EC487 Advanced Microeconomics, Part I: Lecture 3 Leonardo Felli - PowerPoint PPT Presentation

EC487 Advanced Microeconomics, Part I: Lecture 3 Leonardo Felli 32L.LG.04 13 October, 2017

Bordered Hessian Recall that when considering the cost minimization problem in the case of a technology with only two inputs f ( x 1 , x 2 ) we stated that the SOC are: � 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 ∗ ) � � � Why is this the case? Consider the cost minimization problem: min w 1 x 1 + w 2 x 2 { x 1 , x 2 } s . t . f ( x 1 , x 2 ) ≥ y Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 2 / 48

Bordered Hessian (cont’d) In other words: max − ( w 1 x 1 + w 2 x 2 ) { x 1 , x 2 } s . t . f ( x 1 , x 2 ) ≥ y The lagragian function is then: L ( λ, x 1 , x 2 ) = − ( w 1 x 1 + w 2 x 2 ) − λ [ f ( x 1 , x 2 ) − y ] According to the Lagrange method the solution to the cost minimization problem coincides with the solution to: max L ( λ, x 1 , x 2 ) { λ, x 1 , x 2 } Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 3 / 48

Bordered Hessian (cont’d) Notice now that the hessian matrix of the lagrangian H ( x ), where x = ( x 1 , x 2 ) is the bordered hessian   ∂ 2 L ∂ 2 L ∂ 2 L ∂λ 2 ∂λ∂ x 1 ∂λ∂ x 2  0 − f 1 ( x ) − f 2 ( x )    ∂ 2 L ∂ 2 L ∂ 2 L   = − f 1 ( x ) − λ f 11 ( x ) − λ f 12 ( x ) ∂ x 2 ∂ x 1 ∂λ ∂ x 1 ∂ x 2     1   − f 2 ( x ) − λ f 21 ( x ) − λ f 22 ( x )   ∂ 2 L ∂ 2 L ∂ 2 L ∂ x 2 ∂λ ∂ x 2 ∂ x 1 ∂ x 2 2 Therefore the local SOC of cost minimization require that in a neighborhood of x ∗ : |H ( x ∗ ) | < 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 4 / 48

Bordered Hessian (cont’d) Notice now that since λ > 0  � − f 1 ( x ∗ ) − f 2 ( x ∗ ) �  0 � �   � − f 1 ( x ∗ ) − λ f 11 ( x ∗ ) − λ f 12 ( x ∗ ) � sign  = � � � � − f 2 ( x ∗ ) − λ f 21 ( x ∗ ) − λ f 22 ( x ∗ )  � � f 1 ( x ∗ ) f 2 ( x ∗ )  � �  0 � �   � f 1 ( x ∗ ) f 11 ( x ∗ ) f 12 ( x ∗ ) � = − sign � � � f 2 ( x ∗ ) f 21 ( x ∗ ) f 22 ( x ∗ ) �   � � From here our 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 13 October, 2017 5 / 48

Competitive Equilibrium Consider the entire economy, in which three main activities occur: production , consumption and trade . We shall focus first on a pure exchange economy (two activities only: consumption and trade). Consumers are born with endowments of commodities. They can either consume the endowments or trade them. Consider I = 2 consumers and L = 2 commodities. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 6 / 48

Pure Exchange Economy In such case the consumption feasible set for every consumer is X i ∈ R 2 + and consumer i ’s endowment is: � ω i � ω i = 1 ω i 2 The total endowment of commodity ℓ available in the economy is: ω ℓ = ω 1 ℓ + ω 2 ¯ ℓ > 0 ∀ ℓ ∈ { 1 , 2 } An allocation in this economy is then a pair of vectors x such that �� x 1 � x 2 � �� x = ( x 1 , x 2 ) = 1 1 , x 1 x 2 2 2 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 7 / 48

Pure Exchange Economy (cont’d) An allocation is feasible if and only if x 1 ℓ + x 2 ℓ ≤ ¯ ∀ ℓ ∈ { 1 , 2 } ω ℓ An allocation is non-wasteful if and only if x 1 ℓ + x 2 ℓ = ¯ ∀ ℓ ∈ { 1 , 2 } ω ℓ This economy can be represented in an Edgeworth box. In the example below we assume ω 2 1 = ω 1 2 = 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 8 / 48

