General equilibrium

Question 1

Allocation

Answer 1

vector that gives a production plan for each producer and a consumption plan for each consumer

Question 2

Feasible allocation

Answer 2

sum of all the endowments + sum of all production plans=sum of all consumption plans

Question 3

Pareto optimal

Answer 3

allocation x* such that there is no allocation x that is as least as prefered x* by every consumer and there is a consumer that strictly preferes it to x*

Question 4

Competitive equilibrium

Answer 4

Allocation and price vector such that:

• Consumers are maximizing utility subject to their wealth
• Producers are maxmimizing profits
• Allocation is feasible.

Question 5

Competitive equilibrium with transfers

Answer 5

• Allocation
• Price vector
• Monetary wealth assignment for each consumer such that Sum of assignments=value of total endowment + value of total production

Such that:

• Consumers maximize utility subject to their wealth
• Producers maximize profits
• Allocation is feasible

Question 6

Offer curve

Answer 6

set of all walrasian demans for all prossible prices

Competitive equilibria are the set of all the intersections of the offer curves

This set lies in the upper contour set of the initial endowment

Question 7

Contract curve

Answer 7

set of all pareto optima that pareto dominate the initial endowment

Question 8

Alternative definition of competitive equilibrium for pure exchange economis

Answer 8

a price vector and an allocation is a walrasian eq iff:

• y*<=0, py*=0, p*>=0
• x*_i=x_i(p,pw) for all i
• allocation is feasible

Question 9

equilibrium with aggregate excess demand

Answer 9

a price vector defines a competitive equilibrium iff p>=0, z(p)<=0

Question 10

Consequenses of walras law in pure exchange economies (aggregate excess demand)

Answer 10

• z_l(p*)<0 then p_l*=0
• In equilibrium p*>>0 =>z(p*)=0
• If preferences are strictly monotone then p*>>0
• if p*>>0 clears L-1 markets, then p* clears all the markets
• z(p) may not be defined for p>=0

Question 11

Brower fixed point theorem

Answer 11

Let A a convex non-empty compact set. Then if f is continuous on A => f has a fixed point.

Question 12

Existence if a competitive eq

Answer 12

• pure exchage economy
• aggregate demand functions defined for all p>=0.
• z(p) continuous, homogeneous of degree zero and satisfies walras law

Then

There is a p*>=0: z(p*)<=0

Question 13

Existence of a competitve eq. Conditions needed to ensure the existence

Answer 13

• exchange economy
• rational, continuous and stricty convex preferences that satisfy LNS
• endowments strictly positive for all consumers

Then there is a competitive eq.

Question 14

Conditions to ensure existence of comp eq with strictly positive prices, exchange ec

Answer 14

• exchange economy
• rational, continuous, strictly convex and strictly monotone preferences
• endowment strictly positive.

Then there is a p>>0 such that z(p)=0

Question 15

First fundamental theorem of welfare

Answer 15

If there is a competitive eq with transfers and preferences are LNS

then the allocation of this eq is pareto optimal.

Question 16

Second fundamental theorem of welfare

Answer 16

• Consumptions sets are R

Preferences

• rational
• strictly convex
• continous
• strictly monotone

Production sets:

• Yj non empty
• closed
• convex
• free disposal
• There is a production plan such that w+sum of poduction is strictly positive

Then any pareto optimal can be reached as a competitive equilibrium with transfers, p>>0, p diff from zero.