The Equivalence Theorem Arrow-Debreu and Radner Back

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:

p0(x0h - e0h) + qV-1p-0[xh - eh]-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-1p-0 = m -0p-0 = (m 1p1, m 2p2, ..., m sps, ..., m SpS] 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 spis). 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 a Radner equilibrium is equivalent to an Arrow-Debreu equilibrium. In other words:

(i) if consumption demands x* = {xh*}h H Rn(S+1)H, portfolio decisions a* = {ah*}h H RFH, asset prices q* = (q1*, q2*, .., qS*) RF++ and spot prices p* = (p0*, p1*, p2*, .., pS*) Rn(S+1)++ constitute a Radner equilibrium, then there are multipliers (m 1, ..., m S) RS++ such that the allocation x* and the state-contingent commodities price vector (p0*, m 1p1*, ..., m SpS*) Rn(S+1) constitute an Arrow-Debreu equilibrium.

(ii) if the consumption demands x* = {xh*}h H Rn(S+1)H and state-contingent prices p* = (p0*, p1*, .., pS*) Rn(S+1)++ constitute an Arrow-Debreu equilibrium then there is a q* RF+ and portfolio decisions a* = {ah*}h H RFH such that the plans (x*, a*) and prices (q*, p*) constitute a Radner equilibrium.

Proof: (i) (ii) (Radner 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 -0V. We now wish to prove that m p* = (p0*, m 1p1*, m 2p2*, ...., m SpS*) is a set of Arrow-Debreu state-contingent equilibrium prices. To prove this, let us define:

BA = {yh R+n(S+1) | p0*(y0h - e0h) + ・/font> s S m sps*(ysh - esh) 0}

which is the Arrow-Debreu budget constraint with equilibrium state-contingent prices (p0*, m 1p1*, .., m SpS*) and:

BR = {yh R+n(S+1) | p0*(y0h - e0h) -qah, and ps*(ysh - esh) Vsah for all s S, ah RF}

which is the Radner budget constraint with equilibrium spot prices (p0*, p1*, ..., pS*). Our first step is to prove that BA BR, i.e. if xh BA, then xh BR. To see this, suppose xh BA, so:

p0*(x0h - e0h) + ・/font> s S m sps*(xsh - esh) 0

or:

p0*(x0h - e0h) + m -0 p-0*[xh - eh]-0 0

where p-0*[xh - eh]-0 = [p1*(x1h - e1h), p2*(x2h - e2h), ....., pS*(xSh - eSh)] . By completeness, we know that for any V with rank(V) = S, there is a vector ah RF such that:

p-0*[xh - eh]-0 = Vah

and this ah will be unique. Consequently, substituting into our Arrow-Debreu constraint:

p0*(x0h - e0h) + m -0Vah 0

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

p0*(x0h - e0h) + qah 0

Thus, as p-0*[xh - eh]-0 = Vah and p0*(x0h - e0h) -qah, then obviously xh BR. Thus BA BR. Now, all we have to prove is that if the Radner equilibrium allocation xh*, which is a utility-maximizing bundle over BR, is also feasible under the Arrow-Debreu and thus is also utility maximizing under BA. To see this, note that if xh* BR, then:

p0*(x0h* - e0h) -qah*

p-0*[xh* - eh]-0 Vah*

So pre-multiplying the second constraint by m -0, then m -0p-0*[xh* - eh]-0 m -0Vah*. Thus adding this to the first constraint:

p0*(x0h* - e0h) + m -0p-0*[xh* - eh]-0 m -0Vah* - qah*

but as q = m -0V and, post-multiplying by ah*, implies qah* = m -0Vah*, which implies in turn that:

p0*(x0h* - e0h) + m -0p-0*[xh* - eh]-0 0

so xh* satisfies the Arrow-Debreu constraint, i.e. xh* BA. Thus, allocation xh* and prices (p0*, m 1p1*, m 2p2*, .., m SpS*) define an Arrow-Debreu equilibrium. Q.E.D.

