Francesco CARBONE, Antonino SCARELLI, Zoltan VARGA

1 Department for Innovation in Biological, Agro-food and Forest Systems, University of Tuscia, Largo dell’Università(Building E), Viterbo I–01100, Italy

2 Department of Ecology and Biology, University of Tuscia, Largo dell’Università (Building E), Viterbo I–01100, Italy 3 Institute of Mathematics and Informatics, Szent István University, Páter K. u. 1. H-2103, Budapest, Hungary

Mathematical modeling for evaluating gain of logging companies in the timber market

Francesco CARBONE1,?, Antonino SCARELLI2, Zoltan VARGA3

1Department for Innovation in Biological, Agro-food and Forest Systems, University of Tuscia, Largo dell’Università(Building E), Viterbo I–01100, Italy

2Department of Ecology and Biology, University of Tuscia, Largo dell’Università (Building E), Viterbo I–01100, Italy3Institute of Mathematics and Informatics, Szent István University, Páter K. u. 1. H-2103, Budapest, Hungary

We propose a game-theoretic model for a fi rst-price sealed-bid auction (FPSB auction) of standing timber.A FPSB auction means that bidders simultaneously submit sealed bids so that no bidder knows the bid of any other participant. The highest bidder pays the submitted price. The mathematical model has been elaborated considering only two bidders. Each bidder is supposed to know only the probability distribution of the bid of the other bidder. Using appropriate distributions, the adversity and propensity of a bidder to risk can be described. From the view point of a bidder,the conservative solution is calculated for both types of the other bidder. If a standing timber belongs to a public owner,the market has to follow a public evidence procedure, including public auctions. The function obtained is illustrated with a numerical example of a typical standing timber auction from a forest located in central Italy. Considering average costs and prices of the local chestnut timber market, winning bids and expected gains are estimated. Their amounts are comparable with the currently average stumpage price registered in this market.

game theory, standing action, stumpage value

?Author for correspondence (Francesco CARBONE)

E-mail: fcarbone@unitus.it

Introduction

An auction is a process frequently used for buying and selling many goods or services. One of them is the standing timber of private or public forests in many countries of the world (Toivonen,1997). Authors and organisations have reported how these auctions are carried out (Blin and Hendricks, 1997; Paarsch, 1997; Stone and Rideout,1997; Office National des Forest, 2000; Forest Commission, 2002; Zheng, 2009).

Elyakime et al. (1994) and Li and Perrigne(2003) have modeled standing auctions on the basis of independent unique private values where bids concern the price of the lot. Elyakime and Loisel (2005) instead, produced an economic model where they evaluated the fi nancial impact on the seller, whether they know the timber volume of the lot or not and whether to keep the reserve strategic value secret or not.

There are several variations of the basic auction form. A fi rst-price sealed-bid auction (FPSB auction), also known as sealed fi rst-price auction, is the version considered in our study. All bidders simultaneously submit sealed bids but no bidder knows the bid of the other participants. The submitted bids are then compared and the bidder who submits the highest bid buys the standing timber. This is the market price the bidder has to pay to the seller ( McAfee and McMillan, 1987).

For the validation of our mathematical model we developed a numerical example considering the case of a common local market located in central Italy, where chestnut coppice of public owners is usually sold and all the logging companies which operate in this market come from the nearby area. Historically, coppice generally represents an important source of wood in Italy, suggesting that low-input coppice may be an efficient and sustainable biomass production system (Picchio et al., 2009). In this particular context, the chestnut coppice forests are renowned for their wide typologies of market products (from round wood to poles for agriculture or fencing). The chestnut coppice forests of central Italy are also quoted as some of the most productive forests.

In this context, applying a particular gametheoretical approach, we want to find the price the logging companies (bidders) would offer to the forest fi rm (seller) for purchasing the standing timber. The bids and the corresponding expected gains are the fi nancial parameters that will be calculated under the hypotheses of the model, corresponding to their probabilistic bidding behavior.The latter will be modeled for bidders that are either adverse or have a propensity for risk.

The formalization of the conflict situation taking place in an auction mainly depends on the information the bidders are supposed to have about the valuation and the possible bidding strategy of other bidders. In the present study an individual private value action is considered which means that the bidders only know their own valuation of the item in question.

