Example: No Gains from Trade and Forced TradingTrading

e^{−Rtxλs2(µ(τ))dτ}[δ+ζ(1−^πho(x))^∆V_o^s(0)−^ζ(1−^πhn(x))^∆V_n^s(0)] dx≥^0.

(6.10)

Gains from Trade a Long Time after the Shock

The constraint a long time after an aggregate liquidity shock, given no additional shock has occurred, i.e. lim

t→^∞(^∆V_o^s(t)−^∆Vn^s(t))≥^{0, is}

δ ≥^ζ[(1−^πhn(ss))^∆V_n^s(0)−(1−^πho(ss))^∆V_o^s(0)]. (6.11) The analysis of inequalities (6.5), (6.10), and (6.11) is deferred to appendix 6C.

This analysis shows that, in general, it is the function π_hn(t) which provokes a market freeze. In the aggregate liquidity shock model, however, Duffie, Gâr-leanu, and Pedersen (2007) do not take this market freeze into account. Agents are forced to trade despite no gains from trade. I show the effect and related impacts of a forced trading in the following section by means of an example.

6.4 Example: No Gains from Trade and Forced Trading

The existence of a bargaining solution is best highlighted by a numerical example.

To analyze the driving forces, I continue the example of chapter 5. The time it takes for the price to recover from an aggregate liquidity shock is influenced by several factors, as I have stated in section 5.2:

(1.) Search frictions, determined byλandρ, which measure liquidity,

(2.) agents’ individual recovery time, denoted byλ_u,

(3.) the severity of the shock, determined byπ_ho(ss)andπ_hn(ss).

The interaction of all three parameters—condensed within π_ho(t) and π_hn(t)— combined particularly with the possibility of further shocks—expressed withζ—

determines whether the market in this model recovers or whether there are no gains from trade. These effects are analyzed in the following passage.

Meeting Intensities λ and ρ

The meeting intensities λ and ρ are the input parameters initiating a situation with no gains from trade in the aggregate liquidity shock model.

25 0 75 50

125 100

200 300 400 500 600 0

0.5 1

λ ρ

∆V

s o−∆V

s n≥0

Figure 6.1:Region for gains from trade forλ∈[200, 600]andρ∈ [0, 125].

Figure 6.1 plots the observation of gains from trade for various combinations of λand ρ. If∆V_o^s(t)−^∆Vn^s(t) ≥0 is valid for allt≥0, the value 1 is assigned. The value 1 reflects a well-defined bargaining problem and gains from trade. Other-wise, the value 0 is set, i.e. no gains from trade. Figure 6.1 shows that a bargaining solution does not exist for high meeting intensitiesλandρ. There is a sharp edge where an increase inρ must come along with a decrease inλ, so that the condi-tion (6.1) is met further on. Under fairly normal condicondi-tions, increasing meeting intensities reduces search frictions and therefore reduces illiquidity. But within the aggregate liquidity shock model, high meeting intensities might result in no gains from trade. No gains from trade should cause a market freeze with no trading at all, i.e. infinite search frictions.

The shape over time of the difference ∆V_o^s(t)−^∆V_n^s(t) is shown in figure 6.2.

I keep market makers’ meeting intensity constant and vary investors’ meeting

intensity with λ= [200, 600]. The left panel is without market makers, i.e. it is equal to the example of Duffie, Gârleanu, and Pedersen (2007) withρ=0. The right panel is with market makers, i.e. ρ=125. This figure demonstrates the presumption of the aggregate liquidity shock model that all meetings result in a trade—even if there are no gains from trade, i.e. agents are forced to trade. This forced trading is reflected by values below the black plane. However, the market should be frozen during this time—which is not considered within this model.

From an economic perspective, this aggregate liquidity shock model is flawed.

ρ=0 ρ=125

Figure 6.2:Difference∆V_o^s(t)−^∆Vn^s(t)for varyingλ∈[200, 600].

Figure 6.2 shows that situations of no gains from trade but forced trading pre-vail mainly during the time shortly after the shock, i.e. when the selling pressure is large, which is the case for 0≤^t ≤^t∗. If agents are forced to trade despite no gains, the market would endogenously return to normal conditions, i.e. with gains from trade, for sufficiently small values forζ. Equation (6.11) shows this effect, too.

