19 Profit Maximization

# 19 Profit Maximization - CHAPTER 1 9 PROFIT MAXIMIZATION In...

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Unformatted text preview: CHAPTER 1 9 PROFIT MAXIMIZATION In the last chapter we discussed ways to describe the technological choices facing the ﬁrm. In this chapter we describe a model of how the ﬁrm chooses the amount to produce and the method of production to employ. The model we will use is the model of proﬁt maximization: the ﬁrm chooses a production plan so as to maximize its proﬁts. In this chapter we will assume that the ﬁrm faces ﬁxed prices for its in— puts and outputs. We said earlier that economists call a market where the individual producers take the prices as outside their control a competitive market. So in this chapter we want to study the proﬁt—maximization prob- lem of a ﬁrm that faces competitive markets for the factors of production it uses and the output goods it produces. 19.1 Profits Proﬁts are deﬁned as revenues minus cost. Suppose that the ﬁrm produces n outputs (yl, . . . , y“) and uses m inputs (x1, . . . ,mm). Let the prices of the output goods be (131, . . . ,pn) and the prices of the inputs be (101, . . . ,wm). PROFITS 335 The proﬁts the ﬁrm receives, 7r, can be expressed as n m 7T = E piyi — 2 ”(Utah‘- i=1 i=1 The ﬁrst term is revenue, and the second term is cost. In the expression for cost we should be sure to include all of the factors of production used by the ﬁrm, valued at their market price. Usually this is pretty obvious, but in cases where the ﬁrm is owned and operated by the same individual, it is possible to forget about some of the factors. For example, if an individual works in his own ﬁrm, then his labor is an input and it should be counted as part of the costs. His wage rate is simply the market price of his laboruwhat he would be getting if he sold his labor on the open market. Similarly, if a farmer owns some land and uses it in his production, that land should be valued at its market value for purposes of computing the economic costs. We have seen that economic costs like these are often referred to as op- portunity costs. The name comes from the idea that if you are using your labor, for example, in one application, you forgo the opportunity of employing it elsewhere. Therefore those lost wages are part of the cost of production. Similarly with the land example: the farmer has the oppor— tunity of renting his land to someone else, but he chooses to forgo that rental income in favor of renting it to himself. The lost rents are part of the opportunity cost of his production. The economic deﬁnition of proﬁt requires that we value all inputs and outputs at their opportunity cost. Proﬁts as determined by accountants do not necessarily accurately measure economic proﬁts, as they typically use historical costs¥what a factor was purchased for originallyvrather than economic costszhat a factor would cost if purchased now. There are many variations on the use of the term “proﬁt,” but we will always stick to the economic deﬁnition. Another confusion that sometimes arises is due to getting time scales mixed up. We usually think of the factor inputs as being measured in terms of flows. So many labor hours per week and so many machine hours per week will produce so much output per week. Then the factor prices will be measured in units appropriate for the purchase of such ﬂows. Wages are naturally expressed in terms of dollars per hour. The analog for machines would be the rental rate—the rate at which you can rent a machine for the given time period. In many cases there isn’t a very well-developed market for the rental of machines, since ﬁrms will typically buy their capital equipment. In this case, we have to compute the implicit rental rate by seeing how much it would cost to buy a machine at the beginning of the period and sell it at the end of the period. 336 PROFIT MAXIMIZATION (Ch. 19) 19.2 The Organization of Firms In a capitalist economy, ﬁrms are owned by individuals. Firms are only legal entities; ultimately it is the owners of ﬁrms who are responsible for the behavior of the ﬁrm, and it is the owners who reap the rewards or pay the costs of that behavior. Generally speaking, ﬁrms can be organized as proprietorships, partner- ships, or corporations. A proprietorship is a ﬁrm that is owned by a single individual. A partnership is owned by two or more individuals. A corporation is usually owned by several individuals as well, but under the law has an existence separate from that of its owners. Thus a partnership will last only as long as both partners are alive and agree to maintain its existence. A corporation can last longer than the lifetimes of any of its owners. For this reason, most large ﬁrms are organized as corporations. The owners of each of these different types of ﬁrms may have different goals with respect to managing the operation of the ﬁrm. In a proprietor- ship or a partnership the owners of the ﬁrm usually take a direct role in actually managing the day—to—day