe Arburkle is the town council president and owner of AALS, which has never previously conducted business. In preparation for the upcoming mowing season, Arbuckle has contracted with a large regional landscaping service. They will install a dedicated AALS phone line in the headquarters and provide staff who will answer the phones as AALS “employees” to schedule service and deal with billing issues. They will also do the actual mowing, arriving onsite in their trucks (with AALS sign attached to the side). They will collect mowing fees for AALS and forward the net proceeds to AALS. The net proceeds will be the revenue from mowing less 1) a $23 per lawn mowing fee and 2) a $200 per week charge (during the “mowing season”) for the other services noted above.

Arbuckle is aware that in AA, there exists one person who is willing to pay $100 for each weekly mowing. We will refer to that person as C100. No one is willing to pay more than $100. There is a person willing to pay $99 (C99), one willing to pay $98 (C98), one willing to pay ($97), and so forth, with one willing to pay $2 (C2), and one willing to pay $1 (C1).

Single-Price Profit Maximization

Brute-Force Solution

Assume Arbuckle is going to set one price for lawn mowing. We will focus on the weekly revenue and expenses during the mowing season. If he sets the price at $100, only C100 is willing to order lawn service. The revenue is $100, and the regional firm will charge AALS $23 for the mowing plus $200 in the fixed fee. AALS would have a loss of $123 ($100 – $23 – $200). AALS has a contribution margin (revenue minus variable costs) of $77 ($100 – $23); this is also what economists call the producer surplus. C100 is no better or worse off for the presence of AALS, as he is paying exactly his reservation price for the service (it’s worth $100; he pays $100). If having the customer buy even if the price exactly equals their reservation price seems troubling now, we will see later it is not. Of course Arbuckle would rather close down AALS than charge $100 and lose $123.

If Arbuckle sets the price at $99, AALS can obtain two customers: C100 and C99. The revenue is $198 (2 × $99), and the expenses are $46 (2 × $23) for the mowing and $200 for the fixed fee. AALS would have a contribution margin of $152 ($198 – $46) and a loss $48, still not worth doing business, but better than before. We will find it easier in later steps to do the math slightly differently. AALS charges $99 and has variable expenses of $23 per lawn, producing a $76 contribution margin per lawn. With two customers, they have a contribution margin of $152 (2 × $76) and the $48 loss after the fixed cost. AALS has a producer surplus equal to the contribution margin of $152. While C99 is no better or worse off for using AALS, C100 is better off by $1. That is, while C100 would have paid $100, he is paying $99 instead. That $1 is what economists would call the consumer surplus.

Setting the price at $98 yields a contribution margin per lawn of $75 ($98 – $23) and three customers: C100, C99, and C98. The contribution margin is $225 (3 × $75). After subtracting fixed costs, profit is $25. The producer surplus is $225, and the consumer surplus is $3 ({$100 – $98} + {$99 – $98}). A price of $97 yields four customers with a $74 contribution margin per lawn, a contribution margin and producer surplus of $296, a profit of $96, and consumer surplus of $6. A price of $93 produces a contribution margin and producer surplus of $560 (8 × {$93–23}), a profit of $360, and consumer surplus of $28.

It might seem lowering the price keeps making things better: AALS has higher profits and the consumer surplus goes up each time we lower the price. Does this continue? The answers are “not always” and “yes” in that order. Let’s skip ahead and try a price of $23. Obviously, AALS can’t charge less than that (so long as they are charged $23 for each mowing). At a $23 price, there would be 78 customers (C100 down to C23), but with a $0 contribution margin per lawn, total contribution margin (and producer surplus) would also be $0 (78 × $0). This results in a loss $200. There is good news for consumers, though. The consumer surplus is $3,003. C100 has a surplus of $77 ($100 – $23), C99 has a surplus of $76 ($99 – 23), and so forth, with C24 having a surplus of $1 ($24 – $23) and C23 having a surplus of $0 ($23 – $23). This gives us $77 + $76 + $75 + $74 . . . + $2 + $1 + $0, or $3,003. AALS is worse off than the previous best of a $93 price, but consumers have never been (nor could they really ever be) better off.

If we move the price up a notch to $24, we get 77 customers with a $1 contribution margin per lawn, $77 in contribution margin (producer surplus), a loss of $123, and consumer surplus of $2,926. Note that $3,003 – $2,926 gives a $77 loss in consumer surplus. Each of these 77 customers lost $1 in consumer surplus when the price went up by $1. At a $25 price, 76 customers provide a $2 contribution per lawn, netting a $48 loss. AALS would have a $25 profit when 75 customers buy at a price of $26. You may note that both a price of $100 and $24 produce a loss of $123 and the pair of $99 and $25 produces a loss of $48. A $26 price would produce the same $25 profit we saw at a price of $98. The symmetry is no surprise with the assumption we made on reservation prices in AA (a linear, down-sloping demand curve).

