An optimistic search equilibrium

Abstract

We study a market search equilibrium with aggregate uncertainty, private information and heterogeneous beliefs that are initially optimistic. Despite these biased beliefs, it is shown that all optimistic equilibria converge to perfect competition in the limit as the time between matches tends to 0.

Appendix

Proof of Proposition 1

Equations (18) and (19) imply that in a full trade equilibrium, the stock of pessimistic buyers is equal to the stock of sellers,

$$\begin{aligned} B\left( H|H\right) =S\left( H\right) . \end{aligned}$$

(39)

Equations (21) and (22) imply

$$\begin{aligned} B^{0}\left( L|H\right)&=\frac{G_{B}\left( \underline{v}\left( H\right) |H\right) -G_{B}\left( \underline{v}\left( L\right) |H\right) }{1-G_{B}\left( \underline{v}\left( H\right) ,H\right) }B^{1}\left( L|H\right) \nonumber \\&=\beta \cdot B^{1}\left( L|H\right) \end{aligned}$$

(40)

where

$$\begin{aligned} \beta \equiv \frac{G_{B}\left( \underline{v}\left( H\right) |H\right) -G_{B}\left( \underline{v}\left( L\right) |H\right) }{1-G_{B}\left( \underline{v}\left( H\right) |H\right) }>0. \end{aligned}$$

Equation (23) is equivalent to

$$\begin{aligned} \left( B^{1}\left( L|H\right) +B^{0}\left( L|H\right) \right) B^{1}\left( L|H\right) =B\left( H|H\right) ^{2}, \end{aligned}$$

which upon the substitution of (40) for $B^{0}\left( L|H\right)$ can be solved for $B\left( H|H\right)$,

$$\begin{aligned} B\left( H|H\right) =\left( 1+\beta \right) ^{1/2}B^{1}\left( L|H\right) . \end{aligned}$$

(41)

Substituting (40) and (41) into (21) gives us the solution for $B^{1}\left( L|H\right)$ and

$$\begin{aligned} B^{1}\left( L|H\right) =\tau \cdot \left[ 1-G_{B}\left( \underline{v}\left( H\right) ,H\right) \right] \frac{1+\beta +\left( 1+\beta \right) ^{1/2}}{1+\beta }, \end{aligned}$$

(42)

and the other stocks $B^{0}\left( L|H\right)$, $B\left( H|H\right)$ are then determined from (40) and (41). The probability of meeting a pessimistic buyer is

$$\begin{aligned} \theta _{B}\left( H|H\right)&=\frac{B\left( H|H\right) }{B\left( H\right) }\\&=\frac{\left( 1+\beta \right) ^{1/2}}{1+\beta +\left( 1+\beta \right) ^{1/2}}\\&=\left( 1+\sqrt{\frac{1-G_{B}\left( \underline{v}\left( L\right) |H\right) }{1-G_{B}\left( \underline{v}\left( H\right) |H\right) }}\right) ^{-1}. \end{aligned}$$

The entry equation, say (19) in $\mu =H$ is then equivalent to

$$\begin{aligned} \frac{1}{2}\left( \underline{v}\left( H\right) -\bar{c}\left( H\right) \right) =\tau \cdot \kappa \left( 1+\sqrt{\frac{1-G_{B}\left( \underline{v}\left( L\right) |H\right) }{1-G_{B}\left( \underline{v}\left( H\right) |H\right) }}\right) . \end{aligned}$$

(43)

For $\mu =L$ we obtain in parallel

$$\begin{aligned} \frac{1}{2}\left( \underline{v}\left( L\right) -\bar{c}\left( L\right) \right) =\tau \cdot \kappa \left( 1+\sqrt{\frac{G_{S}\left( \bar{c}\left( H\right) |L\right) }{G_{S}\left( \bar{c}\left( L\right) |L\right) }}\right) . \end{aligned}$$

(44)

The marginal types must also satisfy the mass balance conditions (20) and (25), and for $\mu \in \left\{ H,L\right\}$,

$$\begin{aligned} p_{W}\left( \mu \right) \in \left[ \underline{v}\left( \mu \right) ,\bar{c}\left( \mu \right) \right] . \end{aligned}$$