operations of the ﬁrm, so they are in a position to carry out whatever objectives they have in operating the ﬁrm. Typically, the owners would be interested in maximizing the proﬁts of their ﬁrm, but, if they have nonproﬁt goals, they can certainly indulge in these goals instead. In a corporation, the owners of the corporation are often distinct from the managers of the corporation. Thus there is a separation of ownership and control. The owners of the corporation must deﬁne an objective for the managers to follow in their running of the ﬁrm, and then do their best to see that they actually pursue the goals the owners have in mind. Again, proﬁt maximization is a common goal. As we’ll see below, this goal, properly interpreted, is likely to lead the managers of the ﬁrm to choose actions that are in the interests of the owners of the ﬁrm. 19.3 Profits and Stock Market Value Often the production process that a ﬁrm uses goes on for many periods. Inputs put in place at time t pay off with a whole ﬂow of services at later times. For example, a factory building erected by a ﬁrm could last for 50 or 100 years. In this case an input at one point in time helps to produce output at other times in the future. In this case we have to value a ﬂow of costs and a ﬂow of revenues over time. As we’ve seen in Chapter 10, the appropriate way to do this is to use the concept of present value. When people can borrow and lend in ﬁnancial markets, the interest rate can be used to deﬁne a natural price of consumption at different times. Firms have access to the same sorts of PROFITS AND STOCK MARKET VALUE 337 ﬁnancial markets, and the interest rate can be used to value investment decisions in exactly the same way. Consider a world of perfect certainty where a ﬁrm’s ﬂow of future proﬁts is publicly known. Then the present value of those proﬁts would be the present value of the ﬁrm. It would be how much someone would be willing to pay to purchase the ﬁrm. ' As we indicated above, most large ﬁrms are organized as corporations, which means that they are jointly owned by a number of individuals. The corporation issues stock certiﬁcates to represent ownership of shares in the corporation. At certain times the corporation issues dividends on these shares, which represent a share of the proﬁts of the ﬁrm. The shares of ownership in the corporation are bought and sold in the stock market. The price of a share represents the present value of the stream of dividends that people expect to receive from the corporation. The total stock market value of a ﬁrm represents the present value of the stream of proﬁts that the ﬁrm is expected to generate. Thus the objective of the firm—maximizing the present value of the stream of proﬁts the ﬁrm generates—could also be described as the goal of maximizing stock market value. In a world of certainty, these two goals are the same thing. The owners of the ﬁrm will generally want the ﬁrm to choose production plans that maximize the stock market value of the ﬁrm, since that will make the value of the shares they hold as large as possible. We saw in Chapter 10 that whatever an individual’s tastes for consumption at different times, he or she will always prefer an endowment with a higher present value to one with a lower present value. By maximizing stock market value, a ﬁrm makes its shareholders’ budget sets as large as possible, and thereby acts in the best interests of all of its shareholders. If there is uncertainty about a ﬁrm’s stream of proﬁts, then instructing managers to maximize proﬁts has no meaning. Should they maximize ex- pected proﬁts? Should they maximize the expected utility of proﬁts? What attitude toward risky investments should the managers have? It is difﬁ— cult to assign a meaning to proﬁt maximization when there is uncertainty present. However, in a world of uncertainty, maximizing stock market value still has meaning. If the managers of a ﬁrm attempt to make the value of the ﬁrm’s shares as large as possible then they make the ﬁrm’s owners—the shareholdersias well—off as possible. Thus maximizing stock market value gives a well—deﬁned objective function to the ﬁrm in nearly all economic environments. Despite these remarks about time and uncertainty, we will generally limit ourselves to the examination of much simpler proﬁt—maximization prob— lems, namely, those in which there is a single, certain output and a single period of time. This simple story still generates signiﬁcant insights and builds the proper intuition to study more general models of ﬁrm behavior. Most of the ideas that we will examine carry over in a natural way to these more general models. 