Auctioning Risky Conservation Contracts

Abstract

Conservation auctions have the potential to increase the efficiency of payments to farmers to adopt conservation-friendly management practices by fostering competition among them. The literature considers bidders that have complete information about the costs of adoption and optimal bidding behavior reflects this information advantage. Farmers seek information rents and bids decrease when risk aversion increases because farmers are more averse to losing the auction. We contribute to the literature by allowing for cost risk. Our paper shows that farmers must balance the risk of losing the auction (thus foregoing information rent) with the risk of submitting a bid that is not high enough to pay the costs of adopting conservation practices (thus incurring losses). We design an experiment to trade off these two risks and examine how risk aversion affects bidding behavior when participants face different sources and levels of risk. Our experiment contributes to a small literature on experimental auctions with risky product valuations. We find that participants decrease their bids as risk aversion increases, even in auctions with cost risk, suggesting that the risk of losing the auction dominates. These findings uncover new challenges for the practical implementation of conservation auctions as an efficient policy instrument.

Appendix

1.1 Proofs

Proposition 1

In a conservation auction with known adoption costs, the farmer’s optimal bid is inversely related to his degree of risk aversion.

Proof

Applying the implicit function theorem to the first order condition (3) leads to:

$$\begin{aligned} \frac{\partial b^*}{\partial \theta } = - \frac{\frac{\partial U^{\prime }( \cdot )}{\partial \theta } \left[ 1 - F\left( b^*\right) \right] - \frac{\partial U(\cdot )}{\partial \theta }f(b) }{\textit{SOC}}, \end{aligned}$$

(8)

where $\textit{SOC}$ is the second order condition of problem (2) and hence is assumed to be negative. The numerator of (8) must be negative for $\frac{\partial b^*}{\partial \theta } \le 0$. As the terms $\frac{\partial U^{\prime }(\cdot )}{\partial \theta }$ and $\frac{\partial U(\cdot )}{\partial \theta }$ have negative signs, the numerator is negative when Assumption 2 holds. Therefore, if the relative effect of risk aversion is greater than the hazard rate, increases in risk-aversion decrease the optimal bid.

$$\begin{aligned} \frac{\partial b^*}{\partial \theta } \le 0 \iff \frac{\frac{\partial U^{\prime }\left( b^*\right) }{\partial \theta } }{\frac{\partial U\left( b^*\right) }{\partial \theta } }\ge \frac{f\left( b^*\right) }{1 - F\left( b^*\right) }. \end{aligned}$$

(9)

Corollary 1

The efficiency of a conservation auction with known adoption costs is positively associated with the degree of risk aversion of its participants.

Proof

Typically, efficiency of conservation auctions is measured by the level of EGS that the auction is capable of generating for a given budget. In our simple framework, heterogeneity between farmers is due to risk preferences and costs of adoption. Farmers are assumed to be homogeneous in terms of the benefit generated by their adoption, therefore, the number of adopters define the level of EGS. To prove the corollary we will show that an increase in the average level of risk aversion (weakly) increases the provision of EGS.

Let $\mathbf{b}({\varvec{\theta }}$) denote a sorted bid profile; a vector that collects bids such that $b_1(\theta _1) \le b_2(\theta _2) \le \cdots \le b_n(\theta _n)$, where $\varvec{\theta }$ is the auction’s risk aversion profile $(\theta _1, \theta _2, \ldots , \theta _n$) and n is the number of participants in the conservation auction. Let B denote the regulator’s limited budget $\left( B < \sum \nolimits _{i=1}^n b_i(\theta _i)\right)$. Bids are accepted from the lowest one upwards until the budget is exhausted (or insufficient to finance an additional bid). Let $b_k(\theta _k)$ denote the last affordable bid, i.e. the bid of farmer k such that $B \ge \sum \nolimits _{i=1}^k b_i(\theta _i)$ and $B < \sum \nolimits _{i=1}^{k+1} b_i(\theta _i)$ . All farmers $i \le k$ are winners and adopt the conservation technology, therefore, the level of EGS induced by the auction can be measured by k. Consider the auction’s risk aversion profile $\varvec{\theta }^{\prime } \ge \varvec{\theta }$ in which $\theta _i^{\prime } \ge \theta _i$ for all i. Let t denote the winning threshold associated with $\varvec{\theta }^{\prime }$: $B \ge \sum \nolimits _{i=1}^t b_i(\theta _i^{\prime })$ and $B < \sum \nolimits _{i=1}^{t+1} b_i(\theta _i^{\prime })$. From Proposition 1 we know that $\frac{\partial b}{\partial \theta } \le 0$. As a result, $\sum \nolimits _{i=1}^k b_i(\theta _i) =\sum \nolimits _{i=1}^k b_i(\theta _i^{\prime }) + X$, where $X\ge 0$. If $X \ge b_{k+1}(\theta _{k+1}^{\prime })$, then $t >k$. If $X < b_{k+1}(\theta _{k+1}^{\prime })$, then $t = k$. This concludes the proof as we showed that the supply of EGS under $\varvec{\theta }$ is less than (or equal to) the EGS supply under a higher level of risk aversion $\varvec{\theta }^{\prime }$, i.e. $k \le t$. Note that when $k=t$, the same amount of EGS is generated under $\varvec{\theta }$ and $\varvec{\theta }^{\prime }$, however, the cost of provision is smaller under the $\varvec{\theta }^{\prime }$ profile.

