A ppendix to. I soquants. Producing at Least Cost. Chapter

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1 A ppendix to Chpter 0 Producing t est Cost This ppendix descries set of useful tools for studying firm s long-run production nd costs. The tools re isoqunts nd isocost lines. I soqunts FIGURE A0. SHOWS SWANY S PRODUCTION function. The figure highlights tht Swnky cn use three different comintions of lor nd cpitl to produce sweters dy nd two different comintions to produce 0 nd sweters dy. These comintions, nd mny others not shown in the figure, cn e illustrted y using n isoqunt mp. An Isoqunt Mp An isoqunt is curve tht shows the different comintions of lor nd cpitl required to produce given quntity of output. The word isoqunt mens equl quntity iso mening equl nd qunt mening quntity. There is n isoqunt for ech output level. A series of isoqunts is clled n isoqunt mp. Figure A0. shows n isoqunt mp with three isoqunts: one for 0 sweters, one for sweters, nd one for sweters. Ech isoqunt shown is sed on the production function in Fig. A0.. Although ll goods nd services cn e produced y using vriety of lterntive methods of production or techniques, the ese with which cpitl nd lor cn e sustituted for ech other vries from industry to industry. The shpe of the production function reflects the ese with which inputs cn e sustituted for ech other. Therefore the production function cn e used to clculte the degree of sustitutility etween inputs. Such clcultion involves new concept the mrginl rte of sustitution of lor for cpitl.

2 6 A PPENDIX TO CHAPTER 0 PRODUCING AT EAST COST FIGURE A0. Swnky s Production Function FIGURE A0. An Isoqunt Mp Output (sweters per dy) c sweters sweters sweters 0 0 The figure shows how mny sweters cn e produced per dy y vrious comintions of lor nd cpitl inputs. For exmple, y using worker nd mchines, Swnky cn produce 0 sweters dy; nd y using workers nd knitting mchines, Swnky cn produce 0 sweters. The figure illustrtes n isoqunt mp, ut one tht shows only isoqunts those for 0,, nd sweters dy. These curves correspond to the production function shown in Fig. A0.. If Swnky uses mchines nd worker (point ), it produces 0 sweters. If it uses mchines nd worker (point ), it produces sweters. And if it uses mchines nd workers (point c), it produces sweters. Mrginl Rte of Sustitution The mrginl rte of sustitution of lor for cpitl is the decrese in cpitl needed per unit increse in lor so tht output remins constnt. The mrginl rte of sustitution is the mgnitude of the slope of the isoqunt. Figure A0. illustrtes this reltionship. The figure shows the isoqunt for sweters dy. Pick ny point on this isoqunt nd imgine incresing lor y the smllest conceivle mount nd decresing cpitl y the mount necessry to keep output constnt t sweters. As we increse the lor input nd decrese the cpitl input so s to keep output constnt t sweters dy, we trvel down long the isoqunt. The mrginl rte of sustitution of lor for cpitl t point is the mgnitude of the slope of the stright red line tht is tngent to the isoqunt t point. The slope of the isoqunt t point equls the slope of the red line. To clculte tht slope, let s move long the red line from knitting mchines nd no workers to. workers nd no knitting mchines. Cpitl decreses y knitting mchines, nd lor increses y. workers. The mgnitude of the slope is divided y., which equls. Thus when Swnky uses technique to produce sweters dy, the mrginl rte of sustitution of lor for cpitl is. The mrginl rte of sustitution of lor for cpitl t point is the mgnitude of the slope of the stright red line tht is tngent to the isoqunt t point. Along this red line, if cpitl decreses y. knitting mchines, lor increses y workers. The mgnitude of the slope is. knitting mchines divided y, which equls /. Thus when Swnky uses technique to produce sweters dy, the mrginl rte of sustitution of lor for cpitl is /. The mrginl rtes of sustitution we ve just clculted oey the lw of diminishing mrginl rte of sustitution, which sttes tht

