Lecture21-2011

# Lecture21-2011 - Primal Approach to the Proﬁt Function...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Proﬁt Functions: XXI Charles B. Moss1 1 University of Florida November 8, 2011 Charles B. Moss Proﬁt Functions: XXI Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function 1 Primal Approach to the Proﬁt Function Starting with the Cost Function Properties of the Proﬁt Function Netput Vector 2 Hotelling’s Lemma Le Chatelier-Samuelson Principle 3 Dual Proﬁt Function Charles B. Moss Proﬁt Functions: XXI Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Starting with the Cost Function Properties of the Proﬁt Function Netput Vector Primal Approach to the Proﬁt Function Taking as a starting point, the Cobb-Douglas form, we could formulate the proﬁt function as αβ max π = py x1 x2 − w1 x1 − w2 x2 x ∂π y = p y α − w1 = 0 ∂ x1 x1 ∂π y = p y β − w2 = 0 ∂ x2 x2 Charles B. Moss ⇒ α x2 w1 β w1 = ⇒ x2 = x1 β x1 w2 α w2 (1) Proﬁt Functions: XXI Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Starting with the Cost Function Properties of the Proﬁt Function Netput Vector Substituting this result back into the ﬁrst-order conditions α p y α x1 − 1 ⇒ α x1 + β − 1 β w1 x1 α w2 1 = py 1 1− α − β ⇒ x1 = p y 1 1− α − β ⇒ x2 = p y ￿ ￿ w2 β ￿ w2 β ￿ ￿ w2 β ￿ Charles B. Moss ￿β = w1 ￿β ￿ β 1− α − β w 1 ￿1− β α ￿w ￿ 1 α 1− α 1− β − α ￿w ￿ 1 α Proﬁt Functions: XXI 1− β 1− α − β α 1− α − β (2) Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Starting with the Cost Function Properties of the Proﬁt Function Netput Vector Starting with the Cost Function If we use the Cobb-Douglas cost function max π = py y − C (w , y ) y π = p y y − w1 y 1 α+β ￿ w2 α w1 β ￿ β α+β − w2 y 1 α+β ￿ w1 β w2 α ￿ α α+β ￿￿ ￿β ￿ ￿α￿ 1− α − β ∂π 1 w2 α α+β w 1 β α +β = py − y α+β w1 + w2 =0 ∂y α+β w1 β w2 α y = [(α + β ) py ] α +β 1− α = β ￿ w1 Charles B. Moss ￿ w2 α w1 β ￿ β α +β + w2 Proﬁt Functions: XXI ￿ w1 β w2 α ￿ α α+β ￿ αα+β 1 +β − (3) Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Starting with the Cost Function Properties of the Proﬁt Function Netput Vector Properties of the Proﬁt Function However, like our discussion regarding the cost function, we can bypass the primal form and derive the economic concepts of the dual proﬁt function. Properties of the Proﬁt Function π (p , w ) ≥ 0, If p ≥ p , then π (˜, w ) ≥ π (p , w ) - Proﬁt is nondecreasing in ˜ p p, If w ≥ w , then π (p , w ) ≤ π (p , w ) - Proﬁt is nonincreasing in ˜ ˜ w, π (p , w ) is convex and continuous in (p , w ), and π (tp , tw ) = t π (p , w ), t > 0 - Proﬁt is positive linear homogenous. Charles B. Moss Proﬁt Functions: XXI Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Starting with the Cost Function Properties of the Proﬁt Function Netput Vector Property 1 basically states that a producer faced with losing money from production will simply choose not to produce. Speciﬁcally, letting x = 0 implies y = 0, and in the absence of ﬁxed cost π (p , w ) = 0. Property 2 is based on the