The Role of Search Frictions and Trading Ratios in Tradable Permit Markets

Abstract

Search frictions, defined as the costs of finding trading partners, are a common feature of most permit markets. In these markets prices are not publicly available, buyers and sellers need to find their trading partners, trades often take place bilaterally, and there is often no central market-clearing mechanism. In this paper, we study the search and trading decisions of participants in a permit market with search frictions. We build an analytical model of the trading decision in a market with search frictions and show that individuals set a reservation price and an optimal search effort. The model shows that in the presence of search frictions, those who intend to trade (buy or sell) greater quantities search more. This trading behavior is not expected in a market without search frictions. Furthermore, we show that trading ratios affect the probability of trades taking place. We test the predictions of the model using a unique dataset of trades from a groundwater market with trading ratios that was designed to reduce a spatially explicit externality. We find that search frictions are significant so that buyers or sellers who trade greater quantities search more. Furthermore, we show that while the trading ratio system provides incentives to participants to reduce the spatial externality, search frictions reduces the effectiveness of the market by affecting the pool of potential trading partners for participants in the market.

Appendix

1.1 Derivation of Eq. 1

For a potential seller, the payoff of selling the permits equals a one-time payment of $\omega Q$ for the Q number of permits sold and the present value of a stream of payments under production with no permits, $\pi ^{0} Q$, is:

$$\begin{aligned} W_s(\omega )= & {} \omega Q + \delta \pi ^{0} Q + \delta ^2 \pi ^{0} Q + \delta ^3 \pi ^{0} Q + \dots \end{aligned}$$

(18)

where $\delta$ is the discount factor. We can rewrite the equation above as:

$$\begin{aligned} W_s(\omega ) - \omega Q= & {} \delta \pi ^{0} Q + \delta ^2 \pi ^{0} Q + \delta ^3 \pi ^{0} Q + \dots \end{aligned}$$

(19)

We can also write Eq. 18 as:

$$\begin{aligned} W_s(\omega )= & {} \omega Q + \delta \pi ^{0} Q + \delta \Big \{ \delta \pi ^{0} Q + \delta ^2 \pi ^{0} Q + \delta ^3 \pi ^{0} Q + \dots \Big \} \end{aligned}$$

(20)

Combining Eqs. 19 and 20, we get:

$$\begin{aligned} W_s(\omega )= & {} \omega Q + \delta \pi ^{0} Q + \delta \Big \{ W_s(\omega ) - \omega Q \Big \} \end{aligned}$$

(21)

Rearranging the terms in Eq. 21, we get:

$$\begin{aligned} W_s(\omega )= & {} \omega Q + \frac{\delta }{(1-\delta )} \pi ^{0} Q \end{aligned}$$

(22)

$\delta$ is the discount factor. As a result, we can write $\frac{\delta }{(1-\delta )}$ in terms of discount rate: $\frac{\delta }{(1-\delta )} = \frac{1}{r}$ and modify Eq. 22 to:

$$\begin{aligned} W_s(\omega )= & {} \omega Q + \frac{1}{r} \pi ^{0} Q \end{aligned}$$

(23)

Finally, we can write Eq. 23 as:

$$\begin{aligned} rW_s(\omega )= & {} r\omega Q + \pi ^{0} Q \end{aligned}$$

(24)

Which is the Eq. 1.

1.2 Derivation of Eq. 2

The payoff of not selling the permits in the current period is equal to producing with the permits in the next period, $\delta \pi ^{1} Q$, while looking for a trading partner. The search process results in meeting a trading partner with probability $\alpha$. The probability of trade can increase but it’s costly, $g(\alpha )$. After meeting the potential buyer, a price is realized from the distribution $F(\omega )$. This is the price offered by the potential buyer to the seller. If the payoff from selling the price is greater than the payoff of not selling the permits, the seller will sell the permits and the payoff of the next period will be $W_s(\omega )$. Otherwise, if the payoff of selling the permits is less than that of not selling the permits, the seller will wait another period to sell their permits and receive a payoff of $U_s$ in the next period.

$$\begin{aligned} \begin{aligned} U_s =\,&\delta \pi ^{1} Q + \alpha _s \delta \int _0^{\bar{\omega }} \! Max\{U_s, W_s(\omega )\} \, \mathrm {d}F(\omega ) \\&+ \delta (1 - \alpha _s ) U_s - \delta g_s(\theta _s) \end{aligned} \end{aligned}$$

(25)

We can rearrange Eq. 25 as:

$$\begin{aligned} \begin{aligned} (1-\delta ) U_s&= \delta \pi ^{1} Q + \alpha _s \delta \int _0^{\bar{\omega }} \! Max\{U_s - U_s, W_s(\omega ) - U_s\} \, \mathrm {d}F(\omega ) - \delta g_s(\theta _s) \longrightarrow \\ U_s&= \frac{\delta }{(1-\delta )} \pi ^{1} Q + \alpha _s \frac{\delta }{(1-\delta )} \int _0^{\bar{\omega }} \! Max\{0, W_s(\omega ) - U_s\} \, \mathrm {d}F(\omega ) - \frac{\delta }{(1-\delta )} g_s(\theta _s) \longrightarrow \\ r U_s&= \pi ^{1} Q + \alpha _s \int _0^{\bar{\omega }} \! Max\{0, W_s(\omega ) - U_s\} \, \mathrm {d}F(\omega ) - g_s(\theta _s) \end{aligned} \end{aligned}$$