3 I SOCOST 7 INES FIGURE A0. The Mrginl Rte of Sustitution FIGURE A0. Swnky s Input Possiilities MRS = Totl cost = $00 MRS = sweters 0 The mrginl rte of sustitution is mesured y the mgnitude of the slope of the isoqunt. To clculte the mrginl rte of sustitution t point, use the red line tht is tngentil to the isoqunt t point. Clculte the slope of tht line to find the slope of the isoqunt t point. The mgnitude of the slope t point is. Thus t point, the mrginl rte of sustitution of lor for cpitl is. The mrginl rte of sustitution t point is found from the slope of the red line tngentil to the isoqunt t tht point. Tht slope is /. Thus the mrginl rte of sustitution of lor for cpitl t point is /. The mrginl rte of sustitution of lor for cpitl diminishes s the mount of lor increses nd the mount of cpitl decreses. You cn now see tht the lw of diminishing mrginl rte of sustitution determines the shpe of the isoqunt. When the cpitl input is lrge nd the lor input is smll, the isoqunt is steep nd the mrginl rte of sustitution of lor for cpitl is lrge. As the cpitl input decreses nd the lor input increses, the isoqunt ecomes fltter nd the mrginl rte of sustitution of lor for cpitl diminishes. Only curves tht re owed towrd the origin hve this feture; hence isoqunts re lwys owed towrd the origin. Isoqunts re very nice, ut wht do we do with them? The nswer is tht we use them to work out c d Isocost line e 0 For given totl cost, Swnky s input possiilities depend on input prices. If lor nd cpitl cost $ dy ech, for totl cost of $00 Swnky cn employ the comintions of cpitl nd lor shown y the points through e. The line pssing through these points is n isocost line. It shows ll possile comintions of cpitl nd lor tht Swnky cn hire for totl cost of $00 when cpitl nd lor cost $ dy ech. firm s lest-cost technique of production. But to do so, we need to illustrte the firm s costs in the sme sort of figure tht contins the isoqunts. I socost ines A FIRM S COSTS CAN BE IUSTRATED BY ISOCOST lines. An isocost line shows ll the comintions of cpitl nd lor tht cn e ought for given totl cost. For exmple, suppose Swnky is going to spend $00 dy producing sweters. nitting-mchine opertors cn e hired for $ dy, nd knitting mchines cn e rented for $ dy. The points,, c, d, nd e in Fig. A0. show five possile comintions of lor nd cpitl tht Swnky cn employ for $00. For exmple, point shows tht Swnky cn use mchines (costing $7) nd worker (costing

4 8 A PPENDIX TO CHAPTER 0 PRODUCING AT EAST COST $). If Swnky cn employ workers nd mchines for frctions of dy, then ny comintion long the line e will cost Swnky $00 dy. This line is Swnky s isocost line for totl cost of $00. The Isocost Eqution The isocost line cn e descried y n eqution. We ll work out the isocost eqution y using symols tht pply to ny firm nd with numers tht descrie Swnky s sitution. The vriles tht ffect the firm s totl cost (TC ) re the prices of the inputs the price of lor (P ) nd the price of cpitl (P ) nd the quntities of the inputs employed the quntity of lor () nd the quntity of cpitl ( ). In Swnky s cse, we re going to look t the mount of lor nd cpitl tht cn e employed when ech input costs $ dy nd the totl cost is $00. The cost of the lor employed (P ) plus the cost of the cpitl employed (P ) is the firm s totl cost (TC ). Tht is, nd in Swnky s cse, P + P = TC To clculte the isocost eqution, divide the firm s totl cost y the price of cpitl nd then sutrct (P /P ) from oth sides of the resulting eqution. The isocost eqution is It tells us how the firm cn vry its cpitl input s it vries its lor input, holding totl cost constnt. Swnky s isocost eqution is =. This eqution corresponds to the isocost line in Fig. A0.. The Effect of Input Prices Along the isocost line tht we hve just clculted, cpitl nd lor cost $ dy ech. Therefore, to decrese its cpitl input y unit nd keep its totl cost t $00 dy, the firm must increse the lor input y unit. The mgnitude of the slope of the isocost line shown in Fig. A0. is. The slope tells us tht unit of lor costs unit of cpitl. + = 00. = TC/ P ( P / P ) If fctor prices chnge, the slope of the isocost line chnges. If the wge rte rises to $0 dy nd the rentl rte of mchine remins t $ dy, then worker costs mchines nd the isocost line ecomes steeper line B in Fig. A0.. If the wge rte remins t $ dy nd the rentl rte of mchine rises to $0 dy, then mchine costs workers nd the isocost line ecomes less steep line C in Fig. A0.. The higher the reltive price of lor, the steeper is the isocost line. The mgnitude of the slope of the isocost line mesures the reltive price of lor in terms of cpitl tht is, the price of lor divided y the price of cpitl. As the price of either cpitl or lor chnges, so too does the reltive price of lor nd the slope of the isocost line. FIGURE A0. Input Prices nd the Isocost ine or $0 Cpitl $ B or $ Cpitl $ 0 C A or $ Cpitl $0 The slope of the isocost line depends on the reltive input prices. Three cses re shown (ech for totl cost of $00). If the prices of lor nd cpitl re $ dy ech, the isocost line is line A. If the price of lor rises to $0 ut the price of cpitl remins $, the isocost line ecomes steeper nd is line B. If the price of cpitl rises to $0 nd the price of lor remins constnt t $, the isocost line ecomes fltter nd is line C.

