Lecture04-2011 - Outline Single Product Primal Optimization...

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Unformatted text preview: Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment Some Simple Production Mechanics: Lecture IV Charles B. Moss September 2, 2011 Charles B. Moss Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment Single Product Primal Optimization Supply Curve Profit Function Cost Function Multiproduct Primal Functions Assignment Charles B. Moss Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment Supply Curve Profit Function Cost Function Single Product Primal Optimization ￿ Profit Maximization αβ max π = pY x1 x2 − w1 x1 − w2 x2 x 1 ,x 2 ∂π Y = p Y α − w1 = 0 ∂ x1 x1 w1 β ⇒ x2 = x1 w2 α ∂π Y = p Y β − w2 = 0 ∂ x2 x2 Charles B. Moss (1) Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment pY α Y β α − w 1 = 0 ⇒ p Y x1 − 1 x2 − w 1 = 0 x1 ￿ ￿β w1 β pY α x1 = w 1 w2 α ￿ w1 pY α w2 pY ⇒ Supply Curve Profit Function Cost Function ￿ β w2 ∗ x1 ( p Y , w 1 , w 2 ) ￿β ￿ ￿β β α x1 + β − 1 = w 1 α ￿β ￿ (2) w1 ￿β −1 1 = x1 − α − β α 1 1− α − β = pY Charles B. Moss ￿w ￿ 1 α 1− β 1− α − β ￿ β w2 ￿ β 1− α − β Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment Supply Curve Profit Function Cost Function ￿ ￿ β ￿ w ￿ β − 1 ￿ β ￿ 1− α − β 1 w1 β 1 1− α − β 1− α − β x2 = p w2 α Y α w2 1 1− α − β = pY 1+ 1 α β −1 1− α − β ￿ β w2 ￿1+ β 1− α − β β−1 1−α−β+β−1 −α = = 1−α−β 1−α−β 1−α−β 1+ ⇒ ￿ w ￿1+ (3) β 1−α−β+β 1−α = = 1−α−β 1−α−β 1−α−β ∗ x2 ( p Y , w 1 , w 2 ) 1 1− α − β = pY Charles B. Moss ￿ α w1 ￿ α 1− α − β ￿ w2 β ￿ α− 1 1− α − β Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment ￿ Supply Curve Profit Function Cost Function By convention ∗ x1 ( p Y , w 1 , w 2 ) 1 1− α − β = pY 1 1− α − β ∗ x2 ( p Y , w 1 , w 2 ) = p Y Charles B. Moss ￿ ￿ α w1 ￿ β w2 ￿ 1− β 1− α − β 1− α 1− α − β ￿ ￿ β w2 ￿ α w1 ￿ β 1− α − β (4) α 1− α − β Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment Supply Curve Profit Function Cost Function Supply Curve ￿ ￿ 1 1− α − β Y ∗ ( p Y , w 1 , w2 ) = p Y ￿ 1 1− α − β × pY α+β 1 = pY−α−β ￿ ￿ α+β 1− α − β = pY α w1 α w1 ￿ ￿ α w1 α 1− α − β ￿ ￿ 1− β 1− α − β β w2 ￿ α−αβ+αβ ￿ α w1 1− α − β ￿ α 1− α − β Charles B. Moss ￿ ￿ ￿ β w2 1− α 1− α − β ￿ ￿β β w2 ￿ ￿α (5) ￿ β−αβ+αβ β w2 β 1− α − β 1− α − β β 1− α − β Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment Supply Curve Profit Function Cost Function Profit Function ∗ ￿ α+β 1− α − β ￿ α w1 ￿ α 1− α − β ￿β￿ β 1− α − β w2 ￿ β ￿ π ( p Y , w1 , w 2 ) = p Y p Y ￿ ￿ ￿ 1− β ￿ ￿ 1 α 1− α − β β 1 − α − β 1− α − β − w1 p Y w1 w2 ￿ 1α ￿ ￿ α ￿ ￿ 1−−−β ￿ 1 α 1 α 1− α − β β −w2 pY−α−β w1 w2 Charles B. Moss (6) Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment Supply Curve Profit Function Cost Function Cost Function ￿ Cost Minimization min w1 x1 + w2 x2 x 1 ,x 2 αβ s . t . x1 x2 = Y ￿ (7) Lagrangian ∂L ∂ x1 ∂L ∂ x2 ￿ ￿ αβ L = w 1 x 1 + w 2 x2 + λ Y − x1 x2 Y β α = w1 − λαx1 −1 x2 = 0 ⇒ w1 − λα = 0 x1 Y α β −1 = w2 − λβ x1 x2 = 0 ⇒ w2 − λβ = 0 x2 ∂L αβ = Y − x1 x2 = 0 ∂λ Charles B. Moss (8) Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment ￿ Supply Curve Profit Function Cost Function Taking the ratio fo the first two first-order conditions yields Y w1 α x2 w1 x1 = ⇒ = Y w2 β x1 w2 λβ x2 w1 β ⇒ x2 = x1 w2 α ￿ ￿ β α w1 β ⇒ Y − x1 x1 = 0 w2 α ￿ ￿ w1 β β α ⇒ Y − x1 + β =0 w2 α λα Charles B. Moss (9) Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment ￿ Supply