Since we will deal with continuous random variables, representing the bidding of the partner bidder(s), ties have zero probability and therefore can be neglected. Our approach is different from the usual one (Aliprantis and Chakrabarti, 2000)at several points. First, in the usual approach, a bidding strategy is a function of the valuation of the object given by the bidder, while in our model the strategy itself will be a possible choice of a bid value taken from an interval (the market space,MS), determined by bidders. Furthermore, the usual approach provides a Nash equilibrium of the corresponding game, based on the uniform distribution of the valuation of the other players,since it looks rather diff i cult to fi nd a Nash equilibrium if the distribution is not uniform. When the problem is considered from the viewpoint of a distinguished player (Player 1), his bid is a deterministic value of this interval, while this play er only guesses according to a probability distribution describing what the bid of the other bidder(Player 2) would be. We will use non-uniform distribution s, w hich make it possible to include both risky and non-risky bidding behavior in the model. Finally, instead of an equilibrium solution, which implicitly involves a rationality on the part of the players, we will deal with the maximin or conservative solution for a fixed bidder. This solution is based on the idea that Player 1 counts on the worst case, supposing the unknown bid distribution of the other bidders might minimize his expected payoff and he, by his own strategic choice, tries to maximize this m inimum. We emphasize that although both players are logging companies, in our setting, their roles are not symmetric. We consider the bidding conflict strictly from the point of view of Player 1, who plays against Player 2, in the face of incomplete information. Therefore the description of the strategic choice of Player 2 is probabilistic.

Materials and methods

Types of bidding behavior

For simplicity we will suppose throughout the paper that there are only two bidders, considered players, in the auction and we look for a solution from the viewpoint of Player 1. Although for the sake of simplicity we consider that Player 1 plays only against a single player, i.e., Player 2, the conservative solution (based on the maximin or worst case approach mentioned in the Introduction)also makes sense if, apart from the distinguished player there are several bidders; our approach can be easily generalized to this case, without any conceptual diff i culty.

Assume, again for the sake of simplicity, that for both players the bids are rationally limited to a fi xed interval [a,b] witha＜b. We consider the situation where Player 1 has only a vague idea on the bidding behavior of Player 2. More precisely, in the fi rst case Player 1 assumes Player 2 would bid according to a supposed probability distribution in the limiting interval [a,b]. We shall consider three basic types of distributions, each describing different bidding behavior.

Type 1: Uniform distribution expressing a kind of bidding behavior neutral to risk in the sense that higher or lower bids are equally probable.The corresponding density and distribution functions are:

Type 2: The following density and distribution functions describe a bidding behavior averse to risk (higher biddings are less probable):

Type 3: A bidding behavior with a propensity for risk is modeled by the following density and distribution functions:

For a more general model, let us assume Player 2 will behave according to a combination of distributionsF1andF2, orF1andF3, potentially displaying different grades of adversity and propensity to risk.

Conservative solution in bidding games

In our model setup, providing a conservative (in other terms maximin or pessimistic) solution for Player 1, we will proceed in the following way:given a strategyv1of Pl ayer 1, Player 2 is supposed to bidv2according to a distribution belonging to a parametrized family of distributions.This family of distributions is known to Player 1,but he does not know which concrete member of the family Player 2 would play. Player 1 counts on the worst case, i.e., he assumes his payoff(expected gain from the auction) would be minimized by the actual distribution played by Player 2. Therefore, Player 1 will choose a bidv1so as to maximize this minimum.

Playing against a player adverse to risk

Next we show that the coefficient ofλis always positive. To this end, it is enough to prove that the functionp(v1) = (b–v1)(v1–a) –v12+2bv+a2– 2abis positive within th e interval ofv[a,b1]. The solution of the equation (Eq. 8) is th1eand both are located outside the interval [a,b]. Hence we havep(v1) ＞ 0(v1[a,b]). Therefore,is attained atλ= 0, i.e., Eq. 9.

For the maximi zation of the latter, with respect tov1[a,b], it is enough to apply the inequality between the arithmetic and geometric means to the numbers (b–v1) and (v1–a). We obviously obtain Eq. 10, and the equality holds iff.Hence, Eq. 11 was obtained.

Playing against a player with a propensity for risk

Reasoning in a way similar to the previous section, we consider the following family of distributions:Ψμ= (1 –μ)F1+μF1(μ[0, 1]).

