Eric Maskin

Kevin W.S. Roberts

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
On the Fundamental Theorems of General Equilibrium
† 
 
 
 
 
 
 
 
 
 
Eric S. Maskin* 
Kevin W.S. Roberts** 
 
 
 
 
 
November 1980 
Revised August 2006 
 
 
 
 
  *  Institute for Advanced Study and Princeton University 
  ** Nuffield College, Oxford 
 
 
 †   Research support from the NSF is gratefully acknowledged.  The first version 
of this manuscript was written long ago, but, to the best of our knowledge, its 
results are still not widely known.  Moreover, although some textbooks (e.g., 
Varian 1992, pp329 and 336) refer to our Theorem 3, they do not provide 
complete proofs of it.   1
0.  Introduction 
The three fundamental theorems of general equilibrium theory are the 
propositions that, under appropriate hypotheses, (i) a competitive equilibrium exists; (ii) a 
competitive equilibrium is Pareto efficient; and (iii) a Pareto efficient allocation can be 
decentralized as a competitive equilibrium with transfer payments.  Of these theorems, 
assertion (ii) (often called the First Welfare Theorem) is mathematically virtually trivial, 
whereas the existence and decentralization results are usually considered “deeper.” 
  In this paper, we will provide a simple generalization of the existence theorem for 
economies where Walras’ Law (which asserts that the value of excess demand is zero) 
need not be satisfied out of equilibrium.  Assertion (iii) (the Second Welfare Theorem) is 
an almost immediate corollary of this generalization.  Our approach makes it clear that, 
given the existence of equilibrium, the first and second welfare theorems are equally 
“trivial”; indeed, we show that they can be proved in very similar ways. 
  We begin in Section 1 by establishing equilibrium existence for a “generalized 
competitive” mechanism. In Section 2 we apply this result to a “fixed allocation” 
mechanism.  We take up the welfare theorems in Section 3, and Section 4 summarizes 
our findings. 
1.  Generalized Competitive Mechanisms 
  Let the basic data of the economy be given by the specification of 
preferences{ }
h  
 
, endowments { }
h ω , and production sets { }
f Y , where consumers are 
indexed by  1, hH = … , and firms by  1, , f F = … .  For all h, consumer h’s preference   2
ordering 
h  
 
is defined
1 over  + ℜ
  , and his endowment 
h ω  belongs to  + ℜ
  , where  is the 
number of commodities (the assumption that the consumption space is the positive 
orthant is more restrictive than necessary).  For all f, firm f’s production set 
f Y  is a 
subset of ℜ
 . 
A generalized competitive mechanism (GCM) is a rule that, for each vector of 
prices p (in the unit simplex) and each specification { }
f y  of production plans by firms 
(where 
f f y Y ∈  for all f), assigns to each consumer h an income  { } ( ) ,
hf Ip y .  One 
example of a GCM is, of course, the ordinary competitive mechanism, in which 
consumer h is assigned income  ( )
hh f
f
f
p py ωθ ⋅+ ⋅ ∑ , where 
h
f θ  is consumer h’s share in 
firm f’s profit 
f p y ⋅ , and  1
h
f
h
θ = ∑  for all f.  Another example is the mechanism that, 
given some fixed allocation 
2  { } { } { } ,
hf xy     and prices p, gives consumer h income 
h p x ⋅   . 
An equilibrium of a GCM is a price vector p and an allocation  { } { } { } ,
hf xy  such 
that (i) for each  ,
h hx is preference-maximizing in  + ℜ
  , subject to the constraint 
{ } () ,
hh f p xI p y ⋅≤ ; (ii) for each  ,
f f y  is profit-maximizing in 
f Y  given prices p; and 
(iii) 
hfh
hfh
xyω =+ ∑∑∑ . 
                                                 
