The model considered in the previous section is very simplified - notably, it omits all uncertainty. The sequence economy initiated by Roy Radner (1972) is similar in construction, but considers a situation where there are multiple assets, multiple goods, multiple time periods and multiple states (thus uncertainty). However, we shall still maintain, for simplicity, a two-period structure, thus T = {0, 1}. At time t = 0, we have only one state of nature (call it s = 0). Let S be the set of possible states of nature at time t = 1. For simplicity, but abusive of notation, we also denote the number of states as S, i.e. #S = S. Thus conflating both periods, the total number of possible states is S+1. Let there be n physically-differentiated goods, thus the "commodity space" X is a subset of R^{n(S+1)}.
Let there be H households where the preferences of the hth household ｳ _{h} are representable by a nice utility function u^{h}: R^{n(S+1)} ｮ R and each household has a set of endowments, e^{h} = [e_{0}^{h}, e_{1}^{h}, ...., e_{S}^{h}] ﾎ R^{n(S+1)} -- where e_{s}^{h} represents the vector of endowed goods at each state s ﾎ S+1. An economy-wide allocation x is defined as a vector {x^{h}}_{hﾎ H} where bundle x^{h} is the allocation to the hth household and is defined as a vector x^{h} = [x_{0}^{h}, x_{1}^{h}, ..., x_{S}^{h}] ﾎ R^{n(S+1)} where each x_{s}^{h} is the vector of commodities the hth household receives in state s ﾎ S+1. We can restrict allocations and endowments to the positive orthant of R^{n(S+1)}, but that is not strictly necessary.
Notice that so far, everything is as in the Arrow-Debreu state-contingent commodity case. The difference now lies in specifying which markets are open and which are closed. We now introduce the modification that we cannot trade most state-contingent commodities. However, Arrow (1953) claimed and Radner (1972) proved, even if not all state-contingent commodities are available for trading at t = 0, a Pareto-optimal equilibrium may nonetheless be reached when markets re-opened at time t = 1 after the state is realized and trading occurs if the resulting spot prices in the future states are correctly anticipated by agents at t = 0 and we can transfer purchasing power across states.
Let p_{s} = [p_{1s}, ..., p_{ns}] ﾎ R^{n}_{+} be the spot prices for goods in a particular state s ﾎ S+1. Note that these are not Arrow-Debreu state-contingent prices because p_{is} is the price of good i in state s expressed at time t = 1 and not in the initial time period t = 0. However, it is important to recall at this point that we must have "self-fulfilled" or "perfect foresight" expectations. This implies that, in equilibrium, a vector p_{s} represents both expected prices in state s and actual prices in state s. The individual optimization program is going to be maximized at t = 0 and future spot prices for every state {p_{s}}_{sﾎ S} must be correct. If we do not impose this condition, then we are effectively returning to "temporary" equilibrium.
The main issue, however, is transferring purchasing power from t = 0 to t = 1 (and vice-versa). This can be done by the purchasing of real or financial assets - although asset structures, as we shall see, will be a rather complicated issue. The basic idea, nonetheless, is that we do not need a full set of state-contingent forward contracts, but only a subset of them, enough to "span" the state space. Let F be the set of financial assets in an economy which can be bought at t = 0. Abusing notation again, let #F = F, i.e. there are F such assets. Let q = [q_{1},..., q_{F}] ﾎ R^{F} denote the vector of security prices at t = 0. Thus, a particular asset f ﾎ F can be purchased in period t = 0 at price q_{f} which yields a return r_{fs} ﾎ R at t = 1 if state s ﾎ S is realized. Let r_{f} = [r_{f1}, ...., r_{fS}]｢ ﾎ R^{S} be the vector of returns for the fth security - thus a particular security f ﾎ F pays off a variety of returns depending on the state s ﾎ S that is realized in t = 1 (thus r_{f} is a random variable r_{f}: S ｮ R). Conversely, r_{s} = [r_{1s}, r_{2s}, ..., r_{Fs}] ﾎ R^{F} is the set the returns to all assets in a particular state s ﾎ S. For the moment, let us suppose that these returns are paid in some unit of account ("money"). This will cause trouble later, but we shall sweep this under the rug for the moment in order to construct the rudiments of the economy we are dealing with.
Let a_{f}^{h} denote agent h's purchases of asset f at t = 0. Thus, a^{h} = [a_{1}^{h}, ..., a_{F}^{h}] ﾎ R^{F} is the vector of securities acquired at t = 0 by the agent h, i.e. agent h’s "portfolio". Note that we are not restricting a^{h} to the positive orthant: we can allow a_{f}^{h} < 0 for any asset f, what would be regarded as "short-selling" asset f. Given asset prices, q, then qa^{h} = ・/font> _{f} q_{f}a_{f}^{h} is the cost of purchasing a portfolio a^{h} at time t = 0. Thus, ・/font> _{fﾎ F }r_{fs}a_{f}^{h} is the set of returns received from a portfolio a^{h} at time t = 1 in a particular state s ﾎ S. The hth agent then faces the following optimization problem at time t = 0:
max u^{h}[x_{0}^{h}, x_{1}^{h}, ...., x_{S}^{h}]
s.t.