(45)

Equations (43) and (44), together with the mass balance conditions (20) and (25), form a system of four equations for 4 unknowns, now denoted as $\left( \underline{v}_{\tau }\left( H\right) ,\bar{c}_{\tau }\left( H\right) ,\right. \left. \underline{v}_{\tau }\left( L\right) ,\bar{c}_{\tau }\left( L\right) \right)$. For $\tau =0$, these equations imply

$$\begin{aligned} \underline{v}_{0}\left( \mu \right) =\bar{c}_{0}\left( \mu \right) =p_{W}\left( \mu \right) \text {.} \end{aligned}$$

The Implicit Function Theorem implies that $\bar{\tau }>0$ exists such that a solution exists for all $\tau \in [0,\bar{\tau }]$ provided the Jacobian of this system is nonzero. Moreover, as $\tau \rightarrow 0$, the marginal types converge to the corresponding Walrasian prices, $\underline{v}(H),\bar{c}(H)\rightarrow p_{W}\left( H\right)$ and $\underline{v}(L),\bar{c}(L)\rightarrow p_{W}\left( L\right)$.

To evaluate the Jacobian, it is convenient to reduce this system by eliminating $\bar{c}\left( \mu \right)$ from Eqs. (20) and (25):

$$\begin{aligned} \bar{c}\left( \mu \right)&=G_{S}^{-1}\left( 1-G_{B}\left( \underline{v}\left( \mu \right) |\mu \right) |\mu \right) \\&\equiv \phi \left( \underline{v}\left( \mu \right) |\mu \right) ,\end{aligned}$$

where the mapping $\phi \left( \cdot |\mu \right) :\left[ p_{W}\left( \mu \right) ,1\right] \rightarrow \left[ 0,p_{W}\left( \mu \right) \right]$ (smoothly extended to an open $\varepsilon$ neighborhood of $p_{W}\left( \mu \right)$) has the derivative at $p_{W}\left( \mu \right)$ equal to

$$\begin{aligned} \phi ^{\prime }\left( p_{W}\left( \mu \right) |\mu \right) =-\frac{g_{B}\left( p_{W}\left( \mu \right) |\mu \right) }{g_{S}\left( p_{W}\left( \mu \right) |\mu \right) }<0.\end{aligned}$$

(46)

Now the system of Equations for $(\underline{v}\left( H\right) ,\underline{v}\left( L\right) )$ becomes

$$\begin{aligned} \frac{1}{2}\left( \underline{v}\left( H\right) -\phi \left( \underline{v}\left( H\right) |H\right) \right) -\tau \cdot \kappa \left( 1+\sqrt{\frac{1-G_{B}\left( \underline{v}\left( L\right) |H\right) }{1-G_{B}\left( \underline{v}\left( H\right) |H\right) }}\right)&=0, \end{aligned}$$

(47)

$$\begin{aligned} \frac{1}{2}\left( \underline{v}\left( L\right) -\phi \left( \underline{v}\left( L\right) |L\right) \right) -\tau \cdot \kappa \left( 1+\sqrt{\frac{G_{S}\left( \phi \left( \underline{v}\left( H\right) |H\right) |L\right) }{G_{S}\left( \phi \left( \underline{v}\left( L\right) |L\right) |L\right) }}\right)&=0. \end{aligned}$$

(48)

The Jacobian of this system at $\tau =0$ is

$$\begin{aligned}&\left| \begin{array}{ll} \frac{1}{2}\left( 1-\phi ^{\prime }\left( p_{W}\left( H\right) |H\right) \right) &{} 0\\ 0 &{} \frac{1}{2}\left( 1-\phi ^{\prime }\left( p_{W}\left( L\right) |L\right) \right) \end{array}\right| \\&=\frac{1}{4}\left( 1-\phi ^{\prime }\left( p_{W}\left( H\right) |H\right) \right) \left( 1-\phi ^{\prime }\left( p_{W}\left( L\right) |L\right) \right) \\&>0, \end{aligned}$$

where the inequality in the last line follows from (46).   Q. E. D. $\square$

Lemma 1

A threshold $\bar{r}>0$ exists such that, for all $r\in \left[ 0,\bar{r}\right]$ and $\tau <1$, buyers do not have an incentive to deviate by offering prices p that are not acceptable to some active sellers who share the same belief about $\mu$:

$$\begin{aligned} p\in \left[ \tilde{c}\left( 0|\mu _{B}\right) ,\bar{c}\left( \mu _{B}\right) \right) . \end{aligned}$$

