Using the Deep Creek Mining Company example, suppose now that both capital (mea- sured by the maximum brake horsepower [bhp] rating of the equipment) and labor (measured by the number of workers) are variable inputs to the ore-mining process.
The firm can choose to operate the production process using any of the capital-labor combinations shown previously in Table 7.1.
Production Isoquants
A production function with two variable inputs can be represented graphically by a set of two-dimensional production isoquants. A production isoquant is either a geometric curve or an algebraic function representing all the various combinations of the two in- puts that can be used in producing a given level of output. In the Deep Creek example, a
TABLE 7.3 M A R G I N A L R E V E N U E P R O D U C T A N D M A R G I N A L F A C T O R C O S T—D E E P C R E E K M I N I N G C O M P A N Y
L A B O R I N P U TL ( N U M B E R
O F W O R K E R S )
T O T A L P R O D U C T
Q= (TPL) ( T O N S O F
O R E )
M A R G I N A L P R O D U C T O F L A B O R MPL( T O N S
P E R W O R K E R )
T O T A L R E V E N U E TR= P ã Q
( $ )
M A R G I N A L R E V E N U E MRQ= ΔTR ΔQ ( $ / T O N )
M A R G I N A L R E V E N U E P R O D U C T MRPL= MPLã MRQ
($/WORKER)
M A R G I N A L F A C T O R C O S T MFCL
($/WORKER)
0 0 — 0 — — —
1 6 6 60 10 60 50
2 16 10 160 10 100 50
3 29 13 290 10 130 50
4 44 15 440 10 150 50
5 55 11 550 10 110 50
6* 60 5 600 10 &50 &50
7 62 2 620 10 20 50
8 62 0 620 10 0 50
production isoquant An algebraic function or a geometric curve representing all the various combinations of two inputs that can be used in producing a given level of output.
production isoquant shows all the alternative ways in which the number of workers and various sizes of mining equipment can be combined to produce any desired level of out- put (tons of ore). Several of the production isoquants for the ore-mining example are shown in Figure 7.5. For example, an output of 6 tons can be produced using any of three different labor-capital combinations: one worker and 750-bhp equipment, two workers and 500-bhp equipment, or four workers and 250-bhp equipment. Similarly, as seen in the graph, an output of 62 tons can be produced using any one offive different labor-capital combinations.
Although each isoquant indicates how quantities of the two inputs may be substi- tuted for one another, these choices are normally limited for two reasons: first, some input combinations in Figure 7.5 employ an excessive quantity of one input. Just as more than eight workers result in negative marginal returns in choosing a single vari- able input for Deep Creek Mining (see Figure 7.2), so too here with 750-bhp machin- ery crowding effects introduced by the presence of an eighth worker would actually reduce output. Similarly, more than 1,500-bhp machinery would result in negative marginal returns to capital equipment with only five workers. Because all such ineffi- cient mixes of capital and labor increase the input requirements (and therefore costs) without increasing output, they should be eliminated from consideration in making in- put substitution choices.
Second, input substitution choices are also limited by the technology of production, which often involves machinery that is not divisible. Although one can find smaller and larger mining equipment, not every brake horsepower machine listed on the Y axis of Figure 7.5 will be available. The industrial engineering of mining operations often requires that we select from three or four possible fixed proportions production processes involving a particular size drilling machine and a requisite size labor force to run it.
FIGURE 7.5 Production Isoquants—Deep Creek Mining Company
Q = 62
0 1 2 3 4 5 6 7 8
Labor input L (number of workers) 250
9 10
500 1,000 1,250 1,500 1,750 2,000
Q = 55 Q = 29
Q = 6
Capital input K (brake horsepower)
750
The Marginal Rate of Technical Substitution
In addition to indicating the quantity of output that can be produced with any of the var- ious input combinations that lie on the isoquant curve, the isoquant also indicates therate at which one input may be substituted for another input in producing the given quantity of output. Suppose one considers the meaning of a shift from pointAto pointBon the isoquant labeled“Q= 29”in Figure 7.7. At pointA, three workers and a 750-bhp machine are being used to produce 29 tons of output, whereas at point B, four workers and a 500-bhp machine are being used to produce the same amount of output. Thefirst input Example Just What Exactly Is a Refinery, and Why Won ’ t
Anyone Build One?6
New petroleum refineries are under way in Kuwait and in Saudi Arabia that will be capable of processing 600,000 barrels of crude per day (bbl/d) and 450,000 bbl/d, respectively. However, no petroleum refinery has been built in the United States in more than 30 years. Why not? To answer this question requires knowing a little about what a refinery is and what it does.