p_{0}x_{0}^{h} + qa^{h} ｣ p_{0}e_{0}^{h}
p_{1}x_{1}^{h} ｣ ・/font> _{fﾎ F} r_{f1}a_{f}^{h} + p_{1}e_{1}^{h}
p_{2}x_{2}^{h} ｣ ・/font> _{fﾎ F} r_{f2}a_{f}^{h} + p_{2}e_{2}^{h}
........................................p_{s}x_{s}^{h} ｣ ・/font> _{fﾎ F} r_{fs}a_{f}^{h} + p_{s}e_{s}^{h}
........................................
p_{S}x_{S}^{h} ｣ ・/font> _{fﾎ F} r_{fS}a_{f}^{h} + p_{S}e_{S}^{h}
Thus, the first constraint says that the amount of goods and securities demanded at t = 0 must be less than endowment at t = 0, while the rest of the constraints that the amount of goods demanded at t = 1 in any particular state s ﾎ S must be less than the endowment at state s and the returns provided in state s by all the securities purchased earlier. Notice that each constraint in this second set is self-contained: the agent cannot directly borrow from other states other than the initial one; in any particular state, he must make do with the endowment he has in that state and the realized returns of the assets he happens to have carried over from the previous time period.
However, he can borrow from other states "indirectly" by purchasing an appropriate set of assets initially. For instance, suppose there are two possible future states 1 and 2, and he wishes to "transfer" endowment in state 1 to state 2. In this case, at the initial time period, t = 0, he would short-sell an asset which yields returns in state 1 (thus, in state 1, he would be committed to using his state 1 endowment to pay the returns on that asset, thus his budget in state 1 is reduced) and then use the proceeds from that short-sale to buy an asset which yields a positive return in state 1 (thus his budget constraint expands in state 1). Thus, via the indirect means of portfolio choice, he can transfer purchasing power across different states.
Let us now define a "Radner equilibrium" or, as Radner (1972) himself called it, an "equilibrium in plans, prices and price expectations":
Radner Equilibrium: a "Radner equilibrium" is a price pair (p*, q*) ﾎ R^{n(S+1)} ｴ R^{F} and an allocation pair, ({x^{h}*}_{hﾎ H} , {a^{h}*}_{hﾎ H}} ﾎ R^{n(S+1)H} ｴ R^{FH} such that:
(i) for every h ﾎ H, consumption plans x^{h}* and portfolios a^{h}* solve the individual optimization problem given earlier.
(ii) ・/font> _{hﾎ H} a_{f}^{h}* ｣ 0 " f ﾎ F
(iii) ・/font> _{hﾎ H} x_{s}^{h}* ｣ ・/font> _{hﾎ H} e_{s}^{h}* " s ﾎ S
Thus, condition (i) guarantees that every agent is optimizing; condition (ii) implies that asset markets clear while condition (iii) implies that product markets clear in every state.
We are not going to prove the existence of a Radner equilibrium now as there are still a few more things to clear up. However, the intuition behind the equivalence between the Radner equilibrium and the Arrow-Debreu equilibrium can be thought through already at present. In order to see this, let us first modify the notation in the individual optimization problem a little bit. Let us define the matrix V as an (S ｴ F) matrix with typical element r_{sf} ﾎ R^{nSF} denoting the returns in state s ﾎ S of asset f ﾎ F as shown in Table 1 below:
Assets |
r_{s}: F ｮ R (or V_{s}) | |||||||
1 | 2 | ... | f | ... | F | |||
1 | r_{11} | r_{12} | ... | r_{1f} | ... | r_{1F} | ｬ r_{1} | |
States | 2 | r_{21} | r_{22} | ... | r_{2f} | ... | r_{2F} | ｬ r_{2} |
... | ... | ... | ... | ... | ... | ... | ... | |
s | r_{s1} | r_{s2} | ... | r_{sf} | ... | ... | ｬ r_{s} | |
... | ... | ... | ... | ... | ... | ... | ... | |
S | r_{S1} | r_{S2} | ... | r_{Sf} | ... | r_{SF} | ｬ r_{S} | |
ｭ | ｭ | ｭ | ｭ | |||||
r_{f}: S ｮ R (or V_{f}) | r_{1} | r_{2} | ... | r_{f} | ... | r_{F} |
Table 1 - An (S ｴ F)-dimensional matrix of possible returns, V
The matrix V has columns V_{f} = r_{f }delineating the returns the fth asset pays in each state. Also V_{s} = [r_{1s}, ...., r_{Fs}] ﾎ R^{nF} denotes a row of V and it represents the returns to all assets in a particular state s ﾎ S. Thus, we can see that V_{s}a^{h} = ・/font> _{fﾎ F} r_{fs}a_{f}^{h}. Consequently, the optimization problem can be rewritten as:
max u^{h}[x_{0}^{h}, x_{1}^{h}, ...., x_{S}^{h}]
s.t.
p_{0}x_{0}^{h} + qa^{h} ｣ p_{0}e_{0}^{h}
p_{s}x_{s}^{h} ｣ V_{s}a^{h} + p_{s}e_{s}^{h} for all s ﾎ S.