5 I SOCOST 9 INES The Isocost Mp An isocost mp shows series of isocost lines, ech for different totl cost when the price of ech input is constnt. With lrger totl cost, lrger quntities of ll the inputs cn e employed. Figure A0.6 illustrtes n isocost mp. In tht figure, the middle isocost line is the one in Fig. A0.. It is the isocost line for totl cost of $00 when cpitl nd lor cost $ dy ech. The other two isocost lines in Fig. A0.6 re for totl cost of $ nd $7, when the input prices re constnt t $ ech. The nswer cn e seen in Fig. A0.7. The isoqunt for sweters is shown, nd the three points on tht isoqunt (mrked,, nd c) illustrte the three techniques of producing sweters tht re shown in Fig. A0.. The figure lso contins two isocost lines ech drwn for price of cpitl nd price of lor of $. One isocost line is for totl cost of $, nd the other is for totl cost of $00. First, consider point. Swnky cn produce sweters t point y using worker nd mchines. The totl cost when Swnky uses this technique of production is $. Point c, which uses workers nd mchine, is nother technique y The est-cost Technique The lest-cost technique is the comintion of inputs tht minimizes totl cost of producing given output. et s suppose tht Swnky wnts to produce sweters dy. Wht is the lest-cost wy of doing this? FIGURE A0.7 The est-cost Technique of Production FIGURE A0.6 An Isocost Mp TC = $ est-cost technique TC = $ TC = $00 c sweters TC = $00 0 TC = $7 0 This isocost mp shows three isocost lines, one for totl cost of $7, one for $00, nd one for $. For ech isocost line, the prices of cpitl nd lor re $ dy ech. The slope of ech isocost line is equl to the price of lor divided y the price of cpitl constnt. The lrger the totl cost, the frther is the isocost line from the origin. The lest-cost technique of producing sweters is mchines nd workers (point ) nd the totl cost is $00. An output of sweters cn e produced y using mchines nd worker (point ) or mchine nd workers (point c). But with either of these techniques, the totl cost is greter, t $. At, the isoqunt for sweters is tngentil to the isocost line for $00. The isocost line nd the isoqunt hve the sme slope. If the isoqunt intersects the isocost line for exmple, t nd c the lest-cost technique hs not een found. With the lest-cost technique, the mrginl rte of sustitution (slope of isoqunt) equls the reltive price of the inputs (slope of isocost line).

6 0 A PPENDIX TO CHAPTER 0 PRODUCING AT EAST COST which the firm cn produce sweters for cost of $. Next look t point. At this point, Swnky uses mchines nd workers to produce sweters t totl cost of $00. Point is the lest-cost technique or the economiclly efficient technique for producing sweters when knitting mchines nd workers cost $ dy ech. Notice tht lthough there is only one wy in which Swnky cn produce sweters for $00, there re severl wys of producing sweters for more thn $00. Techniques shown y points nd c re two exmples. All the points etween nd nd ll the points etween nd c re lso wys of producing sweters for cost tht exceeds $00 ut is less thn $. Tht is, there re isocost lines etween those shown, for totl costs etween $00 nd $. Those isocost lines cut the isoqunt for sweters t the points etween nd nd etween nd c. Swnky cn lso produce sweters for cost tht exceeds $. Tht is, the firm cn chnge its technique of production y moving long the isoqunt to point ove point or to point to the right of point c. All of these wys of producing sweters re economiclly inefficient. You cn see tht Swnky cnnot produce sweters for less thn $00 y imgining the isocost line for $99. Tht isocost line will not touch the isoqunt for sweters. Tht is, the firm cnnot produce sweters for $99. At $ for unit of ech input, $99 will not uy the inputs required to produce sweters. Mrginl Rte of Sustitution Equls Reltive Input Price At the lest-cost technique point, the slope of the isoqunt is equl to the slope of the isocost line. Equivlently, when firm is using the lest-cost technique of production, the mrginl rte of sustitution etween the inputs equls their reltive price. Recll tht the mrginl rte of sustitution is the mgnitude of the slope of n isoqunt. Reltive input prices re mesured y the mgnitude of the slope of the isocost line. We ve just seen tht producing t lest cost mens producing t point where the isocost line is tngentil to the isoqunt. Becuse the two curves re tngentil, their slopes re equl. Hence the mrginl rte of sustitution (the mgnitude of the slope of isoqunt) equls the reltive input price (the mgnitude of the slope of isocost line). M rginl Product nd Mrginl Cost in the ong Run WHEN ONG-RUN COST IS MINIMIZED, THE MARginl cost of incresing output y using one more unit of cpitl is equl to the mrginl cost of incresing output y using one more unit of lor. To see why, we re first going to lern out the reltionship etween the mrginl rte of sustitution nd mrginl product. Mrginl Rte of Sustitution nd Mrginl Products The mrginl rte of sustitution nd the mrginl products re linked together in formul: The mrginl rte of sustitution of lor for cpitl equls the mrginl product of lor divided y the mrginl product of cpitl. The chnge in output resulting from chnge in inputs is determined y the mrginl products of the inputs. Tht is, Chnge in output = ( MP ) + ( MP ). Tht is, the chnge in output equls the mrginl product of lor, MP multiplied y the chnge in lor,, plus the mrginl product of cpitl, MP, multiplied y the chnge in the cpitl,. Suppose tht Swnky wnts to chnge its inputs ut remin on n isoqunt tht is, it wnts to chnge its inputs of lor nd cpitl ut produce the sme numer of sweters. To remin on n isoqunt, the chnge in output must e zero. We cn mke the chnge of output zero in the ove eqution; doing so yields the eqution MP = MP. Divide oth sides of this eqution y the increse in lor ( ) nd lso divide oth sides y the mrginl product of cpitl (MP ) to give MP / MP = / This eqution tells us tht, when Swnky remins on n isoqunt, the decrese in its cpitl input ( ) divided y the increse in its lor input ( ) is equl to the mrginl product of lor (MP )

