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

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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) ...
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