The Basic Model
3.3 Equilibrium Prices
3.3.2 Bargaining over the Price
Until now, not much has been said about the structure of the price, except that (i) trade occurs only betweenlo-type investors who want to sell andhn-type in-vestors who want to buy, and (ii) they start bargaining over the price as soon as they meet each other.71
It is clear from equation (3.25) that anlo-type investor will only accept a price that is higher than or equal to∆Vl(t) = Vlo(t)−Vln(t), i.e.Vln(t) +P(t)−Vlo(t) ≥0.
This means that the price has to compensate the investor for the change in his utility when selling the asset. Otherwise it would be advantageous for him to keep the asset. ∆Vl(t) is the reservation value or participation constraint of a potential seller. At this point, a potential seller is indifferent to trading.
Equation (3.26) shows that an hn-type investor will only purchase an asset for a price which is lower than or equal to ∆Vh(t) = Vho(t)−Vhn(t), that is Vho(t)−P(t)−Vhn(t)≥0. Otherwise he would be better off without the asset.
∆Vh(t)is the reservation value or participation constraint of a potential buyer.
If these two types of investors, low owner and high non-owner, start bargaining over the price, the result is located somewhere between these two reservation values∆Vl(t)and∆Vh(t).
It is assumed that all agents have complete and symmetric information, and, under this assumption, bargaining theory states that trade happens instantly.
Assume further that anlo-type investor (potential seller) has bargaining power q ∈[0, 1]. The generalized Nash solution, defined in equation (2.15), applies with
fqP(t) =arg max
P(t)
[Vln(t) +P(t)−Vlo(t)]q [Vho(t)−P(t)−Vhn(t)]1−q, (3.33)
subject to (s. t.)
0≤Vln(t) +P(t)−Vlo(t), (3.34)
0≤Vho(t)−P(t)−Vhn(t), (3.35)
71 An introduction into bargaining theory is given in section 2.2.
for all t.72 The agreement point for a potential seller (lo-type) is Vln(t) +P(t), which is the value of a seller if an agreement is reached. The agreement point for a potential buyer (ln-type) is Vho(t)−P(t), which is the value of a buyer if an agreement is reached. In absence of an agreement, the seller keeps the asset and stays with the valueVlo(t), which is his disagreement or threat point. The situation is analogous for the buyer: He remains without an asset and stays with the valueVhn(t) if they fail to reach an agreement. The bargaining situation of buyers and sellers depends on their outside options, which are equal to their threat points in this setting.73 These outside options depend on the availability of suitable counterparties over time,74 since outside options are the expected value of waiting for a new trading partner—unless an intrinsic type switch occurs in the meantime.
The maximization of the argument in (3.33) is carried out over the priceP(t)and is subject to bothVln(t) +P(t) ≥Vlo(t) and Vho(t)−P(t) ≥Vhn(t). When there are gains from trade, the first order condition is
0 =q[Vln(t) +P(t)−Vlo(t)]q−1[Vho(t)−P(t)−Vhn(t)]1−q
−[Vln(t) +P(t)−Vlo(t)]q(1−q)[Vho(t)−P(t)−Vhn(t)](−q). As a result, the price function is
P(t) = (1−q) [Vlo(t)−Vln(t)] +q[Vho(t)−Vhn(t)], (3.36) s. t. Vlo(t)−Vln(t)≤ P(t) ≤Vho(t)−Vhn(t). (3.37) An intuitive interpretation for inequality (3.37) is that the reservation value of a buyer should be higher than a seller’s reservation value. The buyer is of high intrinsic type, whereas the seller is a low-type. Because the buyer has no holding costs, the flow of dividends has a higher value for the buyer than for the seller.