338 PROFIT MAXIMIZATION (Ch. 19) 19.4 The Boundaries of the Firm One question that constantly confronts managers of ﬁrms is whether to “make or buy.” That is, should a ﬁrm make something internally or buy it from an external supplier? The question is broader than it sounds, as it can refer not only to physical goods, but also services of one sort or another. Indeed, in the broadest interpretation, “make or buy” applies to almost every decision a ﬁrm makes. Should a company provide its own cafeteria? Janitorial services? Pho- tocopying services? Travel assistance? Obviously, many factors enter into such decisions. One important consideration is size. A small mom—and—pop video store with 12 employees is probably not going to provide a cafeteria. But it might outsource janitorial services, depending on cost, capabilities, and stafﬁng. Even a large organization, which could easily afford to operate food ser— vices, may or may not choose to do so, depending on availability of alter- natives. Employees of an organization located in a big city have access to many places to eat; if the organization is located in a remote area, choices may be fewer. One critical issue is whether the goods or services in question are exter- nally provided by a monopoly or by a competitive market. By and large, managers prefer to buy goods and services on a competitive market, if they are available. The second-best choice is dealing with an internal monop— olist. The worse choice of all, in terms of price and quality of service, is dealing with an external monopolist. Think about photocopying services. The ideal situation is to have dozens of competitive providers vying for your business; that way you will get cheap prices and high-quality service. If your school is large, or in an urban area, there may be many photocopying services vying for your business. On the other hand, small rural schools may have less choice and often higher prices. The same is true of businesses. A highly competitive environment gives lots of choices to users. By comparison, an internal photocopying division may be less attractive. Even if prices are low, the service could be sluggish. But the least attractive option is surely to have to submit to a single external provider: An internal monopoly provider may have bad service, but at least the money stays inside the ﬁrm. As technology changes, what is typically inside the ﬁrm changes. Forty years ago, rms managed many services themselves. Now they tend to outsource as much as possible. Food service, photocopying service, and janitorial services are often provided by external organizations that spe— cialize in such activities. Such specialization often allows these companies to provide higher quality and less expensive services to the organizations that use their services. SHORT-RUN PROFIT MAXIMIZATION 339 19.5 Fixed and Variable Factors [n a given time period, it may be very difficult to adjust some of the inputs. Typically a ﬁrm may have contractual obligations to employ certain inputs at certain levels. An example of this would be a lease on a building, where the ﬁrm is legally obligated to purchase a certain amount of space over the period under examination. We refer to a factor of production that is in a ﬁxed amount for the ﬁrm as a ﬁxed factor. If a factor can be used in different amounts, we refer to it as a variable factor. As we saw in Chapter 18, the short run is deﬁned as that period of time in which there are some ﬁxed factors—factors that can only be used in ﬁxed amounts. In the long run, on the other hand, the ﬁrm is free to vary all of the factors of production: all factors are variable factors. There is no rigid boundary between the short run and the long run. The exact time period involved depends on the problem under examination. The important thing is that some of the factors of production are ﬁxed in the short run and variable in the long run. Since all factors are variable in the long run, a ﬁrm is always free to decide to use zero inputs and produce zero output—that is, to go out of business. Thus the least proﬁts a ﬁrm can make in the long run are zero proﬁts. In the short run, the ﬁrm is obligated to employ some factors, even if it decides to produce zero output. Therefore it is perfectly possible that the ﬁrm could make negative proﬁts in the short run. By deﬁnition, ﬁxed factors are factors of production that must be paid for even if the ﬁrm decides to produce zero output: if a ﬁrm has a long— term lease on a building, it must make its lease payments each period whether or not it decides to produce anything that period. But there is another category of factors that only need to be paid for if the ﬁrm decides to produce a positive amount of output. One example is electricity used for lighting. If the ﬁrm produces zero output, it doesn’t have to provide any lighting; but if it produces any positive amount of output, it has to purchase a ﬁxed amount of electricity to use for lighting. Factors such as these are called quasi-ﬁxed factors. They are factors of production that must be used in a ﬁxed amount, independent of the output of the ﬁrm, as long as the output is positive. The distinction between ﬁxed factors and quasi-ﬁxed factors is sometimes useful in analyzing the economic behavior of the ﬁrm. 19.6 Short-Run Profit Maximization Let’s consider the short-run proﬁt-maximization problem when input 2 is ﬁxed at some level 52. Let