Likewise, sellers do not have an incentive to deviate by offering prices p that are not acceptable to some of the buyers who share the same belief about $\mu$:

$$\begin{aligned} p\in \left( \underline{v}\left( \mu _{S}\right) ,\tilde{v}\left( 1|\mu _{S}\right) \right] . \end{aligned}$$

Proof

Without loss of generality, let’s assume that the state is $\mu =H$ and focus on the seller. Whenever the corresponding stocks are positive we denote the distribution of active buyer types with belief $\mu _{B}$ when the true state is $\mu$ as $\Phi \left( \cdot |\mu _{B},\mu \right)$, while the distributions of their reservation values $\tilde{v}\left( v|\mu _{B}\right)$ are denoted as $\tilde{\Phi }\left( \cdot |\mu _{B},\mu \right)$.

For notational expedience, from now on we denote the probability that a buyer with belief $\mu _{B}=H$ will meet a seller as

$$\begin{aligned} \ell _{B}^{*}\equiv \frac{S\left( H\right) }{B\left( H\right) }=\frac{B(H|H)}{B(H)},\end{aligned}$$

where the second equality follows from (30).

The proposing strategies must be non-decreasing by standard single-crossing arguments; the proof parallels Lemma 2 in Shneyerov and Wong (2010b) and is omitted. It is therefore sufficient to show that the marginal types will not deviate. First, we focus on the incentives of the sellers (a symmetric argument will apply for the buyers, with obvious changes). The expected profit contingent on proposing $\lambda \ge \underline{v}\left( \mu \right)$ is

$$\begin{aligned} \pi _{S}\left( \bar{c}\left( \mu \right) ,\lambda |\mu \right) =\left( \lambda -\bar{c}\left( \mu \right) \right) \left( 1-\tilde{\Phi }\left( \lambda |\mu ,\mu \right) \right) , \end{aligned}$$

and its slope is

$$\begin{aligned} \frac{\partial \pi _{S}\left( \bar{c}\left( \mu \right) ,\lambda |\mu \right) }{\partial \lambda }&=\left( 1-\tilde{\Phi }\left( \lambda |\mu ,\mu \right) \right) -\left( \lambda -\bar{c}\left( \mu \right) \right) \tilde{\Phi }^{\prime }\left( \lambda |\mu ,\mu \right) \nonumber \\&=-\tilde{\Phi }^{\prime }\left( \lambda |\mu ,\mu \right) \left[ \tilde{J}_{B}\left( \lambda |\mu \right) -\bar{c}\left( \mu \right) \right] \end{aligned}$$

(49)

where $\tilde{J}_{B}\left( \lambda |\mu \right)$ is the “virtual type” that corresponds to the distribution of reservation values $\tilde{\Phi }\left( \cdot |\mu ,\mu \right)$,

$$\begin{aligned} \tilde{J}_{B}\left( \lambda |\mu \right) \equiv \lambda -\frac{1-\tilde{\Phi }\left( \lambda |\mu ,\mu \right) }{\tilde{\Phi }^{\prime }\left( \lambda |\mu ,\mu \right) }. \end{aligned}$$

Notice that $\tilde{\Phi }\left( \lambda |\mu ,\mu \right) =\Phi \left( \tilde{v}^{-1}(\lambda |\mu )|\mu ,\mu \right)$. Contingent on meeting a seller, pessimistic buyers trade with probability 1 regardless of their type. Therefore, their distribution of types in the market is a truncation of the inflow distribution,

$$\begin{aligned} 1-\Phi (v|\mu ,\mu )=\frac{1-G_{B}(v|\mu )}{1-G_{B}(\underline{v}\left( \mu \right) |\mu ))}\quad \left( v\ge \underline{v}\left( \mu \right) \right) . \end{aligned}$$

