Interfaz HMI para el simulador

pro-ducing a pizza. Most obviously, a football game is service, like a play or movie, not a physical object. In addition, output must be produced in concert with another producer (team). Despite these differences, we can still apply the economic theory of production to sports. In this section, we review the basics of production theory, including why these concepts are so important to the study of team sports and the special nature of sports relative to other goods and services.

a note on the definition of output

Before analyzing a market, economists must determine how to measure output.

In some markets, such as the pizza market, defining output (Q) is easy. It is the number of pizzas produced in a given time period. In sports markets, defining and measuring output is more complicated. If we think of output as what a firm sells in order to obtain revenue, we could measure output as attendance or tele-vision appearances. If we focus on production, it may be more useful to mea-sure output as games, because the team must combine inputs to produce games throughout the course of the season. Finally, if a team’s popularity, and hence its revenue, depend on its performance, the appropriate output is wins or win-ning percentage rather than simply games played. Our problem resembles that facing those who study higher education. From the standpoint of revenue, a col-lege or university may define output as the number of students enrolled. From the standpoint of input utilization, it may define output as the amount that its students learn, perhaps measured by their future incomes. Unfortunately, there is no simple resolution to this issue. To force a universal definition of output would cloud the issue as often as it would clarify it. In this text, we address this thorny issue by defining output according to the aspect of the market under consider-ation. For the remainder of this section, we consider output to be the number of wins produced per season.

the production Function

Production transforms inputs into output. A production function shows the relationship between the quantity of inputs used and the quantity of output pro-duced. This relationship is an expression of the technology of production and typ-ically includes labor and capital as inputs because the firm can substitute between them. In sports, however, capital (the need for a field of play, for example) is fixed.

In addition, the number of players is set by rule. Thus, as we consider a team’s attempt to produce wins (Q), it makes more sense to speak in terms of units of tal-ent. The more talent a team has on the roster, the more games it can expect to win.

We can show a production function in which a football team can invest in either offensive talent (T_O) or defensive talent (T_D) as:

Q = f(TO, T_D)

Another term for Q is the total product of labor. In order to evaluate the impact of increases in either input on the number of wins, we must hold the other input constant. Figure 2.10a shows the typical relationship between one variable input, offensive talent in this case, and output while holding the quan-tity of defensive talent constant. As T_O rises from 0 to T_O1, each successive unit of talent adds more to wins than the last. The intuition behind this is fairly sim-ple. If a football team only has one or two talented players out of the starting 11, the team will benefit enormously from additional talent on the field. From T_O1 to T_O2, wins continue to rise but at a decreasing rate. These players still improve the team but not as much as the first few talented players. Beyond T_O2, addi-tional talent might actually cause the team to win less. Is this possible? Perhaps you are familiar with the saying, “there’s only one ball” as a reference to the fact that having too many players want to carry the ball can damage team chemistry and reduce wins.

When coaches and general managers consider adding talent to a roster, they think like economists—by focusing on the margin. If they add one more talented player, how many more games might the team win? To focus on the change in output resulting from a small increase in one input, economists evaluate the play-er’s marginal product. The marginal product of an input indicates the increase in output that results from a one-unit increase in that input, holding the other input constant. For both offense and defense we thus have:

MP_T = ∆Q/∆T

Figure 2.10b shows the marginal product of offensive talent. From 0 to T_O1, as the slope of the total product curve increases, the marginal product increases.

Again, the number of games the team wins increases quickly as the coach adds talent to the roster. From T_O1 to T_O2, the slope of the total product curve decreases, and marginal product falls. The decline in marginal product, known

Q (Q = TP = wins)

TP

0 T_O1 T_O2 OT (offensive

talent) FIgure 2.10a The Total Product Curve

as the law of diminishing returns, is one of the most important concepts in all of economics. The law of diminishing returns states that as a firm (team) con-tinually increases one input while holding the other fixed, the marginal product of that input must eventually fall. This concept explains why, for example, a baseball team with five good starting pitchers does sign yet another starter and why the Giants do not sign both Eli Manning and Tom Brady to play in the same year. Other than for a few trick plays, a team can use only one quarterback at a time, and, barring injury, additional quarterbacks rarely play. Finally, beyond T_O2, where the total product curve is downward sloping, the marginal product

MP (marginal product)

0 OT (offensive

talent) MP

T_O1 T_O2

FIgure 2.10b The Marginal Product Curve

MC ($ marginal cost)

MC

0 Q₁ Q (Units)

FIgure 2.10c The Marginal Cost Curve

curve is negative.¹⁴ If the Giants had signed Eli Manning, Tom Brady, and Drew Brees, the tension created over who plays how much (or the difficulty in trying to play all three) may cause the Giants to win fewer rather than more games. We will use the theory of production, in particular the law of diminishing returns, later in the text as we discuss the need for salary caps, roster limits, and teams’

profit-maximizing strategies.

