Firm Objectives. The Theory of the Firm II. Cost Minimization Mathematical Approach. First order conditions. Cost Minimization Graphical Approach


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1 Pro. Jy Bhttchry Spring 200 The Theory o the Firm II st lecture we covered: production unctions Tody: Cost minimiztion Firm s supply under cost minimiztion Short vs. long run cost curves Firm Ojectives Proit mximiztion: irms choose tht level o output nd tht comintion o inputs which yields the highest level o proits Cost minimiztion: irms choose tht comintion o inputs to produce certin level o output t minimum cost. Cost minimiztion should hold even or nonproit irms. Spring 200 Econ ecture 2 Spring 200 Econ ecture 2 2 Cost Minimiztion Mthemticl Approch min w + r, s. t. = F (, ) ( F( ) ) G = w + r λ, First order conditions F = r λ = 0 r = λmp F = w λ = 0 w = λmp = F(, ) = 0 λ r MP = w MP Spring 200 Econ ecture 2 3 Spring 200 Econ ecture 2 4 Input demnd equtions Solving the three irst order conditions simultneously yields two input demnd equtions: = ) = ) Plugging these ck into the totl input expenditure identity (Expend. = w + r) yields the minimum totl cost curve: r ) = r + w = C, Spring 200 Econ ecture 2 5 Cost Minimiztion Grphicl Approch Decresing costs MP slope = MP w slope = r ( ) = F, Spring 200 Econ ecture 2 6 Econ ecture 2
2 Pro. Jy Bhttchry Spring 200 Firm Optimiztion Tngency Condition increse use o n input until its mrginl product is equl to its rel price i MP /MP > w/r then incresing use o (while decresing use o ) will decrese costs i MP /MP < w/r then decresing use o (while incresing use o ) will decrese costs Technology Constrint Exmple Prolem For CoDouglss Production =, wht re the input demnd unctions or nd? Wht is the minimum cost unction? Set up the grngin: G = r + w λ ( ) Spring 200 Econ ecture 2 7 Spring 200 Econ ecture 2 8 First Order Conditions = r λ = 0 r = λ = w λ = 0 w = λ = = 0 λ r = w Input Demnd Rerrnging the FOC s yields: r = = w Sustituting in nd solving yields: r w + = = w r Spring 200 Econ ecture 2 9 Spring 200 Econ ecture 2 0 Cost Function The demnd or lor is: w = r Plugging the input demnd unctions into the expenditure unction yields the cost unction: ) w = r + w = r r w r w r ) = r + w Spring 200 Econ ecture 2 w + w r Mrginl nd Averge Cost Cost = w + r The cost unction reltes costs to input prices nd to quntity produced. Mrginl Cost equls the extr cost required to produce smll mount or o the good: MC = Averge Cost equls the cost per unit o good produced. AC = Spring 200 Econ ecture 2 2 Econ ecture 2 2
3 Pro. Jy Bhttchry Spring 200 Reltionship Between Mrginl Cost nd Mrginl Product MC is the mount it costs to produce smll mount o extr output. In principle, irm might rdiclly chnge its demnd or inputs in order to increse output while minimizing costs. MP X is the numer o dditionl units o output you get rom n dditionl unit o input, X. In this thought experiment, the mount o ll inputs other thn X re held constnt. Mrginl Costs Intersect Averge Cost Curves t the Minimum Point I MC > AC t some point, then producing the extr unit will drive up verge costs. I MC < AC t some point, then producing the extr unit will drive down verge costs. I MC = AC, then producing the extr unit will reproduce verge costs. Togethe these three sttements imply mrginl cost curves intersect verge costs t the minimum point. Spring 200 Econ ecture 2 3 Spring 200 Econ ecture 2 4 Grphing Mrginl nd Averge Costs Proo Costs MC AC AC=/ To ind the minimum point, ind the irst order condition: AC = = 0 2 MC = = AC Homework: show tht this is minimum point. Spring 200 Econ ecture 2 5 Spring 200 Econ ecture 2 6 Exmple Prolem (continued) Wht is the mrginl cost curve or the CoDouglss production unction? w w w, = r w r + r MC = MC = r + w + r r ( ) w w Spring 200 Econ ecture 2 7 Costs nd Returns to Scle Incresing returns to scle implies decresing mrginl cost unctions. Constnt returns to scle implies constnt mrginl cost unctions. Decresing returns to scle implies incresing mrginl cost unctions. This is esy to see in the CoDouglss exmple. Spring 200 Econ ecture 2 8 Econ ecture 2 3
4 Pro. Jy Bhttchry Spring 200 Fixed Costs So we hve only included vrile costscosts which vry with the mount o output produced Fixed costs do not vry with the mount o output produced. e.g. ctory, piece o equipment Totl Cost() = Fixed Cost + Vrile Cost() Spring 200 Econ ecture 2 9 Short Run vs. ong Run In the long run, irms cn chnge ll o their inputs in response to chnges in input prices or. In the short run, some input my e diicult to djust quickly. These inputs re ixed s r s the irm is concerned. Under the dditionl constrint in the short run, the irm typiclly will not e le to choose the long run optiml input mix. Spring 200 Econ ecture 2 20 Averge Totl vs. Averge Vrile Cost q 2 q q 0 ShortRun Averge Totl Cost (SA) = Vrile Costs( ) + Fixed Costs SA = Averge Vrile Cost (SVC) = Vrile Costs( ) SVC = Spring 200 Econ ecture 2 2 Spring 200 Econ ecture 2 22 Properties o SMC, SA, nd SVC Short Run Mrginl Cost (SMC) = SA SVC Vrile Costs SMC = = = Minimum SA occurs to the right o minimum SVC MC curve cuts through the minimum o oth A nd AVC. Why? ( ) Spring 200 Econ ecture 2 23 Short Run Cost Curves or the Exmple Derive the SA, SMC, nd SVC curves or the CoDouglss production unction, = when is ixed t in the short run. We re constrined to produce, while ixing the level o. This leve no rel choice s to the level o lor required: = = Spring 200 Econ ecture 2 24 Econ ecture 2 4
5 Pro. Jy Bhttchry Spring 200 Short Run Exmple (continued) Plug the lor demnd curve nd the ixed level o cpitl into the expenditure unction to get short run totl costs: S SA (, ) = r + w = r + w (, ) S = (, ) r = + w SVC, SMC in the Exmple SVC does not count ixed costs: ( ) SVC, = w = w SMC: SVC w SMC(, ) = = You should veriy tht the other ormuls or SMC yield the sme outcome. Spring 200 Econ ecture 2 25 Spring 200 Econ ecture 2 26 Homework Red out the reltionship etween short nd long run cost curves in Nicholson. Punchlines: Short run cost curves re lwys greter thn or equl to long run cost curves (since ll inputs djust in the long run). ong nd short run cost curves re equl only when the ixed inputs hppen to e optiml or producing. Spring 200 Econ ecture 2 27 Econ ecture 2 5
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