1 As usual, “
h x y  
 
” means “x is weakly preferred to y by consumer h.” 
2 An allocation is a specification { }
h x  of consumers’ consumption bundles together with a specification 
{ }
f y  of firms’ production plans.  The allocation is feasible if 
h x + ∈ℜ
  for all h, 
f f y Y ∈  for all f, and 
hfh
hfh
xyω ≤+ ∑∑∑ .  The allocation is Pareto efficient if it is feasible if there is no other feasible 
allocation  { } { } { } ,
hf xy    such that 
hhh x x    
 
for h, with strict preference for some h.   3
Notice that, in the first example above, the value of excess demand, 
()
hh f
hf
p xy ω
⎛⎞
⋅− − ⎜⎟
⎝⎠ ∑∑ , is zero, regardless of whether  { } { } { } ( ) ,,
hf px y  constitutes an 
equilibrium, so long as consumers exhaust their income (i.e., 
hh p xI ⋅ = ).  In other words, 
Walras’ Law holds for the ordinary competitive mechanism when preferences are strictly 
monotonic.  It is clear, however, that Walras’ Law generally fails for the second example.  
Although assumptions guaranteeing that Walras’ Law holds are usually invoked to prove 
existence theorems, we shall now show that a rather weaker condition will suffice.  This 
observation will enable us to establish an existence theorem for the second example. 
Given a GCM, let  () Z ⋅  be the corresponding excess demand correspondence.  
That is, for any prices p,  (){ ()
hh f
hf
Z pz z x y ω == − − ∑ ∑ , and 
h x  and 
f y  are such 
that 
h x maximizes 
h  
 
 subject to 
hh p xI ⋅ ≤  
and  
f y maximizes firm f’s profit in 
f Y given prices  } p . 
The following is our basic existence result: 
 
Lemma (Existence): Given a GCM, suppose that  ( ) Z ⋅ is well defined, upper hemi-
continuous, and convex- and compact-valued.  Suppose that if p is such that 0 i p =  for 
some i, then for all  () ,0 i zZ pz ∈> .  Finally, suppose that, for all p and all  () zZ p ∈ ,   4
either (a)  0 z =  or (b) there exist i and j such that  0 i z >  and  0 j z ≤ .  (Note that this last 
hypothesis constitutes a weakening of Walras’ Law).  Then, there exists an equilibrium.
3   
Proof: Because  () Z ⋅  is upper hemi-continuous and, for any p and i, 0 i z >  if  ( ) zZ p ∈  
and  0, there exists  0 i p δ =>  such that, for all p, all i and all  ( ), 0   i f   ii zZ pz pδ ∈ >< .  
Hence, upper hemi-continuity implies that there exists 0 K >  such that for all p, all 
() zZ p ∈ , and all j,  0 jj zK p +> .  Define the correspondence 
   () () () ,
ii
i
zK p
Hp h h z Zp
zK p
⎧⎫
+ ⎪⎪ == ∈ ⎨⎬
+ ⎪⎪
⎩⎭ ∑
. 
The correspondence H() ⋅  takes the unit simplex to itself.  It is upper hemi-continuous 
and convex-and compact-valued because Z( ) ⋅  is.  Therefore, by the Kakutani fixed point 
lemma, there exists  p  such that  ( ) p Hp ∈ .  Choose  ( ) zZ p ∈  such that 
() ii
i
zK p
p
zK p
+
=
+ ∑
.  Then, for all j, 
() ∗     
()
jj
j
ii
i
zK p
p
zK p
+
=
+ ∑
. 
If  0 j p = , for some j then, by hypothesis, 0 j z > , which contradicts ( ) ∗ .  Hence,  0 j p >  
for all j.  Thus, if  0 i
i
z > ∑ , then from ( ),0 j z ∗ >  for all j, contradicting our weakened 
Walras’ Law.  Similarly, if  0 i
i
z < ∑ , then ( ) ∗  implies that  0 j z <  for all j, also a 
                                                 
3 Varian (1981) makes a related observation for excess demand functions.   5
contradiction.  Therefore,  0 i
i
z = ∑ , and so, from ( ),0 j z ∗ =  for all j, i.e.,  p  and the 
allocation corresponding to z  constitute an equilibrium. 
 
