Content Articles in Economics Fixed and Sunk Costs

Content Articles in Economics In this section, the Journal of Economic Education publishes articles concerned with substantive issues, new ideas, and research findings in economics that may influence or can be incorporated into the teaching of economics. HIRSCHEL KASPER, Section Editor

Fixed and Sunk Costs Revisited X. Henry Wang and Bill Z. Yang

Abstract: The authors attempt to clarify the concepts of, and the link between, fixed costs and sunk costs. They argue that the root for possible confusion between fixed costs and sunk costs is the inconsistency in defining the term fixed costs. They de^me, fixed costs uniformly as the costs that are independent of the level of output and suggest that instructors refer to the part of fixed costs that are irrevocably committed as sunk costs. Under these definitions, the statement "there are no long-run fixed costs" is incorrect. Instructors should teach students that in the long run there are no sunk costs, although there may easily be fixed costs. Key words: avoidable cost, fixed cost, sunk cost, variable cost JEL codes: A2, DO "In the long run, as you know, there are no fixed costs" (Hall and Lieberman 1998, 192).' A serious student in microeconomics is probably expected to know this commonsense statement. Unfortunately, this statement is unclear at best and may be a myth. It is indeed incorrect under a commonly used definition Ihat fixed costs are the "costs that do not vary with the quantity of output produced" (Mankiw 1998, 271).^ In this article, we attempt to show why this statement can be false and to clarify the distinction between fixed costs and sunk costs. We address the following questions: Are there long-run fixed costs? What is the linkage between fixed costs and sunk costs? How should fixed costs be defined? Are there long-run fixed costs at all? Every economist as well as every busiX. Henry Wang is an associate professor of economics at the University of Missouri-Columbia (email: [email protected]), and Bill Z. Yang is an assistant professor of economics at Georgia Southern University, Statesboro. The authors wish to thank Andy Daughety, Jennifer Reinganum, Hirschel Kasper, and three anonymous referees for helpful comments. Bill Z. Yang is grateful to Minot State University for research support.

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nessperson may believe that the use of some factors is independent of the production scale, no matter whether these inputs have been committed in the short run or are under consideration for the long-run production plan. For example, a restaurant must pay an annual license fee every year to operate legally. The annual license fee is a fixed cost because it does not depend on how many customers are served each year. Suppose an entrepreneur is planning to open a restaurant next year and has not yet purchased any equipment, paid any fee, or signed any contract. Thus, it is a long-run decision problem and the annual license fee to be paid for the next year will be a long-run fixed cost. Clearly, if fixed costs are the costs that do not vary with the quantity of output produced, there may easily be long-run fixed costs. The statement, "there are no long-run fixed costs," has been handed down unelearly or even incorrectly in economic classes from generation to generation. Clarification and correction are overdue. STANDARDIZING THE DEFINITION OF FIXED COSTS There are two different definitions for fixed costs in the prevailing microeconomics textbooks. Many authors define fixed costs as the costs that do not vary with the quantity of output produced. Other authors use fixed costs as a synonym of sunk costs—the costs that have already been irrevocably committed and cannot be recovered. For instance, Varian (1999, 353) defines fixed costs as the "costs that are independent of the level of output, and, in particular, they must be paid whether or not the firm produces output." When the interpretation of the term fixed costs is switched back and forth between these two different definitions, it can be incorrect to state that there are no long-run fixed costs. In economics, long run means a long enough time horizon over which everything can be (re)adjusted, or nothing is irrevocably committed. By the definitions of long run and sunk costs, therefore, it should follow that in the long run there are no sunk costs. Perhaps because two different commonly used definitions exist for fixed costs, a kind of informational externality among different textbooks leads quite a few authors to be inconsistent when defining fixed costs. We recognize that this inconsistency has caused two problems in economic education. The first problem is specific to the myth that there are no long-run fixed costs. Some textbook authors interpret/(A:eJ costs as the costs that are independent of the scale of production. This is false. This mistake is made in two steps. In the first step, an author correctly notes that there are no long-run sunk costs by stating that there are no long-run fixed costs. In the second step, forgetting that fixed costs meant sunk costs, the author re-interprets fixed costs inconsistently as the costs that are independent of the level of output. Then the statement that "there are no long-run fixed costs" becomes false. It is the inconsistency in defining fixed costs that makes it a myth. The second problem is more general to the teaching of microeconomics. Many students are confused between fixed costs and sunk costs when they are taught these concepts for the first time. More and more textbooks devote pages to clarifying the distinction between the two concepts. Logically, if one uses fixed costs Spring 2001

