The Hart Existence Problem

Magister Ludi



For Radner equilibrium, we need to assume complete asset markets. This does not require that we have a full set of "state-contingent" markets as in Arrow-Debreu, but rather only that the set of assets can "span" the entire state returns space. However, assuming this does not end all difficulties. Recall that rfs = ps rfs so that the return of asset f in state s is equal to the payoff of asset f in state s evaluated at the spot prices in state s. The fact that returns in different states depend on different state prices can lead to some rather unpleasant consequences, as pointed out by Oliver D. Hart (1975).

To see this, suppose, for the sake of argument, that we have only two states S = (1, 2), two goods and two assets, F = (f, g) where the assets pay in bundles of commodities. Suppose they have the following payoff structures:

rf = [rf1, rf2] = [(1, 0), (0.5, 3)]

rg = [rg1, rg2] = [(1, 2), (2, 0)]

thus a unit of asset f pays bundle rf1 = (1, 0) in state 1 and bundle rf2 = (0.5, 3) in state 2 whereas a unit of asset g pays bundle rg1 = (1, 2) in state 1 and bundle rg2 = (2, 0) in state 2. As a result, at any set of spot prices in state 1, p1 = (p11, p21) and spot prices in state 2, p2 = (p12, p22) the set of returns to assets f and g are:

rf = (p1rf1, p2rf2) = [p11, 0.5p12 + 3p22]

rg = (p1rg1, p2rg2) = [p11 + 2p21 , 2p12].

which, as depicted in Figure 3 are two linearly independent vectors in state net transfer space. Now, we can see that any net income transfers, such as t * = (t 1*, t 2*), can be made across both states by constructing portfolios of assets which are linear combinations of assets f and g. As we saw earlier, we can purchase afh units of asset f and agh units of asset g so that we obtain the sum vector afhrf + ahgrg which yields a particular set of returns in both states, etc. Thus, by suitable constructions of portfolios of assets f and g, an agent can undertake any set of transfers of purchasing power between states 1 and 2.

Figure 3 - A Hart Problem

However, as Oliver D. Hart (1975) indicated through a few well-chosen examples, things can go quite awry. The return vectors, after all, are functions of the spot prices, p1 = [p11, p21] in state 1 and p2 = [p12, p22] in state 2. Consequently, there may be vectors of prices that make the vectors of returns in both states collinear. Specifically, consider the case where spot prices in state 1 are p11/p21 = 2 and spot prices in state 2 are p12/p22 = 6. In this case, the return vectors become:

rf = psrf = [p11, p12]

rg = psrg = [2p11, 2p12]

thus, the return vectors are linearly related, as shown in Figure 3. Consequently, when return vectors become collinear, then they can no longer span the entire space. Specifically, when state prices are such, we can make construct all sorts of portfolios with assets f and g, but the possible returns are now restricted to the hyperplane H that passes through both rf and rg . Any other set of returns in the state space (such as t *) are not obtainable by any portfolio. The dimensionality of the space of possible returns is severely reduced.

What Hart (1975) went on to demonstrate, such a situation could lead to non-existence of Radner equilibrium. Heuristically, the possibility of such sudden reductions in dimensionality may cause discontinuities in our demand functions - and this resulting "hole" in the demand function may be precisely where the supply function passes through, consequently, it is quite possible that a Radner equilibrium does not exist.

The discontinuity arises because of the structure of our budget constraint. Recall that we had "consolidated" our budget constraint into p0z0h + qV-1p-0z-0h 0, which implies that excess demand functions for goods in any state, zish, are a function of all state spot prices. Thus, by Berge's Theorem, for the continuity of excess demand functions with respect to state spot prices, we require that this "grand" budget constraint be upper semicontinuous. Let us denote the grand budget constraint by B(p, e) where p represents the set of spot prices in all states and e is the particular agents endowment. If B(p, e) is upper semicontinuous, then if we take a sequence of convergent state-spot prices{pn}, then we can construct a sequence of convergent excess demand vectors {zn}contained within the sequence of budget constraints where the limiting demand vector is in the limiting budget constraint. In other words, pn p and xn x where xn B(pn, e) implies that x B(p, e).