Arbuckle would certainly want to know the price that would maximize AALS profit. If we calculate each possible price by hand (or use a spreadsheet program to get there faster), we can see the maximum profit occurs when the price is $62. At a $62 price, 39 customers (C100 through C62) will buy, providing a contribution margin/producer surplus of $1,521 (39 × $39) and a profit of $1,321. Consumer surplus will be $741. It is interesting to note that both a price of $61 and a price of $63 produce a profit just $1 less at $1,320. The profit function tends to be almost flat near the maximum point.

A Continuous Approximation

We can get more mathematically adventurous (and look more like standard economic presentations). First, we need to specify the demand curve. The demand curve is simply a function that provides us the quantity as a function of price. In this case, the demand curve is , where is the quantity sold (number of customers or lawns) and is the price. So, at a price of 100, AALS sells 1. At a price of 99, they sell 2. At a price of 23, they sell 78. This seems to exactly match what was previous described.

We can determine profit () as revenues minus expenses. For AALS, the weekly revenue is , the weekly variable expenses are (i.e., a $23 per customer charge), and the fixed expenses are . This leads us to a profit function of . We can simplify the profit function to . Essentially, we change from “revenue minus variable expense minus fixed expense” to “contribution margin minus fixed expense.” We can further simplify the profit function by replacing with from the demand curve, resulting in the new profit function of . Two simple algebra steps take us to and then to . Now, we need calculus to solve for the price that maximizes profit. We take the first derivative of with respect to and get: . After setting to 0 and simplifying, we get ( gives ). Note the addition of the asterisk is used to denote an optimal value. (Calculus fans may want to note that shows that this is a maximum, not minimum.)

These calculations arrive at exactly the same price as the one we calculated by hand. It should have been close. The fact it is exact either comes from luck or (as here) careful design. The original problem has a discrete demand curve; AALS can only have a whole number of sales. With the continuous linear, down-sloping demand curve we just used (), we would have cut 26.47 lawns if we set a price of $74.53. However, any price from $75.00 to $74.01 would result in sales to 26 customers in the original setup (we wouldn’t cut 47% of a lawn). Thus, there would be no reason to set the price between $74.99 and $74.01. Still, the continuous approximation of the discrete distribution should get us close, if not spot on. Note that, many firms will face discrete demand curves (even though we can use continuous approximations). One doesn’t think of selling a partial book, a quarter of an auto, two-thirds of a computer tablet, or one-eighth of a jumbo jet.

A General Single-Price Solution

Let’s think about solving a problem like this is a more general way to do some exploration. Let’s assume a simple linear, down-sloping demand curve in the form: . For AALS, and , and the new constants let us consider the general case of linear demand curves. We will use a general purpose profit function in the form: , where the variable represents variable costs per unit (it was 23 for AALS) and represents fixed costs (it was 200). Combine the two to get the new equation: . Simply that to . The first derivative is . Setting to 0 and solving for produces . Double check with the original data (, , and ), and we, again, get 62.

Fixed Costs and Profit Maximization

Note that in double checking our general purpose solution, we did not have to substitute in a value for . Fixed costs are irrelevant for maximizing profit. A firm’s profit can be calculated as contribution margin minus fixed costs. And, if fixed costs are indeed fixed, we can maximize profit simply by maximizing contribution margin. What if the city of Croniesville changed this scenario by adding on a $300 weekly “franchise fee” for the right of AALS to operate in town? The only thing that would happen is that AALS would make $300 less per week.1 Profit (and contribution margin) is still maximized at a price of $62.

You will not the above observation flies in the face of what you might hear from some businesses. “We had to raise prices because our rent went up” is something one wouldn’t be surprised to hear. Assuming the only thing that changed was increased rent, this is a silly conjecture. If they were maximizing contribution margin prior to the rent increase, then raising prices would result in a lower contribution margin and lower profit than if they kept the old prices. Now, if over time the fixed costs of all competitors were going up, we would expect it to affect the competitive environment and the demand curve. But, we are focusing on a monopolist where the environment is fixed.

Why are prices high at airport retailers? One might suggest that is because airports charge high rents, but that is wrong. Airports provide an environment with limited competition for retailers (along with a stream of wealthier than average consumers). Because of this, airport retailers can and do charge higher prices than equivalent non-airport retailers. While we expect higher variable costs associated with operating at an airport, airport shops can still obtain a higher total contribution margin than their non-airport equivalents. Knowing this, airports charge high rents. High rent doesn’t lead to high prices. The existence of consumers who are better-off financially “trapped” leads to higher prices. That allows airports to charge higher rents.

