EC487 Advanced Microeconomics, Part I: Lecture 2 Leonardo Felli 32L.LG.04 6 October, 2017
Properties of the Profit Function Recall the following property of the profit function π ( p , w ) = max p f ( x ) − w x = p f ( x ( p , w )) − w x ( p , w ) x ◮ Hotelling’s Lemma: ∂π ∂π ∂ p = y ( p , w ) ≥ 0 and = − x i ( p , w ) ≤ 0 ∂ w i Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 2 / 49
Hessian of the Profit Function Consider now the Hessian matrix of the profit function π ( p , w ): ∂ 2 π ∂ 2 π ∂ 2 π · · · ∂ p 2 ∂ p ∂ w 1 ∂ p ∂ w h ∂ 2 π ∂ 2 π ∂ 2 π · · · ∂ w 2 ∂ w 1 ∂ p ∂ w 1 ∂ w h H = 1 . . . ... . . . . . . ∂ 2 π ∂ 2 π ∂ 2 π · · · ∂ w 2 ∂ w h ∂ p ∂ w h ∂ w 1 h By Hotelling’s Lemma the matrix H is: ∂ y ∂ y ∂ y · · · ∂ p ∂ w 1 ∂ w h − ∂ x 1 − ∂ x 1 − ∂ x 1 · · · ∂ p ∂ w 1 ∂ w h H = . . . ... . . . . . . − ∂ x h − ∂ x h − ∂ x h · · · ∂ p ∂ w 1 ∂ w h Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 3 / 49
Properties of the Hessian of the Profit Function Result Let F : A → R where A ⊂ R n is a convex open set. Let F ( · ) be twice differentiable. Then the function F ( · ) is convex if and only if the Hessian matrix of F ( · ) is positive semi-definite on A. ◮ Recall that the profit function π ( p , w ) is convex in ( p , w ). ◮ This together with Young theorem implies that H is symmetric and has a non negative main diagonal. ◮ Assume now that π ( p , w ) is defined on an open and convex set S of prices ( p , w ) then convexity of π ( p , w ) is equivalent to the Hessian Matrix H being positive semi-definite . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 4 / 49
Profit Maximization Profit maximization can be achieved in two sequential steps: 1. Given y , find the choice of inputs that allows the producer to obtain y at the minimum cost; this generates conditional factor demands and the cost function ; 2. Given the cost function, find the profit maximizing output level . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 5 / 49
Comments ◮ Notice: step 1 is common to firms that behave competitively in the input market but not necessarily in the output market. ◮ Only in step 2 we impose the competitive assumption on the output market. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 6 / 49
Cost Minimization We shall start from step 1: min w x x s . t . f ( x ) ≥ y The necessary first order conditions are: y = f ( x ∗ ) , w ≥ λ ∇ f ( x ∗ ) and [ w − λ ∇ f ( x ∗ )] x ∗ = 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 7 / 49
Cost Minimization (cont’d) or for every input ℓ = 1 , . . . , h : w ℓ ≥ λ∂ f ( x ∗ ) ∂ x ℓ with equality if x ∗ ℓ > 0. The first order conditions are also sufficient if f ( x ) is quasi-concave (the input requirement set is convex). Alternatively, a set of sufficient conditions for a local minimum are that f ( x ) is quasi-concave in a neighborhood of x ∗ . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 8 / 49
Cost Minimization (cont’d) This can be checked (sufficient condition) by means of the bordered hessian matrix and its minors. In the case of only two inputs f ( x 1 , x 2 ) we have: w ℓ ≥ λ∂ f ( x ∗ ) , ∀ ℓ = 1 , 2 ∂ x ℓ with equality if x ∗ ℓ > 0 and SOC: � � f 1 ( x ∗ ) f 2 ( x ∗ ) 0 � � � � f 1 ( x ∗ ) f 11 ( x ∗ ) f 12 ( x ∗ ) > 0 � � � � f 2 ( x ∗ ) f 21 ( x ∗ ) f 22 ( x ∗ ) � � Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 9 / 49
Cost Minimization (cont’d) In the case the two first order conditions are satisfied with equality (no corner solutions) we can rewrite the necessary conditions as: � � = ∂ f ( x ∗ ) /∂ x 1 � dx 2 = w 1 � � MRTS = � � dx 1 ∂ f ( x ∗ ) /∂ x 2 w 2 � and y = f ( x ∗ ) Notice a close formal analogy with consumption theory (expenditure minimization). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 10 / 49
Conditional Factor Demands and Cost Function This leads to define: ◮ the solution to the cost minimization problem: z 1 ( w , y ) . x ∗ = z ( w , y ) = . . z h ( w , y ) the conditional (to y) factor demands (correspondences). ◮ the minimand function of the cost minimization problem: c ( w , y ) = w z ( w , y ) the cost function. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 11 / 49