  Q.E.D. 
 
2.  Application of the Existence Lemma 
  Theorem 1 (Existence of equilibrium at a Pareto efficient allocation): Let the 
allocation  { } { } { } ,
hf xy     be Pareto efficient, and suppose that, for all h, all components of 
h x    are strictly positive.
4  Suppose that, for all h and p,  { } ( ) ,
hf h I py p x = ⋅   .  Assume that 
preferences are convex, continuous, and strictly monotone, and that production sets are 
convex, closed, and bounded.
5  Then an equilibrium exists. 
Proof: Because the aggregate feasible set 
hf
hf
Y ω +
⎛⎞
+ ℜ ⎜⎟
⎝⎠ ∑∑
  ∩ is bounded, we can 
choose M  big enough so that if each consumer h is limited to the truncated consumption 
set  [ ] , M M + ℜ−
    ∩ , then any allocation  { } { } { } ,
hf xy  for which, for some h 
h x  is on the 
truncation boundary must be infeasible.  Let  ( ) Z ⋅  be the excess demand correspondence 
for the truncated consumption sets.  It is well-defined, upper hemi-continuous, and 
compact-valued because agents’(i.e., consumers’ and firms’) objectives are continuous, 
                                                 
4 This hypothesis can be relaxed using standard methods, as in Debreu (1959). 
5 The boundedness assumption can be dropped if we impose certain conditions on the aggregate production 
set 
f
f
Y ∑ ; see Debreu (1959).   6
their choice sets are closed and bounded, and each 
h x    is strictly positive.  It is convex-
valued because agents’ objectives and their choice sets are convex. 
 Consider  p such that  0 i p =  for some i.  Choose  ( ) zZ p ∈  and write 
()
hh f
hf
zx y ω =− − ∑∑ .  Because preferences are strictly monotone, 
h
i x M =  for all h.  
Hence, from our choice of  , 0 i Mz> . 
Thus, to apply our Lemma, it remains to verify that  ( ) Z ⋅  satisfies our weak 
Walras’ Law.  Suppose, for given p and  ( ) zZ p ∈ , that  0 z ≠ .  Write 
()
f hh
hf
zx y ω =− − ∑∑ . By definition of the GCM, 
hh p xp x ⋅ ≤⋅    for all h.  From profit 
maximization 
f f p yp y ⋅≥ ⋅    for all f.  Therefore,  () 0
f hh
hf
pz p x y ω
⎛⎞
⋅ ≤⋅ − − = ⎜⎟
⎝⎠ ∑∑    , 
where the last equation holds because  { } { } { } ,
hf xy     is Pareto efficient and preferences are 
strictly monotone.  Thus, there exists good j such that  0 j z ≤ .  If  0 z ≤ , then the 
allocation  { } { } { } ,
hf xy  is feasible.  Because consumer h can afford  ,
hhh h x xx     
 