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as a synonym of sunk costs, it is impossible to clarify successfully the distinction between the two. In fact, whenever a textbook defines fixed costs the same as sunk costs, the author(s) always implicitly re-interprets the term fixed costs differendy from the initial definition when attempting to clarify the distinction between the two. More often than not, such discussions have caused more confusion than clarification! We will come back to this point. We propose to standardize the definition for fixed cost for the sake of teaching economics, given that the concept of sunk cost has been well recognized and uniformly defined in almost all microeconomics textbooks. We define fixed cost, variable cost, sunk cost, and avoidable cost as follows': Fixed costs do not vary with the quantity of output produced; variable costs do vary with the quantity of output produced; sunk costs have been irrevocably committed and cannot be recovered; and avoidable costs'* have not been committed or can be recovered. ALL SUNK COSTS MUST BE FIXED, BUT NOT ALL EIXED COSTS ARE SUNK Under the above definitions, how are variable costs, avoidable costs, fixed costs, and sunk costs linked to each other? Sunk costs vs. avoidable costs and fixed costs vs. variable costs classify total costs from two different perspectives. A cost is sunk (over a given time period) if it has been irrevocably committed (as of a given time point), otherwise it is avoidable. Sunk and avoidable costs are therefore concepts related to time and timing, which is a dynamic issue.' On the other hand, a cost is fixed if it is independent of the output level to be produced, whereas a cost is variable if it varies with the production scale. Fixed and variable costs are then concepts determined by technology and the legal system. Nevertheless, a cost-minimization decision is always made at a time point given the quantity of output to be produced over a specific time period. We may classify total costs by asking the following two questions sequentially: Question 1: Is this cost now irrevocably committed over the relevant time period? Yes —> It is a sunk cost. (It is now irrelevant whether the firm intended the costs to be fixed or variable at the time they were irrevocably committed.) No

i It is an avoidable cost. Then, we ask. Question 2: Does this cost vary with the output level to be produced over that time period? No —> It is an (avoidable) fixed cost. Yes —» It is a variable cost (avoidable). Answers to these two questions will determine whether a cost is sunk or avoidable and whether fixed or variable if it is still avoidable (Figure 1). In the long run, there are no sunk costs, because all inputs are avoidable. How180


FIGURE 1 Taxonomies for Total Costs

Avoidable Fixed Costs Total Costs

—

Sunk Costs (Fixed) Variable Costs (Avoidable)

ever, there may easily be long-run fixed costs. Suppose that a potential entrepreneur is planning ahead for his or her future business without committing to the purchase of equipment, lease of office, or payment of annual license fee. In this case, none of these costs are sunk yet. By nature, however, all of them are fixed costs for the future production. In the short run, by definition, at least some costs must be sunk. When a cost is sunk, it cannot be varied at all and hence does not vary with the scale of production. That is, all sunk costs must be fixed. As a corollary, a variable cost must be avoidable, because otherwise it would be fixed. It cannot be overemphasized that even in the short run not all fixed costs are necessarily sunk. Carlton and Perloff (2000, 28 and 59-61) have provided an example to illustrate the existence of short-run, avoidable fixed costs. A lawyer has signed a lease to rent an office from a landlord. Monthly rent is a short-run fixed cost for his or her business. Suppose that the lawyer decides to shut down the office for a couple of months or immediately get out of the business after the lease has been signed. In this case, if he or she can sublet to someone else or pay the financial penalty for breaking the lease, at least some of the monthly rent can be avoided. Only that unavoidable part of the rent or penalty fee is sunk cost. Note that in both the short-run shutdown and the long-run exit problems, it is the avoidable costs that are relevant to the decision. We summarize the linkage among variable costs, fixed costs, avoidable costs, and sunk costs in the following proposition. Proposition (a) Total costs can be categorized as total costs = sunk costs + avoidable costs = sunk costs + avoidable + variable costs (fixed) Fixed costs (avoidable) =