Variable Costs and Profit Maximization

Also note that, in our general purpose solution, the first part () of the profit-maximizing price is a constant. Since and are fixed—that is to say the demand curve is fixed—that fraction is a constant. The second fraction () is half of variable cost per unit. AALS would charge $50.50 per lawn even if they got the lawns mowed for free (i.e., with no variable costs).2 The optimal price will increase $0.50 for every $1.00 increase in variable costs per unit. For example, take the original data and change the cost AALS is charged per lawn to $27. The new optimal price is $64. A $4 increase in variable cost per unit has resulted in a $2 price increase. This would certainly be counterintuitive to some. Note, this 2:1 ratio of variable cost to optimal price is true for any linear, down-sloping demand curve where the variable cost per unit is constant (an economist would describe the latter requirement by saying the marginal cost per unit is constant or there is a flat marginal cost curve). We do not necessarily expected demand curves to be linear or marginal cost per unit to be constant. Indeed most economic models of this have non-linear demand curves and marginal costs that decrease in the short run and increase in the long.

Cost of Prediction Errors

How costly is it if Arbuckle makes incorrect estimates of costs? Consider fixed costs first. The contract with the lawn service company specifies that AALS will have to pay them a fixed fee, previously established at exactly $200 per week. Let’ change the contract to one in which AALS must pay them a percentage of their fixed costs, say 10%. Arbuckle estimates that fee will be $200 per week. AALS sets the price at $62, has 39 customers, and a contribution margin, as predicted, of $1,521. The lawn service company charges them the fixed fee which amounts to $328.

AALS predicted a profit of $1,321 ($1,521 – $200) but earned $1,193 ($1,521 – $328). What is the cost of the prediction error? There is no cost. If Arbuckle knew the fixed fee would be $328, he still would have charged the same price and had the same number of customers. Knowing the actual fixed fee would not have lead Arbuckle to change any decisions, so he couldn’t earn any more money with that foreknowledge. His earnings were $1,193 based on the wrong estimate, and his earnings would have been $1,193 if he had estimate the fixed fee correctly. This is the accounting equivalent to “no harm, no foul.” There is no cost to the prediction error.

Assume the original data. However, the lawn service has not set a fixed, $23 price per lawn. Instead, they calculate the average cost to them of mowing a lawn across all their customers and charge AALS that amount per customer. Arbuckle uses the $23 amount as his best estimate. Based on this, he sets the $62 price and has 39 customers. However, the lawn service charges him $25 per lawn. He expected a contribution margin of $1,521 (39 × {$62 – $23}) and a profit of $1,321. He obtained a contribution margin of $1,443 (39 × {$62 – $25}) and a profit of $1,243. He earned a profit $78 less than expected.

What did it cost Arbuckle to make to have made an incorrect estimate? One might be tempted to use $78, but that is wrong. We have calculated his actual earnings ($1,243). Now, we must calculate what the earnings would have been without the inaccurate estimate. If Arbuckle knew the variable costs would be $25, he would have established $63 as the optimal price.3 His contribution margin would have been $1,444 (38 × {$63 – $25}) with a profit of $1,244. The best estimate would have resulted with AALS making just $1 more than they actually made with an incorrect estimate. The cost of the prediction error is $1.

Again, it is useful to note that there is flatness in this example to the optimal profit curve near the optimal price. Assume Arbuckle uses $23 as the estimate of costs, we see the incorrect estimate only cost him $1 ($1,444 optimal – $1,443 actual) if the actual variable costs are $25. Actual variable costs of $27, $29, $31, $37, and $45 lead to prediction errors of $4, $9, $16, $49, and $121.4 The nature of this environment is such that using an estimate nearly one-half the actual cost ($23 instead of $45), results in a prediction error reduction of contribution margin of only about 15% ($121 less margin, where $784 was available).

It would be a major mistake to draw the wrong conclusions from the above examples. First, it details the important issue of what constitutes a prediction error. It is not unusual for someone to conclude a prediction error is the difference originally predicted and actual net income. As we saw, it is not. It the difference between the net income we would have had if we made the right decision and the actual net income. Second, in this case, a poor estimate of variable costs is not highly costly. Often small mistakes have little cost, but here even much large mistakes had little cost. Part of this most certainly comes from assumptions we made on the demand curve (and to a lesser extent on the cost curve). Do not leave this discussion thinking that even large mistakes in cost estimates aren’t particularly costly in naturally occurring situations. It is true only in the context of this linear demand curve.