f (\$1,562) be the production function for the ﬁrm, let p be the price of output, and let 1111 and 1112 be the prices of the 340 PROFIT MAXIMIZATION (Ch. 19) two inputs. Then the proﬁt-maximization problem facing the ﬁrm can be written as HEX Pf(\$1,E2) “ 1111331" 2.0252. The condition for the optimal choice of factor 1 is not difficult to determine. If x: is the proﬁt-maximizing choice of factor 1, then the output price times the marginal product of factor 1 should equal the price of factor 1. In symbols, pMP1(a:’{,'372): wl. In other words, the value of the marginal product of a. factor should equal its price. In order to understand this rule, think about the decision to employ a little more of factor 1. As you add a little more of it, Am, you produce Ay : M PIAzrl more output that is worth pMPlAacl. But this marginal output costs wlAccl to produce. If the value of marginal product exceeds its cost, then proﬁts can be increased by increasing input 1. If the value of marginal product is less than its cost, then proﬁts can be increased by decreasing the level of input 1. If the proﬁts of the ﬁrm are as large as possible, then proﬁts should not increase when we increase or decrease input 1. This means that at a proﬁt—maximizing choice of inputs and outputs, the value of the marginal product, pMPﬁrLTg), should equal the factor price, wl. We can derive the same condition graphically. Consider Figure 19.1. The curved line represents the production function holding factor 2 ﬁxed at T2. Using 3; to denote the output of the ﬁrm, proﬁts are given by 7r : py — 101231 — 1112552. This expression can be solved for y to express output as a function of \$1: 7T ’11} w y 2 — + Jag + —1:r1. (19.1) p p p This equation describes isoproﬁt lines. These are just all combinations of the input goods and the output good that give a constant level of proﬁt, 11'. As 7r varies we get a family of parallel straight lines each with a slope of 101 / p and each having a vertical intercept of 7r/p+ wgfg / p, which measures the proﬁts plus the ﬁxed costs of the ﬁrm. The ﬁxed costs are ﬁxed, so the only thing that really varies as we move from one isoproﬁt line to another is the level of proﬁts. Thus higher levels of proﬁt will be associated with isoproﬁt lines with higher vertical intercepts. The proﬁt-maximization problem is then to ﬁnd the point on the produc— tion function that has the highest associated isoproﬁt line. Such a point is illustrated in Figure 19.1. As usual it is characterized by a tangency condition: the slope of the production function should equal the slope of COMPARATIVE STATICS 341 OUTPUT [soproﬁt lines slope = w1/p Y=Ifixh 532) ' production * function Y r 12;; p + P x: ' ' " x, Proﬁt maximization. The ﬁrm chooses. the input and output combination that lies on the highest iSoproﬁt line. In this case the proﬁt-maximizing point is (sf, 3;“). the isoproﬁt line. Since the slope of the production function is the marginal product, and the slope of the isoproﬁt line is ml /p, this condition can also be written as MP1 2 3‘1, P which is equivalent to the condition we derived above. 19.7 Comparative Statics We can use the geometry depicted in Figure 19.1 to analyze how a ﬁrm’s choice of inputs and outputs varies as the prices of inputs and outputs vary. This gives us one way to analyze the comparative statics of ﬁrm behavior. For example: how does the optimal choice of factor 1 vary as we vary its factor price w]? Referring to equation (19.1), which deﬁnes the isoproﬁt line, we see that increasing in} will make the isoproﬁt line steeper, as shown in Figure 19.2A. When the isoproﬁt line is steeper, the tangency must occur further to the left. Thus the optimal level of factor 1 must decrease. This simply means that as the price of factor 1 increases, the demand for factor 1 must decrease: factor demand curves must slope downward. Similarly, if the output price decreases the isoproﬁt line must become steeper, as shown in Figure 19.2B. By the same argument as given in the 342 PROFIT MAXIMIZATION (Ch. 19) mum—mm f(x1) Low p High p x1 x1 A B Comparative statics. Panel A shows that increasing w; will reduce the demand for factor 1. Panel B shows that increasing the price of output will increase the demand for factor 1 and therefore increase the supply of output. last paragraph the proﬁt-maximizing choice of factor 1 will decrease. If the amount of factor 1 decreases and the level of factor 2 is ﬁxed in the short run by assumption, then the supply of output must decrease. This gives us another comparative statics result: a reduction in the output price must decrease the supply of output. In other words, the supply function must slope upwards. Finally, we can ask what will happen if the price of factor 2 changes? Because this is a short-run analysis, changing the price of factor 2 will not change the ﬁrm’s choice of factor 2Ain the short run, the level of factor 2 is ﬁxed at E2. Changing the price of factor 2 has no effect on the sl...
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