Cost in the Long Run

In the long run, a firm has much more flexibility. It can expand its capacity by expanding existing factories or building new ones; it can expand or contract its labor force, and in some cases, it can change the design of its products or intro- duce new products. In this section, we show how a firm can choose its combi- nation of inputs to minimize its cost of producing a given output. We will also examine the relationship between long-run cost and the level of output. We begin by taking a careful look at the cost of using capital equipment. We then show how this cost, along with the cost of labor, enters into the production decision.

1. The User Cost of Capital

Firms often rent or lease equipment, buildings, and other capital used in the production process. On other occasions, the capital is purchased. In our analy- sis, however, it will be useful to treat capital as though it were rented even if it was purchased. An illustration will help to explain how and why we do this. Let’s suppose that Delta Airlines is thinking about purchasing a new Boeing 777 airplane for $150 million. Even though Delta would pay a large sum for the air- plane now, for economic purposes the purchase price can be allocated or amor- tized across the life of the airplane. This will allow Delta to compare its revenues and costs on an annual flow basis. We will assume that the life of the airplane is 30 years; the amortized cost is therefore $5 million per year. The $5 million can be viewed as the annual economic depreciation for the airplane.

So far, we have ignored the fact that had the firm not purchased the airplane, it could have earned interest on its $150 million. This forgone interest is an opportunity cost that must be accounted for. Therefore, the user cost of capital— the annual cost of owning and using the airplane instead of selling it or never buying it in the first place—is given by the sum of the economic depreciation and the interest (i.e., the financial return) that could have been earned had the money been invested elsewhere.6 Formally,

User Cost of Capital = Economic Depreciation + (Interest Rate) (Value of Capital)

In our example, economic depreciation on the airplane is $5 million per year. Suppose Delta could have earned a return of 10 percent had it invested its money elsewhere. In that case, the user cost of capital is $5 million  + (.10) ($150 million −  depreciation). As the plane depreciates over time, its value declines, as does the opportunity cost of the financial capital that is invested in it. For example, at the time of purchase, looking forward for the first year, the user cost of capital is $5 million + (.10)($150 million) = $20 million. In the tenth year of ownership, the airplane, which will have depreciated by $50 million, will be worth $100 million. At that point, the user cost of capital will be $5 million + (.10)($100 million) = $15 million per year.

We can also express the user cost of capital as a rate per dollar of capital:

r = Depreciation rate + Interest rate

For our airplane example, the depreciation rate is 1 > 30 =  3.33 percent per year. If Delta could have earned a rate of return of 10 percent per year, its user cost of capital would be r = 3.33 + 10 = 13.33 percent per year.

As we’ve already pointed out, in the long run the firm can change all of its inputs. We will now show how the firm chooses the combination of inputs that minimizes the cost of producing a certain output, given information about wages and the user cost of capital. We will then examine the relationship between long-run cost and the level of output.

2. The Cost-Minimizing Input Choice

We now turn to a fundamental problem that all firms face: how to select inputs to produce a given output at minimum cost. For simplicity, we will work with two vari- able inputs: labor (measured in hours of work per year) and capital (measured in hours of use of machinery per year).

The amount of labor and capital that the firm uses will depend, of course, on the prices of these inputs. We will assume that because there are competitive markets for both inputs, their prices are unaffected by what the firm does. (In Chapter 14 we will examine labor markets that are not competitive.) In this case, the price of labor is simply the wage rate, w. But what about the price of capital?

THE PRICE OF CAPITAL In the long run, the firm can adjust the amount of cap- ital it uses. Even if the capital includes specialized machinery that has no alterna- tive use, expenditures on this machinery are not yet sunk and must be taken into account; the firm is deciding prospectively how much capital to obtain. Unlike labor expenditures, however, large initial expenditures on capital are necessary. In order to compare the firm’s expenditure on capital with its ongoing cost of labor, we want to express this capital expenditure as a flow—e.g., in dollars per year. To do this, we must amortize the expenditure by spreading it over the life- time of the capital, and we must also account for the forgone interest that the firm could have earned by investing the money elsewhere. As we have just seen, this is exactly what we do when we calculate the user cost of capital. As above, the price of capital is its user cost, given by r = Depreciation rate + Interest rate.

THE RENTAL RATE OF CAPITAL As we noted, capital is often rented rather than purchased. An example is office space in a large office building. In this case, the price of capital is its rental rate—i.e., the cost per year for renting a unit of capital.

Does this mean that we must distinguish between capital that is rented and capital that is purchased when we determine the price of capital? No. If the capi- tal market is competitive (as we have assumed it is), the rental rate should be equal to the user cost, r. Why? Because in a competitive market, firms that own capital (e.g., the owner of the large office building) expect to earn a competitive return when they rent it—namely, the rate of return that they could have earned by investing their money elsewhere, plus an amount to compensate for the depre- ciation of the capital. This competitive return is the user cost of capital.

Many textbooks simply assume that all capital is rented at a rental rate r. As we have just seen, this assumption is reasonable. However, you should now understand why it is reasonable: Capital that is purchased can be treated as though it were rented at a rental rate equal to the user cost of capital.

