We are nearing the point we wish to achieve: namely, a proof of the equivalence that the allocations in a Radner equilibrium would be equivalent to an Arrow-Debreu equilibrium. We have already given the intuition for this equivalence earlier - namely, the collapsing of the S+1 constraints of the Radner problem into a single constraint for the Arrow-Debreu problem:

p

_{0}(x_{0}^{h}- e_{0}^{h}) + qV^{-1}p_{-0}[x^{h}- e^{h}]_{-0}｣ 0

From the fundamental theorem of asset pricing, we see that under complete
asset markets, qV^{-1} = m _{-0} = [m _{1}, m _{2}, .., m _{S}] are the marginal values of the states. Thus, qV^{-1}p_{-0}
= m _{-0}p_{-0} = (m
_{1}p_{1}, m _{2}p_{2}, ..., m _{s}p_{s}, ..., m _{S}p_{S}]
will represent the Arrow-Debreu state-contingent prices (i.e. the price of good i in state
s in a state-contingent market is m _{s}p_{is}).
Notice that the inversion of V^{-1} is fundamental here - and, consequently, V
must be of full rank. In other words, asset markets must be *complete* for the
equivalence of Radner and Arrow-Debreu equilibrium. We turn to the theorem now:

Theorem: (Equivalence Theorem) if the asset structure is complete (i.e. rank(V) = S), then aRadnerequilibrium is equivalent to anArrow-Debreuequilibrium. In other words:(i) if consumption demands x* = {x

^{h}*}_{hﾎ H}ﾎ R^{n(S+1)H}, portfolio decisions a* = {a^{h}*}_{hﾎ H ﾎ }R^{FH}, asset prices q* = (q_{1}*, q_{2}*, .., q_{S}*) ﾎ R^{F}_{++}and spot prices p* = (p_{0}*, p_{1}*, p_{2}*, .., p_{S}*) ﾎ R^{n(S+1)}_{++}constitute aRadnerequilibrium, then there are multipliers (m_{1}, ..., m_{S}) ﾎ R^{S}_{++}such that the allocation x* and the state-contingent commodities price vector (p_{0}*, m_{1}p_{1}*, ..., m_{S}p_{S}*) ﾎ R^{n(S+1)}constitute anArrow-Debreuequilibrium.(ii) if the consumption demands x* = {x

^{h}*}_{hﾎ H}ﾎ R^{n(S+1)H}and state-contingent prices p* = (p_{0}*, p_{1}*, .., p_{S}*) ﾎ R^{n(S+1)}_{++}constitute anArrow-Debreuequilibrium then there is a q* ﾎ R^{F}_{+}and portfolio decisions a* = {a^{h}*}_{hﾎ H}ﾎ R^{FH}such that the plans (x*, a*) and prices (q*, p*) constitute aRadnerequilibrium.

Proof: (i) ﾞ (ii) (Radner ﾞ
Arrow-Debreu) Let (x*, a*, q*, p*) denote a Radner equilibrium. Then by our fundamental
theorem of asset pricing, there is consequently as set of semi-positive multipliers m = (1, m _{1}, m _{2}, ..., m _{S}) such
that q = m _{-0}V. We now wish to prove that m p* = (p_{0}*, m _{1}p_{1}*,
m _{2}p_{2}*, ...., m
_{S}p_{S}*) is a set of Arrow-Debreu state-contingent equilibrium prices.
To prove this, let us define:

B

_{A}= {y^{h}ﾎ R_{+}^{n(S+1)}| p_{0}*(y_{0}^{h}- e_{0}^{h}) + ・/font>_{sﾎ S}m_{s}p_{s}*(y_{s}^{h}- e_{s}^{h}) ｣ 0}

which is the Arrow-Debreu budget constraint with equilibrium
state-contingent prices (p_{0}*, m _{1}p_{1}*,
.., m _{S}p_{S}*) and:

B

_{R}= {y^{h}ﾎ R_{+}^{n(S+1)}| p_{0}*(y_{0}^{h}- e_{0}^{h}) ｣ -qa^{h}, and p_{s}*(y_{s}^{h}- e_{s}^{h}) ｣ V_{s}a^{h}for all s ﾎ S, a^{h}ﾎ R^{F}}

which is the Radner budget constraint with equilibrium spot prices (p_{0}*,
p_{1}*, ..., p_{S}*). Our first step is to prove that B_{A} ﾍ B_{R}, i.e. if x^{h} ﾎ
B_{A}, then x^{h} ﾎ B_{R}. To see
this, suppose x^{h} ﾎ B_{A}, so:

p

_{0}*(x_{0}^{h}- e_{0}^{h}) + ・/font>_{sﾎ S}m_{s}p_{s}*(x_{s}^{h}- e_{s}^{h}) ｣ 0

or:

p

_{0}*(x_{0}^{h}- e_{0}^{h}) + m_{-0 }p_{-0}*[x^{h}- e^{h}]_{-0}｣ 0

where_{ }p_{-0}*[x^{h} - e^{h}]_{-0}
= [p_{1}*(x_{1}^{h} - e_{1}^{h}), p_{2}*(x_{2}^{h}
- e_{2}^{h}), ....., p_{S}*(x_{S}^{h} - e_{S}^{h})]｢ . By completeness, we know that for any V with rank(V) = S, there
is a vector a^{h} ﾎ R^{F} such that:

p

_{-0}*[x^{h}- e^{h}]_{-0}= Va^{h}

and this a^{h} will be unique. Consequently, substituting into our
Arrow-Debreu constraint:

p

_{0}*(x_{0}^{h}- e_{0}^{h}) + m_{-0}Va^{h}｣ 0

but as q = m _{-0}V by the fundamental
theorem of asset pricing, then this implies:

p

_{0}*(x_{0}^{h}- e_{0}^{h}) + qa^{h}｣ 0

Thus, as p_{-0}*[x^{h} - e^{h}]_{-0} = Va^{h}
and p_{0}*(x_{0}^{h} - e_{0}^{h}) ｣ -qa^{h}, then obviously x^{h} ﾎ
B_{R}. Thus B_{A} ﾍ B_{R}. Now, all we
have to prove is that if the Radner equilibrium allocation x^{h}*, which is a
utility-maximizing bundle over B_{R}, is also feasible under the Arrow-Debreu and
thus is also utility maximizing under B_{A}. To see this, note that if x^{h}*
ﾎ B_{R}, then:

p

_{0}*(x_{0}^{h}* - e_{0}^{h}) ｣ -qa^{h}*

p

_{-0}*[x^{h}* - e^{h}]_{-0}｣ Va^{h}*

So pre-multiplying the second constraint by m _{-0},
then m _{-0}p_{-0}*[x^{h}* - e^{h}]_{-0}
｣ m _{-0}Va^{h}*.
Thus adding this to the first constraint:

p

_{0}*(x_{0}^{h}* - e_{0}^{h}) + m_{-0}p_{-0}*[x^{h}* - e^{h}]_{-0}｣ m_{-0}Va^{h}* - qa^{h}*

but as q = m _{-0}V and,
post-multiplying by a^{h}*, implies qa^{h}* = m
_{-0}Va^{h}*, which implies in turn that:

p

_{0}*(x_{0}^{h}* - e_{0}^{h}) + m_{-0}p_{-0}*[x^{h}* - e^{h}]_{-0}｣ 0

so x^{h}* satisfies the Arrow-Debreu constraint, i.e. x^{h}*
ﾎ B_{A}. Thus, allocation x^{h}* and prices (p_{0}*,
m _{1}p_{1}*, m _{2}p_{2}*,
.., m _{S}p_{S}*) define an Arrow-Debreu
equilibrium. Q.E.D.

(ii) ﾞ (i): (Arrow-Debreu ﾞ
Radner) The converse is only a little bit trickier and we need to employ the numeraire
good (which we take to be good 1, "gold"). Let p* = (p_{0}*, p_{1}*,
p_{2}*, .., p_{S}*) denote the equilibrium Arrow-Debreu state-contingent
prices. As we have an Arrow-Debreu equilibrium, then it must be that x^{h}* ﾎ B_{A｢ }, now defined as:

B

_{A｢ }= {y^{h}ﾎ R_{+}^{n(S+1)}| p_{0}*(y_{0}^{h}- e_{0}^{h}) + ・/font>_{sﾎ S}p_{s}*(y_{s}^{h}- e_{s}^{h}) ｣ 0}

