3.1 THE 2 2 EXCHANGE ECONOMY

Transcription

1 Essential Microeconomics THE 2 2 EXCHANGE ECONOMY Private goods economy 2 Pareto efficient allocations 3 Edgewort box analysis 6 Market clearing prices and Walras Law 14 Walrasian Equilibrium 16 Equilibrium and Efficiency 22

2 Essential Microeconomics -2- Private goods excange economy Consumer (or ouseold), = 1,..., Has strictly increasing preferences over n X + =. We assume tat te basic preference axioms are satisfied so tat tese are represented by a continuous utility function U (). 1 Were it is elpful we will assume tat U is continuously differentiable ( U () ). Endowments: Te initial allocation of commodities is { ω } = 1. Feasible allocations: Te final allocation { x } H is feasible if = 1 x = 1 = 1 H H H ω

3 Essential Microeconomics -3- Pareto-efficient allocations An allocation { x ˆ } 1 is Pareto efficient if tere is no oter feasible allocation in wic at least one H = consumer is strictly better off and no consumer is worse off. Consider an alternative allocation { x } H 1 Tat is, U ( x ) U ( xˆ ) Uˆ = in wic consumers 2,,H are all at least as well off. Since { x ˆ } it cannot be te case tat ˆ = 1 U ( x ) > U ( x ). H Terefore U ( x ) U ( xˆ ) and so { x ˆ } H = 1 solves te following maximization problem. H H ˆ arg { ( ) ( ) ˆ x = Max U x U x U, = 2,..., H, } x ω { x } = 1 = 1

4 Essential Microeconomics -4- Two commodity 2 consumer case (Alex and ev) For te special 2 2 case, Alex and ev must sare te aggregate endowment ω = ( ω1, ω2). Let x ˆ be te allocation to ev and let ˆ be te set of allocations tat ev prefers over x ˆ. Tis is depicted in Figure For any te allocation to Alex is x A x ˆ, = ω x. Tus te best possible allocation to Alex tat leaves ev no worse off is Alex s utility maximizing allocation in ˆ. Figure 3.1-2: ev s upper contour set

5 Essential Microeconomics -5- Edgewort box diagram Since preferences are strictly increasing a PE allocation uses all te endowment A A x + x = ω = ω + ω In te diagram te sum of te two consumption vectors is te vector ( ω1, ω 2), tat is, te rigt and corner of te Edgewort box. Figure 3.1-3: Edgewort box Diagram

6 Essential Microeconomics -6- For Pareto-efficiency, tere can be no mutually preferred alternative. One PE allocation is depicted in Figure As long as an allocation A A { xˆ, xˆ = ω xˆ } is in te interior of te Edgewort box, a necessary condition for te allocation to be PE is tat te slopes of te two indifference curves must be equal. Tus te grap of te PE allocations is te set of allocations to Alex (and ence ev) satisfying Figure 3.1-4: PE allocations MRS A A U A U ( xˆ ) ( xˆ ) A x1 x1 ( xˆ ) = = A U A U ( xˆ ) ( xˆ ) x x 2 2 A, were xˆ + xˆ = ω.

7 Essential Microeconomics -7- Example: Identical CES Preferences If preferences are identical and CES wit an elasticity of substitution σ, te MRS for Alex and ev are σ x 2 MRS ( x ) = k x. 1 1/ For a PE allocation in te interior of te Edgewort box, te indifference curves of te two consumers must ave te same slope, tat is, 1/ σ 1/ σ A 2 x2 = A 1 x1 x x A x2 x2 ence =. A x x 1 1 Figure 3.1-4: PE allocations wit identical CES preferences

8 Essential Microeconomics -8- * Example: Identical CES Preferences If preferences are identical and CES wit an elasticity of substitution σ, te MRS for Alex and ev are σ x 2 MRS ( x ) = k x. 1 1/ For a PE allocation in te interior of te Edgewort box, te indifference curves of te two consumers must ave te same slope, tat is, 1/ σ 1/ σ A 2 x2 = A 1 x1 x x a1 b1 a1+ b1 Ratio Rule: = = a b a + b A x2 x2 ence =. A x x 1 1 a1 b1 Proof: If = = k ten a1 = ka2 and b 1 = kb 2 and so a1+ b1 = ka ( 2 + b2). a b a1+ b1 Hence a + b = k. Figure 3.1-4: PE allocations wit identical CES preferences

9 Essential Microeconomics -9- Appealing to te Ratio Rule and ten setting demand equal to supply, x x x + x x x x x ω. ω A A = = = A A 1 1 a Tus, in a PE allocation eac consumer is allocated a fraction of te aggregate endowment. It follows tat for eac consumer te marginal rate of substitution is ω 2 MRS ( xˆ ) = k ω1 1/ σ. (3.1-1) Te PE allocations are depicted in Figure Figure 3.1-4: PE allocations wit identical CES preferences

10 Essential Microeconomics -10- Walrasian Equilibrium for an Excange Economy Let p 0 be a price vector of tis excange economy. In a WE eac consumer is a price taker. We write te set of consumers as H= {1,..., H}. ***

11 Essential Microeconomics -11- Walrasian Equilibrium for an Excange Economy Let p 0 be a price vector of tis excange economy. In a WE eac consumer is a price taker. We write te set of consumers as H= {1,..., H}. We assume tat preferences are strictly convex so consumer as is a unique most preferred consumption vector, x ( p, ω ). x ( p, ω ) solves { Max U ( x) p x p ω }. x **

