The Equivalence Theorem Arrow-Debreu and Radner

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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(x₀^h - e₀^h) + qV^-1p_-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^-1p_-0 = m _-0p_-0 = (m ₁p₁, m ₂p₂, ..., m _sp_s, ..., m _Sp_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 _sp_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 a Radner equilibrium is equivalent to an Arrow-Debreu equilibrium. In other words:

(i) if consumption demands x* = {x^h*}_{hÎ H} Î R^n(S+1)H, portfolio decisions a* = {a^h*}_{hÎ H Î} R^FH, asset prices q* = (q₁*, q₂*, .., q_S*) Î R^F₊₊ and spot prices p* = (p₀*, p₁*, p₂*, .., p_S*) Î R^n(S+1)₊₊ constitute a Radner equilibrium, then there are multipliers (m ₁, ..., m _S) Î R^S₊₊ such that the allocation x* and the state-contingent commodities price vector (p₀*, m ₁p₁*, ..., m _Sp_S*) Î R^n(S+1) constitute an Arrow-Debreu equilibrium.

(ii) if the consumption demands x* = {x^h*}_{hÎ H} Î R^n(S+1)H and state-contingent prices p* = (p₀*, p₁*, .., p_S*) Î R^n(S+1)₊₊ constitute an Arrow-Debreu equilibrium then there is a q* Î R^F₊ and portfolio decisions a* = {a^h*}_{hÎ H} Î R^FH such that the plans (x*, a*) and prices (q*, p*) constitute a Radner equilibrium.

Proof: (i) Þ (ii) (Radner Þ 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 _Sp_S*) is a set of Arrow-Debreu state-contingent equilibrium prices. To prove this, let us define:

B_A = {y^h Î R₊^n(S+1) | p₀*(y₀^h - e₀^h) + å _{sÎ S} m _sp_s*(y_s^h - e_s^h) £ 0}

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

B_R = {y^h Î R₊^n(S+1) | p₀*(y₀^h - e₀^h) £ -qa^h, and p_s*(y_s^h - e_s^h) £ V_sa^h for all s Î S, a^h Î R^F}

which is the Radner budget constraint with equilibrium spot prices (p₀*, p₁*, ..., p_S*). Our first step is to prove that B_A Í B_R, i.e. if x^h Î B_A, then x^h Î B_R. To see this, suppose x^h Î B_A, so:

p₀*(x₀^h - e₀^h) + å _{sÎ S} m _sp_s*(x_s^h - e_s^h) £ 0

or:

p₀*(x₀^h - e₀^h) + m _-0 p_-0*[x^h - e^h]_-0 £ 0

where p_-0*[x^h - e^h]_-0 = [p₁*(x₁^h - e₁^h), p₂*(x₂^h - e₂^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 Î 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:

p₀*(x₀^h - e₀^h) + m _-0Va^h £ 0

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

p₀*(x₀^h - e₀^h) + qa^h £ 0

Thus, as p_-0*[x^h - e^h]_-0 = Va^h and p₀*(x₀^h - e₀^h) £ -qa^h, then obviously x^h Î B_R. Thus B_A Í 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* Î B_R, then:

p₀*(x₀^h* - e₀^h) £ -qa^h*

p_-0*[x^h* - e^h]_-0 £ Va^h*

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

p₀*(x₀^h* - e₀^h) + m _-0p_-0*[x^h* - e^h]_-0 £ m _-0Va^h* - qa^h*

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

p₀*(x₀^h* - e₀^h) + m _-0p_-0*[x^h* - e^h]_-0 £ 0

so x^h* satisfies the Arrow-Debreu constraint, i.e. x^h* Î B_A. Thus, allocation x^h* and prices (p₀*, m ₁p₁*, m ₂p₂*, .., m _Sp_S*) define an Arrow-Debreu equilibrium. Q.E.D.

(ii) Þ (i): (Arrow-Debreu Þ 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*, .., p_S*) denote the equilibrium Arrow-Debreu state-contingent prices. As we have an Arrow-Debreu equilibrium, then it must be that x^h* Î B_A¢ , now defined as:

B_A¢ = {y^h Î R₊^n(S+1) | p₀*(y₀^h - e₀^h) + å _{sÎ 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¢ Í 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 Î H, then B_A¢ implies at equilibrium prices p*:

p₀*(x₀^h - e₀^h) + å _{sÎ 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 ´ F matrix with typical element r_fs Î R₊ and let M be some S ´ S dimensional diagonal matrix with typical element p_1s Î R₊ along the diagonal, i.e.

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 Î S, that we would be able to find a F´ 1 vector a^h = [a₁^h, a₂^h, ..., a_f^h, .., a_F^h] Î R^F such that p_s*(x_s^h - e_s^h) = å _{fÎ F} p_1sr_fsa_f^h. Or, letting p_-0*[x^h - e^h]_-0 denote an S´ 1 vector [p₁*(x₁^h - e₁^h), p₂*(x₂^h - e₂^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] = å _{fÎ F}p_s1r_fsa_f^h which represents the sth row of MVa^h. Letting us define a 1 ´ S "summation" vector e = [1, 1, .., 1], vector, note that eMVa^h = ep_-0*[x^h - e^h]_-0 = å _{sÎ S} p_s*(x_s^h - e_s^h). Thus, our Arrow-Debreu equilibrium becomes:

p₀*(x₀^h - e₀^h) + eMVa^h = 0.

Let us now turn to the initial state, s = 0. Let us define q_f = å _{sÎ S} p_1s r_fs where p_1s is the price of gold in state s Î S, or simply q = eMV where e = [1, 1, .., 1]. Post multiplying by a^h, then qa^h = eMVa^h. Thus:

p₀*(x₀^h - e₀^h) + qa^h = 0

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

B_R¢ = {y^h Î R₊^n(S+1) | p₀*(y₀^h - e₀^h) £ -qa^h, and p_s*(y_s^h* - e_s^h) £ å _{fÎ F} p_1sr_fsa_f^h for all s Î S, a^h Î 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, å _{hÎ 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:

å _{hÎ H} p*(x^h* - e^h) = 0

Now, for any particular h Î 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 (a¹*, a²*, .., a^H-1*) that fulfills these conditions. Consequently, we know that p_-0*(x^H* - e^H) = - å _h=1^H-1 p_-0*(x^H* - e^H), but then p_-0*(x^H* - e^H) = -å _h=1^H-1 MVa^h* = -MVå _{hÎ 1}^H-1a^h*, thus defining a^H* = -å _h=1^H--1a^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:

å _{hÎ H} p_-0*(x^h* - e^h) = 0 Þ å _{hÎ H} MVa^h* = 0 Þ å _{hÎ 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 Î B_R¢ , then x^h Î B_A¢ , so B_R¢ Í B_A¢ . To see this, recall that if x^h Î B_R¢ , then

p₀*(x₀^h - e₀^h) £ -q*a^h*

p_s*(x_s^h - e_s^h) £ å _{fÎ F} p_1sr_fsa_f^h* for all s Î S

summing up over states:

p₀*(x₀^h - e₀^h) + å _{sÎ S} p_s*(x_s^h - e_s^h) £ -q*a^h* + å _{sÎ Så fÎ F} p_1sr_fsa_f^h*

But recall that å _{sÎ Så fÎ F} p_1sr_fsa_f^h* = e·MVa^h* and as we know, by the definition of q*, e·MVa^h* = q*a^h*, thus:

p₀*(x₀^h - e₀^h) + å _{sÎ 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¢ Í 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) Þ (ii), (ii) Þ (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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