Basic Price Discrimination

Price Discrimination to the Extreme

Can Arbuckle do better than we have already calculated? We assumed that AALS would set one price that would be offered to all 100 homeowners. Let’s try a different plan. AALS sends homeowner C100 and offer to cut grass at a price of $100. C100 is willing to pay $100 and accepts the offer. C99 is offered a price of $99, C98 is offered a price of $98, and so on until C24 is offered a price of $24. AALS has a contribution margin of $77 on C100, $76 on C99, $75 on C98, and $1 on C24 (AALS could sell mowing to C23 at $23 without increasing or decreasing their total contribution margin). The contribution margin is $3,003, with a net profit of $2,803. The single price solution ($62) had a contribution margin of $1,521 and a profit of $1,321. AALS has captured all the previous producer surplus of $1,521, converted the consumer surplus of $741 to producer surplus, and captured $741 not previously either producer or consumer surplus (the difference between the reservation price of homeowners C61 through C24 {or C23} and $23). AALS has done what economists refer to as walking down the demand curve. This is an example of price discrimination.

AALS can extract a profit of $1,321 by setting a single price, but can obtain $2,803 if they price discriminate and charge each potential customer the maximum that customer is willing to pay. By the way, if it bothers you that the price-discrimination solution has 77 (or 78 if they sell to C23 at $23) customers paying the exact price that makes them indifferent between buying from AALS and cutting their own lawn, there is a simple solution. What if we increase every customer’s value for the service by one cent? Thus, we could offer to sell to C67 (whose value is now $67.01) at $67. AALS would make the same amount, but C67 has a $0.01 consumer surplus (as would every customer). Alternatively, we could keep the same reservation price (e.g., C67 still is at $67), and just subtract $0.01 from the price offer. AALS would make $0.77 (the 77 customers, which would not include C23, pay $001 less) less than $2,803. Our math is easier if we assume that customers will purchase even if they are indifferent (i.e., no better off buying from us). However, our solution is within 99.97% if we give away a small amount of producer surplus to avoid indifference. Thus, we continue with the simplifying, but monetarily trivial, assumption that customers are willing to purchase at their reservation price).

Another Type of Price Discrimination: Tiered Prices

AALS has abandoned the notion of individual prices for each customer. Does that mean price discrimination is out? Consider the following. It is known that the 38 customers C2, C4, C6, C8, …, C72, C74, and C76 are all senior citizens. AALS could price discriminate by offering one price to non-seniors and another to seniors. The continuous approximation of the demand curve for seniors is . We cannot specify an accurate profit function for seniors, as we have no basis on which to split the fixed cost between groups. However, we can solve for the contribution margin () for seniors: . We need only maximize that. Combine the two equations and we get . This simplifies to . Take the first derivative and solve for 0, and we get . The contribution (and profit) maximizing price is $50.5. If we solve this by hand, we discover we can obtain the largest contribution margin of $378 by setting a price of $50 and selling to 14 seniors. Unlike our previous case, the continuous approximation was not exact, but it was still close. We will continue with the more accurate, latter calculations.

For the non-seniors, the demand curve is not linear. They gain one customer for each $1 drop in price from $100 to $77 and gain one customer for every $2 drop in price thereafter. If we solve this by hand, we find that a price of $75 leads to sales of 25 and a contribution margin of $1,300. We could also get the same $1,300 by charging $73 and servicing 26 lawns. Putting this together, we can establish a price of $50 for seniors and a price of $75 (or $73) to non-seniors and service 39 (or 40) lawns for a total contribution margin of $1,678. This is better than the $1,521 contribution margin generated with a single price for all of $62. We will use the $73 price for consideration in upcoming discussions.

The fixed costs were $200 in the original setup. If the fixed costs stayed $200 with the “senior-discount” case, we would be $157 better off with this price discrimination. However, what if it is more expensive to operate the two-tiered pricing structure? It might be costly to verify the senior status, to target advertise, or for some other reason. If offering these senior discounts raises fixed costs by more than $157, AALS would be better off with a single price.

Issues in Price Discrimination

One concern that develops with a solution with price discrimination is “enforcing” the different prices. What if this example was not lawn service, but a weekly box of fresh fruit delivered to each customer’s door? If non-seniors pay $73 for a fruit box, what is to stop a senior from buying two (or three, or four, etc.) boxes at $50 and selling one (or two, or three, etc.) to one (or two, or three, etc.) non-senior neighbor(s) for less than $73.

We could limit the fruit boxes to one per address. Still that leaves 24 seniors who aren’t willing to buy for personal consumption at $50 who would be willing to buy at $50 if they could resell it for more in a “gray market.” What if those 24 seniors bought at $50 and sold them to 24 of the non-seniors who would have bought at $73 at a price between $50 and $73? Since there are 26 who’s next best alternative is to pay $73 and 24 whose cost is $50, we would expect a “sellers’ market,” and the price would be just below $73. We would lose 24 sales at $73. We would gain 24 senior sales a per box contribution margin of $27, increasing contribution margin by $648. But we lose 24 sales that would have been at contribution margin per box of $50, or $1,200 in total. The contribution margin drops $552 (or 24 sales had their sales price dropped by $23 each) down to $1,126. This is worse than the $1,377 contribution margin that would have obtained if $50 had been charged from the start. It is worse because a single price of $50 suggests 51 in sales, but only 38 sales can occur at $50 (there are 38 seniors). The sales to seniors generate $1,026 in contribution margin. There is an additional $100 in contribution margin from the two non-seniors that still must buy at $73.

Let’s look at this another way. There are 11 non-seniors who are willing to pay more than the $50 but less than $73. If 11 seniors who would not buy at $50 for personal consumption buy at $50 and resell to those 11 non-seniors, what effect does this have on the fruit version of AALS? In isolation, this causes AALS to sell 11 more fruit boxes at the $27 contribution margin, increasing the contribution margin by $297. This is an example of gray market sales that help the firm. Now, we cannot lose site of the 13 remaining seniors who are not buying for personal use or resale. They could buy and resell to 13 non-seniors currently buying at $73. This would cause a $299 (13 × ($73 – $50) drop in contribution margin. In this second gray market example, this is just a net $2 decrease in contribution margin.

A summary of the three most recent scenarios is useful. All three use a $73 price for non-seniors and a $50 price for seniors. With a pure separation of non-seniors and seniors, we obtain a $1,521 contribution margin and sales of 26 and 14, respectively. In the second case, separation breaks down and we obtain a $1,126 and sales of 2 and 38, respectively. Case two simply has the same number of sales moving from the higher to lower price, so it must be worse off. The third case has a contribution margin of $1,519 from sales of 13 and 38, respectively. The loss of 13 units at the higher price was almost offset by the increase of 24 units at the lower price. We could construct an example where the second gray-market version is the best of the three.

An important proviso is necessary here. The two gray-market scenarios are incomplete: we have not permitted them to fully evolve (or more technically, hit equilibrium). We would expect other competitive pressures between the parties to cause further change. The reselling seniors would try to get as high of a price they can. The non-seniors would jockey for the best deal between buying from a reselling senior and AALS. Non-seniors with higher reservation values would pull up the prices. Some seniors (e.g., C52, C54, and C56) who were happy to purchase at $50 and consume are now even happier as the buy at $50 and resell for more than their reservation price. At equilibrium, one would expect that they two gray-market scenarios would have the same result.

In the gray-market cases, there would be no sales at $73. The sales would be 38 at $50, all to seniors. There would be 7 seniors (C76 to C64), who consume the box of fruit they purchase. The remaining 31 seniors (C62 to C2) would sell their boxes of fruit to 31 non-seniors (C100 through C63) at a price of $63. AALS would make $1,026 in contribution margin. Well we wouldn’t expect that would last for long, as they were better off with a $62 price for all with a contribution margin of $1,521. The $62 “one-price-for-all” price and the $63 internal transfer price from seniors to non-seniors are close. But, in the former case AALS makes $39 on each of the 39 units sold. In the latter case, it is just $27 on each of 38 units sold.

If we allow seniors to buy as many boxes of fruit as they want, what will happen? They would sell 14 boxes to seniors with a reservation value of at least $50 to consume (C76 to C50). Then, all remaining sales to seniors would be for resale. The 37 non-seniors with reservation values of at least 50 (C100 to C51) would buy on box from a senior. There is limited demand of 37 at the minimum price of $50, and the seniors can buy an unlimited amount. Thus we’d expect the selling price from seniors to non-seniors to be very close to $50. AALS would sell 41 boxes at $50 for a contribution margin of $1,107. Again, they would abandon the two-price structure in favor of a single, $62 price.

We could institute quantity limits here, but such things are not so easily implemented in many cases. Senior discounts on lawn mowing, haircuts, public transport, and car repairs (for vehicles registered to a senior) are easy to limit to seniors. A senior discount on groceries, watches, or most anything that can be resold is difficult to limit.5 We need not be talking of price discrimination on age; we could just as easily have different prices for college students and others, veterans and others, and so on.