. Thus 
{ } { } { } ,
hf xy  is Pareto efficient.  But since  0 z ≠ , there exists j such that  0 j z < , i.e., 
there exists a Pareto efficient allocation with excess supply, a contradiction of strictly 
monotone preferences.  Thus, there exists i such that  0 i z >  and so all the hypotheses of 
the Lemma hold.  Hence, there exists an equilibrium  { } { } { } ˆˆ ˆ ,,
hf p xy  when consumers are 
confined to their truncated consumption sets.  Because the allocation is feasible, no  ˆ
h x  
can lie on the truncation boundary.  Suppose, for consumer h, there exists  ˆ ˆ
h x  outside his   7
truncated consumption set such that he strictly prefers  ˆ ˆˆ to 
hh x x  and  ˆ ˆˆ ˆ
h p xp x ⋅ ≤⋅   .  But 
then strict monotonicity and preference convexity imply that any strict convex 
combination of  ˆ ˆˆ  and 
hh x x  is strictly preferred to  ˆ
h x , which implies that there exists a 
consumption bundle in the truncated consumption set strictly preferred to  ˆ
h x , a 
contradiction.  We conclude that, for all h,  ˆ
h x  globally maximizes consumer h’s 
preferences subject to his budget constraint.  That is,  { } { } { } ˆˆ ˆ ,,
hf p xy  is a full-fledged 
equilibrium. 
  Q.E.D. 
3.  The Welfare Theorems 
  The first welfare theorem asserts that a competitive equilibrium is Pareto efficient.  
The natural generalization to our framework is the following 
Theorem 2: (First Welfare Theorem): If preferences are strictly monotone, then any 
equilibrium of a GCM is Pareto efficient. 
Proof: Suppose that prices  ˆ p  and allocation  { } { } { } ˆˆ ,
hf xy  constitute an equilibrium of a 
GCM.  Suppose, contrary to the Theorem, there exists a feasible allocation  { } { } { } ,
hf xy  
that Pareto dominates  { } { } { } ˆˆ ,
hf xy .  By definition of equilibrium and from strictly 
monotone preferences, we have 
(1)   ˆˆ  and 
hhf hhf
hhf hhf
x yx y ωω =+ =+ ∑∑∑ ∑∑∑  
and 
(2)   ˆˆ ˆ
hh p xp x ⋅≥ ⋅ for all h,   8
with strict inequality for those consumers h who strictly prefer 
h x  to  ˆ
h x .  Thus, summing 
(2) across consumers, we obtain 
(3)   ˆˆ ˆ
hh
hh
p xp x ⋅> ⋅ ∑∑ . 
From profit maximization, we have  ˆˆ ˆ
f f p yp y ⋅ ≤⋅  for all f, and so 
(4)   ˆˆ
f f
ff
p yp y ⋅≤ ⋅ ∑∑ . 
Subtracting (4) from (3) and also subtracting endowments, we obtain  
(5)   () () ˆˆ ˆ ˆ
hh f hh f
hf h
p xy p x y ωω
⎛⎞ ⎛⎞
⋅− −< ⋅− − ⎜⎟ ⎜⎟
⎝⎠ ⎝⎠ ∑∑ ∑ , 
which contradicts (1).  Thus, the equilibrium allocation is Pareto efficient after all. 
  Q.E.D. 
  The proof of Theorem 2 will be recognized as identical to that usually given for 
the first welfare theorem.  We have included it here primarily for comparison with the 
proof of the decentralization theorem: 
Theorem 3: (Decentralization of a Pareto efficient allocation):  Assume that preferences 
are strictly monotone.  Suppose that allocation  { } { } { } ,
hf xy     is Pareto efficient.  Consider 
the GCM in which, given prices p, consumer h receives income 
hh I px = ⋅   .  Then, if an 
equilibrium of this GCM exists,  { } { } { } ,
hf xy     is an equilibrium allocation. 
Proof: Suppose that  ˆ p  is an equilibrium price vector and  { } { } { } ˆˆ ,
hf xy  is a 
corresponding equilibrium allocation for the GCM described.  From Theorem 2, 
{ } { } { } ˆˆ ,
hf xy  is Pareto efficient.  Therefore, because  { } { } { } ,
hf xy     is also Pareto efficient 
and consumer h can afford 
h x   , he must be indifferent between 
h x    and  ˆ
h x , and so, from   9
preference monotonicity,  ˆˆ ˆ
hh p xp x ⋅= ⋅   .  Because firms are profit maximizing, 
ˆˆ ˆ
f f p yp y ⋅≥ ⋅    for all f.  If, for some firm f, the inequality is strict, then 
ˆˆˆ
f f
ff
p ypy ⋅> ⋅ ∑∑ , and so 
(6)   ˆˆ ˆˆ
hf hf
hf hf
p xy p xy
⎛⎞ ⎛⎞
⋅− < ⋅− ⎜⎟ ⎜⎟
⎝⎠ ⎝⎠ ∑ ∑∑ ∑    . 
But, since  { } { } { } ,
hf xy     and  { } { } { } ˆˆ ,
hf xy  are both Pareto efficient and preferences are 
