Daniel l. Rubinfeld

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218 Part 2 Producers, Consumers, and Competitive Markets ii%& Capital per Year - * ¥§6M ..... per Year Chapter 7 The Cost of Production 219 Labor per Year Isocost curves describe the combination of inputs to production that cost the same amount to the firm. Isocost curve C 1 is tangent to isoquant Ql at A and sho'ws that output Ql can be produced at minimum cost with labor input Ll and capital input K 1 . Other input combinations-L 2 , K2 and L 3 , K 3 -yield the same output at higher cost. K1 I I I ---------IL-- I I I I I I I I I I I I I I Labor per Year Facn: g an iSocos: curve C~, the firm produces output Ql at point A using L units of labor and K1 U~l1ts of capIt~L When the price of labor increases, the isoco~t curves bLeco~le steeper. Output Ql IS now produced at point Bon isocost curve Co by using 2 ruuts of labor and K2 units of capital. - Choosing Inputs Suppose we wish to produce at an output le\-el Ql' How can we do so at minimum cost Look at the firm's production isoquant, labeled Q1I 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 Co on inputs. Unfortunately, no combination of inputs can be purchased for expenditure Co that will allm\' the finn to achieve output Q1' However, output Ql can be achie\'ed 'with the expenditure of C 2 , either by using K2 units of capital and units of labor or by using K3 units of capital and L3 units of labor. But C 2 is not the minimum cost. The same output Ql can be produced more cheaply, at a cost of C 1 , by using Kl units of capital and Ll w1its of labor. In fact, isocost line C 1 is the lowest isocost line that allo-ws output Q1 to be produced. TIle point of tangency of the isoquant Ql and the isocost line C1 at point A tells us the cost-minimizing choice of inputs, L1 and K]I \Nhich 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 labOl~ were to increase. In that case, the slope of the isocost line - (w/r) would increase in rnagnitude and the isocost line vvould become steeper. Figure 7.4 shows this. Initiallv, the isocost line is C and t1. 1e fl'r '" . t f . d . - I, 1 m mlIl1rruzes Its cos s a pro UCl~g output Q1 at A by using L1 units of labor and K units of ca i- tal. \~Then the pnce of labor increases, the isocost line becomes st~e . Th . P ~ost lm.e ~2 :'efle~ts the higher price of labor. Facing this higher pricePo~\abo~ l::~ rm rnmU11lzes Its cost of producing output Q1 bv producinO" at B llSU' 10" L ' 't of labor a . d K . t - . - b' bOurn S I b _ 11 . 2 U~11 S at ~apItaL The firm has responded to the higher p~ice of a or by substrtutmg capItal for labor in the production process. in How does ~he isocost line relate to the firm's production process Recall that of ~u~a:1alysIs o~ prduction technology, \'le showed that the marginal rate of ec . 11cal substrtu~lOn MRTS of labor for capital is the negative of the slo e ca;~:l~soquant and IS equal to the ratio of the marginal products of labor a~d MRTS = - ':'K/':'L = MPdMP K (7.3) ~~~~~hwe n~t~d th~t.th: isocost line ~as a slope of tlK/tlL = -w/r. It follows 10" en a .f~Im 111lI11mlZes the cost of producing a particular output the fa 1- 1\ 111g condItion holds: ' MPdMP.J..: = Iu/r In §6.2, we explain that the M~TS is the amount by which the input of capital can be reduced ·when one extra unit of labor is used, so that output remains constant.