Price Discrimination in a Continuous Distribution

Let’s consider another case with price discrimination. Leo Bloom is an accountant who runs the charity Accounting Power. Bloom feels accountants are viewed as too mild mannered. Accounting Power seeks to remedy that by funding tattoos for accounting majors, placing “debit” on the right arm and “credit” on the left. Bloom has engineered an interesting charity fundraiser. Cookie Cutter Cinemas (CCC) has agreed to let Accounting Power use some of their theaters nationwide to show a film for one week to raise funds. My Fair Fellow is a 2005 motion picture that retells the Pygmalion story with Robin Williams as Larry Doolittle and Paris Hilton as Henrietta Higgins. Curiously, it had never been released.

CCC will promote the charity event, screen the film, and collect the box-office revenue. They will return all the revenue to Accounting Power reduced by their fixed costs and $1.20 per ticket sold. Accounting Power will have to pay the owner of the film $1.00 per ticket sold and (a fixed) reimbursement for the cost of making prints. Bloom’s task is to determine the ticket price(s) that will maximize profit (i.e., return to charity). We will assume all the fixed costs referred to above are fixed if the project is undertaken, regardless of pricing decisions.

Bloom knows that the demand curve for adults is , and for children is , where the subscripts differentiate between groups. This implies that if we charge adults and children the same price (i.e., ), the demand curve would be (with the subscripts removed as there is only one demand curve). If Bloom sets a single price, he wants to maximize . Substituting the demand curve produces . Simplification leads to . The first derivative is , the maximum is at , and the single, profit-maximizing price is $11.5167, with a contribution margin of $520,801.67. For convenience, and to avoid fractional-cent ticket prices, Bloom sets the price at $11.50 for each ticket. This produces a contribution margin of $520,800. One should note this small difference between actual and optimal price has a very small effect on profitability.

To maximize contribution margin from adults, Bloom maximizes . Using the adult demand curve, he obtains , simplifies it to , and takes the first derivative and sets it to zero to obtain . The optimal adult ticket price is $13.60, which would generate $519,840 in contribution margin. To maximize contribution margin from children, Bloom maximizes . Using the child demand curve, he obtains , simplifies it to , and takes the first derivative and sets it to zero to obtain . The optimal child ticket price is $7.35, which generate $53,045 in contribution margin.

Bloom now has an easy choice. If he sets one price for all at $11.50, the contribution margin will be $520,800. If he sets a price of $13.60 for adults and $7.35 for children, the contribution margin becomes $572,885 ($519,840 + $53,045). To maximize profit, Bloom will choose the two-tiered pricing system. One would expect he would do the project if fixed costs are less than $572,885.

What if the variable costs were $3? Bloom would calculate optimal price of = $11.9167, = $14.00, and = $7.75, with contribution margins of $477,042, $484,000, and $45,125, respectively. In this case, Bloom can obtain a higher contribution margin from just the adults at $14.00 than he can from the entire group at the optimal single price. Of course, he would still maximize contribution margin with the two-tiered pricing.

What if the variable costs were $5? Bloom would calculate optimal price of = $12.9167, = $15.00, and = $8.75, with contribution margins of $376,042, $400,000, and $28,125, respectively. However, there is an issue here. At a single price of $12.9167, the combined demand curve () suggests that 47,500 patrons will buy tickets. The adults demand curve () suggests 48,333 adults will buy tickets, and there will be –833 children (from ). But, of course, there will not be a negative number of children. So a price of $12.9167 would cause sales of 48,333, not 47,500, and a contribution margin of $382,639. The contribution maximizing solution is obvious: set a price of $15 for adults and $8.75 for children. However, it may not be obvious that they cannot add two demand curves and trust the result without rechecking.

A simply proviso about the above conclusions are important here. We had demand curves for adults and children that were independent.6 For example, if we set the adult price at $13.60 and the child price at $7.35, there would be 45,600 and 10,300 tickets sold, respectively. And, if the price for children were raised to, say, $8.00, the number of children would drop to 9,000. But, since the adult price is still $13.60, we would see no change in adult ticket purchases. Imagine, instead, an increase in child ticket prices would lead not only to fewer children, but less sales to adults who might bring those children. Or, the opposite is a conceptual possibility. Imagine that a reduction in price to children not only might lead to more children, but fewer adults who now find the experience less appealing.

We could arrive at the conclusions we did because adult and child attendance was independent. It is difficult to imagine that adult and child ticket sales are independent for Disneyland. Higher child ticket prices on a particular airline would likely reduce sales to children and those adult ticket sales to their parents, while greatly increasing the interest of nonparents in flying that airline. If the movie screening were limited to those above 18, and the pricing groups were college students and nonstudents, the notion that the demand curves are independent seems more plausible.