But this upper semicontinuity, under the Hart problem, may not be true. Because collinear returns reduce the dimensionality of income transfers across states, then the budget constraint defined by the prices which yield those collinear returns will be drastically different from the budget constraint defined at prices which yield linearly-independent returns. Now, as we can envision from Figure 3, we can take a sequence of prices {pn} which yield linearly independent returns rf and rg and approach a limiting set of prices p which yield the collinear returns, rf and rg . We denote the budget constraint formed by the former B(pn, e) and the budget constraint formed by the latter B(p, e). Taking a sequence of excess demand vectors {zn} in B(pn, e), such that zn z, we can imagine how the drastic reduction in the possibility of state returns at the limiting prices p can easily imply that z B(p, e).

For instance, suppose that at every set of prices in {pn}, we demand the same constant amount so zn = z for all n and thus the desired transfer of income across states is t * is the same for every set of prices, pn. Now, in Figure 3, t * is off the hyperplane defined by the collinear returns rf , rg as t * can be constructed by the linearly-independent returns implied at every pn in the sequence. But if this is true, then this transfer t * is no longer available in the limit when prices are p and returns are collinear and thus the budget set is B(p, e). Thus, as zn implies t * and z implies t *, then x B(p, e), i.e. pn p, zn z where zn B(pn, e) does not imply z (B, p). The budget constraint is not upper semicontinuous and thus the demand functions are not necessarily continuous - and we have a serious problem that can undermine the existence of a Radner equilibrium.

There are (roughly) three answers to the Hart problem. The first is that, in general, Hart counterexamples are rare. As was shown for the complete markets case by Repullo (1986) and Magill and Shafer (1990) - and in the incomplete markets case by Duffie and Shafer (1985, 1986) and Hirsch, Magill and Mas-Colell (1987) - Radner equilibria exist generically, with Hart-type counterexamples being rather exceptional. A second answer, proposed initially by Kenneth J. Arrow (1953) but more forcefully forwarded by John Geanakoplos and Herakles Polemarchakis (1986), is to express all payoffs in terms of a numeraire good ("gold"). A third answer, pursued by David Cass (1984), Jan Werner (1985) and Darrell Duffie (1987) was to propose that we construct a nominal unit of account ("cash") which is independent of prices and have asset payoffs expressed in these.

The first genericity argument is quite involved and we refer elsewhere (e.g. Magill and Shafer , 1991; Magill and Quinzii, 1996) for more extensive reviews. That leaves the numeraire and unit of account options. Both of these, effectively, eliminate real assets - i.e. assets that pay bundles of commodities - in favor of assets that only pay one type of "thing".

Of the two types of reductions, the Arrow-Geanakoplos-Polemarchakis normalization via a numeraire good is the easiest to handle. Let us take good 1 as the numeraire good (call it "gold") and assume that all asset payoffs are in units of "gold". This simplifies our problem tremendously as the return to asset f in state s is merely rfs = p1srfs where rfs is the number of units of gold asset f pays out in state s while p1s is the price of gold in state s. With no loss of generality, we normalize the price of gold to 1 in each states, i.e. p1s = 1 for all s S. Thus, spot price vector in any state is ps/p1s = [1, p2s/p1s, p3s/p1s, .., pns/p1s]. Notice that as the spot price of gold may be different in different states, then the normalization depends on the state of nature that emerges. The main result of the gold normalization is that returns can be written rfs = rfs which is merely a number. As a result, the Hart problem will not arise: all the dimensionality problems are sidestepped: prices between different real goods in a given state can change all they wish - they will not affect the relative returns of assets in that state.