For the remainder of this chapter, we will therefore assume that a firm rents all of its capital at a rental rate, or “price,” r, just as it hires labor at a wage rate, or “price,” w. We will also assume that firms treat any sunk cost of capital as a fixed cost that is spread out over time. We need not, therefore, concern ourselves with sunk costs. Rather, we can now focus on how a firm takes these prices into account when determining how much capital and labor to utilize.7

3. The Isocost Line

We begin by looking at the cost of hiring factor inputs, which can be represented by a firm’s isocost lines. An isocost line shows all possible combinations of labor and capital that can be purchased for a given total cost. To see what an isocost line looks like, recall that the total cost C of producing any particular output is given by the sum of the firm’s labor cost wL and its capital cost rK:

C = wL + rK                                                         (7.2)

For each different level of total cost, equation (7.2) describes a different isocost line. In Figure 7.3, for example, the isocost line C0 describes all possible combi­nations of labor and capital that cost a total of C0 to hire.

If we rewrite the total cost equation as an equation for a straight line, we get

K = C/r – (w/r)L

It follows that the isocost line has a slope of AK/ AL = -(w/r), which is the ratio of the wage rate to the rental cost of capital. Note that this slope is similar to the slope of the budget line that the consumer faces (because it is determined solely by the prices of the goods in question, whether inputs or outputs). It tells us that if the firm gave up a unit of labor (and recovered w dollars in cost) to buy w/r units of capital at a cost of r dollars per unit, its total cost of production would remain the same. For example, if the wage rate were $10 and the rental cost of capital $5, the firm could replace one unit of labor with two units of capital with no change in total cost.

4. Choosing Inputs

Suppose we wish to produce at an output level q1. How can we do so at minimum cost? Look at the firm’s production isoquant, labeled q1, in Figure 7.3. The problem is to choose the point on this isoquant that minimizes total cost.

Figure 7.3 illustrates the solution to this problem. Suppose the firm were to spend C0 on inputs. Unfortunately, no combination of inputs can be purchased for expenditure C0 that will allow the firm to achieve output q1. However, output q1 can be achieved with the expenditure of C2, either by using K2 units of capital and L2 units of labor, or by using K3 units of capital and L3 units of labor. But C2 is not the minimum cost. The same output q1 can be produced more cheaply, at a cost of C1, by using K1 units of capital and L1 units of labor. In fact, isocost line C1 is the lowest isocost line that allows output q1 to be produced. The point of tangency of the isoquant q and the isocost line C1 at point A gives us the cost-minimizing choice of inputs, L1 and K1, which can be read directly from the diagram. At this point, the slopes of the isoquant and the isocost line are just equal.

When the expenditure on all inputs increases, the slope of the isocost line does not change because the prices of the inputs have not changed. The intercept, how­ever, increases. Suppose that the price of one of the inputs, such as labor, were to increase. In that case, the slope of the isocost line -(w/r) would increase in magnitude and the isocost line would become steeper. Figure 7.4 shows this. Initially, the isocost line is C1, and the firm minimizes its costs of producing out­put q1 at A by using L1 units of labor and K1 units of capital. When the price of labor increases, the isocost line becomes steeper. The isocost line C2 reflects the higher price of labor. Facing this higher price of labor, the firm minimizes its cost of producing output q1 by producing at B, using L2 units of labor and K2 units of capital. The firm has responded to the higher price of labor by substituting capital for labor in the production process.

How does the isocost line relate to the firm’s production process? Recall that in our analysis of production technology, we showed that the marginal rate of tech- nical substitution of labor for capital (MRTS) is the negative of the slope of the isoquant and is equal to the ratio of the marginal products of labor and capital:

Above, we noted that the isocost line has a slope of K/L = -w/r It follows that when a firm minimizes the cost of producing a particular output, the following condition holds:

We can rewrite this condition slightly as follows:

MPt/w is the additional output that results from spending an additional dol­lar for labor. Suppose that the wage rate is $10 and that adding a worker to the production process will increase output by 20 units. The additional output per dollar spent on an additional worker will be 20/10 = 2 units of output per dol­lar. Similarly, MPK/r is the additional output that results from spending an addi­tional dollar for capital. Therefore, equation (7.4) tells us that a cost-minimizing firm should choose its quantities of inputs so that the last dollar’s worth of any input added to the production process yields the same amount of extra output.

Why must this condition hold for cost minimization? Suppose that in addi­tion to the $10 wage rate, the rental rate on capital is $2. Suppose also that add­ing a unit of capital will increase output by 20 units. In that case, the additional output per dollar of capital input would be 20/$2 = 10 units of output per dol­lar. Because a dollar spent for capital is five times more productive than a dollar spent for labor, the firm will want to use more capital and less labor. If the firm reduces labor and increases capital, its marginal product of labor will rise and its marginal product of capital will fall. Eventually, the point will be reached at which the production of an additional unit of output costs the same regardless of which additional input is used. At that point, the firm is minimizing its cost.

Source: Pindyck Robert, Rubinfeld Daniel (2012), Microeconomics, Pearson, 8th edition.

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