Table of Contents
In economics, profit maximization is the short run or long run process by which a firm determines the price and output level that returns the greatest profit. There are several approaches to this problem. The total revenue–total cost perspective relies on the fact that profit equals revenue minus cost and focuses on maximizing this difference, and the marginal revenue–marginal cost perspective is based on the fact that total profit reaches its maximum point where marginal revenue equals marginal cost.
Economists assume that firms select prices and output levels that maximize their profits. When economists discuss profits, however, they are referring to the concept of economic profit, defined as:
Economic costs include all opportunity costs, regardless of whether these costs are explicit or implicit. An explicit cost is a cost in which a payment is actually made. An implicit cost, on the other hand, is a cost in which no money changes hands. Suppose that you borrow money from a bank to acquire capital to open a business. In this case, the interest payments on the loan would be an explicit cost. If, on the other hand, you use your own savings to finance this capital, you do not have to pay interest to someone else for the use of these funds. In this case, however, the opportunity cost would be the implicit cost of the interest that you could have received had you placed this money in an interest-bearing asset instead of buying this capital.
Economic costs are different than accounting costs. Accounting costs, for the most part, include only explicit costs. (The only exception is that accounting cost includes a measure of depreciation, which is an implicit cost. But, even in this case, the accounting measure of depreciation is based on the historical price of capital, and not based on its opportunity cost.) The reason for this distinction, of course, is that accounting systems are designed to provide a record of a firm's receipts and expenditures. For such a record to be meaningful to tax authorities and the owners of a firm, each receipt and expenditure must be accompanied by some verifiable record of transactions. Implicit costs are not directly observed (and provide no "receipts" that can be used to verify accounts).
Since economic costs include both implicit and explicit costs while accounting costs consist (almost exclusively) of explicit costs, economic costs are virtually always greater than accounting costs. The difference between these two measures of cost is the opportunity cost of resources supplied by the firm's owner. The opportunity cost of these owner-supplied resources is called normal profit. The owners of corporations (the shareholders) must receive a rate of return on their stock that is equivalent to what they could receive in their next-best alternative. So, normal profit or normal accounting profit is an economic cost that is not counted as an accounting cost. Accounting profit is defined as:
A comparison of the definitions for economic and accounting profits indicates that accounting profits will virtually always exceed economic profits. Let's take a simple example. Suppose that the owner of a firm could receive $90,000 a year using the labor, capital, and other resources that she uses to operate her own business. If she receives $70,000 in accounting profits, she would actually have suffered $20,000 in economic losses, since she is earning $20,000 less than she could receive with an alternative employment of these resources.
If the owners of a firm are receiving economic profits, this means that they are receiving a rate of return on the use of their resources that exceeds that which can be received in their next-best use. In this situation, we'd expect to see other firms entering the industry (unless barriers to entry exist).
If a firm is receiving economic losses (negative economic profits), the owners are receiving less income than could be received if their resources were employed in an alternative use. In the long run, we'd expect to see firms leaving the industry when this occurs.
If the owners of a typical firm receive zero economic profits, this means that they are receiving an income that is just equal to what they could receive in their next-best alternative. In this case, there would be no incentive for firms to either enter or leave this industry. Zero economic profits occur only if the owners are receiving accounting profits equal to normal profit.
A few terms to keep in mind:
The diagram below illustrates the profit-maximizing levels of price and output for a firm facing a downward sloping demand curve. As we'll see below, the profit-maximizing level of output occurs at the point at which MR = MC. This occurs at an output level of Qo in the diagram, the level of output at which the MR and MC curves intersect. The price that firms can charge to sell this much output is given by the demand curve (D). In this example, the price equals Po.
The shaded area in the diagram above represents the level of economic profits received by this firm. The height of this rectangle equals the difference between the price of the good (P) and average total cost (ATC). This vertical distance is equal to the profit received per unit of output. The base of the rectangle is equal to the quantity of output sold by the firm. The area of the rectangle (the shaded area) equals the profit per unit of output x, the number of units of output. This product is equal to total economic profit.
