One of the central features ofsequence economiesare financial markets. Before proceeding onto defining a full sequential equilibria, it might be worthwhile to spend a few moments concentrating on financial asset market equilibrium.
An asset is a financial instrument which carries a current purchase price and a future payoff — or, rather, an entire series of future payoffs depending on the states that emerge in the future. The simplest kind of asset is a bond, which pays a sure monetary payoff in the future regardless of the state that actually occurs in the future (net of default risk). With stocks, of course, the payoff is not certain and depends on corporate profits which will in turn depend on the state of nature that emerges in the future. Other assets, of course, have different payoff structures.
Nonetheless, despite these differences in detail, in general, financial assets are transfers of purchasing power across time periods and states of nature.Buyingan asset implies we are transferring purchasing power from the present to some future state;sellingorshort-sellingan asset implies we are transferring purchasing power from some future state to the present. Regardless of the type of financial asset – stocks, bonds, options, etc. – all of them involve at least these time-and-state spanning properties.
An asset market is, naturally, the place where assets are traded and, consequently, purchase prices are determined. The determination of asset prices is one of the main concerns of financial theory and there are a handful of competing theories. However, one of the central features of asset pricing theory, or, as some claim, the one concept that unifies all of finance (Dybvig andRoss, 1987), is the stipulation that, in equilibrium, asset prices are such that arbitrage is not possible.
It might be useful to clarify at this point what is meant by arbitrage. Intuitively, many people would associate arbitrage with buying low and selling high. More formally, an arbitrageur purchases a set of financial assets at a low price and sells them at a high pricesimultaneously. This timing element is important: namely, because of simultaneity, arbitrageurs requirenooutlay of personal endowment but only need to set up a set of simultaneous contracts such that the revenue generated from the selling contract pays off the costs of the buying contract, i.e. construct a portfolio consisting of purchased assets and short-sold assets which yields positive returns with no commitment. The simultaneity ensures that the arbitrageur carriesnorisk as none of his own personal resources are ever on the line.
One can argue that, in the real world, there is rarely a case ofpurearbitrage. Indeed, most of what the financial community calls arbitrage is really just some very fast or short-term speculation. When speculating, agents usually purchase the assets first and sell them afterwards (or short-sell the assets first and purchase them afterwards), thus they must commit some of their own resources, at least temporarily — and they still run the risk that they will not get to dispose of the second half of their operation at the anticipated price. Pure arbitrage, in contrast, is simultaneous and riskless – the quintessential free lunch.
The reason why pure arbitrage is not commonly observed is precisely the reasoning for the no-arbitrage assumption in financial market equilibrium: if therewerearbitrage opportunities, these would be eliminated immediately. Specifically, if asset prices allow for arbitrage opportunities, then because of strong monotonicity of preferences and no bounds to short-selling, agents would immediately hone in on a portfolio position that yielded arbitrage profits. The no-commitment nature of arbitrage opportunities imply that agentscanreplicate this arbitrage portfolioinfinitelywith no personal resource constraint. If this begins to happen, then at some point (i.e. almost instantly), the price differences which enabled the arbitrageur to hold such a position would close. Thus:
Assuming no arbitrage is compelling because the presence of arbitrage is inconsistent with equilibrium when preferences increase with quantity. More fundamentally, the presence of arbitrage is inconsistent with the existence of an optimal portfolio strategy foranycompetitive agent who prefers more to less, because there is no limit to the scale at which an individual would want to hold the arbitrage position. Therefore, in principle, absence of arbitrage follows from individual rationality of a single agent. One appeal of results based on the absence of arbitrage is the intuition that few rational agents are needed to bid away arbitrage opportunities, even in the presence of a sea of agents driven by `animal spirits. (P.H. Dybvig and S.A.Ross, 1987)
The concept of no-arbitrage asset prices in financial theory stretches over naturally into sequential general equilibrium theory where asset market equilibrium is part of the structure. In order to visualize this, let us begin with the simplest model possible. Suppose we have a two-period economy with T = (0, 1) with one (consumption) good and one financial asset which yields a known, riskless return (say, a bond). Let H be the set of consumers, where, abusing notation, H = H. Each agent hﾎH has an endowment of the consumption good in each of the two periods, eh= e0h, e1h and has preferencesｳhdefined over the amount of the consumption good consumed each period. Thus, an agent can receive a bundle xh= x0h, x1hﾎR2+denoting the amount of consumption good he consumes in period 0 and 1 respectively. As preferencesｳhare defined over R2+, then with enough assumptions, we can define a nice utility function uh: R2+ｮR with all the desirable properties. Let p = p0, p1 denote the prices for the consumer good in periods 0 and 1 respectively (superfluous in this simple one-good example, but we shall maintain them nonetheless).