We want to do two things: firstly, deduce asset prices q and portfolios a^{h}
that will permit us to construct a Radner budget constraint B_{R｢
}from this. Secondly, we must prove that B_{R｢ }ﾍ B_{A｢ }so that if x^{h}*
is utility-maximizing over B_{A｢ }, it must be
utility-maximizing over B_{R｢ }. Finally, we must
ensure that asset markets clear for the B_{R｢ }constructed
at equilibrium. Now, we begin with the Arrow-Debreu situation so that, for a particular h ﾎ H, then B_{A｢ }implies at
equilibrium prices p*:

p

_{0}*(x_{0}^{h}- e_{0}^{h}) + ・/font>_{sﾎ S}p_{s}*(x_{s}^{h}- e_{s}^{h}) = 0.

which is our Arrow-Debreu budget constraint for the single individual at
equilibrium. Now, let V be some S ｴ F matrix with typical
element r_{fs} ﾎ R_{+} and let M be some S ｴ S dimensional *diagonal *matrix with typical element p_{1s}
ﾎ R_{+} along the diagonal, i.e.

p_{11} |
0 | ... | 0 | |

M = | 0 | p_{12} |
... | 0 |

... | ... | ... | ... | |

0 | 0 | ... | p_{1S} |

Note that p_{1s} is the price of good 1 (the numeraire good,
"gold") in state s. We would like it that for any s ﾎ
S, that we would be able to find a Fｴ 1 vector a^{h} =
[a_{1}^{h}, a_{2}^{h}, ..., a_{f}^{h}, ..,
a_{F}^{h}] ﾎ R^{F} such that p_{s}*(x_{s}^{h}
- e_{s}^{h}) = ・/font> _{fﾎ
F} p_{1s}r_{fs}a_{f}^{h}. Or, letting p_{-0}*[x^{h}
- e^{h}]_{-0} denote an Sｴ 1 vector [p_{1}*(x_{1}^{h}
- e_{1}^{h}), p_{2}*(x_{2}^{h} - e_{2}^{h}),
....., p_{S}*(x_{S}^{h} - e_{S}^{h})]｢ , then we would write our hypothesis as:

p

_{-0}*[x^{h}- e^{h}]_{-0}= MVa^{h}

Is there such a vector a^{h}? As asset markets are complete, then
rank(V) = S, and so rank(MV) = S, thus there is indeed such a unique solution a^{h}
to this system. Note that this implies that p_{s}*[x_{s}^{h} - e_{s}^{h}]
= ・/font> _{fﾎ F}p_{s1}r_{fs}a_{f}^{h}
which represents the sth row of MVa^{h}. Letting us define a 1 ｴ
S "summation" vector e = [1, 1, .., 1], vector, note that eMVa^{h} = ep_{-0}*[x^{h}
- e^{h}]_{-0} = ・/font> _{sﾎ
S} p_{s}*(x_{s}^{h} - e_{s}^{h}). Thus, our
Arrow-Debreu equilibrium becomes:

p

_{0}*(x_{0}^{h}- e_{0}^{h}) + eMVa^{h}= 0.

Let us now turn to the initial state, s = 0. Let us define q_{f} =
・/font> _{sﾎ S} p_{1s }r_{fs}
where p_{1s} is the price of gold in state s ﾎ S, or
simply q = eMV where e = [1, 1, .., 1]. Post multiplying by a^{h}, then qa^{h}
= eMVa^{h}. Thus:

p

_{0}*(x_{0}^{h}- e_{0}^{h}) + qa^{h}= 0

Thus, from these results, we can construct the Radner budget constraint B_{R｢ }as follows:

B

_{R｢ }= {y^{h}ﾎ R_{+}^{n(S+1)}| p_{0}*(y_{0}^{h}- e_{0}^{h}) ｣ -qa^{h}, and p_{s}*(y_{s}^{h}* - e_{s}^{h}) ｣ ・/font>_{fﾎ F}p_{1s}r_{fs}a_{f}^{h}for all s ﾎ S, a^{h}ﾎ R^{F}}

Thus, for any p* and x^{h}, we can find a^{h} and q^{h}
in order to build a set B_{R｢ }.

