Chapter 8
In This Chapter
Examining whether a firm is a profit maximizer
Looking at how a firm maximizes profits in the short and long run
Understanding how firms minimize costs
Economists tend to begin their analysis of firms with the assumption that the firm is a profit maximizer — that is, the firm’s ultimate aim is making the most profit it can. Although microeconomics doesn’t stop there (and has produced a number of more in-depth analyses of managerial motivation and results that don’t rely on this assumption), the profit-maximizing firm is the basic building block of microeconomic analysis.
In this chapter we justify the idea that a firm is in practice a profit maximizer. We also look at profit maximization more closely, showing how it rests at the heart of economists’ conception of efficiency and how profit-maximizing firms choose the optimum amount of stuff to produce. In addition, we also discuss the opposite approach to achieving the same end: minimizing costs while maintaining the desired output.
Take a moment to catch your breath. You have quite a ride ahead!
In most cases, economists assume that firms want to maximize their profits and that, as a result, they make their decisions on this basis.
Suppose you know nothing about a company and you don’t know that all those things are the stuff of working life. How would you then make a representation of a firm? What goals, behaviors, or practices would you guess that this representative firm would engage in?
None of this, of course, means that profit maximization is always the goal of the organization. As a simple example, think of a company whose shareholders want to receive the highest possible level of returns, whereas the management of the company wants to hold on to their jobs or take as many of the potential returns for themselves. If the managers have sufficient bargaining power over the shareholders, they can affect the decisions of the firm in their own interests.
When a firm maximizes its profits effectively, it’s acting efficiently. Economists like efficiency and deplore waste. Whatever you do in business, they prefer that you make efficient use of your capital and other resources. If you take the lowest point of the average cost curve — where the marginal cost curve crosses the average cost curve — you’ve found the most efficient level of production. In addition, profit maximization occurs at the point where marginal revenue equals marginal cost (that is, MR = MC). See Chapter 7 for much more on this. Let’s take a moment now to explore — ahem, efficiently — why these two efficiency conditions are so important.
These two definitions are distinct, meaning that satisfying allocative efficiency without satisfying productive efficiency is possible. To see why, we look at the two concepts in a bit more detail.
In Chapter 13, you can use the concept of productive efficiency to tell you a lot about how a market is operating. One application we mention here is in considering how society should treat natural monopolies — those companies that yield sufficient economies of scale relative to the size of the total market that they’re unlikely to ever face a direct competitor. One thing economists notice is that these companies tend to operate inefficiently; that is, that they don’t tend to operate at the lowest possible cost (and that consumers are consequently hurt by this, as inefficiencies get pushed on to the consumer in the form of lower quality or quantity and/or higher prices).
Figure 8-1 summarizes productive efficiency: The two shaded areas reveal how the firm can become better off by making itself more productive.
In the context of production, when a firm is operating at lowest possible cost, it’s also allocating efficiently its budget for inputs between capital and labor. This occurs — you guessed it! — when the average cost of the firm is at a minimum.
Chapter 12 uses this concept of efficiency when it talks about welfare in general. For the moment, we just want to introduce the idea that when all firms operate at their minimum cost, welfare in society is maximized.
Economists distinguish between the long- and short-run positions of a firm. They do so because a firm can find itself, in the short run, in a number of positions where it is constrained. It can’t fully react to change immediately and therefore makes slightly different decisions than it would if there were no constraints — in other words, different than it would in the long run if it were fully capable of reacting to whatever change (pricing, technological, demand, and so on) was taking place.
As an example, imagine that a firm employs ten people to do a job working on ten machines. And suppose the wage it pays its workers has become more expensive, so that the firm would be better off employing only eight machines and eight people. But the lease for the machines is not up until the end of the year, whereas it can lay off workers at any time:
Figure 8-2 illustrates this condition. Remember that a short-run marginal cost curve goes through the minimum of each of the SRATC curves, and the long-run marginal cost (LRMC) is more elastic — price and cost responsive — than in the short run.
Now it’s time to find out a little more about the profit-maximizing process. To do so, we need to make a few little extensions to the model of the firm used in Chapter 7 to take account of how the firm changes its decisions when the costs of inputs change. And to do that, we’re going to express output in terms of two inputs — capital and labor — and their respective prices, the cost of capital and the wage rate.
Firms can use two methods to work out how to use their inputs to make outputs: the profit-maximizing approach of this section and the cost-minimizing approach (see the later section “Slimming Down: Minimizing Costs”). Although eventually they lead to the same place — assuming you do the relevant calculations right! — they arrive via slightly different journeys. Wherever you’re going, though, you need a place to start, and we begin with a production function.
Here, f is the firm’s production function, and x1 is the amount of an input — for example, labor. If you know the amount of the input x1 and the shape of the function f, then you know how much output the firm will produce. We now add the second input — holding it constant because we’re going to look at the short run first — and then use a little math to figure out the relationship between output and costs.
We call the production function f, the two inputs x1 and x2, and their respective costs w1 and w2. We begin by spelling out the production function:
That shows a relationship between inputs and outputs but not profit: The f part of the expression explains that there is a relationship between the amount of output produced and the level of inputs engaged in production, the x1 and the x2. For the moment, we’re not spelling out exactly what that relationship is mathematically. To know about the firm’s profits, we need to know about the revenue gained from selling the output and the cost of obtaining the inputs. The revenue side is very simple if the firm sells its output all at the same price p. Then if you know the output level, multiply it by price (p) and you have total revenue (TR):
And for total costs (TC):
Because we’re holding the second input constant in the short run, we call it x2. It won’t be changing, and so we can treat it as a constant.
From the original production function, we can combine the two preceding equations to make a profit function that we want to maximize:
We now substitute output, y, for the production function for a moment and use that to draw a picture. Again, we’ll use the Greek letter π for profit. The equation is now like this:
This equation shows that the terms on the right-hand side and specifically all those terms that don’t include an x1, are constant. The intercept is where the curve intercepts the y axis. The expression on the right, referring to x1, gives the slope of the line, .
Now we turn to the concept of the marginal — the incremental change (see Chapter 7). In this case, the incremental change we’re interested in is the change in production at the margin. The slope of the production function is the measure of how production changes as x1 changes. In general, the marginal change in output is called marginal product, but here the relationship only exists between x1 and output, because we kept x2 constant.
Figure 8-3 illustrates maximization using the short-run production function f(x1,x2) with x2 held fixed, and three isoprofit curves. Only isoprofit 2 is possible and optimal. Why?
At isoprofit 2, the marginal product (MP, slope of the production function) is equal to the cost of the input used divided by the price received at market for your output. Or to put it more succinctly:
To discover what this means, we ask Jeph the Joiner, proprietor of Jeph’s Joinery. He has a job at the moment making chair legs for an interior designer. He wants to know, given that all his equipment is fixed in the short run, how many people to employ at a given wage in order to make as much profit as possible. His cousin Emma the Economist takes a look at his figures and says that he should employ up to the point where the contribution of the marginal worker is such that multiplying the output produced by the marginal worker by price equals the wage that Jeph will pay.
Jeph knows his hardwoods but not his margins and asks for the answer with less jargon. Emma says: “Suppose you’re making your chair legs and you know that the next one — the marginal one — is going to yield $10 in revenue. Now suppose you have to hire someone for a cost of $11 to make the leg. If the cost is greater than the marginal revenue, you make yourself better off by not producing that unit ($11 is greater than $10 and you’d lose $1 on the output). If the cost of hiring is only $9, though, you’d make a surplus of $1. You can make yourself better off still, assuming that you can sell the product, by hiring up until the cost of hiring equals the value of the marginal product — that is, the marginal revenue — you yield from selling the output.”
Here’s a quick question for you: What’s the difference between the short and the long run to an economist? If you say that in the long run all factors are variable, you’re correct! If not, take a look at the earlier section “Talking about efficiency, in the long and short run” before reading on. When you’re clear on this issue, you can go on to the next section, safe in the knowledge that extending the model to two inputs isn’t so difficult.
I wonder if, after reading the preceding section, you want to say, “Hang on a minute; that situation’s unrealistic — only one factor changes!” You’re right, because in the example Jeph’s Joinery is considering only the short run. What happens in the long run?
Here, * denotes that these are the optimal levels of inputs given the cost of inputs.
The preceding section discusses a firm maximizing its profit by choosing a level of inputs that allows it to produce a given level of profit. Now we rearrange the problem slightly and assume that the firm wants to reduce its costs to the minimum level while producing a desired level of output. To do so, the firm chooses how much to use of two inputs, called x1 and x2, as defined in the earlier section “Understanding a production function.” But unlike in that section, we want to choose a way of minimizing the cost.
Economists write this problem in a new way, using two equations to represent it:
The min means choose values of x1 and x2 that make everything in the equation (w1x1 + w2x2) as small as possible. The values we’re choosing for the inputs are optimal values, and we call those x*1 and x*2, with the * meaning an optimal value.
Rearrange this so that you can put the input x2 on the vertical axis in the graphs (see Figure 8-4), and you get this:
In this rearrangement of the equation, the quantity of x2 used is a function of constant level of cost and the relative price of x1. Given that the price of x1 is w1, the relative price of x1 is w1/w2. Now, when plotted on a graph, it becomes a straight line with a slope of –w1/w2 and an intercept on the vertical axis of C/w2.
The final step is to put the isoquant and isocost together. We restate the problem as finding a point on an isoquant line with the lowest possible isocost associated with it. This happens at a point of tangency between the isoquant and isocost, which is a more mathematical way of saying that at the optimal point or input combination, the slopes of the isocost and the isoquant are equal (see Figure 8-5).
You can derive a cool further condition from knowing about the slopes of the isoquant and isocost and how they must match. Remember that the slope of something is generally an indication of how much it’s changing at a given point. Economists are interested in what the changes along the isoquant reveal about the marginal product (MP) of the two inputs x1 and x2, and what the ratio of the marginal products says about the technology a firm is using.