The law of diminishing marginal returns has direct implications for a firm’s costs. As the marginal product of an input falls, the firm must use more and more of that input to achieve a given increase in output. Thus, as the marginal product falls, marginal cost rises. Marginal cost is the additional cost associated with an increase in output.

MC = ∆C/∆Q

As Figure 2.10c shows, the marginal cost curve is essentially the inverse of the marginal product curve (though the axes are different). Marginal cost falls at first because additional units of talent are highly productive. Beyond Q₁, marginal costs begin to rise as the input is now subject to diminishing returns. The concept of marginal cost is critical to economic theories of firm decision making as we will see later in this chapter and throughout Chapters 3 and 4.

price ceilings and the economics of Scalping

Today, fans wanting to buy or sell tickets to sporting events at the last minute can easily do so by accessing Web sites such as Stubhub. In the past, a University of Michigan football fan who wanted to see the Wolverines play archrivals Ohio State or Michigan State often had to participate in a strange ritual. Students with tickets to the game could be found walking in front of the Michigan Student Union with their tickets in one hand and a pencil in the other. When someone offered to buy the ticket, the student would agree to do so—but only if the potential buyer also bought the pencil.

The key to understanding such an odd sales arrangement lies in the state of Michigan’s antiscalping laws. According to the law, no one can sell tickets for more than the value printed on the ticket (its face value). The face value of the ticket, however, was well below what a free market would dictate. In economic terms, the law placed a price ceiling on tickets, keeping their price far below equi-librium. If the face value of a ticket is $15, and no sales are permitted above this price, the price ceiling (p^c) is $15. Such a ceiling is shown in Figure 2.11.

A price ceiling creates two problems for buyers and sellers. First, the price ceiling (p^c = +15 in Figure 2.10) creates excess demand for tickets, since the quan-tity of tickets demanded (Q_d) is much greater than the quantity of tickets supplied (Q_s). To make matters worse, there is no guarantee that the people who place the greatest value on tickets can get them. By limiting price to p^c, we know only that all buyers are willing and able to pay at least p^c to see a Michigan football game.

14The formula for marginal product is identical to the formula one would use to compute the slope of the total product (ΔQ/ΔL).

If price does not serve as an allocation mechanism, someone who is just willing to pay the face value for a ticket might get one while someone who values it far more highly might not. Many colleges and universities have a persistent excess demand for tickets. This frequently leads to scenes of students camped outside the ticket office for days at a time to be sure that they have a seat for the big game. Thus, when prices do not ration tickets, some other limited resource, in this case time, typically does.

Universities set low prices for one of several reasons. For example, they might do so out of a sense of fairness to students with limited incomes. Recognizing that athletics are a student activity, athletic departments might want to be sure that all (or at least most) students can afford to see “their” team play.

If those with tickets could sell freely to those without, a mutually beneficial trade could be arranged. Suppose, for example, that Daniel is a rabid Michigan fan who is willing to pay $100 for a ticket to see Michigan play Michigan State.

Melanie—the lucky recipient of a ticket—thinks a ticket is worth only $15. If Daniel pays Melanie $70 for the ticket, he would pay $30 less than the ticket is worth to him while Melanie would receive $55 more than the ticket is worth to her. Daniel and Melanie would both benefit from such an exchange, yet the law prohibits it.

That is why Melanie can be found on State Street in Ann Arbor, offering her ticket for the face value of $15, but only to those who are willing to pay $55 for her pencil and why Daniel is happy to pay so much for a pencil!¹⁵

D S

p^c = 15

Q_s Q_d

p ($ per ticket)

Q (quantity of tickets in thousands) p^e = 100

price ceiling

0

FIgure 2.11 The Effect of a Price Ceiling A price ceiling creates excess demand of Q_d- QS.

15A note of caution: It is doubtful that this practice is legal in most areas—and thus it is not one that we would advocate or condone.

2.3 market StructureS: From perFect