general equilibrium theory a general economy m consumers n producers (n goods) resources m x n...
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General Equilibrium Theory
A General Economy
• m consumers• n producers
(n goods)• Resources
• m X n demand equations
• n supply equationsPrices
A Pure Exchange Economy An economy in which there is no production. A special case of a general economy in which economic activities consist only of trading and consuming.
The simplest form of a pure exchange economy is the two-agent, two-good exchange economy, which may be illustrated graphically using the Edgeworth – Bowley Box.
A 1A
2A
1x
2x
The “Edgeworth Box”: a pure exchange economy
0AU
0BU
·
The “Core”: A set of feasibleallocations that cannot be improved upon.
The “Contract Curve”: The set of all Pareto efficient points.
w
B 1x
2x
1B
2B
·x
A
B
1A
2A w
1x
1x
2x
2x
1B
2B
The “Edgeworth Box”: a pure exchange economy
0AU
0BU
·-p1/p2
-p1/p2
-p1*/p2
*
·x
Assumptions:Pure exchange economyTwo goods: and Two agents, A and B, with …
Identical preferences:
Arbitrarily determined, but different, endowments:
1x 2x
10 ,
,12121
12121
xxxxU
xxxxU
B
A
2121 , ,
,
BBBAAA
BA
Equilibrium is defined as a consumption bundle Where aggregate excess demands are zero in both markets:
2121 ,;, BBAA xxxxx
0,
0,
212
211
ppz
ppz Hence, we are seeking a setof prices, , thatsatisfies these equilibrium conditions.
21, pp
The Algebra of Equilibrium
0AU
0BU
A
B
31 A
221 A w ·
1x
1x
2x
2x
61 B
62 B
Pure Exchange and Equilibrium
1AU
1BU
51 Ax
215 Ax
41 Bx
122 Bxx
·
2
1P
P 13-=
1. All market equilibria are Pareto efficient.
First and Second Theorems of Welfare Economics
“With such a definition it is almost self-evident that this so-called maximum [Pareto-optimality] obtains under free competition … But this is not to say that the result of production and exchange will be satisfactory from a social point of view or will, even approximately, produce the greatest possible social advantage.”
Knut Wicksell, “On the Problem of Distribution” (1902)
What are the implicit assumptions underlying this result?
No consumption externalities
Agents behave “competitively” (large # of agents)
A competitive equilibrium actually exists
1.
2.
3.
0AU
0BU
A
B
31 A
221 A w’ ·
1x
1x
2x
2x
61 B
62 B
51 Ax
215 Ax
41 Bx
122 Bx
21
A
71
B
1AU
1BU
x’ ·
25.13 Ax 5.132
Bx
5.41
Ax
5.41
Bx
w ·
x ·
Pure Exchange and Redistribution
1AU
0AU
1BU
0BU
A
B
31 A
221 A
w’’ ·
1x
1x
2x
2x
61 B
62 B
51 Ax
215 Ax
41 Bx
122 Bxx’’
·
′=18 2
A 9=′2
B
25.13 Ax 5.132
Bx
5.41
Ax
5.41
Bx
w ·
x ·
Pure Exchange and Redistribution
Every Pareto efficient allocation can be achieved as a competitive equilibrium (given an appropriate initial endowment and convexity of preferences).
2.
First and Second Theorems of Welfare Economics
“[Pareto optimality] does not define, uniquely, a best situation in any sense of the word … Other criteria – roughly speaking, those we associate with the term ‘distributive justice’ – have to be called into play.”
Kenneth Arrow, The Limits of Organization (1974)
What is the important implication of this result?
Don’t manipulate prices to achieve equity-related goals – use lump sum transfers instead.
“However, there are practical matters involved …”