One of the central features of sequence economies are 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. *Buying*
an asset implies we are transferring purchasing power from the present to some future
state; *selling* or *short-selling* an 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 and Ross, 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 price *simultaneously*. This
timing element is important: namely, because of simultaneity, arbitrageurs require *no*
outlay 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 carries *no* risk 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
of *pure* arbitrage. 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
there *were* arbitrage 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 agents *can* replicate this arbitrage
portfolio *infinitely* with 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 for

anycompetitive 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, e^{h} = {e_{0}^{h},
e_{1}^{h}} and has preferences ｳ _{h}
defined over the amount of the consumption good consumed each period. Thus, an agent can
receive a bundle x^{h} = {x_{0}^{h}, x_{1}^{h}} ﾎ R^{2}_{+} denoting the amount of consumption good
he consumes in period 0 and 1 respectively. As preferences ｳ _{h}
are defined over R^{2}_{+}, then with enough assumptions, we can define a
nice utility function u^{h}: R^{2}_{+} ｮ
R with all the desirable properties. Let p = {p_{0}, p_{1}} 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 a^{h} of 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:

max u

^{h}(x^{h})

s.t.

p

_{0}x_{0}^{h}+ qa^{h}｣ p_{0}e_{0}^{h}

p

_{1}x_{1}^{h}｣ p_{1}e_{1}^{h}+ ra^{h}

thus, he chooses a consumption plan x^{h}* = (x_{0}^{h}*,
x_{1}^{h}*) and the amount of the asset to purchase a^{h}*. Note
that that a^{h} can be positive or negative: if a^{h} > 0, then the
agent is buying an asset, in which case he must surrender qa^{h} at t = 0 in order
to gain ra^{h} at t = 1. If a^{h} < 0 then he is
"short-selling" an asset, in which case he is receiving qa^{h} at t = 0
but must pay ra^{h} at t = 1.

Note that this model is basically the same in structure as that of Irving Fisher's (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 a^{h ﾎ }R such that qa^{h}｣ 0 and ra^{h}> 0.

Arbitrage Opportunity II: there is an "arbitrage opportunity of the second kind" if there is an a^{h}ﾎ R such that qa^{h}< 0 and ra^{h}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, qa^{h} ｣ 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 a *positive*
return in the future, ra^{h} > 0. In an arbitrage opportunity of the second
kind, qa^{h} < 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, ra^{h} 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 a^{h}ﾎ R such that qa^{h}｣ 0 and ra^{h}> 0.

i.e. there is no portfolio a^{h} such that an agent can make a
non-positive commitment (qa^{h} ｣ 0) that yields a
positive return (ra^{h} > 0).

In Figure 2, we can see the meaning of this quite clearly in a net trade
diagram for the hth household. Let t _{0} denote net
trades in the initial period (t = 0) and t _{1} denote
net trades in the future period (t = 1). Thus, a particular net allocation t = (t _{0}, t
_{1}) exchanges some amount of the current consumption for future return or
vice-versa, i.e. t _{0} = p_{0}(x_{0} -
e_{0}) and t _{1} = p_{1}(x_{1}
- e_{1}) (again, we could drop consumer goods prices out if we wished). The origin
(t _{0} = 0, t _{1}
= 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 are t _{0}
= -q and t _{1} = r. As a^{h} is the number of
bonds an agent purchases or short-sells, then if he buys a^{h} bonds, his net
trades are t _{0} = -qa^{h} and t _{1} = ra^{h}. Let us define the following:

W = {(t

_{0}, t_{1}) ﾎ R^{2 }| t_{0}= - qa^{h}, t_{1}= ra^{h}, a^{h}ﾎ R}

Thus, W is a line (or hyperplane) in R^{2} that 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 a^{h} > 0 then he is buying bonds, so that t
= (-qa^{h}, ra^{h}) lies on the hyperplane W in the northwest quadrant. If
a^{h} < 0, then he is short-selling bonds so that t
= (-qa^{h}, ra^{h}) lies on the hyperplane in the southeast quadrant
Finally, if a^{h} = 0 (so t is 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 -qa^{h}+ ra^{h }= 0 for all a^{h ﾎ
}R, thus current commitments (-qa^{h}) are equal to future gains (ra^{h}).
Consequently, *any* point on the hyperplane W *or* below it represents a
situation of *no arbitrage* as defined above.