The consumer can purchase some amount ahof the asset in period t = 0. A unit of the asset can be bought at price q in period t = 0 and will yield a return r in period t = 1 (formally, the return to an asset is defined as the final payoff divided by the initial price, but we shall jump straight into returns here). Thus, the consumer wishes to fulfill the following program:
thus, he chooses a consumption plan xh* = (x0h*, x1h*) and the amount of the asset to purchase ah*. Note that that ahcan be positive or negative: if ah 0, then the agent is buying an asset, in which case he must surrender qahat t = 0 in order to gain rahat t = 1. If ah 0 then he is short-selling an asset, in which case he is receiving qahat t = 0 but must pay rahat t = 1.
Note that this model is basically the same in structure as that of IrvingFishers (1930) famous two-period model – and thus its solution is too well-known to be worth mulling over. Nonetheless, it can be used to obtain a intuitive understanding of the meaning of arbitrage-free prices and asset market equilibrium. The first step is to define an arbitrage opportunity which can be stated in this simple context as follows:
Arbitrage Opportunity I: there is an arbitrage opportunity of the first kind if there is an ahﾎR such that qah｣0 and rah 0.
Arbitrage Opportunity II: there is an arbitrage opportunity of the second kind if there is an ahﾎR such that qah 0 and rahｳ0.
The principle distinction is that arbitrage of the first kind guarantees positive returns with non-positive commitments (think of borrowing and lending at different rates) whereas the second kind guarantees non-negative returns with negative commitments (a free lunch). Thus, in an arbitrage opportunity of the first kind, qah｣0, thus the agent is not buying any asset at time t = 0, thus makes no commitment, or outlays none of his endowment at time t = 0 to purchase assets (although he can sell assets). However, notice that while although he has sacrificed no endowment, he is nonetheless making apositivereturn in the future, rah 0. In an arbitrage opportunity of the second kind, qah 0, he is short-selling the asset and thus has negative commitments, i.e. he is increasing his endowment or purchasing power in period t = 0. However, in this case, rahｳ0, i.e. he makes no future payments and possibly even makes positive gains. Thus an arbitrage opportunity in general can be thought of as a series of trades where an agent commits none of his own endowment and yet makes positive gains along the line. Thus, we can define a no arbitrage or arbitrage free asset price q as follows (using the definition of arbitrage of the first kind):
Arbitrage-free: asset price q is arbitrage free if there is no ahﾎR such that qah｣0 and rah 0.
i.e. there is no portfolio ahsuch that an agent can make a non-positive commitment (qah｣0) that yields a positive return (rah 0).