(ii) (i): (Arrow-Debreu 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* = (p0*, p1*, p2*, .., pS*) denote the equilibrium Arrow-Debreu state-contingent prices. As we have an Arrow-Debreu equilibrium, then it must be that xh* BA , now defined as:

BA = {yh R+n(S+1) | p0*(y0h - e0h) + ・/font> s S ps*(ysh - esh) 0}

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

p0*(x0h - e0h) + ・/font> s S ps*(xsh - esh) = 0.

which is our Arrow-Debreu budget constraint for the single individual at equilibrium. Now, let V be some S F matrix with typical element rfs R+ and let M be some S S dimensional diagonal matrix with typical element p1s R+ along the diagonal, i.e.

 p11 0 ... 0 M = 0 p12 ... 0 ... ... ... ... 0 0 ... p1S

Note that p1s is the price of good 1 (the numeraire good, "gold") in state s. We would like it that for any s S, that we would be able to find a F 1 vector ah = [a1h, a2h, ..., afh, .., aFh] RF such that ps*(xsh - esh) = ・/font> f F p1srfsafh. Or, letting p-0*[xh - eh]-0 denote an S 1 vector [p1*(x1h - e1h), p2*(x2h - e2h), ....., pS*(xSh - eSh)] , then we would write our hypothesis as:

p-0*[xh - eh]-0 = MVah

Is there such a vector ah? As asset markets are complete, then rank(V) = S, and so rank(MV) = S, thus there is indeed such a unique solution ah to this system. Note that this implies that ps*[xsh - esh] = ・/font> f Fps1rfsafh which represents the sth row of MVah. Letting us define a 1 S "summation" vector e = [1, 1, .., 1], vector, note that eMVah = ep-0*[xh - eh]-0 = ・/font> s S ps*(xsh - esh). Thus, our Arrow-Debreu equilibrium becomes:

p0*(x0h - e0h) + eMVah = 0.

Let us now turn to the initial state, s = 0. Let us define qf = ・/font> s S p1s rfs where p1s is the price of gold in state s S, or simply q = eMV where e = [1, 1, .., 1]. Post multiplying by ah, then qah = eMVah. Thus:

p0*(x0h - e0h) + qah = 0

Thus, from these results, we can construct the Radner budget constraint BR as follows:

BR = {yh R+n(S+1) | p0*(y0h - e0h) -qah, and ps*(ysh* - esh) ・/font> f F p1srfsafh for all s S, ah RF}

Thus, for any p* and xh, we can find ah and qh in order to build a set BR .

However, we must ensure if we have the Arrow-Debreu equilibrium allocation xh*, then markets clear. Let q* and ah* be the asset prices and portfolios deduced by the previous means from the equilibrium allocation xh*. Thus, we can construct a corresponding BR from this. Now, if xh* is an Arrow-Debreu equilibrium, then all state-contingent markets must clear, i.e. summing up over households, ・/font> h H (xh* - eh) = 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 H p*(xh* - eh) = 0

Now, for any particular h H, we know that p-0*[xh* - eh]-0 = MVah* by our previous construction. Doing this for all households but one (i.e. all h = 1, 2, ..., H-1), then we obtain a set (a1*, a2*, .., aH-1*) that fulfills these conditions. Consequently, we know that p-0*(xH* - eH) = - ・/font> h=1H-1 p-0*(xH* - eH), but then p-0*(xH* - eH) = -・/font> h=1H-1 MVah* = -MV・/font> h 1H-1ah*, thus defining aH* = -・/font> h=1H--1ah*, then obviously p-0*(xH* - eH) = MVaH*. Thus summing up over households, we obtain by the Arrow-Debreu market-clearing conditions:

・/font> h H p-0*(xh* - eh) = 0 ・/font> h H MVah* = 0 ・/font> h H ah* = 0

so if portfolios ah* are derived from the Arrow-Debreu equilibrium xh*, thus asset markets clear as well.

We are nearly finished: all that remains is to prove that if xh is feasible under Radner, then it is also feasible under Arrow-Debreu, i.e. if xh BR , then xh BA , so BR BA . To see this, recall that if xh BR , then

p0*(x0h - e0h) -q*ah*

ps*(xsh - esh) ・/font> f F p1srfsafh* for all s S

summing up over states:

p0*(x0h - e0h) + ・/font> s S ps*(xsh - esh) -q*ah* + ・/font> s S・/font> f F p1srfsafh*

But recall that ・/font> s S・/font> f F p1srfsafh* = eｷMVah* and as we know, by the definition of q*, eｷMVah* = q*ah*, thus:

p0*(x0h - e0h) + ・/font> s S ps*(xsh - esh) 0

so that xh must also be in the Arrow-Debreu budget constraint, BA . In sum, BR BA . Consequently, we see that this implies that if the Arrow-Debreu equilibrium xh* is a utility-maximizing allocation over BA , then it is also utility-maximizing over a suitably-constructed BR . Thus, an Arrow-Debreu equilibrium pair (x*, p*) implies a Radner equilibrium quadruplet (x*, a*, p*, q*). Q.E.D.

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

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).

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