Figure 2 -No Arbitrage Opportunities

How might we represent a situation *with* arbitrage 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 the *same* (r_{A} = r_{B}
= r) but their respective purchase prices is *different* (thus q_{A} ｹ q_{B}). Letting q_{A} > q_{B}, then our
diagram changes so we now have two hyperplanes W_{A} and W_{B }(Figure 3).
Notice that 1_{A} and 1_{B} represent the purchase of a unit of the bond
issued by A and the bond issued by B respectively.

Now, as both W_{A} and W_{B} go through the origin, it may
*seem* as both q_{A} and q_{B} are "arbitrage free".
Individually this is true: we cannot buy and sell a single asset (say asset A) and make
arbitrage gains. However, considered together, we *do* have 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 a_{A} (< 0) so that he gains -q_{A}a_{A}.
He can use part of the proceeds of this short-sale to purchase amount a_{B} of the
low-priced bond B, thus outlaying -q_{B}a_{B}. We can see, diagramatically
in Figure 3, the consequences of such a procedure: namely, the "net gains" of
the arbitrageur are d _{0} = (-q_{A}a_{A})
- (-q_{B}a_{B}) > 0 and d _{1} = r(a_{B}
+ a_{A}) > 0 and thus net gain vector d = d _{0} + d _{1} lies in
the *positive* orthant of R^{2}. Of course, the agent need not limit himself
to merely selling a_{A} and purchasing a_{B} but, as long as the prices q_{A}
and q_{B} are 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.

Figure 3 -Arbitrage Opportunities

What enables this to happen? Namely, there is a "non-convexity"
which is occasioned by *different* prices for what is effectively the same asset. Let
us define Z(A) as the area under the hyperplane W_{A}, thus Z(A) = {t ﾎ R^{2 }| t
_{0} ｣ -q_{A}a^{h}, t _{1} ｣ r_{A}a^{h}
, a^{h} ﾎ R}. Similarly, let Z(B) be the area under
hyperplane W_{B}, i.e. Z(B) = {t ﾎ
R^{2 }| t _{0} ｣ -q_{B}a^{h},
t _{1} ｣ r_{B}a^{h}
, a^{h} ﾎ R}. Then consider Z(A) ﾈ
Z(B), the union of the areas under hyperplanes W_{A} and W_{B} - which is
the shaded area in Figure 3. Thus, we can see immediately that *arbitrage gains are
possible* if 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 R^{2}_{++} ｹ ﾆ .

The *absence* of arbitrage opportunities thus requires two things:
that no W_{i} intersects 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" in production 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 -qa^{h} + ra^{h} > 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 r_{A}
= r_{B}, then it must be that q_{A} = q_{B} (and so W_{A}
= W_{B}), 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 R^{2} | yW = 0}
and, consequently there is some m ﾎ
W^ such that m = [1, m _{1}], where m _{1} is
to be determined (see Figure 4). But, as m ﾎ
W^ , then we know by orthogonality that m
t = 0 for all t ﾎ
W. Consequently, consider the point [-q, r] ﾎ W (i.e. when we
purchase a unit of the bond). Then, [1, m _{1}]ｷ[-q,
r]｢ = 0 implies that -q + rm _{1}
= 0 or simply m _{1} = 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, m _{1} represents the discounted
future value (i.e. present value) of a unit of future income. As r > 0 and q > 0,
then m _{1} > 0.