In Figure 2, we can see the meaning of this quite clearly in a net trade diagram for the hth household. Lett0denote net trades in the initial period (t = 0) andt1denote net trades in the future period (t = 1). Thus, a particular net allocationt= (t0,t1) exchanges some amount of the current consumption for future return or vice-versa, i.e.t0= p0(x0- e0) andt1= p1(x1- e1) (again, we could drop consumer goods prices out if we wished). The origin (t0= 0,t1= 0) represents the situation where the agent consumes his endowment in each period and makes no intertemporal trades (i.e. neither purchases nor sells assets). If the agent purchases merely a single unit of the bond, denoted as 1 in Figure 2, then he must pay q and gets r in the future state, thus his net trades aret0= -q andt1= r. As ahis the number of bonds an agent purchases or short-sells, then if he buys ahbonds, his net trades aret0= -qahandt1= rah. Let us define the following:
Thus, W is a line (or hyperplane) in R2that passes through the point (-q, r) which we have labeled 1 to represent a unit of the bond. As -q and r are given, then the slope of the hyperplane is constant at -r/q and passes through the origin. As noted, if ah 0 then he is buying bonds, so thatt= (-qah, rah) lies on the hyperplane W in the northwest quadrant. If ah 0, then he is short-selling bonds so thatt= (-qah, rah) lies on the hyperplane in the southeast quadrant Finally, if ah= 0 (sotis at the origin), the agent is not doing either. For the moment, there are no upper or lower limits on the amount of assets purchased or short-sold. Notice that in our diagram, as W passes through the origin, then -qah+ rah= 0 for all ahﾎR, thus current commitments (-qah) are equal to future gains (rah). Consequently,anypoint on the hyperplane Worbelow it represents a situation ofno arbitrageas defined above.
How might we represent a situationwitharbitrage opportunities? This is shown in Figure 3. Let us suppose that we have two issuers of bonds, say A and B so that returns on each of the bonds they issue are thesame(rA= rB= r) but their respective purchase prices isdifferent(thus qAｹqB). Letting qA qB, then our diagram changes so we now have two hyperplanes WAand WB(Figure 3). Notice that 1Aand 1Brepresent the purchase of a unit of the bond issued by A and the bond issued by B respectively.
Now, as both WAand WBgo through the origin, it mayseemas both qAand qBare arbitrage free. Individually this is true: we cannot buy and sell a single asset (say asset A) and make arbitrage gains. However, considered together, wedohave an arbitrage opportunity immediately available to an arbitrageur. To see this, suppose an agent short-sells the high-priced bonds (i.e. bond A) by the amount aA( 0) so that he gains -qAaA. He can use part of the proceeds of this short-sale to purchase amount aBof the low-priced bond B, thus outlaying -qBaB. We can see, diagramatically in Figure 3, the consequences of such a procedure: namely, the net gains of the arbitrageur ared0= (-qAaA) – (-qBaB) 0 andd1= r(aB+ aA) 0 and thus net gain vectord=d0+d1lies in thepositiveorthant of R2. Of course, the agent need not limit himself to merely selling aAand purchasing aBbut, as long as the prices qAand qBare fixed and bonds of each type are supplied without end by the market (i.e. perfect competition), then the arbitrageur can do infinite amounts of such operations and thus increase his arbitrage gains to infinity.
What enables this to happen? Namely, there is a non-convexity which is occasioned bydifferentprices for what is effectively the same asset. Let us define Z(A) as the area under the hyperplane WA, thus Z(A) = tﾎR2t0｣-qAah,t1｣rAah, ahﾎR. Similarly, let Z(B) be the area under hyperplane WB, i.e. Z(B) = tﾎR2t0｣-qBah,t1｣rBah, ahﾎR. Then consider Z(A)ﾈZ(B), the union of the areas under hyperplanes WAand WB- which is the shaded area in Figure 3. Thus, we can see immediately thatarbitrage gains are possibleif there is a non-convexity in the set Z(A)ﾈZ(B). Alternatively, there are arbitrage opportunities if the convex hull of this set intersects the strictly positive orthant beyond the origin, i.e. co(Z(A)ﾈZ(B))ﾇR2++ｹﾆ.
Theabsenceof arbitrage opportunities thus requires two things: that no Wiintersects the strictly positive orthant and that there is a law of one price for assets with the same return. The first condition is equivalent to the old Debreuvian condition of no land of Cockaigne inproduction theory(Debreu, 1959: Ch. 3) – namely, you cannot purchase and short-sell a single asset and make a gain in the process, i.e. it cannot be that -qah+ rah 0. However, as Figure 3 illustrates, this is not enough. We also need the second condition — which is interesting because we now can conceive of the link between the formal definition of absence of arbitrage and the better-known law of one price. This law forces all assets with the same riskless return to have the same price, i.e. that if rA= rB, then it must be that qA= qB(and so WA= WB), only then do we return to the no-arbitrage case of Figure 2 – where we can see that the implied convex hull of Z(A)ﾈZ(B) does not intersect the strictly positive orthant.
