Examples of Cost and Production Functions

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1 Dvso of the Humates ad Socal Sceces Examples of Cost ad Producto Fuctos KC Border October 200 v 20605::004 These otes sho ho you ca use the frst order codtos for cost mmzato to actually solve for cost fuctos The basc steps are these: Solve each of the frst order codtos for x terms of the parameters ages ad output level ad the Lagrage multpler 2 Substtute these expressos for x to the producto fucto ad solve for the Lagrage multpler 3 Substtute ths expresso for the Lagrage multpler to the expresso for x 4 Multply each x by ts age, ad sum to get the cost Admttedly all the examples here are specally chose to be ameable to ths approach Oe thg these examples mae clear s t there s ofte a dualty betee famles of cost ad producto fuctos For stace, the cost fucto assocated th a Leoteff producto fucto s lear, hle the cost fucto assocated th a lear producto fucto s Leoteff I all cases, assume y > 0 ad 0 vo Neuma Leoteff Producto Fucto I ths costat returs to scale producto fucto, the puts must be used exactly the rght proportos or the excess s asted { x y = m,, x }, α α here each α > 0 Note that the producto fucto fals to be dfferetable at ay terestg pot, so you caot actually apply the Lagrage multpler theorem Nevertheless, smple reasog shos that the codtoal factor demads must satsfy y = x α for each, so ˆx y, = α y So the cost fucto s cy, = y α Note that the cost fucto s qute smooth eve though the producto fucto s ot Ths s to be expected sce for 0 the cost mmzg put combato s uque Recall the support fucto theorem Ad just as predcted by the support fucto theorem, cy, = α y = ˆx y,

2 KC Border Examples of Cost ad Producto Fuctos 2 2 Lear Producto Fucto Wth ths costat returs to scale producto fucto, all puts are perfect substtutes for each other provded uts are chose properly y = α x + + α x, here each α > 0, =,, The Lagragea for the cost mmzato problem s x λ α x y ad the aïve frst order codtos are L x = λα = 0 =,,, hch tae at face value mply α = = α, hch s ulely sce these are all exogeous Ths s a red flag that sgals that the oegatvty costrats are bdg ad that you eed to exame the Kuh Tucer frst order codtos They are ad λα 0, =,,, x > 0 = λα = 0 ad λα > 0 = x = 0 I addto, λ 0 ad λ α x y = 0 Thus α λ =,, The questo s, ca e have strct equalty for each? The aser s o, as that ould mply x = 0 for each ad the output ould be zero, ot y > 0 So the soluto must satsfy ˆλ = m Let satsfy ˆλ = α That s, s a factor that maxmzes bag per buc The codtoal factor demad s gve by: y α = ˆx = 0 mmzes cost, ad the cost fucto s cy, = y m α {,, } α α Ths s the cost fucto eve f s ot uque, but he there s more tha oe such, the codtoal factor demad s o loger a uque put vector, but rather a set of cost mmzg put vectors I fact, the set of cost mmzg put vectors s the covex set: co { y e : = α α ˆλ = m j j α j } v 20605::004

3 KC Border Examples of Cost ad Producto Fuctos 3 Note that eve though the producto fucto s very smooth, the cost fucto fals to be dfferetable for 2 Ths s to be expected sce the bordered Hessa of the producto fucto s gve by f f f = f f f f f 0 hch s sgular for 2 It has ra 2 3 Cobb Douglas Producto Fucto Ths producto fucto s gve by y = γx α xα, 0 0 α 0 0 α α α 0 here each α > 0, =,, It s homogeeous of degree Lagragea: α = α x λγx α xα y Frst order codtos, usg the bdg costrat y = γx α xα : So But y = γx α x α, so y = γ L y = λα = 0 =,, x x, x = λα y =,, y α λα = γλ α y α α α Solvg ths for λ gves ˆλ = [ γy α ] α /α α = γ /α y α/α α α /α α /α v 20605::004

4 KC Border Examples of Cost ad Producto Fuctos 4 To smplfy otato a bt, set so b = γ α β = α α, ˆλ = by α/α α β, β Substtutg ths for λ gves the codtoal factor demads for j =,, So the cost fucto s ˆx j y, = by α/α β α y j = α j j by /α cy, = αby /α hch s a Cobb Douglas fucto of th costat returs to scale Note that cy, = by α/α β = y ˆλ, ad cy, j β, β, j = α β j by α/α β = ˆx j y, j 4 Geeralzed Arro Cheery Mhaus Solo Producto Fucto y = γ α x here α > 0, <, 0 Ths producto fucto exhbts costat returs to scale Trust me, ad rerte the costrat as γ α x y = 0 The Lagragea s x λ γ α x y The frst order codtos are: so λγ α x = 0, x = λα γ / v 20605::004

5 KC Border Examples of Cost ad Producto Fuctos 5 or x = λα γ / / 2 To solve for λ, e eed to substtute ths bac to the producto fucto, hch I ll do steps Solvg for ˆλ: x = λα γ / / x = λα γ / / α x = α λα γ / / α x = λγ / α α / / γ α x = γ λγ / α α / / y = γ λγ / α α / / y = γ λγ / / α α / / = λγ / α ˆλ = y γ α Substtutg 3 to 2 gves ˆx y, = y γ α Thus / } {{ } = y γ α ˆx = y γ α λ ˆx = y γ α ˆx = y γ α α γ α α α v 20605::

6 KC Border Examples of Cost ad Producto Fuctos 6 Or settg β = α ad σ = gves the cost fucto as cy, = yγ β σ σ Ths has the form of a ACMS fucto th parameter σ stead of You ca verfy that cy, = ˆx y, usg 4 But ote that cy, y ˆλ Why ot? v 20605::004

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