From (34), $\tilde{v}\left( v|\mu \right)$ is a linear function with the slope

$$\begin{aligned} \tilde{v}^{\prime }\left( v|\mu \right) =\frac{1-R_{\tau }}{1-R_{\tau }+R_{\tau }\ell _{B}^{*}\left( \mu \right) } \end{aligned}$$

Since $\tilde{v}\left( \underline{v}\left( \mu \right) |\mu \right) =\underline{v}\left( \mu \right)$, we can explicitly solve for the responding strategy,

$$\begin{aligned} \tilde{v}\left( v|\mu \right) =\frac{\left( 1-R_{\tau }\right) v+R_{\tau }\ell _{B}^{*}\left( \mu \right) \underline{v}\left( \mu \right) }{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}\left( \mu \right) }. \end{aligned}$$

(50)

From (34), the inverse responding strategy is

$$\begin{aligned} \tilde{v}^{-1}\left( \lambda \right) =\frac{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}}{1-R_{\tau }}\lambda -\frac{R_{\tau }\ell _{B}^{*}\underline{v}\left( \mu \right) }{1-R_{\tau }}. \end{aligned}$$

Then

$$\begin{aligned} 1-\tilde{\Phi }\left( \lambda |\mu ,\mu \right) =\frac{1-G_{B}\left( \tilde{v}^{-1}\left( \lambda |\mu \right) |\mu \right) }{1-G_{B}\left( \underline{v}\left( \mu \right) |\mu \right) }, \end{aligned}$$

$$\begin{aligned} \tilde{\phi }\left( \lambda |\mu ,\mu \right)&=\frac{d\tilde{v}^{-1}\left( \lambda |\mu \right) }{d\lambda }\frac{g_{B}\left( \tilde{v}^{-1}\left( \lambda |\mu \right) |\mu \right) }{1-G_{B}\left( \underline{v}\left( \mu \right) |\mu \right) }\\&=\frac{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}}{1-R_{\tau }}\frac{g_{B}\left( \tilde{v}^{-1}\left( \lambda |\mu \right) |\mu \right) }{1-G_{B}\left( \underline{v}\left( \mu \right) |\mu \right) },\end{aligned}$$

and

$$\begin{aligned} \tilde{J}_{B}\left( \lambda |\mu \right) \equiv&\,\lambda -\frac{1-\tilde{\Phi }\left( \lambda |\mu ,\mu \right) }{\tilde{\phi }\left( \lambda |\mu ,\mu \right) }\\ =&\,\lambda -\frac{1-R_{\tau }}{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}}\frac{1-G_{B}\left( \tilde{v}^{-1}\left( \lambda |\mu \right) |\mu \right) }{g_{B}\left( \tilde{v}^{-1}\left( \lambda |\mu \right) |\mu \right) }\\ =&\,\lambda -\frac{1-R_{\tau }}{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}}\tilde{v}^{-1}\left( \lambda |\mu \right) \\&+\,\frac{1-R_{\tau }}{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}}\left( \tilde{v}^{-1}\left( \lambda |\mu \right) -\frac{1-G_{B}\left( \tilde{v}^{-1}\left( v|\mu \right) |\mu \right) }{g_{B}\left( \tilde{v}^{-1}\left( \lambda |\mu \right) |\mu \right) }\right) \\ =&\,\frac{R_{\tau }\ell _{B}^{*}\underline{v}\left( \mu \right) }{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}}+\frac{1-R_{\tau }}{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}}J_{B}\left( \tilde{v}^{-1}\left( \lambda |\mu \right) |\mu \right) .\end{aligned}$$

Equivalently,

$$\begin{aligned} \tilde{J}_{B}\left( \lambda |\mu \right) =&\,\frac{1}{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}}\nonumber \\&\times \,\left( \left( 1-R_{\tau }\right) J_{B}\left( \tilde{v}^{-1}\left( \lambda |\mu \right) |\mu \right) +R_{\tau }\ell _{B}^{*}\underline{v}\left( \mu \right) \right) . \end{aligned}$$