There is one further thing we can decipher. Consider the no arbitrage case where W passes through the origin. In this case, there is an orthogonal hyperplane W^which also intersects the origin, i.e. W^= yﾎR2yW = 0 and, consequently there is somemﾎW^such thatm= [1,m1], wherem1is to be determined (see Figure 4). But, asmﾎW^, then we know by orthogonality thatmt= 0 for alltﾎW. Consequently, consider the point [-q, r]ﾎW (i.e. when we purchase a unit of the bond). Then, [1,m1]ｷ[-q, r]｢= 0 implies that -q + rm1= 0 or simplym1= q/r. Recall that the bond which costs q pays r units of income in the future, so q/r is the price of a unit of income in the future. Thus,m1represents the discounted future value (i.e. present value) of a unit of future income. As r 0 and q 0, thenm1 0.
Note that when we maximize the earlier optimization problem, if uhis differentiable, quasi-concave and all that, we can collapse both constraints via ahto yield a single constraint that we can place in a Lagrangian:
L = uh(x0, x1) -lh[p0(x0h- e0h) + p1(x1h- e1h)q/r]
wherelhis the Lagrangian multiplier, whose first order condition yields the familiar tangency condition:
which, knowing (p0/p1), r and q, we can then solve for (x0*, x1*), which we can then use to solve for the chosen assets, ah*. For a quick way to decipher what ah* would be, input the indirect utility functionyh(p0,p1, m0h, m1h) where m0h= p0e0his current income and m1h= p1e1h+ rah* is future income. Thus, the Lagrangian can be rewritten as:
so that the future marginal utility of the asset is proportional to the asset price, q. Lettingmh= (1/lh)(ｶyh/ｶm1h), then this becomes:
So ahis chosen untilmh= q/r. But notice thatmhis the marginal utility of the future income for agent hﾎH, weighted by the personal proportionality factor (1/lh), which is the inverse of the marginal utility of current income. Thus,mhis the ratio of the utility of one extra unit of income in the future (ｶyh/dm1h) to the actual utility of an extra unit of income in the present (lh), i.e. it is merely the negative of the slope of the agents indifference curve in the net trade space! As we can see heuristically in Figure 4, ah* is chosen wheremh= q/r implies that if we draw out agent hs indifference map in (t0,t1) space, then we can find the solution at the tangency between hyperplane W (which has slope – q/r) and the highest indifference curve of agent hﾎH (with slope -mh). Furthermore, we can notice that ifmh= q/r for agent hﾎH, then it will be true foranyagent as q and r are given. Thus, we can definem=mh=mh= q/r for all h, h｢ﾎH. It is a simple matter to note that thismand our earliermare thusidentical.
As a final note, an asset market equilibrium is defined if・/font>
hﾎHah= 0, i.e. total demand for the asset by purchasers must equal total supply of it by short-sellers. Thus, if an equilibrium exists, then net trade must net out. For instance, suppose there are only two agents, h, kﾎH. Then, an asset market equilibrium is defined if ah* + ak* = 0, i.e. one agents demand for assets to buy is equal to another agents short-selling of them. The equilibrium price for the asset, q*, will be such that q* =mhr =mkr =m*r and ah* + ak* = 0 so both indifference curves lie tangent to the equilibrium hyperplane W* at diametrically opposed positions. This is shown in Figure 4, where agent h purchases ah* of the asset – thereby conducting net tradesth= (t0h,t1h) which are then picked up by agent k who sells him the same amount ak* = -ah* and thus conducts the converse net trades, i.e.tk= (t0k,t1k) = (-t0h, -t1h). This is a simple version of a financial market equilibrium.