Proposition 2

Farmers increase their bids in response to increases in cost risk.

Proof

The relationship between the farmer’s optimal bid and the level of cost risk is obtained through the implicit function theorem as follows.

$$\begin{aligned} \frac{\partial b^{**}}{\partial x} = - \frac{\left( - \frac{\partial U^{\prime }(\cdot )}{\partial x} +\frac{\partial U^{\prime }(\cdot )}{\partial x} \right) \left[ 1 - F(\cdot )\right] - f(\cdot ) \left( - \frac{\partial U(\cdot )}{\partial x}+\frac{\partial U(\cdot )}{\partial x}\right) }{\textit{SOC}}, \end{aligned}$$

(10)

where SOC is the second order condition of problem (5) and hence is assumed to be negative. Assumption 3 is needed to analytically derive the sign of Eq. (10). If farmers are prudent, then $\left( - \frac{\partial U^{\prime }(b^{**}-c- x| \theta )}{\partial x} +\frac{\partial U^{\prime }(b^{**}-c+ x| \theta )}{\partial x}\right) \ge 0$. If farmers are risk-averse, then $\left( - \frac{\partial U(b^{**}-c- x| \theta )}{\partial x}+\frac{\partial U(b^{**}-c+ x| \theta )}{\partial x}\right) \le 0$. It follows that $\frac{\partial b^{**}}{\partial x} \ge 0$.

Corollary 2

The conservation auction efficiency is negatively associated with the level of adoption cost risk.

Proof

The proof is similar to that of Corollary 1. Cost risk is introduced through a mean preserving spread of adoption costs, therefore, the level of risk is determined by x. Let $\mathbf{b (x)}$ denote a sorted bid profile, where $\mathbf x$ is the corresponding cost risk profile. Let $\mathbf x^{\prime }$ denote a cost risk profile such that $x_i^{\prime } \le x_i$ for all i. We use the notation of the proof of Corollary 1. Let k and t index the threshold bids such that $B \ge \sum \nolimits _{i=1}^k b_i(x_i)$ and $B < \sum \nolimits _{i=1}^{k+1} b_i(x_i)$, and $B \ge \sum \nolimits _{i=1}^t b_i(x_i^{\prime })$ and $B < \sum \nolimits _{i=1}^{t+1} b_i(x_i^{\prime })$. From Proposition 2 we know that $\frac{\partial b}{\partial x} \ge 0$. It follows that $\sum \nolimits _{i=1}^k b_i(x_i) =\sum \nolimits _{i=1}^k b_i(x_i^{\prime }) + X$, where $X\ge 0$. As a result, $k \le t$, where k is the level of EGS under x and t is the EGS level under $\mathbf x^{\prime }$.

Corollary 3

A risk-neutral farmer does not change his bidding strategy in response to mean preserving spreads of the cost distribution.

Proof

The risk-neutral farmer’s problem is to maximize expected payoff from auction participation:

$$\begin{aligned} \max \limits _b \left( \frac{1}{2} (b-c-x) + \frac{1}{2}(b-c+x) \right) [1 - F(b)] = (b-c)[1 - F(b)] , \end{aligned}$$

(11)

and the optimal bid $b^{n}$ is determined by the following first order condition:

$$\begin{aligned} b^{n} = c + \frac{1-F\left( b^{n}\right) }{f\left( b^{n}\right) } \end{aligned}$$

(12)

This implicit function that determines optimal bid for risk-neutral farmers is equivalent to that reported by Latacz–Lohmann and Van der Hamsvoort [1997, equation (5)]. The optimal bid consists of the opportunity cost of adoption plus an auction participation premium, and it does not depend on the variance of the cost distribution. $\square$

1.2 Experimental Instructions

1.3 Eckel–Grossman Risk Task

1.4 Screen Shoots

See Figs. 3 and 4.

Fig. 3

Example of experimental auction participation choice and bid choice page in Z-tree

Fig. 4

Example of experimental auction result page in Z-tree

1.5 Nonlinear Effects of Expected Costs on Bids

Thus far, our analyses are based on regressions with cost as a linear term. However, as expected costs increase, bids may increase at a decreasing rate if high cost bidders decrease markups as they compete to win the auction. Using all experimental data, we estimate bidding models with a quadratic specification to explore nonlinear effects of expected costs on bids. Results are presented in Table 7.

Table 7 Nonlinear effects of expected costs on bids

Fig. 5

Bid curve

First, note that the main findings of the paper remain unchanged. Controlling for nonlinear effects of expected costs on bids, we find that: (i) bids are higher in auctions with cost risk (see column 2); (ii) there is a negative overall effect of risk aversion on bids, i.e. the winning bid incentive dominates (see column 3); and (iii) participants respond to both incentives and the effect of risk aversion on bids is attenuated in auctions with cost risk (see column 4).

Next, note that in all models the coefficient of Expected Cost is positive and significant and the coefficient of Expected Cost Squared is negative and significant. These estimates confirm the hypothesis that bids increase with expected costs at a decreasing rate. We use the estimates of column 1 to predict a bid curve. Figure 5 shows the bid curve and provides a visual illustration of the demand for information rents (i.e., the difference between the bid curve and the 45$^{\circ }$ line). As we move along the bid curve, from low cost to high cost, we find that information rent requests are small when cost is low (e.g. approximately $0.5 when cost is equal to $1), increase to $4 when expected cost is $19, and decreases to $2.5 when expected cost is $51.