It takes factors of production to produce a good or service – no matter what the good is. Factors of production are the resource inputs – land, labor, capital and entrepreneurship – used to produce goods and services.
The production function (discussed previously) is a relationship that expresses the maximum quantity of a good that can be produced from different combinations of factor inputs. The purpose of a production function is to tell us just how much output we can produce with varying amounts of input.
The productivity of any factor of production depends on the amount of other resources available to it. Productivity is defined as output per unit of input ... for example, output per labor hour or output per acre of land. The production function represents maximum efficiency.
Remember that labor is the variable input that determines how much output we get from our fixed inputs (land and capital). In general, as the amount of labor used increases, output also increases.
Efficiency is the maximum output of a good from the resources used in producing it. There is an opportunity cost to inefficiency. When production is inefficient, society either gets fewer goods than it should, or gives up too many other goods and services in order to get the good.
The total amount of output produced by a firm is a function of the levels of input usage by the firm. In the short run, a simplified version of this relationship is provided by a firm's total physical product (TPP) (also known as total product) function. This function captures the relationship that exists between the maximum level of output that can be produced by a firm and its level of labor use, holding other inputs and technology constant. (Remember, the short run is defined as the period of time in which capital cannot be changed.) The table below contains an example of a possible total product function.
The table above indicates that output initially increases more rapidly as the level of labor use increases, but ultimately increases by smaller and smaller increments. In the example illustrated above, output even declines at higher levels of labor use (note that output declines from 275 to 270 when the level of labor use increases from 40 to 45). Economists argue that equal increases in the level of labor use will ultimately result in progressively smaller increases in output in virtually all production processes. This is a consequence of the law of diminishing returns.
According to the law of diminishing returns, the MPP of a variable input declines as more of it is employed with a given quantity of other (fixed) inputs. At some point, the ratio of labor to other factors decreases. As more labor is hired, each unit of labor has less capital and land to work with. Output begins to rise more and more slowly as more workers are hired. You can add more workers but there’s only so much they can do if you cannot also add available space or electricity (for example).
The relationship between the level of input use can also be represented through the average physical product (APP) of labor. The average physical product is defined as the ratio of total physical product to the quantity of labor. The average physical product for the firm described above has been added to the table below. Notice how the value of APP is equal to the ratio of TPP to the quantity of labor in each row of this table. As in this example, economics expect that the APP may initially rises but will ultimately decline as a result of the law of diminishing returns. The average physical product of labor is what is meant when economists talk about labor productivity. When you hear references to rising or declining labor productivity it’s about changes in APP.
The marginal physical product (MPP) (also known as just marginal product) is another useful and important concept. MPP is defined as the additional output that results from the use of an additional unit of a variable input, holding other inputs constant. It is measured as the ratio of the change in output (TPP) to the change in the quantity of labor used. In mathematical terms, this can be expressed as:
When the MPP of labor improves (MPPL >0), then total output increases. Improving the ratio of labor to other factors increases the MPP of labor.
The table below continues the estimated MPP for each of the reported intervals. Be sure that you understand how the MPP is computed from the information contained in the first two columns of this table. For example, consider the interval between 10 and 15 units of labor. Note that since TPP increases by 60 (from 120 to 180) when the quantity of labor increases by 5, the MPP of labor in this interval equals 60/5 = 12
As the table above indicates, the MPP is positive when an increase in labor use results in an increase in output. The MPP is negative when an increase in labor use results in a decrease in output.
The TPP, APP and APP curves can also be illustrated using a graph. The diagram below contains a graph of a possible TPP curve. As was true in the table above, this diagram suggests that output initially rises more rapidly as labor use increases. Beyond some point, however, TPP starts to rise by less and less with each additional unit of labor. It is possible (as in the example here) that TPP may eventually fall when too many workers are present.
