Daniel l. Rubinfeld

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214 Part 2 Producers, Consumers, and Competitive Markets A Variable costs that are constant for all output levels Electricity $316 Alumina 369 Other raw materials 125 Plant power and fuel 10 Subtotal $820 B Variable costs that increase when output exceeds 600 tons/day Labor $150 Maintenance 120 Freight 50 Subtotal $320 Total operating costs $1140 but only when the plant operates two shifts per day- To increase outpu.t above 600 tor{s per day, a third shift would be necessary and would r~sult 111 a 50- percent increase in the per-ton costs of labor, ma~1tenance, and frelght. The short-run maro-inal cost and average vanable cost curves for the smelting plant are shown bin Figure 7.2. The marginal and average variable cost Cost (dollars per ton) 1300 ~ 1200 :v1C ", I;' " , I 14 1 I 1 1 1 1 1 1 1 HO H ___+____-r--r-'-" ,I 1 'f~AVC 1 + I i I i I T I T I T -1 .' I 1100 fL ____,L____--:-::-,-;:-___--;~I;;:;--- 300 600 900 Output (tons per day) curves are horizontal at a cost of 51140 per ton for outputs up to 600 tons per day, which represents the maximum output achievable with two shifts per day of production, As we mow to the increased production of aluminum by means of a third shift, the marginal cost of labor, maintenance, and freight increases from 5320 per ton to 5480 per ton, vvhich causes marginal cost as a whole to increase from 51140 per ton to S1300 per ton, As the figure shows, the increases in marginal costs cause average costs to increase as well. Finally, when output reaches 900 tons per day, an absolute capacity constraint is reached, at which point the marginal and average costs of production become infinite. - 7.3 In the long n111, the firm can change all its inputs. In this section we show how a firm chooses the combination of inputs that minimizes the 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 firm's cost of using capital equipment We then show how this cost, along with the cost of labor, enters into the production decision. 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 analysis, howe\'er, it will be useful to heat capital as though it v,'ere rented, even if it was, in fact, purchased. An illustration will help to explain how and why we do this. Follmving on our previous example, let's suppose that Delta Airlines is thinking about purchasing a new Boeing 777 airplane for 5150 million, Even though Delta would pay a large sum for the airplane now, for economic purposes the purchase price can be allocated or alllorti:ed across the life of the airplane, This will allow Delta to compare its revenues and costs on an al1l11wl flow basis. We will assume that the life of the airplane is 30 years; the amortized cost is therefore 55 million per year. The 55 million can be viewed as the aJl111wl ecollomic depreciatioll 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 5150 million. This forgone interest is an opportunity cost that must be accounted for. Therefore, the user cost of capital-the annual cost of O\\·ning and using the airplane instead of selling it or ne\,er buy ~ng it in the first place-is giyen by the slim of tile ecollomic depreciatioll alld the IIlterest (i,e" the fillallcial retlll'1l) that cOlild have beell ea/'lled fwd the IllOllel/ beell invested elsewhere." Formally, ' User Cost of Capital = Economic Depreciation + (Interest Rate)(Value of Capital) 7 The Cost of Production 2 5 user cost of capital Sum of the aIUlual cost of owning and using a capital asset, equal to economic depreciation plus forgone interest. The short-run averaae variable cost of smelting is constant for output levels usin,g up to two labor shifts. VVhen the third shift is added, marginal cost and average vanable 5 More preciseh', the financial return should renect an il\\'eshnent with similar risk The interest rate, therefore, should include a risk premium. We discuss this point in Chapter 15
216 Part 2 Producers, Consumers, and Competitive Markets 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 im'ested its money elsewhere. In that case, the user cost of capital is $5 million + ClO)($150 millidn - 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 + ClO)(S150 million) = S20 million, In the tenth year of o\vnership, the airplane, which will have depreciated by S50 million, will be worth $100 million. At that point, the user cost of capital will be $5 million + (.10)(5100 million) = S15 million per year. We can also express the user cost of capital as a rate per dollar of capital: I' 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 I' = 3.33 + 10 = 13.33 percent per year. In the long run, the firm can change all its inputs. We will now show hm,>' the firm chooses the combination of inputs that minimizes the cost of producing a given output. We will then exam.ine the relationship between long-run cost and the level of output. The Cost-Minimizing Input Choice We now hm1 to a fundamental problem that all firms face: hmu to select il1pllts to produce a given outPllt at minimum cost. For simplicity, we will work with hvo variable 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 there are competitive rnarkets for both inputs, so that 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 simp Iv the waoe rate, IU. But what about the price of , • 0 capital In the long nm, the firm can adjust the amount of capital it uses, (Even capital includes specialized machinery that has no alternative 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 ~f labor, we want to express this capital expendihll'e as a flow-e.g., in dollars per year. To do this, we must amortize the expendihue by spreading it o\'er the lifetime of the capital, and \\'e must also account for the forgone interest that the firm could have earned by investirw the mone" elsewhere. As we have just seen, • 1:> J this is exactly what we do ,\'hen we calculate the user cost of capital. As above, the price of capital is its user cost, given by r = Depreciation rate + Interest rate. Does this mean that we must distinguish between capital that is rented and capital that .is purcha~~d when we determine th~ ~l'ice of capital No. If the capital market IS cornpehhve (as we have assumed It IS), the rental rate should be equal to the lIser cost, 1'. Why Because in a competitive market, firms that own capital (e.g., the owner of the large office building) expect to earn a competitive rehm1 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 depreciation of the capitaL This cOll1petitive 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. HO'wever, you should now understand wiry it is reasonable: Capital that is purchased call be treated as though it were rellted at a relltall'l1te equal to the user cost of capital. For the remainder of this chapter~ we will therefore assume that the 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 can now focus on how a firm takes these prices into account when determining how much capital and labor to utilize. 6 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 Co describes all possible combinations of labor and capital that cost a total of Co to hire. If we rewrite the total cost equation as an equation for a straight line, we get K = C/r - (wlr)L It follows that the isocost line has a slope of f::,.K/ilL = (wlr), 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 ~y the prices of the goods in question, whether inputs or outputs). It tells us th;t if the firm gave up a unit of labor (and recovered w dollars in cost) to buv wlr units.of capital at a cost of r dollars per unit, its total cost of production ,\:ould remam the same. For example, if the ,·\,age rate were $10 and the rental cost of capital S5, the firm could replace one unit of labor with h,>'o lmits of capital with no change in total cost. Chapter 7 The Cost of Production 217 isocost line Graph showing all possible combinations of labor and capital that can be purchased for a given total cost rental rate Cost per year of renting one unit of capital Sometimes capital is 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. !t is possible, of course, that input prices might increase with demand because of overtime or a rel ;:ve shortage of capital equipment We discuss the possibility of a relationship between the price of ctor I!1puts and the quantitIes demanded by a firm in Chapter 14 ..
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Daniel l. Rubinfeld

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