Thus, as we see, an individual is maximizing a single objective function with n(S+1) arguments subject to (S+1) constraints. The first constraint, p_{0}x_{0}^{h} + qa^{h} ｣ p_{0}e_{0}^{h}, says that the amount of goods and securities demanded at t = 0 must be less than endowment at t = 0. The second set of constraints, p_{s}x_{s}^{h} ｣ V_{s}a^{h} + p_{s}e_{s}^{h} for all s ﾎ S, says that the amount of goods demanded at t = 1 and state s ﾎ S must be less than the endowment at state s and the returns provided in state s by all the securities purchased earlier.
Now, if the matrix V has full rank, i.e. the number of linearly independent securities equals the number of states (F = S) and so V is an (Sｴ S) matrix, a situation known as "complete markets", our task is quite facilitated as we can consolidate the several constraints into one single grand constraint. To see this, let [px^{h} - pe^{h}]_{-0} ﾎ R^{S }be an S-dimensional vector with typical element p_{s}x_{s}^{h} - p_{s}e_{s}^{h}, then we obviously have it that we can consolidate the second set of constraints into the following singe term:
[px^{h} - pe^{h}]_{-0} = Va^{h}
where, note that [px^{h} - pe^{h}]_{-0} excludes the initial state s = 0. Thus within this vector, prices p_{-0} = [p_{1}, ..., p_{S}] are future prices exclusively; similarly, x_{-0}^{h} = [x_{1}^{h}, ..., x_{S}^{h}] are future bundles exclusively and analogously for endowments e_{-0}^{h}. Now, if V is full rank and |V| ｹ 0, then we can invert it to solve for the vector a^{h}, i.e.
V^{-1}[px^{h} - pe^{h}]_{-0} = a^{h}
thus, plugging this into our first constraint, we obtain the following:
(p_{0}x_{0}^{h} - p_{0}e_{0}^{h}) + qV^{-1}[px^{h} - pe^{h}]_{-0} ｣ 0
Thus, the consumer now faces the following consolidated optimization problem:
max u^{h}[x_{0}^{h}, x_{1}^{h}, ...., x_{S}^{h}]
s.t.
(p_{0}x_{0}^{h} - p_{0}e_{0}^{h}) + qV^{-1}p_{-0}^{ }[x^{h} - e^{h}]_{-0} ｣ 0
This is, in effect, the maximization problem of the standard Arrow-Debreu model where, at time t = 0, agent h is purchasing vectors of future state-contingent commodities, x_{s}^{h}. The vector of Arrow-Debreu state-contingent prices is the term qV^{-1}p_{-0}. Now, if we can find a set of multipliers such that qV^{-1} = m _{-0} = [m _{1}, m _{2}, ..., m _{S}], (which we can by the concept of arbitrage-free asset prices as we shall see later), then this consolidated constraint can be rewritten as:
p_{0}(x_{0}^{h} - e_{0}^{h}) + m _{-0}p_{-0}[x^{h} - e^{h}]_{-0} ｣ 0
where m _{-0}p_{-0} = [m _{1}p_{1}, m _{2}p_{2}, m _{3}p_{3}, ..., m _{S}p_{S}]. The claim, then, is that m _{s}p_{s} is a vector of state-contingent prices, i.e. "Arrow-Debreu" prices. Thus, the Arrow-Debreu state-contingent price of good i in state s is merely m _{s}p_{is}, i.e. the spot price of good i in state s multiplied by the marginal value of state s, m _{s}.
We are not going to prove the existence of a Radner equilibrium now. Instead, we are going to wait until we establish the condition for equivalence of Radner equilibrium and Arrow-Debreu equilibrium and thereafter appeal to existence in the latter to establish existence in the former. We go through various steps to this end. Firstly, we establish, through the "fundamental theorem of asset-pricing" that to every Radner equilibrium, there corresponds a vector of no-arbitrage asset prices, q. We consequently show that the existence of these no-arbitrage asset prices implies the existence of a vector of semi-positive multipliers m = (1, m _{1}, .., m _{S}), where m _{s} represents the marginal value of state s. Finally, we demonstrate that we subsequently multiply the spot price vectors p_{1}, ..., p_{S} of the Radner equilibrium by their corresponding multipliers, then the result we obtain is the set of state-contingent price vectors for an Arrow-Debreu equilibrium, i.e. m _{s}p_{s} constitutes the state-contingent price vector for state s ﾎ S in an Arrow-Debreu equilibrium. We also prove this in reverse: namely, if there exist a set of state-contingent price vectors which form an Arrow-Debreu equilibrium, we can deduce no-arbitrage asset price vectors and portfolio choices which imply a Radner equilibrium.
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