12 Essential Microeconomics -12- Walrasian Equilibrium for an Excange Economy Let p 0 be a price vector of tis excange economy. In a WE eac consumer is a price taker. We write te set of consumers as H= {1,..., H}. We assume tat preferences are strictly convex so consumer as is a unique most preferred consumption vectr, x ( p, ω ). x ( p, ω ) solves { Max U ( x) p x p ω }. Total endowment vector: x ω = H ω Total or market demand: x( p) = x ( p, ω ) H Excess demand: z( p) = x( p) ω. *

13 Essential Microeconomics -13- Walrasian Equilibrium for an Excange Economy Let p 0 be a price vector of tis excange economy. In a WE eac consumer is a price taker. We write te set of consumers as H= {1,..., H}. We assume tat preferences are strictly convex so consumer as is a unique most preferred consumption vector, x ( p, ω ). x ( p, ω ) solves { Max U ( x) p x p ω }. Total endowment vector: x ω = H ω Total or market demand: x( p) = x ( p, ω ) H Excess demand: z( p) = x( p) ω. Definition: Market Clearing Prices Let zj ( p ) be te excess demand for commodity j at te price vector p 0. Te market for commodity j clears if zj ( p) 0 and pz j j( p ) = 0.

14 Essential Microeconomics -14- Walras Law: If preferences satisfy local non-satiation and all markets but one clear ten te remaining market must also clear. If preferences satisfy local non-satiation, ten a consumer must spend all of is income. Wy is tis? **

15 Essential Microeconomics -15- Walras Law: If preferences satisfy local non-satiation and all markets but one clear ten te remaining market must also clear. If preferences satisfy local non-satiation, ten a consumer must spend all of is income. Wy is tis? Ten for any price vector p te market value of excess demands must be zero. p z( p) = p ( x ω) = p ( ( x ω )) = ( p x p ω ). H ecause all consumers spend teir entire wealt te rigt and expression is zero. Hence n p z( p) = pz( p) + p z ( p) = 0. i i j j j= 1 j i H *

16 Essential Microeconomics -16- Walras Law: If preferences satisfy local non-satiation and all markets but one clear ten te remaining market must also clear. If preferences satisfy local non-satiation, ten a consumer must spend all of is income. Wy is tis? Ten for any price vector p te market value of excess demands must be zero. p z( p) = p ( x ω) = p ( ( x ω )) = ( p x p ω ). H ecause all consumers spend teir entire wealt te rigt and expression is zero. Hence n p z( p) = pz( p) + p z ( p) = 0. i i j j j= 1 j i Terefore if all markets but market i clear ten market i must clear as well. H Definition: Walrasian Equilibrium price vector Te price vector p > 0is a WE price vector if all markets clear.

17 Essential Microeconomics -17- Edgewort box example In a Walrasian equilibrium consumers coose te best point in teir budget sets given a price vector p= ( p1, p2). In te figure tese are te dotted orange and green triangles. N Figure 3.1-5: Excess supply of commodity 1 **

18 Essential Microeconomics -18- Edgewort box example In a Walrasian equilibrium consumers coose te best point in teir budget sets given a price vector p= ( p1, p2). In te figure tese are te dotted orange and green triangles. It is very important to note tat consumers consider only teir budget sets. In te case depicted, bot of tese budget sets extend beyond te boundary of te N Edgewort box (te set of feasible allocations). Figure 3.1-5: Excess supply of commodity 1 *

19 Essential Microeconomics -19- Edgewort box example In a Walrasian equilibrium consumers coose te best point in teir budget sets given a price vector p= ( p1, p2). In te figure tese are te dotted orange and green triangles. It is very important to note tat consumers consider only teir budget sets. In te case depicted, bot of tese budget sets extend beyond te boundary of te N Edgewort box (te set of feasible allocations). Te eavily saded triangles indicate te desired trades Figure 3.1-5: Excess supply of commodity 1 of te two consumers. As depicted, Alex wants to trade from te endowment point N to is most preferred desired consumption x A, wereas ev wises to trade from N to x. Tus, tere is excess supply of commodity 1.

20 Essential Microeconomics -20- y lowering te price of commodity 1 (relative to commodity 2) te budget line becomes less steep until eventually supply equals demand. Te Walrasian equilibrium E is depicted in Figure N Figure 3.1-6: Walrasian equilibrium

21 Essential Microeconomics -21- Class Exercise: Wic (if any) of tese figures depicts a Walrasian equilibrium? In te left figure te budget line is tangential to ev s indifference curve at x ˆ A. In te middle figure slope of te budget lies between te slopes of te two consumers at x ˆ A. In te rigt figure te budget line is tangential to Alex s indifference curve at x ˆ A.

22 Essential Microeconomics -22- Equilibrium and Efficiency In Figure te WE allocation is in te interior of te Edgewort box. Tus te marginal rates of substitution must bot be equal to te price ratio: A U A U ( x ) ( x ) x p x MRS x = = = = MRS x ( x ) ( x ) x x A A ( ) ( ) A U A p2 U 2 2 N Since te MRS are equal, it follows tat te WE allocation must be PE. Figure 3.1-6: Walrasian equilibrium