Curve Profit Function Cost Function ￿ ￿β 1 ∗ x ( Y , w 1 , w 2 ) = Y α+β w2 α α+β 1 wβ ⇒ ￿ 1 ￿ ααβ 1 ∗ x ( Y , w 1 , w 2 ) = Y α+β w 1 β + 2 w2 α (10) Substituting these results into the cost function yields C ( Y , w1 , w 2 ) = w1 Y 1 α+β ￿ w2 α w2 β Charles B. Moss ￿ β 1− α − β + w2 Y 1 α +β ￿ w1 β w2 α ￿ α α +β (11) Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment ￿ Supply Curve Profit Function Cost Function Note that the profit function can be derived from the cost function max π (Py , w1 , w2 ) = pY Y − C (Y , w1 , w2 ) Y ∗ Y ( p Y , w1 , w2 ) ⇒ Y ￿ : p Y − Charles B. Moss ∂ C ( Y , w1 , w2 ) =0 ∂Y (12) Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment Multiproduct Primal Functions ￿ ￿ ￿ ￿ As discussed in class, the production function can be extended to multiple outputs. However, closed form functions are somewhat limited. In this section, I want to briefly discuss the theoretical application of the multproduct primal function within the context of a planting problem. Specifically, assume that there exists a multivariate production function, f(y,x), where y is a vector of (two) outputs, and x is a vector of (two) inputs. The profit function for this formulation can be formulated as max y 1 ,y 2 , x 1 ,x 2 p 1 y1 + p 2 y2 − w 1 x1 − w 2 x2 f (y , x ) = 0 Charles B. Moss (13) Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment L = p1 y1 + p2 y2 − w1 x1 − w2 x2 − λ (f (y , x )) ∂L ∂ f (.) ∂L ∂ f (.) = p1 − λ ≤0; = p2 − λ ∂ y1 ∂ y1 y2 ∂ y2 ∂L ∂L y1 = 0 ; y2 = 0 ∂ y1 ∂ y2 (14) ∂L ∂ f (.) ∂ f (.) ∂ f (.) = − w1 − λ ≤0; = − w2 − λ ≤0 ∂ x1 ∂ x1 ∂ x2 ∂ x2 ∂L ∂L x1 = 0 ; x2 = 0 ∂ x1 ∂ x2 Charles B. Moss Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment ￿ ￿ Taken together, these conditions imply that the value of marginal product of each input equals the input price, if positive quantities of each output are produced and positive quantities of inputs are used. These conditions admit three possible solutions. ￿ First, if only y1 is produced ∂L ∂ f (.) = p1 − λ = 0 , y1 > 0 ∂ y1 ∂ y1 (15) ∂L ∂ f (.) = p2 − λ < 0 , y2 = 0 ∂ y2 ∂ y2 Charles B. Moss Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment ￿ First order conditions ￿ Second, only y2 could be produced ∂L ∂ f (.) = p1 − λ < 0 , y1 = 0 ∂ y1 ∂ y1 (16) ∂L ∂ f (.) = p2 − λ = 0 , y2 > 0 ∂ y2 ∂ y2 ￿ Third, both outputs could be produced ∂L ∂ f (.) = p1 − λ = 0 , y1 > 0 ∂ y1 ∂ y1 (17) ∂L ∂ f (.) = p2 − λ = 0 , y2 > 0 ∂ y2 ∂ y2 Charles B. Moss Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment p1 = p2 ∂ f (.) ∂ y2 ∂ y1 = ∂ f (.) y1 λ ∂ y2 λ Charles B. Moss (18) Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment y1 y2 Charles B. Moss Some Simple Production Mechanics: Lecture IV Outline Single Product Primal Optimization Multiproduct Primal Functions Assignment ￿ Derive the cost and profit function for a Cobb-Douglas technology and three inputs. ￿ In the above solution, assume that one of the inputs is quasi-fixed (so the level of the input cannot be choosen each year). Rederive the input demands for the first two inputs (assuming the third input is quasi-fixed) and the cost function. Charles B. Moss Some Simple Production Mechanics: Lecture IV ...
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This note was uploaded on 02/01/2012 for the course AEB 6184 taught by Professor Staff during the Fall '09 term at University of Florida.

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