The diagrams below illustrate the APP and MPP curves associated with this TPP curve. As in the table above, APP initially rise and then falls. MPP rises in the range in which TPP is increasing at a more rapid rate and declines in the range in which TPP increases at a declining rate. MPP equals zero at the point at which TPP reaches a maximum and is negative when TPP declines.
As the diagram above indicates, the MPP and APP curves intersect at the maximum level of APP. The reason for this makes sense. For levels of labor use below Lo, MPP is greater than APP. This means that an additional worker adds more to output than the average worker is producing. In this case, the average has to increase. Suppose your grade in a class at any point in time is formed by taking the average of all of the grades that you have achieved up to that point in time. If your score on an additional test (this may be thought of as a "marginal grade") exceeds your average your average grade will rise. If your marginal grade is less than your average grade your average will decline. In the same manner, the average physical product of labor will decline when the marginal physical product of labor is less than the average physical product of labor.
An inspection of the diagram above indicates that APP increases whenever the level of labor use is less than Lo. APP declines, however, when the level of labor use is greater than Lo. Since APP increases up to this point and declines after this point, APP must reach a maximum when Lo workers are employed (at the point at which MPP = APP).
A production function tells us how much a firm can produce but not how much it should produce. The most desirable rate of output is the one that maximizes total profit. Profit is the difference between total revenue and total cost.
The output decision has to be based not only on the capacity to produce (the production function) but also on the costs of production (the cost function).
The shapes of the cost curves are mirror-image reflections of the corresponding productivity curves. When one is increasing, the other is decreasing. When one is at a maximum, the other is at a minimum. The dollar costs of production are directly related to the underlying production function.
PREVIEW OF COST CONCEPTS
In the short run, total costs (TC) consist of two categories of cost: total fixed costs and total variable costs. Total fixed costs (TFC) are costs that do not vary with the level of output. The level of total fixed costs is the same at all levels of output (even when output equals zero).
Examples of such fixed costs include rent, annual license fees, mortgage payments, interest payments on loans and monthly connection fees for utilities (this includes only fixed monthly charges, not the portion of utility fees that varies with the level of use).
Total variable costs (TVC) are costs that vary with the level of output. Labor costs, raw material costs and energy costs are examples of variable costs. Variable costs are equal to zero when no output is produced and increase with the level of output.
The table below contains a listing of a hypothetical set of total fixed cost and total variable cost schedules. Total fixed costs are the same at each possible level of output. Total variable costs are expected to rise as the level of output rises.
As the table below indicates, we can use the TFC and TVC schedules to determine the total cost schedule for this firm. At each level of output, TC = TFC + TVC.
The diagram below contains a graph of a total fixed cost curve. Since total fixed costs are the same at all levels of output, a graph of the total fixed cost curve is a horizontal line.
The total variable cost curve increases as output increases. Initially, it is expected to increase at a decreasing rate (since marginal productivity increases initially, the cost of additional units of output declines). As the level of output rises, however, variable costs are expected to increase at an increasing rate (as a result of the law of diminishing marginal returns). The diagram below contains a possible total variable cost curve.
Since total cost equals the sum of total variable and total fixed costs, the total cost curve is just the vertical summation of the TFC and TVC curves. The diagrams below illustrate this relationship.
Combining TVC with TFC to Get TC
AVERAGE AND MARGINAL COST
Average fixed cost (AFC) is defined as: AFC = TFC / Q. An average fixed cost schedule has been added to the diagram below. Note that average fixed costs always decline as the level of output increases. As the rate of output increases, AFC decreases as the fixed cost is spread over more output (economies of scale). Any increase in output lowers average fixed cost.
Average variable cost (AVC) is defined as: AVC = TVC / Q. An average variable cost schedule has been added to the table below. It is expected that average variable costs will initially decrease as output increases but will eventually increase as output continues to rise. AVC will eventually rise as the rate of output increases. AVC rises because of diminishing returns in the production process. If each additional worker adds progressively less additional output, the average cost of the additional output must eventually increase.