fixed

costs + variable costs

(b) Fixed costs consist of sunk costs and avoidable fixed costs. Spring 2001

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(c) Avoidable costs include avoidable fixed costs and variable costs. (d) In the long run, there are no sunk costs, but there may easily be fixed costs. (e) In the short run, at least some costs are sunk, and there may still be avoidable fixed costs. WHY SHOULD FIXED COSTS BE DIFFERENTIATED FROM SUNK COSTS? Earlier we proposed to standardize the definition for fixed costs as the costs that are independent of the production scale. In the literature, sunk costs are uniformly defined as the costs that have been irrevocably committed. Under these definitions for fixed costs and sunk costs, we have clarified the distinction between the two concepts, whose connection can be summarized as follows: fixed costs = sunk costs + avoidable fixed costs.

(1)

The alternative definition for fixed costs adopted in many microeconomics textbooks uses the term fixed costs as a synonym of sunk costs—the costs that have been irrevocably committed. This alternative definition for fixed costs can potentially confuse students and instructors as well as some authors. To our knowledge, almost all of the prevailing microeconomics textbooks (from principles to graduate levels) introduce the concept of sunk costs and define it clearly and uniformly, no matter how they define fixed costs. As a matter of fact, many microeconomics, managerial economics, and industrial organization textbooks dedicate pages to clarifying the distinction between the two concepts.* This reflects a common belief by economists that sunk cost is an important concept and is different from fixed cost in economic theory and in business practice. For example, in entry deterrence it is sunk costs, not fixed costs, that have a strategic effect resulting from their commitment value. Also, in both the long-run exit and the short-run shutdown problems, decisions are based on all avoidable costs (including variable costs and avoidable fixed costs), because sunk costs are irrelevant. Now that the concept of sunk costs is so important and has been clearly defined uniformly, why is it also called fixed costs to confuse the students? How about replacing the term sunk costs with fixed costs so that the term sunk costs need not arise at all? Some textbooks do adopt this approach. For example, Varian (1999, 353) defines fixed costs as the "costs that are independent of the level of output, and, in particular, they must be paid whether or not the firm produces output." Clearly, it is the very sunk cost. Varian also recognizes the existence of our avoidable fixed costs and uses the term "quasi-fixed costs" for the "costs that are also independent of the level of output, but only need to be paid if the firm produces a positive amount of output" (1999, 353). It is the existence and the importance of sunk costs and avoidable fixed costs per se that require two different terms for them, no matter how they are named. Let us rewrite equation (1) in Varian's terms and compare it with ours. 182


Our definition: fixed costs = sunk costs -i- avoidable fixed costs. Varian's terms: no name = fixed costs -i-quasi-fixed costs. So far, Varian did not name our fixed costs and hence saved a term. His definitions for fixed costs and quasi-fixed costs are so clear that the distinction between them is self-clarified. If a student (say, in a principles class) has never heard the term sunk costs at all, Varian's treatment (so far) is self-consistent. However, Varian does know that sunk costs is an important concept in economics and many other competing textbooks have discussed the concept of sunk costs. He could not help but add a new section titled Sunk Costs in the most recent edition of his intermediate text (1999, 353-54), attempting to clarify the distinction between fixed costs and sunk costs. What he actually does is discuss the difference between his fixed costs and his quasi-fixed costs, leaving the term sunk costs undefined.' Not surprisingly, this treatment can only cause more confusion than clarification between fixed costs and sunk costs. Given that the concept of sunk costs has been well recognized and clearly defined in the literature and all microeconomics texts, using fixed costs as a synonym of sunk costs is what makes students confused between the two concepts. Finally, we point out that it is incorrect to assume that fixed costs are also sunk,* because it is easy to find avoidable fixed costs. Also, because sunk costs must be fixed, it is redundant to coin a term for the costs that are independent of the level of output, and, in particular, they must be paid whether or not the firm produces output. If we are referring to sunk cost, why not simply refer to it as sunk cost? In conclusion, it would benefit students of economics and avoid unnecessary confusions if fixed costs and sunk costs were defined uniformly as follows. Fixed costs are independent of the level of output; sunk costs have been irrevocably committed and cannot be recovered. Under these definitions, the statement "there are no long-run fixed costs" is false. We should teach our students that in the long run there are no sunk costs because all costs are avoidable, although there may easily be long-run avoidable fixed costs. A FORMAL TREATMENT In this section, we provide a formal treatment for our discussions, using the notations employed in Varian (1992). A short-run cost minimization problem can be formally described as minwx