If the demand curve for adults was , this would suggest the non-independence referred to above. Alternatively, it could be in the form of . Both of these suggest that adult sales are highest when child sales are highest (and thus, child prices are lowest).

Price Discrimination and Quantity Discounts

Catherine Gunther runs Pâtisserie Française, a French-themed bakery. She has developed a new pastry she calls PDS (a variation from the original recipe for pets de soeurs). She bakes them once a week. Howard Brubaker enjoys these pastries and is deciding how many he should buy. They have a one-week shelf life and freeze poorly. Brubaker determines he likes the first one at the equivalent of 90 cents. He likes the second one each week a bit less, and values that at 80 cents. This process continues as Brubaker figures his values for the first, second, third, fourth, fifth, sixth, seventh, eight, and ninth are 90, 80, 70 60, 50, 40, 30, 20, and 10 cents, respectively. Thus, if Gunther prices these at, say, 60 cents, she will sell Brubaker four each week. As it turns out, all PDS fans are exactly like Brubaker. So, if Gunther can determine a pricing scheme to maximize her contribution margin from Brubaker, she has the first step (and maybe the only step) to maximize PDS contribution margin. The cost of producing each PDS is 27 cents.

Just like before, we can consider a single, profit-maximizing price. At 90 (cents), Brubaker buys one and PF nets 63 (cents). Prices of 80, 70, 60, 50, 40, and 30 produce contribution margins of 106 (2 × 53), 129 (2 × 43), 132 (3 × 33), 115 (5 × 23), 78 (6 × 13), and 21 (7 × 3), respectively. If PF sets the price to 60, they will sell 4 each week to Brubaker and every PDS fan and maximize the contribution margin.

PF would like to improve on this. They establish the price of 60, but offer the following promotion: “if you buy 4 at the regular price, you can buy up to 4 more for 50 cents each.” Brubaker would buy the first four, since the cost of each, 60, is no less than his reservation values (90, 80, 70, and 60). His desire for a fifth PDS is 50, and it is offered at 50. He will buy a fifth PDF. PF has the previous 132 contribution margin from Brubaker, plus an addition 23 (50–27) for a total of 155. Instead, if the deal were stated as “if you buy 4 at the regular price, you can buy up to 4 more for 40 cents each,” Brubaker would buy six. PF would make 132 from the first four and 26 from the next two (2 × (40 – 27)) for 158. You can confirm yourself that if the second price was 30 instead of 40 or 50, PF would make 141 (132 + 9).

PF considers further tweaking of the pricing structure. After reading the previous material here, Gunther realizes she needs to walk down the demand curve. So, PF sets a price of 90 cents for the first PDS, 80 for the second, 70 for the third, 60 for the fourth, 50 for the fifth, 40 for the sixth, and 30 for the seventh. This could be packaged in others ways. For example, PF could offer 1/90 (one for 90 cents), 2/170, 3/240, 4/300, 5/350, 6/390, or 7/420. Brubaker would buy 7, pay 420, and PF would have a contribution margin of 231. PF has extracted Brubaker’s entire consumer surplus.

The pricing structure above is unnecessarily complicated if all PDS fans are like Brubaker. PF could simply offer one price, 7 PDS for 420, since that is all anyone would purchase. Indeed, they could just go ahead and sell them only prepackaged at 420. Instead, try this as new pricing structure. Brubaker is invited to join a PDS fan club at a cost of 231 per week. This allows him to purchase all the PDS he wants at 27. We get to the same place. Brubaker buys 7 and PF gets 420 (231 + (7 × 27)) and the entire consumer surplus. Of course, the same basic problem arises for PF in the above examples: customers can foil the attempt to extract the entire consumer surplus with reselling. A group of 7 friends get together and buy three, 7 packs for 1260. Each pays 180 and gets 3 PDS. They value three PDS at 240 (90 + 80 + 70), thus netting a consumer surplus of 60 each.

Practical Issues of Monopolies

Forms of Competition

We have looked at pricing issues in a monopoly situation because that offers us a unique backdrop in which we can explore some important issues. Pricing issues don’t exist in pure competition. A producer can sell all production output at the prevailing market price and would sell none at any higher price. There are no pricing decisions in pure competition. In an oligopoly, pricing must take into account strategic behaviors of the others in the oligopoly. What we have explored here does parallel monopolistic competition. In monopolistic competition, there are many firms with differentiated products. Firms ignore the prices of competitors. The pricing process is similar to what we saw here under monopolies.

