Thetheoryofthefirm: Cost minimization
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1 Thetheorotherm Cost mmzto Letures Mroeoom Theor Fll 9 Prt 4 99 GB Ashem ECON4335 #4 Cost mmzto Motto A lterte to osder the ehor o rotmmzg rm tht tkes outut d tor res s ge A to lze the ehor o rotmmzg rm tht tkes tor res s ge ut luee outut res 99 GB Ashem ECON4335 #4 Cost mmzto Assume retkg ehor tor mrkets Cost uto m suh tht Short ru ost uto m suh tht d 99 GB Ashem ECON4335 #4 3
2 Codtos or ost mmzto Cost uto Frst order odtos m suh tht or Prolems th rstorder roh Deretlt? Iteror soluto? Glol mmum? 99 GB Ashem ECON4335 #4 4 Codtol tor demd uto rg m st Proertes Com stts t or or Cost uto or the CoDougls teh Produto Cost uto A m st A A A A A A 99 GB Ashem ECON4335 #4 5 Wht hes he tor re reses? I or ll the mles Wek om o ost mmzto 99 GB Ashem ECON4335 #4 6
3 Cost uto Proo o roert Assume or ll uts Let e ost mmzg t so tht d Let e ost mmzg t so tht d B ost mmzt o Se or ll uts Hee Cost uto Proo o roert Let e ost mmzg t so tht d Hoeer the t t Hee s ost mmzg t t Cost uto Proo o roert 3 Let e ost mmzg t Let e ost mmzg t Let e ost mmzg t here t t B ost mmzto t t t t t t Addg these to equltes together t t Suose s ost mmzg tor etor t Dee the uto g The g d g or Produto uto A Cost m st A A A A A A Produto uto A Cost m st A A A A A A Cost uto m st Proertes No ssumtos out the teholog eeded No deresg Homogeeous o 3 Coe degree t t or ll t Cost uto or the CoDougls teh Sloe 4 Cotuous 99 GB Ashem ECON4335 #4 7 Shehrd s Lemm Proo o Shehrd s Lemm Sloe Assume deretlt Ioke ot Cost uto or the CoDougls teh 99 GB Ashem ECON4335 #4 8 More o Shehrd s Lemm Assume deretlt Sloe Ioke oug s theorem 99 GB Ashem ECON4335 #4 9 3
4 4 99 GB Ashem ECON4335 #4 A dett Hee Puu s equto 99 GB Ashem ECON4335 #4 Prot mmzg resose greter th ost mmzg resose he tor re hges? Hotellg s Lemm The resose the tor s o re hges d se Alg Puu s equto 99 GB Ashem ECON4335 #4 Logru d shortru ost utos log ru I the L ru totl ost Log / L ost ru erge Log / L ru mrgl ost Log d I the short ru ru osts ru d short log etee Coeto
5 Short Produto ru ost k Short uto k k k m k st k k k k k k ru erge rle ost Short ru erge ed ost Short ru mrgl ost k k k More o shortru ost utos Shortru CoDougls ost uto Short ru totl ost Short ru erge ost Short ru rle ost Short ru erge r ost k Short ru erge ost Fed ost S S / SVC SAVC FC Short ru ed erge ost SAFC Short ru mrgl ost / / S / 99 GB Ashem ECON4335 #4 3 Reltosh etee d d d d d s deresg d ol < 99 GB Ashem ECON4335 #4 4 Geometr o osts I 99 GB Ashem ECON4335 #4 5 5
6 Geometr o osts II Logru Shortru 99 GB Ashem ECON4335 #4 6 Geometr o osts III 99 GB Ashem ECON4335 #4 7 Cost uto Proo o roert Assume or ll uts Let e ost mmzg Let e ost mmzg t so tht d t so tht d B ost mmzto Se or ll uts Hee 99 GB Ashem ECON4335 #4 8 6
7 Cost uto Proo o roert Let e ost mmzg t so tht d Hoeer the t t Hee s ost mmzg t t 99 GB Ashem ECON4335 #4 9 Cost uto Proo o roert 3 Let e ost mmzg t Let e ost mmzg t Let e ost mmzg B ost mmzto t here t t t t t t Addg these to equltes together t t t t 99 GB Ashem ECON4335 #4 Proo o Shehrd s Lemm Suose s ost mmzg tor etor t Dee the uto g The g d g or 99 GB Ashem ECON4335 #4 7
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