Now the payoff of Player 1 is Eq. 12.

The coefficient ofμis negative fo r allv1[a,b], since for the functionq(v1) = (v1–a)2+ (a–v1)(b–a). We clearly haveq(a) =q(b) = 0 andq(v1) ＜ 0 (v1[a,b]). Hence for allv1[a,b] the equality is Eq. 13.

A straightforward calculati o n shows that the maximum of (b–v1)(v1–a)2is attained at. This is now the conservative solution for Player 1 and corresponding expected gain is Eq.14.

Timber market

General aspects

The sale of standing timber is one of the crucial phases of forest management. It involves two different types of enterprises, forest firms (FF) and logging companies (LC). Both are characterized by a strong complementarity to the mutual benef i t of each other, ensuring the completion of the production process in the fi rst part of the forestwood chain.

A FF manages the forest during the long period of growth and its relevant factors are land, standing timber and farm management. Major management decisions are: when to cut the stand and which LC would buy the standing timber.

The FF does not harvest directly its standing timber that should be cut at the end of the rotation or at any intermediate time. Standing timber is sold to a LC which owns factors of production such as manpower, machines and other facilities,capital and management, that includes know-how necessary to transform the standing timber into products for the market and it also has the knowl-edge necessary for their optimum allocation in the market.

Between these two enterprises a market will be established: FF supports the market supply, LC supports the market demand, while standing timber is the product of the market. Bargaining between the parties takes place in different ways. It depends on the legal status of the FF: a market is carried out vis a vis, if FF is privately owned without any public knowledge of the way the price is def i ned; otherwise, if FF is a public owner, the market has to follow a public evidence procedure.This includes a public call, open to all interested LCs, which specifies the conditions, the procedure, the timing of the public auction of standing timber and its minimum price. This latest price is denoted stumpage price and it is a function of the value of the marketable timber products, reduced of total production costs (Picchio et al., 2011), except the price for purchasing the standing timber.

Lower and upper limit calculation in the chestnut timber market

We developed a numerical example in which is considered the case of a local market located in central Italy where standing chestnut coppice of public owners are usually sold and all LCs operating in this market come from the nearby area. In this area chestnut forests are managed as coppice,with the fi nal cutting made at age 20–25 years.

The mathematical model has been implemented using average costs and prices which characterizes the LCs in this market. The goals are to determine the lower and upper limitsa,bfor the bids and the expected prof i t of a LC according to the two differently inclined types of risk.

In this standing market each LC can produce different types of marketable products according,first of all, to the equipment and other facilities they have. Usually, ordinary LCs maximize the production for the agricultural sector (e.g. poles);advanced LCs, instead, maximize round wood production, for a market with high standards of quality.

The first step is to determine the lower bid limit, i.e., parameteraof the models. There is no specif i c and direct market where standing timber is quoted, and consequently, its value is obtained indirectly from the estimate by the consultant of the FF. Its main goal is to evaluate the standing trees so as to guarantee the maximum number of LCs participating in the auction. To this end, the FF consultant considers a set of marketable products produced by most LCs, which have more or less the same manpower, equipment and other facilities, capital and management. The LC, representative of most local LCs, is referred to as the ordinary LC and denoted by LCord.

The stumpage value (SV), before tax, is the total market value of marketable products, minus their total production cost net of the standing timber price SP. Formally, fori=1,…,m, let PMIibe the most frequent market price of thei-th marketable product, taken into consideration by the FF consultant;Qiis the amount of thei-th marketable product produced and hypothesized by the FF consultant;kiis the transformation cost of the standing timber into thei-th product, net of purchasing cost of the standing timber.

Then the lower limit for the bid is

here, the stumpage value used in the model is the mean value with which the public owners have recently placed the standings of chestnut coppice on the local market. Expectations of the FF are that the bid is as high as possible, at best equal to the upper limit (Fig. 1). The upper limit is the maximum bid that the LCs are willing to pay for the standing timber. According to the capacity of its own factors of production, LCs define the most convenient types and quantities of products when calculating the maximum bidb. Considering a specific LC＊, we assume it has at least one factor of production different from LCord.

Fig. 1 Market space and bid

Given its advanced equipment and other facilities, LC＊ could produce more higher valued market products. From an economic point of view,the production process guarantees minor production costs and/or major revenues.