fact that we could at least produce the same bundle of outputs π (˜, w ) ≥ p y − c (w , y ) ≥ py − c (w , y ) = π (p , w ) p ˜ (4) The ﬁrst inequality can be argued based on the concavity of the cost function π (˜, w ) = max p − c (w , y ) p ˜ y π (˜, w ) = [py − c (w , y )] + [p − ∇y c (w , y )] (˜ − p ) p p 1 − (˜ − p ) ∇2 c (w , y ) (˜ − p ) ≥ 0 p p xx 2 Charles B. Moss Proﬁt Functions: XXI (5) Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Starting with the Cost Function Properties of the Proﬁt Function Netput Vector Property 3 follows a similar proof as property 3 π (p , w ) = py − c (w , y ) ≤ py − c (w , y ) ≤ π (p , w ) ˜ ˜ (6) Similarly, the last inequality can be demonstrated by the concavity of the cost function. Property 4 can actually be seen in our discussion of property 2. Starting with the development of the property that proﬁt is nondecreasing in output prices π (˜, w ) = [py − c (w , y )] + [p − ∇y c (w , y )] (˜ − p ) p p 1 2 (˜ − p ) ∇yy c (w , y ) (˜ − p ) ≥ 0 ￿: p − ∇y c (w , y ) = 0 for a max p p 2 1 ⇒ π (ˆ, w ) − [py − c (w , y )] − (ˆ − p ) ∇2 c (w , y ) (ˆ − p ) p p p yy 2 2 2 ⇒ ∇pp π (p , w ) = −∇yy c (w , y ) (7) Charles B. Moss Proﬁt Functions: XXI Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Starting with the Cost Function Properties of the Proﬁt Function Netput Vector Expanding the discussion to the concept of a netput vector. Let us rewrite the proﬁt function as ￿ p w 1￿ 1 ￿ ￿ π (p , w ) = α0 + α p + p Ap + β w + w ￿ Bw + p ￿ Γw ￿ 2 ￿￿ ￿2 ￿ 1￿ A Γ α π (z ) = α0 + z+ z z β Γ￿ B 2 π ( p , w ) = π ( z ) ￿: z = Charles B. Moss ￿ Proﬁt Functions: XXI (8) Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Starting with the Cost Function Properties of the Proﬁt Function Netput Vector Thus, the Hessian matrix is 2 ∇pp π (p , w ) = A ￿ ￿ AΓ 2 ∇2 π ( p , w ) = B πzz (p , w ) = ⇒ ww Γ￿ B ∇2 π ( p , w ) = Γ wp Charles B. Moss Proﬁt Functions: XXI (9) Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Le Chatelier-Samuelson Principle Hotelling’s Lemma The above discussion leads into a discussion of Hotellings lemma (a proﬁt function counterpart to Shephards lemma). This proof is a similar envelope proof to the ﬁrst proof we established for Shephards lemma π (p , w ) ≥ p y − c (w , y ) ˜ ˜ ˜˜ ⇒ L (p , w , y ) = π (p , w ) − p y + c (w , y ) ˜˜ ˜ ˜ ˜˜ ⇒ ∇p L∗ (p , w , y ) = ∇p π (˜, w ) − y = 0 ˜˜ p˜ ˜ ∗ ⇒ y ( y , w ) = ∇p π ( p , w ) = ∂π (p , w ) ∂p ⇒ ∇w L∗ (p , w , y ) = ∇w π (˜, w ) + ∇w c (w , y ) = 0 ˜˜ p˜ ˜˜ ⇒ x (p , w ) = ∇w c (w , y ) = −∇w π (p , w ) Charles B. Moss Proﬁt Functions: XXI (10) Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Le Chatelier-Samuelson Principle Recall the simple proof of the Slutsky decomposition from demand theory x (p , y ) = x (p , e (p , U )) ⇒ dx (p , y ) ∂ x (p , e (p , U )) ∂ x (p , e (p , U )) ∂ e (p , U ) = + dp ∂p ∂ e (p , U ) ∂p ⇒ dx (p , y ) ∂ (p , y ) x (p , y ) H = + x (p , U ) dp ∂p ∂y (11) The same eﬀect occurs in the input demand equations x (p , w ) = x (w , y (p , w )) ∂ x (p , w ) ∂ x (w , y ) ∂ x (w , y ) ∂ y (p , w ) = + ∂w ∂w ∂y ∂w Charles B. Moss Proﬁt Functions: XXI (12) Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Le Chatelier-Samuelson Principle x1 Ly Ly x2 Charles B. Moss Proﬁt Functions: XXI Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Le Chatelier-Samuelson Principle Relying on the Hotellings lemma results, we can then see ∂ y (p , w ) = ∂p ∂π (p , w ) ∂ 2 π (p , w ) ∂p = ≥0 ∂p ∂ p2 ∂π (p , w ) ∂ xI ( p , w ) ∂ 2 π (p , w ) ∂ wi =− =− ≤0 ∂ wi ∂ wi ∂ wi2 ∂ xj ( p , w ) xi ( p , w ) ∂ 2 π (p , w ) = =− ∂ wj ∂ wi ∂ wi ∂ wj ∂ y (p , w ) ∂ xi ( p , w ) ∂ 2 π (p , w ) =− = ∂ wi ∂y ∂ wi ∂ p Charles B. Moss Proﬁt Functions: XXI (13) Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Le Chatelier-Samuelson Principle Like the Slutsky results x (p , w ) = x (p , y (p , w )) ⇒ ∂ x (p , w ) ∂ x (p , y (p , w )) ∂ y (p , w ) = ∂p ∂ y (p , w ) ∂p ￿: ∂ x (p , w ) ∂ y (p , w ) =− ∂p ∂w ∂ y (p , w ) ∂ x (p , y (p , w )) ∂ y (p , w ) ⇒− = ∂w ∂ y (p , w ) ∂p ∂ y (p , w ) ∂ x (p , y ) ∂w ⇒− = ∂ y (p , w ) ∂y ∂p Charles B. Moss Proﬁt Functions: XXI (14) Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Le Chatelier-Samuelson Principle xi (p , w ) = xi (w , y (p , w )) ∂ xi ( p , w ) ∂ xi ( w , y ∗ ) ∂ xi ( w , y ∗ ) ∂ xj ( p , w ) = − ∂ wj ∂ wj ∂y ∂p ￿ ￿￿ ￿ ∂ xj ( p , w ) ∂ y (p , w ) ∂ xi ( w , y ∗ ) ∂ wi ∂p = + ∂ y (p , w ) ∂ wj ∂p ￿ ￿￿ ￿ ∂ xj ( p , w ) ∂ xi ( p , w ) ∂ xi ( w , y ∗ ) ∂p ∂p = − ∂ y (p , w ) ∂ wj ∂p Charles B. Moss Proﬁt Functions: XXI (15) Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Le Chatelier-Samuelson Principle Le Chatelier-Samuelson Principle With the last equation suﬃcient to demonstrate the Le Chatelier-Samuelson principle ∂ xi ( p , w ) ∂ xi ( w , y ∗ ) ≤ ∂ wi ∂ wi Charles B. Moss Proﬁt Functions: XXI (16) Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function Dual Proﬁt Function In our previous discussion of the dual, we showed that a cost function obeying the standard properties implied the existence of a production function that obeyed the properties we set out for a production technology. The proﬁt function dual is similar to the cost function. Speciﬁcally, we want to show that the properties of the proﬁt function imply the existence of a cost function with the properties outlined earlier c ∗ (w , y ) ≥ 0, w > 0, y > 0, If w ≥ w , the c ∗ (w , y ) ≥ c ∗ (w , y ), ˜ ˜ c ∗ (w , y ) is concave and continuous in w , c ∗ (tw , ty ) = tc ∗ (w , y ), t > 0, If y ≥ y , then c ∗ (w , y ) ≥ c ∗ (w , y ), ˜ ˜ c ∗ (w , 0) = 0, and c ∗ (w , y ) is convex and continuous in y . Charles B. Moss Proﬁt Functions: XXI Primal Approach to the Proﬁt Function Hotelling’s Lemma Dual Proﬁt Function To demonstrate these characteristics, we use the inverse of the relationship that has been useful above π (p , w ) = py − c (w , y ) ⇒ c (w , y ) = py − π (p , w ) Charles B. Moss Proﬁt Functions: XXI (17) ...
View Full Document

Ask a homework question - tutors are online