(26)

1.3 Derivation of Reservation Price

The reservation price, R, is the price where the payoff of selling the permits is equal to the payoff of not selling the permits and wait another period, i.e., at the reservation price, Eq. 1 is equal to Eq. 2:

$$\begin{aligned} \begin{aligned} rW_s(R)&= rU_s \longrightarrow \\ rR Q + \pi ^{0} Q&= \pi ^{1} Q + \alpha _s \int _0^{\bar{\omega }} \! Max\{0, W_s(\omega ) - U_s\} \, \mathrm {d}F(\omega ) - g_s(\theta _s) \longrightarrow \\ rR Q&= \Delta \pi Q + \alpha _s \int _0^{\bar{\omega }} \! Max\{0, W_s(\omega ) - U_s\} \, \mathrm {d}F(\omega ) - g_s(\theta _s) \end{aligned} \end{aligned}$$

(27)

where $\Delta \pi = \pi ^{1} - \pi ^{0}$ is the profit differential. We can also write:

$$\begin{aligned} \begin{aligned} W_s(R)&= U \longrightarrow \\ rW_s(\omega ) - rU&= r \omega Q + \pi ^0 Q - rRQ - \pi ^0 Q \longrightarrow \\ rW_s(\omega ) - rU&= r \omega Q - rRQ \longrightarrow \\ W_s(\omega ) - U&= \omega Q - RQ \longrightarrow \\ W_s(\omega ) - U&= (\omega - R) Q \end{aligned} \end{aligned}$$

(28)

replacing $W_s(\omega ) - U$ with $(\omega - R) Q$, in Eq. 27, we get:

$$\begin{aligned} rR Q&= \Delta \pi Q + \alpha _s Q \int _R^{\bar{\omega }} \! (\omega - R) \, \mathrm {d}F(\omega ) - g_s(\theta _s) \end{aligned}$$

(29)

Rearranging the terms above, we get the reservation price for a potential seller:

$$\begin{aligned} R = \frac{\Delta \pi }{r} + \frac{\alpha _s}{r} \int _R^{\bar{\omega }} \! (\omega - R) \, \mathrm {d}F(\omega ) - \frac{g_s(\theta _s)}{rQ} \end{aligned}$$

(30)

We can further simplify the integral in Eq. 30 as:

$$\begin{aligned} \begin{aligned}&\int _R^{\bar{\omega }} \! (\omega - R) \, \mathrm {d}F(\omega ) =\\&\int _R^{\bar{\omega }} \! \omega \, \mathrm {d}F(\omega ) - R \int _R^{\bar{\omega }} \! \, \mathrm {d}F(\omega ) \end{aligned} \end{aligned}$$

(31)

Using integration by parts, we can write the above equation as:

$$\begin{aligned} \int_{R}^{{\bar{\omega }}} (\omega - R){\mkern 1mu} {\text{d}}F(\omega ) & = \\ \{ \bar{\omega }F(\bar{\omega }) - RF(R)\} - \int_{R}^{{\bar{\omega }}} F(\omega ){\mkern 1mu} {\text{d}}\omega - \{ RF(\bar{\omega }) - RF(R)\} & = \\ \{ F(\bar{\omega })(\bar{\omega } - R)\} - \int_{R}^{{\bar{\omega }}} F(\omega ){\mkern 1mu} {\text{d}}\omega & \\ \end{aligned}$$

(32)

Given that $F(\bar{\omega }) = 1$, we can write $\int _R^{\bar{\omega }} \! (\omega - R) \, \mathrm {d}F(\omega )$ as $\int _R^{\bar{\omega }} \! (1- F(\omega )) \, \mathrm {d}\omega$. We can then write the reservation price for a seller as (Tables 4 and 5):

$$\begin{aligned} R = \frac{\Delta \pi }{r} + \frac{\alpha _s}{r} \int _R^{\bar{\omega }} \! (1- F(\omega )) \, \mathrm {d}\omega - \frac{g_s(\theta _s)}{rQ} \end{aligned}$$

(33)

Which is Eq. 3. The effort level that maximizes the seller’s present payoff (Eq. 6) can be derived as:

$$\begin{aligned} g'_s(\theta _s^*) = Q \alpha _s'(\theta _s^*; \eta _s) \int _R^{\bar{\omega }} \! (1- F(\omega )) \, \mathrm {d}\omega \end{aligned}$$

(34)

Table 4 The effect of quantity traded on distance between the seller and the buyer. Distance is a proxy for search effort

Table 5 The effect of quantity traded on distance between the seller and the buyer for trades that include only trades with single transaction. Distance is a proxy for search effort