Properties of the Cost Function 1. c ( w , y ) is non-decreasing in y . Proof: Suppose not. Then there exist y ′ < y ′′ such that (denote x ′ and x ′′ the corresponding solution to the cost minimization problem) w x ′ ≥ w x ′′ > 0 If the latter inequality is strict we have an immediate contradiction of x ′ solving the cost minimization problem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 12 / 49
Properties of the Cost Function (cont’d) If on the other hand w x ′ = w x ′′ > 0 then by continuity and monotonicity of f ( · ) there exists α ∈ (0 , 1) close enough to 1 such that f ( α x ′′ ) > y ′ and w x ′ > w α x ′′ which contradicts x ′ solving the cost minimization problem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 13 / 49
Properties of the Cost Function (cont’d) 2. c ( w , y ) is non-decreasing with respect to w ℓ for every ℓ = 1 , . . . , h . Proof: Consider w ′ and w ′′ such that w ′′ ℓ ≥ w ′ ℓ but w ′′ k = w ′ k for every k � = ℓ . Let x ′′ and x ′ be the solutions to the cost minimization problem with w ′′ and w ′ respectively. Then by definition of c ( w , y ) c ( w ′′ , y ) = w ′′ x ′′ ≥ w ′ x ′′ ≥ w ′ x ′ = c ( w ′ , y ) . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 14 / 49
Properties of the Cost Function (cont’d) 3. c ( w , y ) is homogeneous of degree 1 in w . Proof: The feasible set of the cost minimization problem f ( x ) ≥ y does not change when w is multiplied by the factor t > 0. Hence ∀ t > 0, minimizing ( t w ) x on this set leads to the same answer as minimizing w x . Let x ∗ be the solution, then: c ( t w , y ) = ( t w ) x ∗ = t c ( w , y ) . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 15 / 49
Properties of the Cost Function (cont’d) 4. c ( w , y ) is a concave function in w . w = t w + (1 − t ) w ′ for t ∈ [0 , 1]. Proof: Let ˆ Let ˆ x be the solution to the cost minimization problem for ˆ w . Then x + (1 − t ) w ′ ˆ c ( ˆ w , y ) = ˆ w ˆ x = t w ˆ x ≥ t c ( w , y ) + (1 − t ) c ( w ′ , y ) by definition of c ( w , y ) and c ( w ′ , y ) and f (ˆ x ) ≥ y . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 16 / 49
Properties of the Conditional Factor Demands 5. Shephard’s Lemma: If z ( w , y ) is single valued with respect to w then c ( w , y ) is differentiable with respect to w and ∂ c ( w , y ) = z ℓ ( w , y ) ∂ w ℓ Proof: By constrained Envelope Theorem and the definition of cost function c ( w , y ) = w z ( w , y ) = w z ( w , y ) − λ ( w , y ) [ f ( z ( w , y ) − y ] Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 17 / 49
Properties of the Conditional Factor Demands (cont’d) 6. z ( w , y ) is homogeneous of degree 0 in w . Proof: By Shepard’s Lemma and the following result. Result If a function G ( x ) is homogeneous of degree r in x then ( ∂ G /∂ x ℓ ) is homogeneous of degree ( r − 1) in x for every ℓ = 1 , . . . , L. Proof: Differentiate with respect to x ℓ the identity that defines homogeneity of degree r : G ( k x ) ≡ k r G ( x ) ∀ k > 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 18 / 49
Properties of the Conditional Factor Demands (cont’d) We obtain: k ∂ G ( k x ) = k r ∂ G ( x ) ∀ k > 0 ∂ x ℓ ∂ x ℓ or ∂ G ( k x ) = k ( r − 1) ∂ G ( x ) ∀ k > 0 ∂ x ℓ ∂ x ℓ This is the definition of homogeneity of degree ( r − 1) of the function ∂ G ( k x ) . ∂ x ℓ Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 19 / 49
Properties of the Conditional Factor Demands (cont’d) Notice that 6. The Lagrange multiplier of the cost minimization problem is the marginal cost of output: ∂ c ( w , y ) = λ ( w , y ) ∂ y Proof: By constrained Envelope Theorem applied to the cost function: c ( w , y ) = w z ( w , y ) − λ ( w , y ) [ f ( z ( w , y ) − y ] Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 6 October, 2017 20 / 49
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