Individual Recovery

If agents know that low intrinsic types do not have to stay in this unfavorable low state for a long time—that meansλu is relatively high—then the market re-covers quickly from an aggregate liquidity shock. Hence, a forced trading in the aggregate liquidity shock model can be avoided if investors have access to easy refunding conditions.

Severity of the Shock and the Risk of Further Shocks

The matching functions presume that all meetings actually result in a trade.

These matching functions are contained in the flow equations of masses µ_σ(t),

which affect the evolution of probabilitiesπ_ho(t)andπ_hn(t). With these probabil-ities, however, the termζ(1−^πhn(t))^∆V_n^s(0)of inequality (6.10) can temporarily outweigh the termζ(1−^πho(t))^∆V_o^s(0). No gains from trade result.

The third crucial parameter I have announced above is the severity of a shock, represented by the probabilities of high agents switching to a low state due to the shock and based on steady state values, i.e.π_ho(ss)andπ_hn(ss). These probabili-ties control for the starting condition of agents’ massesµ_σ(0), as seen in equations (4.3) and (4.4), These starting conditions determine the evolution of masses over time. The probabilitiesπ_ho(t)andπ_hn(t)in turn depend both on the starting con-ditions of agents’ massesµ_σ(0)and on the type distribution µ_σ(t) at timet ≥^0, as seen by equations (4.1) and (4.2).

Figure 6.3 shows the probability for a high owner switching to a low owner upon a shock, i.e. π_ho(t) as a function of time and meeting intensity λ∈ [200, 600]. This figure illustrates that the meeting intensity λ has an inferior effect on the evolution ofπ_ho(t). In general, the type distribution of high ownersµ_ho(t) mono-tonically increases after a shock, i.e. µ_ho(0) ≤^µho(t). This effect can be seen in figure 5.5. The increase in the fraction ofho agents is due to an elevated quantity of misallocated assets right after the shock. This misallocation alleviates over time. The change in the fraction of high owners is mainly due to trade and up-shifts, while down-shifts are a secondary influence. Generally, the condition 0≤^πho(t)≤^πho(ss)holds.

Figure 6.3:Evolution ofπho(t)withπho(ss) =0.5 andρ=125.

Figure 6.4 shows the probability for a high owner switching to a low non-owner, i.e.π_hn(t)as a function of time and meeting intensityλ∈ [200, 600]. The pattern of the probabilityπ_hn(t)differs significantly from the pattern of the

prob-Figure 6.4:Evolution ofπ_hn(t)withπ_hn(ss) =0.5 andρ =125.

ability π_ho(t). For high values of λ (and ρ), the value π_hn(t) drops sharply be-low zero shortly after the shock. As a result, π_hn(t) loses its characteristic as a probability. The reason is as follows: The fraction of potential buyers µ_hn(t) further decreases immediately after a shock and the market becomes one-sided.

The higher π_ho(ss) and λare and the lower π_hn(ss) is, the more severe and the faster the reduction of potential buyers after the shock. This reduction is due to an immediate absorption of nearly all remaining potential buyers in this buy-ers’ market, when search frictions are low, i.e. meeting intensities are high—

and when all meetings actually result in a trade, even by force. The matching functions Mλ(t) = 2λµ_hn(t)µ_lo(t) and Mρ(t) =ρmin{^µlo(t),µ_hn(t)} ^precisely assume this ensured trade. With this, the fraction µ_hn(t) temporarily drops al-most to zero and leads to 0≥^πhn(t). Thereafter,µ_hn(t) recovers and moves to-wards its long run level, given no further shock occurs. Due to this fact, the term ζ(1−^πhn(t))^∆V_n^s(0) can temporarily outweigh the term ζ(1−^πho(t))^∆V_o^s(0) so that inequality (6.10) can become invalid for high meeting intensities λ (and ρ).

The next section presents some ideas for preventing the aggregate liquidity shock model from forcing agents to trade despite no gains from trade.