However, we must ensure if we have the Arrow-Debreu *equilibrium*
allocation x^{h}*, then markets clear. Let q* and a^{h}* be the asset
prices and portfolios deduced by the previous means from the equilibrium allocation x^{h}*.
Thus, we can construct a corresponding B_{R｢ }from
this. Now, if x^{h}* is an Arrow-Debreu equilibrium, then *all*
state-contingent markets must clear, i.e. summing up over households, ・/font>
_{hﾎ H} (x^{h}* - e^{h}) = 0. Now,
this is also necessary for Radner equilibrium - which is nice. But, for Radner, we also
have the further requirement that *asset* markets must clear. This can actually be
derived from the former. Pre-multiplying the market-clearing conditions by p*, note that:

・/font>

_{hﾎ H }p*(x^{h}* - e^{h}) = 0

Now, for any particular h ﾎ H, we know that p_{-0}*[x^{h}*
- e^{h}]_{-0} = MVa^{h}* by our previous construction. Doing this
for all households but one (i.e. all h = 1, 2, ..., H-1), then we obtain a set (a^{1}*,
a^{2}*, .., a^{H-1}*) that fulfills these conditions. Consequently, we
know that p_{-0}*(x^{H}* - e^{H}) = - ・/font>
_{h=1}^{H-1} p_{-0}*(x^{H}* - e^{H}), but then p_{-0}*(x^{H}*
- e^{H}) = -・/font> _{h=1}^{H-1} MVa^{h}*
= -MV・/font> _{hﾎ 1}^{H-1}a^{h}*,
thus defining a^{H}* = -・/font> _{h=1}^{H--1}a^{h}*,
then obviously p_{-0}*(x^{H}* - e^{H}) = MVa^{H}*. Thus
summing up over households, we obtain by the Arrow-Debreu market-clearing conditions:

・/font>

_{hﾎ H}p_{-0}*(x^{h}* - e^{h}) = 0 ﾞ ・/font>_{hﾎ H}MVa^{h}* = 0 ﾞ ・/font>_{hﾎ H}a^{h}* = 0

so if portfolios a^{h}* are derived from the Arrow-Debreu
equilibrium x^{h}*, thus asset markets clear as well.

We are nearly finished: all that remains is to prove that if x^{h}
is feasible under Radner, then it is also feasible under Arrow-Debreu, i.e. if x^{h}
ﾎ B_{R｢ }, then x^{h}
ﾎ B_{A｢ }, so B_{R｢ }ﾍ B_{A｢
}. To see this, recall that if x^{h} ﾎ B_{R｢ }, then

p

_{0}*(x_{0}^{h}- e_{0}^{h}) ｣ -q*a^{h}*

p

_{s}*(x_{s}^{h}- e_{s}^{h}) ｣ ・/font>_{fﾎ F}p_{1s}r_{fs}a_{f}^{h}* for all s ﾎ S

summing up over states:

p

_{0}*(x_{0}^{h}- e_{0}^{h}) + ・/font>_{sﾎ S}p_{s}*(x_{s}^{h}- e_{s}^{h}) ｣ -q*a^{h}* + ・/font>_{sﾎ S・/font> fﾎ F}p_{1s}r_{fs}a_{f}^{h}*

But recall that ・/font> _{sﾎ
S・/font> fﾎ F} p_{1s}r_{fs}a_{f}^{h}*
= eｷMVa^{h}* and as we know, by the definition of q*, eｷMVa^{h}* = q*a^{h}*,
thus:

p

_{0}*(x_{0}^{h}- e_{0}^{h}) + ・/font>_{sﾎ S}p_{s}*(x_{s}^{h}- e_{s}^{h}) ｣ 0

so that x^{h} must also be in the Arrow-Debreu budget constraint,
B_{A｢ }. In sum, B_{R｢ }ﾍ B_{A｢ }. Consequently, we see
that this implies that if the Arrow-Debreu equilibrium x^{h}* is a
utility-maximizing allocation over B_{A｢ }, then it is
also utility-maximizing over a suitably-constructed B_{R｢ }.
Thus, an Arrow-Debreu equilibrium pair (x*, p*) implies a Radner equilibrium quadruplet
(x*, a*, p*, q*). Q.E.D.

In view of (i) ﾞ (ii), (ii) ﾞ (i), then we have proved that under complete markets, a Radner equilibrium is equivalent to an Arrow-Debreu equilibrium.ｧ

The equivalence theorem between Arrow-Debreu economy and a Radner economy is useful in many respects. The most obvious, of course, is that any existence theorem that applies to the former will consequently apply to the latter, thus we can omit that part by appealing to the general existence proofs we have in conventional general equilibrium models. Those who wish to see a direct proof of existence of Radner equilibrium are referred to Radner (1972) or Magill and Quinzii (1996).

Back |
Top |
Notation | Selected References |