Average total cost (ATC) is defined as: ATC = TC / Q. The table below includes an ATC schedule. ATC can also be measured as:
In addition to these average cost measures, it is also useful to measure the cost of an additional unit of output. The cost of an additional unit of output is called marginal cost (MC). Marginal cost can be measured as:
A marginal cost schedule has been added to the table below. Be sure that you understand how marginal cost is computed in this table. Consider, for example, the interval between 10 and 20 units of output. In this case, total costs increase by 20 (from 40 to 60) when 10 additional units of output are produced, so in this interval, marginal cost is 20/10 = 2. Remember that marginal cost refers to the change in total costs associated with one more unit of output. Diminishing returns in production cause MC to increase as the rate of output is expanded.
We can also represent these average and marginal cost relationships using diagrams. The diagram below contains a graph of a typical AFC curve. Note that AVC declines as output increases.
The diagram below contains a graph of the ATC, AVC and MC curves for a typical firm. The vertical distance between the ATC and the AVC curve is equal to AFC (since AFC + AVC = ATC). The MC curve intersects the AVC and the ATC curves at their respective minimum points. To see this, note that whenever marginal costs are less than average costs, the average cost must decline. When marginal costs exceed average costs, the average must rise. The MC curve must cross each of these average cost curves at their respective minimum points.
The Marginal-Average Rule: An extremely useful mathematical relationship between an average measure and its corresponding marginal measure, for example average product and marginal product, or average total cost and marginal cost. When the marginal measure is greater than the average measure, then the average measure increases. Alternatively, when the marginal measure is less than the average measure, then the average measure decreases. In addition, when the marginal measure is equal to the average measure the average measure doesn't change. So...
LONG RUN COST
The short run is characterized by costs that cannot be changed (fixed costs). In the short run, it pays to sell to any customer who'll pay marginal cost. Even if you’re losing money overall, you're losing less than if you had turned down the sale.
There are no fixed costs in the long run. The long run is a period of time long enough for all inputs to be varied (no fixed costs). In the long run, when you can get out of your fixed cost, you shut down if your average price is not more than average cost. To make their long run decisions, firms look at costs of various inputs and the technologies available for combining these inputs. They choose the combination which offers the lowest cost.
The long-run average cost curve is the curve that traces the lowest cost per unit at which a firm can produce any level of output and can build any desired plant size.
In the long run, all inputs are variable. As the firm changes the amount of capital it uses, it will shift from one short-run average total cost curve (SRATC) to another. The diagram below illustrates this relationship. As a firm acquires more capital, the minimum point on its average total cost curve is associated with a higher level of output. In this diagram, SRATC4 represents a firm with a relatively high level of capital while SRATC1 represents a firm with a low level of capital.
The long-run average total cost curve (LRATC curve) represents the lowest level of average cost that can occur in the long run at each possible level of output. It is a summary of our best short run cost possibilities. It is assumed that firms producing any given level of output in the long run would always select the size of firm that has the lowest short-run average total costs at that level of output. In the diagram above, a firm would select a level of capital that places it on the short-run average total cost curve SRATC2 if it were to produce Qo units of output. The long run marginal costs (LRMC) curve intersects our long run average total cost (LRATC) curve at its lowest point.
The law of diminishing marginal productivity does not apply in the long run. It is often argued that the long-run average cost curve has a shape similar to the diagram below. At low levels of output, it is suggested that economies of scale result in a decrease in long-run average costs as output increases. Economies of scale are factors that result in a reduction in LRATC as output rises. These factors include gains from specialization and division of labor, indivisibilities in capital, and similar factors. Diseconomies of scale, factors that result in higher levels of LRATC as output increase, are believed to be important at high levels of output. These factors include the increased cost of managing and coordinating a firm as the size of the firm rises. Constant returns to scale occur when LRATC does not change when the firm becomes larger or smaller. It is believed that this happens over a relatively large range of output (as illustrated in the diagrams below). Constant returns to scale are increases in plant size that do not affect minimum average cost – minimum per-unit costs are identical for small plants and large plants.
Typical Long Run Average Total Cost Curves
The diagram above also illustrates the concept of minimum efficient scale (MES). The minimum efficient scale of a firm is the lowest output level at which LRATC are minimized. The MES is important in determining the market structure for a particular output market. Competition among firms forces firms to produce at a level of output at which LRATC is minimized. If the MES is large, relative to the quantity of output demanded in a market, only a small number of firms can profitably coexist. If, for example, the MES is 10,000 and a quantity of only 20,000 units of output is demanded, at most two firms can survive in the market.
MARGINAL REVENUE AND MARGINAL COST
What happens to a firm's profits when it produces an additional unit of output? Remember that its economic profits are defined as:
When a firm produces an additional unit of output its revenue rises (in all practical situations) and its costs rise as well. Profits rise if revenue rises by more than costs and fall if costs rise by more than revenue. The additional revenue resulting from the sale of an additional unit of output is called marginal revenue (MR). The additional cost associated with the production of an additional unit of output is marginal cost (MC).
Consider a firm's decision about whether to produce more or less output. If marginal revenue exceeds marginal cost, the production of an additional unit of output adds more to revenue than to costs. In this case, a firm is expected to increase its level of production to increase its profits. If, instead, marginal cost exceeds marginal revenue, the production of the last unit of output costs more than the additional revenue generated by the sale of this unit. In this case, firms can increase their profits by producing less. So a profit-maximizing firm will produce more output when MR > MC and less output when MR < MC. If MR = MC, however, the firm has no incentive to produce either more or less output. In fact, the firm's profits are maximized at the level of output at which MR = MC.
Marginal revenue is an important part of a firm's decision concerning how much output to produce. Marginal revenue is defined as:
If a firm faces a perfectly elastic demand curve, the price of the good is the same at all levels of output. In this case, marginal revenue is simply equal to the market price. Suppose, for example, that corn sells for $1 per dozen. The marginal revenue received by a farmer from the sale of an additional dozen ears of corn is simply the price of $1. This is illustrated in the diagram below.
Suppose, however, that a firm faces a downward sloping demand curve. In this case, it must lower the price if it wishes to sell additional units of this good. In this case, marginal revenue is less than the price. Let's use an example to see why this is true. Consider the situation described in the diagram below. When the price is $6, the firm can sell 4 units of output while receiving a total revenue equal to $6 x 4 = $24. If it wishes to sell the 5th unit of output, it must lower the price to $5. Its total revenue in this case will equal $25. Marginal revenue in this case equals: change in total revenue / change in quantity = $1 / 1 = $1. As this example illustrates, marginal revenue will always be less than the price of the good when the firm faces a downward sloping demand curve. This is because the firm has to lower the price not just on the last unit sold but instead on all units that it sells. In this case, the firm received an additional $5 in revenue from the same of the 5th unit, but it lost $4 in revenue when it lowered the price on the first 4 units by $1. Thus, total revenue increased by only $1 when the 5th unit is sold.
The diagram below illustrates the relationship that exists between the marginal revenue curve and the demand curve. The demand curve provides the price that can be charged at each level of output. Since we know that MR is less than the price, the marginal revenue curve must lie below the demand curve. We can also see that marginal revenue is positive in the elastic section of the demand curve (since a price decrease results in an increase in total revenue in this case), is zero when demand is unit elastic (since total revenue remains unchanged when the price falls when demand is unit elastic) and is negative when demand is inelastic (since total revenue declines when the price falls in this portion of the demand curve.)
Think about each of the following statements carefully and make certain you understand WHY each is true!