= WJATJ + W2X2

where y is the quantity of output to be produced;/(xi, 0:2) is the production function, M' = (wj, W2) is a vector of factor prices with wj and W2 being its subvectors; and X = (xi, Xz) is a vector of all inputs, consisting of two subvectors JCi and X2. In this problem, y, w, and A:2O are said to be exogenous parameters that are predeterSpring2001

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mined in the program, and all elements in subvector Xi are said to be endogenous variables, meaning that they are to be determined by solving the problem. Because inputs, X2 = JC20. are exogenously given, their costs, W2 JC20, are predetermined, meaning that "they are irrevocably committed and cannot be recovered." These costs are called sunk costs. On the other hand, because inputs JCi are to be endogenously determined in this problem, their costs, H-IJTI, can be avoidable if JCi = 0. The optimal solution of x*i will be a function of all the exogenous variables, jc'i = Xi(y, w\ Xzo). However, depending on whether an element in x is a degenerated (constant) or nondegenerated function of y, we can decompose costs wiXi(y, w; JC20) into two parts: w{x{{w; X20) + w\x\ (y, w; X20). Here, the costs w{x{(w, JC20) are said to be avoidable fixed costs; they are independent of the value of y for all y>0, and they can be avoided if >> = 0. And the costs w'lX'i (y,w; Xjo) are called variable costs because they vary with the value of parameter y. It is worth noting that sunk costs w-^iczo are independent of y for all >> > 0, and hence are also fixed costs. The classification of total costs and the linkage among fixed costs, variable costs, sunk costs, and avoidable costs are summarized and can be clearly seen in the following expressions:^ Total costs Fixed costs Sunk costs Avoidable fixed costs Avoidable costs Variable costs

= = = — =

M'2 JC20 + w{x{ (w; X20) + w\x\ (y, w\ JC20). W2 X20 + w{x{ (w; X20). M'2 ac2oWiXi(w,X2o). wiXi{w; X20) + »«'W (y, w; w'lX'i (y, w; 0:20)-

Finally, note that a long-run cost function can be obtained in the special case when Xi=x OTX2 = 0. Clearly in the long run, there are no sunk costs. But there may be avoidable fixed costs if w^x^(w) > 0. NOTES 1, Similar statements can also be found in Case and Fair (1996, 196) and Varian (1992 and 1999), among other microeconomics textbooks from principles to graduate levels, 2, The same definition can also be found in Baumol and Blinder (1997, 156), Carleton and Perloff (2000, 51), Hall and Lieberman (1998, 155), and Pindyck and Rubinfeld (1995, 198), among others, 3, Note that our definitions basically follow the standard usage in the literature, except that no standard definition for fixed costs exists in the literature, 4, For example, Carleton and Perloff (2000, 28) employed such a term for nonsunk costs, 5, We are grateful to an anonymous referee for this point, 6, For example, Baumol and Blinder (156), Carleton and Perloff (28), Hall and Lieberman (155), Mankiw (291), Pindyck and Rubinfeld (195), Samuelson and Marks (1999,262-63), Tirole (1988, 307-08), and Varian (1999, 353-54) all have attempted to clarify the distinction between fixed costs and sunk costs, 7, Note that Varian (1998, 353-54) does not clearly define sunk costs, he only vaguely calls it "another kind of fixed costs," Obviously, when Varian attempts to show how fixed costs are different from sunk costs (i.e,, his fixed costs), he implicitly interprets the term fixed costs as the

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"costs that are independent of the level of output," which follows our definition instead of his own, 8, Baumol and Blinder "assume that the fixed costs are also what economists call sunk costs, meaning that the firm has already spent the money in question or signed a contract to do so" (1997, 156, note 4), 9, Technically speaking, avoidable fixed costs also vary with parameter y, only discretely: they are 0 when y = 0, and for all y > 0, they are a constant

Logically, it is sensible to refer to the avoidable costs, , w\ X2o) + w{x{(w\ Xio)

as variable costs by definition. But it can be problematic to refer to its complement, ii'2 J:2O. as fixed costs. It may cause confusion when attempting to discuss the distinction between fixed costs [= H'2 JC20 +