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This article uses the forest management problem under uncertainty to derive the optimal reservation price when a standing timber is to be auctioned. Theoretically, the resulting optimal reservation price that considers the harvesting decision is an extended version of Laffont and Maskin’s and Riley and Samuelson’s reservation price, which is suboptimal in the forestry context.

In forestry, the selling of standing timber is conducted either through direct negotiation between the forest owner and the exploiting firm or by auctions. The former is commonly used in Scandinavian countries (Finland, Sweden, and Norway) whereas the latter is popular in countries such as Canada, Great Britain, and the US. For example, the US government agency the United States Forest Service (USFS) uses both first price sealed-bid auctions and ascending auctions for the sale of standing timber. Over the past decade, theoretical and empirical works have focused intensively on these two auction formats by analysing the binding behaviour and the minimum price that must be bid optimally (the optimal reservation price) under various assumptions. Using an independent private value paradigm, [

This article combines both the auctions and the forest management problem to derive theoretically the optimal reservation price in forestry when the forest owner sells its timber through auctions under uncertainty regarding the stumpage price of timber^{1}. The bidder’s valuation is endogenous as it depends on the total supply of timber which depends on the optimal cutting age of the tree. Thus, the fixation of a reservation price must take into account the harvesting decision.

In the forest management problem under uncertainty, [

This article contributes theoretically to the literature of forest auctions by extending the results of Laffont and Maskin as well as Riley and Samuelson in the context of forestry management.

The article is organized as follows. In Section 2, I first present the theoretical model that combines auctions and the forest management problem under uncertainty when stumpage prices follow an autoregressive process. Section 3 then determines the optimal bidding strategies, the optimal cutting age and the optimal reservation price. Finally, Section 4 concludes the article.

In this section, I use the forest management framework under uncertainty to derive the theoretical optimal mechanism when a standing timber is sold through auctions. The optimal mechanism consists of the optimal bidding strategy, the optimal cutting age (optimal rotation), and the optimal reservation price. The forest owner is interested in selling the standing timber through auctions because, contrary to direct negotiations, auctions involve competition among buyers, thereby increasing the forest owner’s expected revenues, as shown by [

Consider a stand of trees with one species. The trees can be the same age or different ages. Let t be the current period, and

where ^{2}. Following [^{3}. I will discuss the case of a random walk process follow. The process is given by:

The stochastic term

Let’s assume that at each period t there are N potential risk-neutral firms competing for the possession of the standing timber. Prior to the auction, the forest owner announces the total volume of timber ^{4}. It is assumed that each firm

^{5}. Suppose that each

where

^{6}

The dominant strategy for each firm in an ascending auction is to reveal its private value

In the first price auctions, the expected payment of a firm with cost parameter

where

forest owner can now be written as:

where K is the planting cost,

To solve problem (8), I first derive the optimal rotation

As

Equation (10) is an extension of [^{7}

The price

Now let’s find the optimal rotation and the optimal reservation price when the auction takes place in other words, the forest owner finds it optimal to allow the harvest of the forest stand. This decision occurs when

order condition for optionality given by ^{8}

Denoted by

Denoted by

Clearly, in the context of forestry, Laffont and Maskin’s and Riley and Samuelson’s result is suboptimal. Finally, the optimal rotation

Equation (14) means that, as the stumpage price increases, the new rotation is shorter than the old rotation. This leads to “over-mature” timber that will be cut, thereby increasing the volume of timber and hence the supply.

The previous results were obtained under the assumption that the stumpage price process follows a stationary autoregressive process. If, for example, stumpage prices follow a random walk without drift^{9}, the best prediction of the future price is the current price. It follows from the arbitrage equilibrium condition (9) that the optimal rotation is independent of the reservation price and satisfies the well-known Wicksell rule that states the timber will be cut when its relative growth equals the interest rate (

This article combines auctions and the forest management problem to analyse the optimal reservation price in a forest when auctions are used to sell standing timber under uncertain prices by assuming that the price process follows an autoregressive model. The analytical results show that the decision of the forest owner is to auction the standing timber when the current price is higher than a minimum price level that is invariant over time but depends on the parameters of the process. Therefore, to maximise profits, the forest owner will set a reservation that takes into account the optimal harvest time. This optimal reservation price is an extended version of Laffont and Maskin and Riley and Samuelson reservation price ( [

We thank the editor and the referee for their comments.

Francis Didier Tatoutchoup, (2016) Theoretical Reserve Price in Forestry. Theoretical Economics Letters,06,761-767. doi: 10.4236/tel.2016.64080

Recall that the optimal reservation price is

where

Let

tion (10) I obtain,

I also have

It follows from Equation (18) and Equation (17) that

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