In essence, refineries are enormous chemical plants that begin by superheating various grades of crude oil in large vessels and then pass the vapors that boil off through fractional distillation columns where they phase change back into various liquids as the distillates cool (see Figure 7.6). Crude oil contains literally hundreds of hydrocarbons, and the distillates run the gamut from lubricants and grease that
“cool”to a liquid at a whopping 450°C near the bottom of the distillation column to propane at the top. Jet fuel, diesel, and some gasoline liquefy at 250°C with the help of a catalytic cracking process in a separate converter, and farther up the col- umn naphtha distillate yields gasoline after passing through a reformer. Kerosene, butane, and polyethylene (the basic building block for plastic) also distill out.
Refining is a classic variable proportions production process. The chemical cracking process of breaking long chains of hydrocarbons can use more or less pressure, more or less heat, more or less high-quality but expensive light sweet crude or sulfurous heavy crude. From one 42-gallon barrel of crude, the input mix can be optimized to achieve about 20 gallons of gasoline and 10 gallons of diesel and heating oil. Much of the equipment involved is 10 stories high and expensive.
The fixed-cost investment today for a major refinery totals $2 billion. This cost must be recovered from just 22 percent of the final product price of gasoline, whereas crude itself commands 54 percent of the final product price. Profit margins in refining come to just a few cents per gallon, much like the thin margins in gaso- line retailing. Oil exploration and development are much more profitable than re- fining in part to compensate for the extraordinary capital investment risks involved.
P E R C EN T O F F I N A L P R O D U C T P R I C E O F G A S O L I N E B Y S T A G E O F P R O D U C T I O N ( 2 0 0 6 )
Exploration and development and extraction of crude oil 54%
Federal and state excise taxes 16
Refining 22
Distribution and retail 8
6Based on“Working Knowledge: Oil Refineries,”Scientific American(June 2006), pp. 88–89.
mix is capital-intensive like Chrysler’s highly robotic Ontario minivan plant; the second is more labor intensive, like hand-tooled auto assembly. In moving from input mix A to input mixB, we substituted one additional unit of labor for 250 units of capital. The rate at which capital has been replaced with labor in producing the given output is equal to 250/L or 250 units of capital per unit of labor. The rate at which one input may be FIGURE 7.6 Crude Oil Is Made Into Different Fuels from Distillation/Cracking/Reformer Processes
A refinery’s most important processes Products made from a barrel of crude oil (42 gal)
Other products
Liquefied petroleum gas (LPG) Heavy fuel oil
Jet fuel
Diesel fuel & heating oil Gasoline
7.61.7 1.14 10 19.6 gal
LPG
Gasoline Jet fuel Diesel fuel LPG Gasoline Motor gasoline Jet fuel Diesel fuel Gasoline
Vapors LPG Naphtha
Kerosene Diesel distillate Medium- weight gas oil Heavy gas oil
Residuum
Industrial fuel Asphalt base End products
Reformer
Alkylation unit
Coker Cracking
units
Distillation column
Source: Energy Information Administration, U.S. Department of Energy.
substituted for another input in the production process, while total output remains constant, is known as themarginal rate of technical substitution (MRTS).
MRTSis given by the slope of the curve relatingKtoL—that is, the slope of the iso- quant. The slope of theABsegment of the isoquant in Figure 7.7 is equal to the ratio of ACtoCB.Algebraically,AC=K1−K2andCB=L1−L2; therefore, the slope is equal to (K1–K2) ÷ (L1–L2). Because the slope is negative and one wishes to express the substi- tution rate as a positive quantity, a negative sign is attached to the slope:
MRTS =−K1 −K2
L1 −L2
=−ΔK
ΔL [7.8]
In the Deep Creek Mining Company example,ΔL= 3−4 =−1,ΔK= 750−500 = 250.
Substituting these values into Equation 7.8 yields MRTS =−250
−1 =250
Therefore, along Q= 29 between input combinationsAand B, 250 bhp substituted for one worker.
It can be shown that theMRTSis equal to the ratio of the marginal products ofLand Kby using the definition of the marginal product (Equation 7.2). This definition yields ΔL=ΔQ/MPLandΔK=ΔQ/MPK. Substituting these expressions into Equation 7.8 (and dropping the minus sign) yields
MRTS = ΔQ=MPK
ΔQ=MPL
MRTS = MPL
MPK [7.9]
FIGURE 7.7 The Production Isoquant Curve—Deep Creek Mining Company
1
0 2 3 4 5 6 7 8
Labor input L (number of workers) 250
9 10
500 750 1,000 1,250 1,500 1,750 2,000
Q = 29 C
A B K1
K2
L1 L2
Capital inputK (brake horsepower) Captial-intensiveC
Labor-intensive D
marginal rate of technical substitution (MRTS)The rate at which one input may be substituted for another input in producing a given quantity of output.