Formally, forj=1,…,n, let PM Ij＊ be the market price of thej-th marketable product, taken into consideration by LCs.Qj＊is the amount of thej-th marketable product produced a ndkj＊the transformation cost of the standing timber into thej-th product, net of purchasing cost of standing timber.

The maximum bid is def i ned as:

where the upper limit amount has been calculated as the mean of the local market. Assuming the value of the best products obtainable from the standing of chestnut coppice, it has been reduced by the total production costs that LC should sustain. In contrast to the FF, the expectation of the LC is to buy the standing offering a bid very close to the lower limit (Fig. 1).

As shown in Fig. 1, the lower and upper limits define the market space of the forest auction,where the bidding strategies are included. The different directions of the axis indicate the opposite expectations of FF and LC.

Results

In Table 1 an example of a public standing auction is presented using values expressed for an ordinary plot per hectare. In the forest context,chestnut forests are renowned for their wide typologies of market products (from round wood topoles for agriculture or fences), where the chestnut forests of central Italy are also noted as some of the most productive forests. Differences in management can inf l uence the production, but except for a particular situation, usually private and public landowners have a guaranteed high management standard in order to supply the market with quality products. Differences in assortments are, instead, strictly a function of the organization,mechanisation and the skill of the workers of LCs.Most local LCs manage to produce ordinary products from chestnut timber, with production evaluated at 10,397.95 €·ha–1, while more advanced LCs can obtain products with a upper market value of 11,895.20 €·ha–1. Two different production processes should be implemented to obtain these results, the costs of which are 6,080.07 and 4,884.09€·ha–1, respectively. According to these production values and transformation costs, the stumpage values can be evaluated. These are 4,317.89 €·ha–1,marking the start, i.e., the lower limit of the market space, while 7,011.12 €·ha–1is the end, its upper limit. For the LC with a risk adverse attitude,the winning bid is 5,664.50 €·ha–1and the expected gains are 673.31 €·ha–1. In the case of a LC with a propensity to risk, the winning bid is higher while the expected gain is lower, i.e., 6,113.37 and 399.00 €·ha–1respectively.

Table 1 Determination of a conservative solution against a logging company with a different inclination to risk, related to the standing timber market of a chestnut coppice of central Italy

Discussion and conclusion

According to the mathematical model, the amplitude of the market space, the bids and the expected revenues (gain) for the distinguished bidder are calculated, according to the more risky or less risky bidding behavior of the other LCs.

Comparing the results we see that in the considered context of a first-price sealed-bid auction, Player 1 should bid a lower value (a+b)/2 against a Player 2 averse to risk, than against a Player 2 with a propensity for risk, i.e., (a+ 2b)/3;his expected gain from the auction is higher in the first case, (b–a)/4 and lower, 4(b–a)/27 in the second case.

The situation, when Player 1 plays against several players, can be handled similarly. It is also possible to build into the model conditions where some players play more risky bidding strategies,while others are averse to risk. If we assume that the parameters of the corresponding families of distributions can be independently attributed to different players, then for the conservative solution the minimization should be carried out over all possible parameter vectors and Player 1 then maximizes this minimum by his appropriate bidding choice.

Finally, we emphasize that, unlike Aliprantis and Chakrabarti (2000), in our approach the different bidding behavior is modeled by nonuniform distributions appropriate to express both less risky and more risky types of behavior.

The mathematical model developed is not su itable to the single sales of standing timber. In fact, our mathematical model is probabilistic and every single sale is a random realization of this model. On the one hand, the choice of strategy of Player 1, obtained from the model, is valid for the average revenue gained from several subsequent auction games played against Player 2 having the same adversity/propensity to risk. On the other hand, in a given market, sales of standing timber strongly depend on positive and negative characteristics of the standing timber, of the specific characteristics of LCs (manpower, machinery,etc.) and their business strategies, where this latest is not always rational from an economic standpoint.

However, the model has its best performance when it is applied to a timber market, using average values. The implementation in the numerical example relates to a coppice chestnut forest auction in a local market of central Italy and provides results consistent with current average market prices, where stumpage prices fl uctuate between 5,000 and 7,500 €·ha–1.

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10 May 2012; accepted 27 July 2012