In contrast, the Cass-Werner "unit of account" or "cash" structure is more complicated. In a cash economy, the resulting equilibrium is not strictly a "Radner equilibrium" and the Arrow-Debreu equivalence does not really hold precisely and, perhaps most significantly, there is a severe indeterminacy problem. Why this is so can be conceived in the following way: let g fs be the cash return of asset f in state s. Then, the real return of asset f in state s is rfs = gs/Ps where Ps represents the "price level" in state s (some index of the spot prices, ps). To fix ideas fully, we could make it comparable to our previous case by letting Ps = p1s so only good 1 ("gold") is included in the price-index. Thus, the real return of a cash-paying asset f in state s is rfs = g s/p1s. Thus note now that returns are dependent on the price-level p1s. This, of course, should come as no surprise. In a non-cash situation - as in a "gold" or real asset economy - a rise in prices in any state implies that returns rise; but if we restrict payoffs to units of account, then the value of the return will fall as prices rise, i.e. there is an "inflation" element in cash economies.

The problem with cash economies, as first noted by Cass (1985) and proved by Geanakoplos and Mas-Colell (1989) and Balasko and Cass (1989), is that Radner equilibria will be severely indeterminate. We can follow the "Walras-Cassel" test of counting equations and unknowns to see intuitively why this arises. Let us have H agents, n physical goods, S+1 states and F assets. Consequently, let us count the equations of this model. Then the number of demand functions for goods is H[n(S+1)], the number of demand functions for assets is HF. We also have n(S+1) goods markets clearing conditions and F asset market clearing conditions. Thus the total number of equations in the economy is (H+1)[n(S+1) + F]. However, recall that there are H(S+1) budget constraints met thus, in aggregate, S+1 Walras' Law constraints exist which permit us to drop S+1 market-clearing conditions (the "nth" in each state s S+1). Thus, we now have (H+1)[n(S+1) + F] - (S+1) equations.

How about unknowns? We need to determine how much of each good in each state will be allocated to each household, xih, thus we have H(n(S+1)) of these; we need to determine the assets allocated to each household, afh, thus we have HF of these. We have n(S+1) goods prices pis to discover and F asset prices qf to determine. Thus, the total number of unknowns is (H+1)[n(S+1)+F]. Now, conventionally, we would just suspect that normalizing prices would allow us to strike out another S+1 expressions: as we only need relative prices in each state, then normalizing one of the prices pis in each state s S and one of the qf in state s = 0. In this case, it is easy to notice that the number of unknowns would be (H+1)[n(S+1) + F] - (S+1), which would be equal to the number of equations.

However, while this last normalization step is indeed true for the real-asset or numeraire asset case, it is not true when we have a cash economy. Price normalizations do affect real demands. The intuition is effectively the following: suppose spot prices doubled in a particular state s but not in any other state. Obviously, then, relative spot prices within state s have not changed; but the purchasing power of the payoff to an asset which delivers cash in state s has declined. Consequently, one would expect the agent to make changes in his net transfers across states to compensate for the rise in spot prices in state s. Thus, a rise in ps is not "neutral" if payoffs are in cash: it forces agents to shift their transfers between states. The "grand" budget constraint, then, is affected by changes in the price level in state s.

Of course, if all prices in all states double (and the nominal price of asset halves in compensation), then there will be no shifting about - in this case, changes in the price level is neutral. Similarly, if price levels double in the initial states, s = 0, there will also be no change. Thus, in these two exceptional cases, the budget constraint will not be affected - thus we are at least permitted to remove two of the unknowns in our system by appropriate normalization - e.g. set ・/font> s S p1s = 1 and set p01 = 1 to deal with the first and second cases respectively. However, that leaves us with (H+1)[n(S+1) + F] - 2 unknowns whereas we still have (H+1)[n(S+1) + F] - (S+1) equations - thus we have S-1 more unknowns than we have equations. Thus, as we can anticipate, there will be indeterminacy of Radner equilibrium in a cash economy.

Some economists (e.g. Magill and Quinzii, 1992, 1996) have jumped on this indeterminacy as an avenue by which one may "reintegrate" monetary theory into general equilibrium theory - a task long left dangling since the early 1970s. While an interesting area of research, this takes us a bit far off our trajectory here and shall refer to the standard references for further details.

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