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Appendix 2

A bidding model for multi-unit conservation auctions with a budget constraint (Latacz-Lohmann and van der Hamsvoort, 1997)

Let us consider that landowners or farmers hold private information about their own farm income, and let p ₀ be the associated profits. Let p ₁ be the profit remaining after a landowner has given up a proportion of his land, exclusive of any compensation payments by government. More precisely:

p ₀ = profits from business-as-usual land management or farming

p ₁ = profits with a new, conservation-oriented land management

Note that p ₁ may include income from employment outside farming. p ₁ = 0 if the farmer gives up all of his land and has no alternative employment prospects.

In order for the landowner or farmer to participate in the scheme, the payment he receives must be at least equal to ( p ₀ - p ₁), his or her opportunity cost of participation. If he or she submits a bid b that is accepted, utility will be U(p ₁ + b), where U(·) is a monotonically increasing, twice differentiable von Neumann-Morgenstern utility function. If the bid is rejected, the bidder's utility is U(p ₀), the reservation utility.

Now let us consider that landowners' bidding strategies are predicated on the belief that the government agency will decide on a maximum acceptable bid, or payment level, ß , a common practice when the agency is subject to a constrained budget. This maximum bid is determined ex post, after all bids have been received, as the last (highest) bid accepted within the available budget. In other words, no individual bids above ß will be accepted. ß represents a reserve price per unit of environmental service, unknown to potential bidders. A landowner will tender a bid b if the expected utility in case of participation exceeds his or her reservation utility, as shown in equation (12), where p stands for probability:

(1)

Bidders do not know the value of the bid cap ß, but they will form expectations about it, which can be characterized by the density function f(b) and by the distribution function F(b). The probability that a bid is accepted can then be expressed as

(2)

where represents the upper limit of the bidder's expectations about the bid cap, or the maximum expected bid cap. Substituting (2) in (1) yields

(3)

The essence of the bidding problem is to balance out net payoffs and probability of acceptance. This means determining the optimal bid which maximizes the expected utility (on the left hand side of (3)) over and above the reservation utility (on the right hand side of (3)). Let us assume that there are no costs in bid preparation and implementation, and that payment is only a function of the bid. We also assume that bidders are risk neutral ⁱ .

A risk-neutral bidder simply maximizes expected payoff, so that (3) can be rewritten as

(4)

The optimal bid b^* is then obtained by maximizing (4) through the choice of b:

(5)

To gain further insights, one must specify the distribution function F(b). The simplest case is where bidders' expectations about the bid cap ß are uniformly distributed ⁱⁱ in the range , where the lower and upper bounds represent the bidder's minimum and maximum expected bid cap. For example, if a landowner believes that the cut-off point will lie somewhere between $X and $Y per hectare, then and . Note that these bidder's expectations are exogenous to the model.

The density and distribution functions of a uniform (rectangular) distribution are:

Of course, there is no sense in the bidder bidding below (this would not increase the acceptance probability) or above (his chances of winning would be nil).

With this specification of f(b) and F(b), one obtains an explicit optimal bid formula for a risk-neutral bidder:

(7)

Expression (7) shows that the optimal bidding strategy of a risk-neutral bidder increases linearly with both the bidder's opportunity costs ( p ₀ - p ₁) and his or her expectations about the bid cap, and . Thus, a bidder's bid conveys information about his or her opportunity costs, which are private information unknown to government. The information asymmetry is thus reduced, but not completely: indeed, the auction's cost revelation property is blurred by the fact that the bid also reflects the bidder's beliefs about the bid cap chosen by the agency. This creates room for bidders to bid above their true opportunity costs and thereby to secure themselves an information rent - area CBDG in Figure 1.

Figure 1 - Bid and opportunity cost curves

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