7 M ARGINA PRODUCT AND MARGINA COST IN THE ONG RUN divided y the mrginl product of cpitl (MP ). The decrese in cpitl divided y the increse in lor when we remin on given isoqunt is the mrginl rte of sustitution of lor for cpitl. Wht we hve discovered, then, is tht the mrginl rte of sustitution of lor for cpitl equls the rtio of the mrginl product of lor to the mrginl product of cpitl. Mrginl Cost When the lest-cost technique is employed, the slope of the isoqunt nd the isocost line re the sme. Tht is, MP / MP = P / P Rerrnge the ove eqution in the following wy. First, multiply oth sides y the mrginl product of cpitl nd then divide oth sides y the price of lor. We then get MP / P = MP / P This eqution sys tht the mrginl product of lor per dollr spent on lor is equl to the mrginl product of cpitl per dollr spent on cpitl. In other words, the extr output from the lst dollr spent on lor equls the extr output from the lst dollr spent on cpitl. This mkes sense. If the extr output from the lst dollr spent on lor exceeds the extr output from the lst dollr spent on cpitl, it will py the firm to use less cpitl nd use more lor. By doing so, it cn produce the sme output t lower totl cost. Conversely, if the extr output from the lst dollr spent on cpitl exceeds the extr output from the lst dollr spent on lor, the firm cn lower its cost of producing given output y using less lor nd more cpitl. A firm chieves the lest-cost technique of production only when the extr output from the lst dollr spent on ll the inputs is the sme. Mrginl cost with fixed cpitl nd vrile lor equls mrginl cost with fixed lor nd vrile cpitl. To see this proposition, simply flip the lst eqution over nd write it s P / MP = P / MP. Expressed in words, this eqution sys tht the price of lor divided y its mrginl product must equl the price of cpitl divided y its mrginl product. But wht is the price of n input divided y its mrginl product? The price of lor divided y the mrginl product of lor is mrginl cost when the cpitl input is held constnt. To see why this is so, first recll the definition of mrginl cost: Mrginl cost is the chnge in totl cost resulting from unit increse in output. If output increses ecuse one more unit of lor is employed, totl cost increses y the cost of the extr lor, nd output increses y the mrginl product of the lor. So mrginl cost is the price of lor divided y the mrginl product of lor. For exmple, if lor costs $ dy nd the mrginl product of lor is sweters, then the mrginl cost of sweter is $.0 ($ divided y ). The price of cpitl divided y the mrginl product of cpitl hs similr interprettion. The price of cpitl divided y the mrginl product of cpitl is mrginl cost when the lor input is constnt. As you cn see from the ove eqution, with the lest-cost technique of production, mrginl cost is the sme regrdless of whether the cpitl input is constnt nd more lor is used or the lor input is constnt nd more cpitl is used.