(51)

Substituting (51) in the slope formula (49), we obtain

$$\begin{aligned} \frac{\partial \pi _{S}\left( \bar{c}\left( \mu \right) ,\lambda |\mu \right) }{\partial \lambda } =&\,-\tilde{\Phi }^{\prime }\left( \lambda |\mu ,\mu \right) \{\frac{1}{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}}\nonumber \\&\times \,(\left( 1-R_{\tau }\right) J_{B}\left( \tilde{v}^{-1}\left( \lambda |\mu \right) |\mu \right) +R_{\tau }\ell _{B}^{*}\underline{v}\left( \mu \right) )-\bar{c}\left( \mu \right) \}. \end{aligned}$$

(52)

Clearly, a deviation to $\lambda <\underline{v}\left( \mu \right)$ is not profitable, so we only need to consider $\lambda >\underline{v}\left( \mu \right)$. A necessary condition for such a deviation to be not profitable is that $\partial \pi _{S}\left( \bar{c}\left( \mu \right) ,\lambda |\mu \right) /\partial \lambda \le 0$ at $\lambda =\underline{v}\left( \mu \right)$, i.e. the expression in the brackets on the right-hand side of Eq. (52) is non-negative when $\lambda =\underline{v}\left( \mu \right)$. This is also sufficient because of the assumed monotonicity of $J_{B}\left( \cdot |\mu \right)$ (Assumption 1). This gives us the inequality

$$\begin{aligned} \frac{\left( 1-R_{\tau }\right) J_{B}\left( \underline{v}\left( \mu \right) |\mu \right) +R_{\tau }\ell _{B}^{*}\underline{v}\left( \mu \right) }{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}}-\bar{c}\left( \mu \right) \ge 0. \end{aligned}$$

We now show that this inequality is satisfied for small r. We can rewrite it as

$$\begin{aligned} \underline{v}\left( \mu \right) -\bar{c}\left( \mu \right) -\frac{\left( 1-R_{\tau }\right) }{\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}}\frac{1-G_{B}\left( \underline{v}\left( \mu \right) |H\right) }{g_{B}\left( \underline{v}\left( \mu \right) |H\right) }\ge 0. \end{aligned}$$

(53)

From either (18) or (19) we have $\underline{v}\left( \mu \right) -\bar{c}\left( \mu \right) =2\tau \kappa /\ell _{B}^{*}$. Substituting this into (53) and replacing $\frac{1-G_{B}\left( \underline{v}\left( \mu \right) |H\right) }{g_{B}\left( \underline{v}\left( \mu \right) |H\right) }$ with an upper bound $1/\underline{g}_{B}$, and $\left( 1-R_{\tau }\right) +R_{\tau }\ell _{B}^{*}$ with $R_{\tau }\ell _{B}^{*}$, we have a stronger inequality that is sufficient for no deviation:

$$\begin{aligned} \frac{2\tau \kappa }{\ell _{B}^{*}}-\frac{\left( 1-R_{\tau }\right) }{R_{\tau }\ell _{B}^{*}}\frac{1}{\underline{g}_{B}}\ge 0. \end{aligned}$$

Alternatively,

$$\begin{aligned} \frac{1-e^{-r\tau }}{\tau e^{-r\tau }}\le 2\kappa \underline{g}_{B}. \end{aligned}$$

(54)

The l.h.s. of the above equation, $\left( e^{r\tau }-1\right) /\tau$, is an increasing function of $\tau$ because $e^{r\tau }-1$ is a convex, increasing function of $\tau$ taking value 0 at $\tau =0$. Therefore, if $\tau \le 1$, it is sufficient to require

$$\begin{aligned} e^{r}-1\le 2\kappa \underline{g}_{B}, \end{aligned}$$

or equivalently

$$\begin{aligned} r\le \log \left( 1+2\kappa \underline{g}_{B}\right) . \end{aligned}$$

A parallel argument applied to marginal sellers.   Q. E. D. $\square$