220 Part 2 Producers, Consumers, and Competitive Markets We can rewrite this condition slightly as follows: Chapter 7 The Cost of Production 22 (7.4) ] MPdw is the additional output that results from spending an additional dollar for labor. Suppose that the wage rate is $10 and that adding a ,yorker to the production process will increase output by 20 units. The additional output per dollar spent on an additional worker \·vill be 20/10 = 2 units of output per dollar. Similarly, MPdr is the additional output that results fronl spending an additional dollar for capital. Therefore, equation (7.4) tells us that a cost-mini_ mizing finn should choose its quantities of inputs so that the last dollar's :vorth of any input added to the production process yields the same amount ot extra output \,\ihy must this condition hold for cost minimization Suppose that in addition t~ the 510 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 20/52 = 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 finn reduces labor and increases capital, its marginal product of labor ,vill rise and its marginal product of capital vl'ill fall. Eventually, the point will be reached where the production of an additional unit of output costs the same regardless of which additional input is used. At that point the finn is minimizing its cost. ~eel plan~s are often bui~t on or near riv~rs. A river offers r~adily ava,ilable, S mexpensl\'e h'ansportahon for both the Iron ore that goes mto the production process and the finished steel itself. Unfortunately, it also pro\'ides a cheap disposal method for by-products of the production process, called eff7uent. For example, a steel plant processes its iron ore for use in blast furnaces by grinding taconite deposits into a fine consistency. During this process, the ore is extracted bv a ma£cnetic field as a flow of water and fine ore passes through the • b d plant. One by-product of this process-fine taconite particles-can be dumpe in the river at relatively little cost to the firm. Alternative removal methods Of private h'eahnent plants are relatively expensive. Because the taconite particles are a nondegradable waste that can harm vegetation and fish, the Elwironmental Protection Agency (EPA) has imposed an effluent fee-a per-unit fee that the steel finn must pay for the effluent that goes into the river. How should the manager of a steel plant deal with the imposition of this effluent fee to minirnize the costs of production Suppose that without regulation the plant is producing 2000 tons of steel ~er month, using 2000 machine-hours of capital and 10,000 gallons of water (wluch contains taconite particles when returned to the river). The manager estimates that a machine-hour costs $40 and dumping each gallon of waste\','ater in the river costs 510. The total cost of production is therefore 5180,000: 580,000 for capital and 5100,000 for ·wastewater. How should the manager respond to an EPAimposed effluent fee of 510 per gallon of waste\\'ater dumped The manager (machinehours per month) 1,000 20,000 Wastewater (gallons per month) When th~ firm is not ch~rged for dumping its wastewater in a rivel~ it chooses to produc.e a gwen output usmg 10,000 gallons of 'wastewater and 2000 machine-hours of capItal, at A. Howevel~ an effluent fee raises the cost of wastewatel~ shifts the isocost curve h'om Fe to DE, and causes the finn to produce at B-a process that results in much less effluent. ~10WS that there is sOI1:e He:,ibility in the production process. If the firm puts mto place more expensIve etHuent treatment equipment, the finn can achieve the same output with less wastewater" F~ur: ~.5 shO\\:s the cost-minimizing response. The vertical axis measures the hrm s rnput ot capital in machine-hours per month-the horizontal axis measures the quantity of wastewater in gallons per month. First, consider the level at w~lich th~ firm produces when there is no eft1uent fee. Point A represents the rnput ot capital and the level of wastewater that allows the firm to produce its quota of steel at minimum cost. Because the firm is minimizina cost: A lies O~l th~ isocost line Fe, which is tangent to the isoquant. The slope of the ISCoSt lme IS equal to $10/540 = - 0.25 because a unit of capital costs four times more than a unit of wastewater. When the eft1uent fee is imposed, the cost of wastewater increases from 510 per gallon to $20: For e\'ery gallon of wastewater (which costs 510), the firm has to pay the, gO\'ernment an additional 510. The eft1uent fee therefore increases the cost ot wastewater relati\'e to capital. To produce the same output at the ~~,~es~ Possib2e ~os~' the l:nanager must chose the isoco:,t line with a slope a S-~(540.- O"J .that IS tange:'t to the Isoquant. In FIgure 7.5, DE is the :PlO~I1ate IS0COSt lme, and B glyeS the appropriate choice of capital and 1\ aste\\~ter" The move from A to B shows that with an eft1uent fee the use of an alternatll'e production technology that emphasizes the use of capital (3500
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Daniel l. Rubinfeld

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