When the buyer switches to a low valuation type some day, he has the same trading possibilities—in steady state—as the seller had at that time. As a result, trade should be efficient.75
72 The alternating-offer bargaining model of Rubinstein (1982) (see section 2.2.3) leads to a com-parable result. The probability of the seller making the first offer equals the bargaining power qif there is a positive probability of a breakdown while waiting for a counteroffer. See Duffie, Gârleanu, and Pedersen (2007), pp. 1871 and the model of Rubinstein and Wolinsky (1985).
73 See Binmore, Shaked, and Sutton (1989) for a discussion about disagreement points and outside options.
74 See Duffie, Gârleanu, and Pedersen (2005), p. 1820.
75 See Vayanos and Wang (2007), p. 73.
When investors and market makers bargain over the price, bid or ask prices re-sult. An investor sells to the market maker at the bid price B(t) and buys from the market maker at the ask price A(t), with A(t) ≥B(t). Both prices can be calculated analogously to the bargaining between two investors. Bid and ask prices depend both on buyers’ and sellers’ outside options, i.e. the availability of suitable counterparties over time. Market makers can trade the asset in the inter-dealer market, so that their outside option is to trade at the interinter-dealer priceM(t). The bargaining power of market makers is defined withz∈ [0, 1]. To specify the ask price, the generalized Nash solution is
fzA(t) = arg max
A(t)
[A(t)−M(t)]z [Vho(t)−A(t)−Vhn(t)]1−z, (3.38)
subject to
0≤ A(t)−M(t), (3.39)
0≤Vho(t)−A(t)−Vhn(t). (3.40) The generalized Nash solution for the bid price is
fzB(t) = arg max
B(t)
[M(t)−B(t)]z [Vln(t) +B(t)−Vlo(t)]1−z, (3.41)
subject to
0≤ M(t)−B(t), (3.42)
0≤Vln(t) +B(t)−Vlo(t). (3.43) Maximization of the argument in (3.38) over A(t)and in (3.41) over B(t)results in the following askA(t)and bidB(t)prices, where
A(t) = (1−z)M(t) +z[Vho(t)−Vhn(t)], (3.44) s. t. M(t) ≤ A(t) ≤Vho(t)−Vhn(t), (3.45) and
B(t) = (1−z)M(t) +z[Vlo(t)−Vln(t)], (3.46) s. t. Vlo(t)−Vln(t)≤B(t) ≤M(t). (3.47) Equilibrium requires that supply and demand of the asset are balanced. This
con-dition affects the interdealer price M(t): It depends on whether market makers meet an equal number of buyers and sellers (µlo(t) = µhn(t)), or whether there is an imbalance between potential sellers and potential buyers (µlo(t) ≶µhn(t)).
In the case thatµlo(t)<µhn(t), more potential buyers than potential sellers meet market makers. Not all potential buyers have the possibility to buy an asset. Since not all potential buyers are able to trade, market makers and buyers must be in-different to trading. The interdealer priceM(t)must be equal to buyers’ reserva-tion valueVho(t)−Vhn(t)and is therefore set equal to the ask priceM(t) = A(t). In the caseµlo(t)>µhn(t), more potential sellers than potential buyers meet mar-ket makers. Hence, marmar-ket makers and potential sellers must be indifferent to trading. The interdealer price M(t) must be equal to sellers’ reservation value Vlo(t)−Vln(t), implyingM(t) = B(t).
In the rare event that the buy and sell side is balanced, i.e. µlo =µhn, the inter-dealer price is located somewhere between the bid and ask price.
As a result, the general interdealer price can be stated as follows:
M(t) = (1−eq(t)) [Vlo(t)−Vln(t)] +eq(t) [Vho(t)−Vhn(t)], (3.48) with
eq(t)
=1 ifµlo(t) <µhn(t)
=0 ifµlo(t) >µhn(t)
∈[0, 1] ifµlo(t) = µhn(t),
(3.49)
where the caseqe(t) ∈[0, 1] ifµlo(t) = µhn(t) denotes that, initially,qe(t)can arbi-trarily be chosen from[0, 1], but is then a constant for all casesµlo(t) =µhn(t).76