You will note that AALS had a monopoly because a governmental unit granted them one. A resident of the subdivision’s only recourse to hiring AALS was to cut his own lawn. The government also grants monopolies through the issuance of patents and copyrights. Only Pfizer has a legal right to sell Viagra until 2019 in the United States. This is a monopoly, but not a pure monopoly. There exist two other patented drugs to treat similar conditions owned by other pharmaceutical companies. So, Pfizer must consider the pricing of the alternative drugs in pricing their product. The potential monopolist must wrestle with issues of barriers to entry and substitute goods.

Difficulties in Single-Price Monopolies

Consider the initial case of AALS. When you have all the information we assumed Arbuckle had, deriving a single, profit-maximizing price was quite easy. Now, what would Arbuckle really most likely know? He would know everything buy the formula of the demand curve. There is no reason for him to believe it is linear. As appealing as having a monopoly might be, think of the daunting task facing Arbuckle. Without knowledge of the actual demand curve, how can you determine the price to offer? In laboratory experiments on this, subjects in the monopolist position are not particularly good at setting a single, profit-maximizing price. And, higher levels of success requires the subject change prices regularly to learn enough about the market to establish the desired price. That is, they search for the demand curve. However, we don’t see that type of behavior (big price swings) from many firms.7

Difficulties in Price Discrimination

Price discrimination is easy here, but how easy is it to implement in the naturally occurring markets? All we need to do is determine the most someone is willing to pay (i.e., the reservation price) and charge them that. For our convenience in the mowing scenario, we knew that C83 was willing to pay $83, and viola, we know the price to charge him. This is a serious challenge outside of work like this. People don’t come with their reservation price tattooed to their forehead. A customer has every incentive to hide their willingness to pay. The seller has a desire to discover the potential customer’s reservation price.

In automobile sales, price is typically established through face-to-face negotiations. The salesperson desires to ascertain the maximum price the customer is willing to pay. The customer seeks to learn the lowest price the salesperson will accept. A talented negotiator finds ways to ferret out this information. Car dealerships certainly do price discriminate, but nobody imagines they are able to extract all consumer surplus.

Price discrimination happens in higher education, and the universities have a particularly useful tool. Universities typically operate like car dealerships; they price discriminate by having a “sticker price” and offering discounts. The full tuition amounts might be $20,000 per year, but then the university offers financial aid or scholarships that reduce the net tuition bill. The university has a very useful tool in formulating those reductions: financial aid information. Students fill out forms detailing their and their parents’ financial situation. This is a very useful tool for the university to price discriminate.

In our lives, we run into a relative few cases where a seller is attempting to extract as much consumer surplus as it can. Simply, how many situations are like buying a car or applying to university? How often do you negotiate a price? The most extreme versions of price discrimination are rare. This is not surprising. While the rewards can be high, it is very costly to implement.

We do see a good amount of other types of price discrimination. We see discounts offered for seniors, college students, children, and veterans.8 There is possibly no easier place to find price discrimination than in the seats of an airplane. Airlines are masters of multi-tiered pricing. A ticket is less expensive if bought 14 days before the flights. A vacation traveler (with a lower reservation price) will typically be able to plan that far ahead. A business traveler (with a high reservation price) is less likely to have that much lead time. Tickets are cheaper in they include a Saturday night stay. Again, this eliminates fewer vacationers than business travelers. An important element for the airlines’ ability to price discriminate is that there is no gray market.

1 In this case, AALS was making more than $300 after all the previous expenses. The town could charge a high enough fee that would cause AALS to abandon business. Even so, the highest profit (i.e., smallest loss) is earned with the $62 price.

2 This is a case were the continuous approximation is close, but not exact. Using the discrete distribution, we calculate AALS would charge either $51 or $50 per lawn, as both provide the same contribution margin. Charging $50.50 would result in a lower contribution margin.

3 You can recalculate the discrete distribution by hand or use the continuous approximation. Note, a $2 higher variable cost translated into a $1 optimal price.

4 You may note that when AALS uses $23 to obtain optimal price and the actual variable costs are , the prediction error is: .

5 Note a big issue here on limiting the consumption of the item purchased to the purchaser depends on where and how the product or service is consumed. If you can’t watch the product being consumed, resale is much easier. You can watch an in-restaurant senior meal or a haircut get consumed. You can’t watch who consumes the groceries or watches sold at a senior discount.

6 This is easily determined. The demand curve for adults had no variable in it relating to the price charged children or the quantity of children’s tickets sold. The demand curve for children did not have any variable relating to adults in it.

7 You may note you see big price swings in gas prices. This is not evidence of their search for the demand curve. Oil is a commodity, and their prices do vary regularly. Price changes in airfares are more likely a search for the demand curve.

8 Not all cases of multi-tiered pricing can be traced to simple price discrimination. Offering children a lower price at an all-you-can-eat buffet can simply reflect they have lower variable costs because they tend to eat less.

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