Edgeworth Box 2 ✛ ✻ u 2 ( x 1 , x 2 ) x 2 u 1 ( x 1 , x 2 ) ω ✲ q ❄ 1 x 1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 9 / 48

Edgeworth Box (cont’d) Notice that in such an environment the income of each consumer is the market value of the consumer endowment: m i = p ω i where however p is determined in equilibrium. The budget set of consumer i is then: x i ∈ R 2 + | p x i ≤ p ω i � B i ( p ) = � For a vector of equilibrium prices p the budget sets of both consumers are two complementary sets in the Edgeworth box (slope of the separating line − p 1 p 2 ). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 10 / 48

Edgeworth Box (cont’d) 2 ✛ ✻ . . . . . . . . u 2 ( x 1 , x 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . x 2 . . . . . . . . . . . . u 1 ( x 1 , x 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ω . . . . . ✲ . . q ❄ 1 x 1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 11 / 48

Edgeworth Box (cont’d) The preferences of the two consumers are represented by two maps of indifference curves. For any given level of prices we can represent the offer curve of each consumer: the consumption bundle that represent the optimal choice for each consumer. The offer curve necessarily passes through the endowment point. Indeed the allocation �� ω 1 � ω 2 � �� ω = ( ω 1 , ω 2 ) = 1 1 , ω 1 ω 2 2 2 is always affordable hence each consumer must choose an optimal consumption bundle that makes him/her at least as well off as at ω . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 12 / 48

Edgeworth Box (cont’d) 2 ✛ ✻ . . . . . . . . u 2 ( x 1 , x 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . x 2 . . . . . . . . . . . . u 1 ( x 1 , x 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ω . . . . . ✲ . . q ❄ 1 x 1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 13 / 48

Edgeworth Box (cont’d) 2 ✛ ✻ . . . . . . . . u 2 ( x 1 , x 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . x 2 . . . . . . . . . . . . u 1 ( x 1 , x 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ω . . . . . ✲ . . q ❄ 1 x 1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 14 / 48

Edgeworth Box (cont’d) Given the preferences of the two consumers the only candidate to be an equilibrium price vector (if it exists) is a unique price vector that defines a unique budget constraint in the Edgeworth box tangent to indifference curves of both consumers. However if the tangency occur at two distinct points on the budget constraint then there will exist excess supply in one good, say ℓ = 2 and excess demand in the other good, say ℓ = 1. The allocation represented by the two tangency point is then not feasible. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 15 / 48

Edgeworth Box (cont’d) 2 ✛ ✻ . . . . . . . . u 2 ( x 1 , x 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . x 2 . . . . . . . . . . . . u 1 ( x 1 , x 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ω . . . . . ✲ . . q ❄ 1 x 1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 16 / 48

Competitive Equilibrium We define a market equilibrium as a situation in which markets clear , the consumers fulfil their desired purchases and the allocation obtained is feasible. Definition A Walrasian (competitive) equilibrium for the Edgeworth box economy is a price vector p ∗ and an allocation x ∗ = ( x 1 , ∗ , x 2 , ∗ ) such that ∀ x i ∈ B i ( p ∗ ) u i ( x i , ∗ ) ≥ u i ( x i ) and x 1 , ∗ + x 2 , ∗ = ¯ ω ℓ ∀ ℓ ∈ { 1 , 2 } ℓ ℓ This corresponds to an intersection of the offer curves: a point where the indifference curves of the two consumers are tangent to the unique budget constraint: the equilibrium allocation. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 17 / 48

Competitive Equilibrium (cont’d) 2 ✛ ✻ . . . . . . . . u 2 ( x 1 , x 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . x 2 . . . . . . . . . . . . u 1 ( x 1 , x 2 ) . . . q E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ω . . . . . ✲ . . q ❄ 1 x 1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 13 October, 2017 18 / 48