Efficiency

This section considers the efficiency of dynamic trading with a shared signal and hence the value of this additional information for the traders. Note that, with positive but not necessarily equal-sized gains from trade in low quality and in high quality, λ > 0 and 1−λ >0 respectively, the total surplus depends on the length of time it takes to trade in different qualities. Since the traders are paired each period, the market is efficient if every

17The average quality is ”high”; we have not got a lemons market for the high costc_h=λis equal to the low utilityu_l=λ.

18This result is robust to change in the order of the moves: it arises whether we have a signaling game or a screening game.

match results in trade; inefficiency is manifested in decreased liquidity. The standard welfare measure for stationary equilibria is the weighted sum of the values to a cohort of buyers and sellers who enters the market at any given point in timet∈Z¹⁹

S :=V_b+_hV_h+_lV_l≤_h(1−λ) +_lλ.

Now, one of our most striking findings is the non-existence of efficient equilibria that obtains when the shared signal is added to the game form; otherwise, there does exist a continuum of efficient equilibria for the payoffs we consider. The reason is bifold: First, it is not possible to trade everything at once, as required by efficiency, without sharing any of the surplus with the buyers or with the high quality sellers. In dynamic markets, one of these groups is hence bound to have a positive search option. Second, what the buyers or what the high quality sellers thus make would have to vary according to the shared signal; the lower the signal, the less they can expect to get. For a low enough signal realization, they should expect to make less if they trade than if they execute their positive search option. Consequently, there is no trade for a low enough signal. This is so irrespective of who is making the price offer as long as the buyer and the seller have this piece of correlated information:

Proposition 1 Consider any stationary or non-stationary²⁰equilibrium.

1. IfV_h= 0, thenV_b>0. IfV_b= 0, thenV_h>0.

2. Suppose V_b>0. Then, ∃s:∀s < s:E(u|s)−p < δV_beven for (the minimal price the high quality sellers can sell for) p=c_h and the buyers cannot trade for these prices.

3. Suppose V_h>0. Then,∃s:∀s < s:p−c_h< δV_h even for (the maximal price the buyer can buy for)p=E(u|s)and the sellers of high quality cannot trade for these prices.

Corollary 1 Any equilibrium is inefficient.

Proof. Note that, after a signals∈[0,1] is viewed but without any further revelation as that would cost, the maximal price a buyer is willing to accept is E_γ(u|s)−δV_b (to compensate the buyer for the loss of the search option) and the minimal price a seller of high quality would be willing to offer isc_h+δV_h (to compensate the seller for the cost and the loss of the search option). Therefore, to guarantee that there would exist such a pricep(s)∈[c_h+δV_h, E_γ(u|s)−δV_b], for almost alls, even for the case in whichE_γ(u|s) is close to c_h it must be that (i) the search option for the buyers δV_b = 0 and, thus,

19Note that, although it is standard and oft-used, the measure ignores the surplus related to the potential transition period needed to reach the stationary equilibrium.

20For simplicity, we omit the time indexes.

p(s) =E_γ(u|s), for almost alls, as the use of any price below it would raiseV_bover zero and (ii) the search option for the high quality sellersδV_h = 0 and, thus, p(s) = c_h, for almost alls, as the use of any price above it would raiseV_hover zero. But (i) and (ii) are clearly incompatible becausec_h < E_γ(u|s) for alls∈(0,1). Observe also that, if we are to trade all goods in the first match, we cannot change the buyers’ beliefs by signaling or screening from where they are taken by the signal,E_γ(u|s), because that would necessarily create some waste and delay in trade. Hence, there exists no equilibrium where everything is traded in the first match. Any equilibrium is inefficient.

Since it is impossible to trade all goods in the first match without giving a fraction of the rents to the buyers or to the high quality sellers, it is also impossible to trade all goods for the lowest signals, which would give them almost no rents. With a positive search option, it is better to wait for higher signals than to trade for the lowest signals.

Someone always becomes too picky to trade for whatever the signal. Thus, some rationing must occur for the lowest signals. We do not even have to specify which party becomes too picky to trade; that is likely to depend on the particularities of the game form. It suffices to know that it is impossible to maintain mutually beneficial trade for all the signals over time. Some rents must to be payed in terms of delay in trade to shroud the signal information when it is unfavorable. This is noteworthy as the primitives of this game are such that all qualities could be traded efficiently in a static Walrasian market;

it would not be a lemons market:

Remark 1 Consider a static Walrasian market in which the fraction γ of the sellers has a high quality good and the fraction1−γ of the sellers has a low quality good. Then, there exist a continuum of efficient Walrasian equilibriap∈[λ, E_γ(u)]where every buyer and every seller trades instantaneously. The buyer value isV_b=E_γ(u)−pand the seller values areV_h=p−λandV_l=p.²¹

Also, with either fully asymmetric or fully symmetric information, all the gains could be realized:

Lemma 1 Consider a market as described in Section 2.2 but where the shared signal is white noise. Then, there exist a continuum of efficient equilibriap∈[λ, E_γ(u)] in which everything is traded in the first match. The buyer value is V_b=E_γ(u)−p≥0and the seller values areV_h=p−λ≥0andV_l=p >0.

Lemma 2 Consider a market as described in Section 2.2 but where the shared signal is perfectly revealing. Then, there exist a unique of efficient equilibriump_h = 1andp_l=λ in which everything is traded in the first match. The buyer value isV_b= 0and the seller

21To specify, any price below the average quality could be the Walrasian equilibrium, the prices above high cost sustain only low quality trade, the prices below high cost sustain trade in both low and high quality. In other words, there would exist a continuum of efficient Walrasian equilibria and a continuum of inefficient Walrasian equilibria.

values areV_h= 1−λandV_l=λ.

Corollary 2 A shared signal can be welfare-reducing.