Figure 4- Asset Market Equilibrium

Note that when we maximize the earlier optimization problem, if u^{h}
is differentiable, quasi-concave and all that, we can collapse both constraints via a^{h}
to yield a single constraint that we can place in a Lagrangian:

L = u

^{h}(x_{0}, x_{1}) - l^{h}[p_{0}(x_{0}^{h}- e_{0}^{h}) + p_{1}(x_{1}^{h}- e_{1}^{h})q/r]

where l ^{h} is the Lagrangian
multiplier, whose first order condition yields the familiar tangency condition:

(ｶ u/ｶ x

_{0})/(ｶ u/ｶ x_{1}) = (p_{0}/p_{1})ｷ(r/q)

which, knowing (p_{0}/p_{1}), r and q, we can then solve
for (x_{0}*, x_{1}*), which we can then use to solve for the chosen
assets, a^{h}*. For a quick way to decipher what a^{h}* would be, input
the indirect utility function y ^{h}(p_{0, }p_{1},
m_{0}^{h}, m_{1}^{h}) where m_{0}^{h} = p_{0}e_{0}^{h}
is current income and m_{1}^{h} = p_{1}e_{1}^{h} +
ra^{h}* is future income. Thus, the Lagrangian can be rewritten as:

L = y

^{h}(p_{0, }p_{1}, m_{0}^{h}, m_{1}^{h}) - l^{h}[qa^{h}]

which yields the first order condition:

dL/da

^{h}= (ｶ y^{h}/ｶ m_{1}^{h})(ｶ m_{1}^{h}/ｶ a^{h}) - l^{h}q = 0

or, as ｶ m_{1}^{h}/ｶ a^{h} = r, we can rewrite this as:

q = (1/l

^{h})(ｶ y^{h}/ｶ m_{1}^{h})r

so that the future marginal utility of the asset is proportional to the
asset price, q. Letting m ^{h} = (1^{ }/l ^{h})(ｶ y
^{h}/ｶ m_{1}^{h}), then this becomes:

q = m

^{h}r

So a^{h} is chosen until m ^{h}
= q/r. But notice that m ^{h} is the marginal utility
of the future income for agent h ﾎ H, weighted by the personal
proportionality factor (1/l ^{h}), which is the inverse
of the marginal utility of current income. Thus, m ^{h}
is the ratio of the utility of one extra unit of income in the future (ｶ
y ^{h}/dm_{1}^{h}) to the actual
utility of an extra unit of income in the present (l ^{h}),
i.e. it is merely the negative of the slope of the agent's indifference curve in the net
trade space! As we can see heuristically in Figure 4, a^{h}* is chosen where m ^{h} = q/r implies that if we draw out agent h's
indifference map in (t _{0}, t
_{1}) 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 - m ^{h}). Furthermore, we can notice
that if m ^{h} = q/r for agent h ﾎ
H, then it will be true for *any* agent as q and r are given. Thus, we can define m = m ^{h} = m
^{h'} = q/r for all h, h｢ ﾎ
H. It is a simple matter to note that this m and our earlier m are thus *identical*.

As a final note, an asset market "equilibrium" is defined if ・/font> _{hﾎ H}a^{h} = 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 a^{h}* + a^{k}* = 0, i.e. one agent's
demand for assets to buy is equal to another agent's short-selling of them. The
"equilibrium" price for the asset, q*, will be such that q* = m ^{h}r = m ^{k}r = m *r and a^{h}* + a^{k}* = 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 a^{h}* of the asset - thereby
conducting net trades t ^{h} = (t
_{0}^{h}_{ }, t _{1}^{h})
which are then picked up by agent k who sells him the same amount a^{k}* = -a^{h}*
and thus conducts the converse net trades, i.e. t ^{k}
= (t _{0}^{k}, t _{1}^{k})
= (-t _{0}^{h}, -t _{1}^{h}).
This is a simple version of a financial market equilibrium.

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