strictly monotone,  ˆˆ
hfhf h
hf hf h
xy xyω −=−= ∑∑∑∑∑    , contradicting (6).  Thus, 
ˆˆ ˆ
f f p yp y ⋅= ⋅    for all f.  Since consumers and firms both are indifferent between the 
hatted and the tildaed allocations, we conclude that  { } { } { } ,
hf xy     is itself an equilibrium 
allocation. 
  Q.E.D. 
  The proof of Theorem 3, like that of Theorem 2, is a simple revealed preference 
argument: given that existential problems can be ignored, agents stay at the pre-trade 
allocation unless they can make themselves better off.  But if the pre-trade allocation is 
Pareto efficient, improvement is impossible.  It is worth emphasizing that Theorem 3 
requires no convexity assumptions.  The theorem illustrates that convexity in 
decentralization theorems is needed only to show that equilibrium exists; it is not 
required to show that the equilibrium occurs at the Pareto efficient allocation.  Indeed, it 
follows directly that if a Pareto efficient allocation cannot be supported as an equilibrium, 
then starting at this allocation, no equilibrium can exist.   10
  Finally, if existence can be guaranteed without the use of convexity—as in large 
nonatomic economies—Theorem 3 ensures that Pareto efficient allocations can be 
decentralized. 
  The usual second welfare theorem follows immediately from Theorems 1 and 3: 
Theorem 4: (Second Welfare Theorem): Suppose that preferences and production sets 
satisfy the hypotheses of Theorem 1.  Then if  { } { } { } ,
hf xy     is Pareto efficient and 
h x    is 
strictly positive for all h, there exist prices  ˆ p  and balanced transfers { }
h T  (i.e., summing 
to zero) such that  { } { } { } ,
hf xy     is an equilibrium allocation with respect to the 
mechanism that, for each p, gives consumer h the income 
hh f h
f
f
p py T ωθ ⋅ +⋅ + ∑   . 
Proof: Under the hypotheses, Theorem 1 implies the existence of an equilibrium of the 
GCM in which, for each p, consumer h is assigned income 
h p x ⋅   .  Theorem 3 then 
implies that  { } { } { } ,
hf xy     is such an equilibrium together some price vector  ˆ p .  To 
complete the argument, set  ( ) ˆˆ ˆ
hh h f
f Tp xp p y ωθ =⋅ − ⋅+ ⋅ ∑   . 
  Q.E.D. 
4.  Concluding Remarks 
  This paper has reconsidered the principal theorems of general equilibrium theory.  
We have attempted to show that 
1)  There are interesting general equilibrium models in which Walras’ Law fails to hold 
out of equilibrium.  However, these models may satisfy a weakened version of Walras’ 
Law.   11
2)  To prove the existence of an equilibrium, Walras’ Law in its strong form can be 
replaced by a weakened version. 
3)  If existence is taken for granted, the second welfare theorem—and not just the first 
theorem—follows from a simple revealed preference analysis.  The usual statement of the 
second welfare theorem involves an existential statement that is the reason behind its 
mathematical “difficulty.”  Separation of the theorem into two parts—the existence part 
invoking the weakened Walras’ Law—makes clear that the standard convexity conditions 
may be needed for existence but not for that part of the theorem that constitutes its real 
substance.   12
 
References 
 
Debreu. G. (1959), Theory of Value, New Haven: Yale University Press. 
Varian, H. (1981), “Dynamical Systems with Applications to Economics,” in K. Arrow 
and M. Intriligator (eds), Handbook of Mathematical Economics, vol. I, 
Amsterdam: North Holland. 
 
Varian, H. (1992), Microeconomic Analysis, Third Edition, New York: Norton. 

Microeconomic Analysis, Third Edition,

Theory of Value,

On the Fundamental Theorems of General Equilibrium

http://www.sss.ias.edu/publications/papers/econpaper74.pdf

On the Fundamental Theorems of General Equilibrium

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics