Optimal Forestry Contract with Interdependent Costs

Francis Didier Tatoutchoup; Paul Samuel Njiki

doi:10.1515/bejte-2016-0058

Article

Optimal Forestry Contract with Interdependent Costs

Francis Didier Tatoutchoup and Paul Samuel Njiki

Published/Copyright: June 9, 2018

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal The B.E. Journal of Theoretical Economics Volume 20 Issue 1

Abstract

This article determines an optimal forestry contract when firms’ harvesting costs are interdependent. The value of the optimal allocation depends on the private signals of all firms. We show that the optimal rotation of the winning firm must satisfy a modified version of the usual Faustmann rule, which holds under perfect information. This modification is necessary in order to induce the revelation of private signals on the part of all participating firms. We find conditions under which the optimal mechanism can be implemented as a second-price auction. The optimal rotation period and the reservation price are derived. Theoretically and numerically, we show that the predicted forest owner surplus is considerably misestimated under the independent private value paradigm and the predicted forest owner profit is more affected when the interdependence is negative.

Keywords: auctions; faustmann rule; forest rotation; forestry contract; interdependent costs; mechanism design

Appendix

Proof of condition (14)

gi∗(θ)=KX′(ti∗)−rX(ti∗)λ(ti∗)=S(ti∗), , where S(t)=K(X′(t)−rX(t))/λ(t) and S′(t)=K[X′′(t)−rX′(t)]λ(t)−λ′(t)[X′(t)−rX(t)]λ(t)2. satisfies

(27)t

Recall that S′(t)<0 . Using eq. (27) we can rewrite gi∗(θ)=gj∗(θ) as

S(ti∗)=S(tj∗)

where s(.) . This means

ti∗=tj∗

For all ∂gi∗(θ)∂θi=−∂iCi(θ)X(ti∗)erti∗−1−(1−αi){hi′(θi)∂iCi(θ)+∂iiCi(θ)hi(θi)}X(ti∗)erti∗−1∂gj∗(θ)∂θi=−∂iCj(θ)X(tj∗)ertj∗−1. satisfying eq. (27), gi∗(θ)=gj∗(θ) . Therefore, ti∗=tj∗ is equivalent to ∂iCi(θ)>∂iCj(θ) . Since ∂gi∗(θ)∂θi<∂gj∗(θ)∂θi is strictly decreasing, it follows that (i) . By the envelope theorem we get:

gi(θ,ti)

If (ψi(θ)X(ti)−Kerti)/(erti−1) then Ti∗(θ) and given that ∂gi(θ,ti)/∂ti=(ψi(θ)λ(ti)−rK)/(erti−1)2 it follows from the preceding equations that ∂gi(θ,ti)/∂ti=0 .

Proof of Lemma 2

ψi(θ)λ(ti)−rK=0 We can rewrite ψi(θ)λ(Ti∗(θ))−rK=0. as θih>θil . For interior solutions, ψi(θih,θ−i)λ(Ti∗(θih,θ−i))=ψi(θil,θ−i)λ(Ti∗(θil,θ−i))=rK>0∂ψi(θ)∂θi=−(1+(1−αi)hi′(θi))∂iCi(θ)−(1−αi)hi(θi)∂iiCi(θ)<0. satisfies^[10]:

(28)λ(Ti∗(θih,θ−i))>λ(Ti∗(θil,θ−i))

Assume λ′(ti)=(1−e−rti)[−X′′(ti)+rX′(ti)]>0 , then

Hence, λ is decreasing in Ti∗(θih,θ−i)>Ti∗(θil,θ−i) which implies that (ii) and therefore gi∗(.,θ−i)−gj∗(.,θ−i) . Observe that v because [θ_,θ‾] is increasing and strictly concave. Thus, v(θ)=0⇒v′(θ)<0 is increasing and we may conclude that v(θh)>0⇒v(θ)>0∀θ<θh . gi∗(θih,θ−i)>max{0,maxj≠igj∗(θih,θ−i)} follows from the application of lemma 3 below to the function gi∗(θih,θ−i)>0 .

Lemma 3.

If gi∗(θih,θ−i)−gj∗(θih,θ−i)>0 is differentiable in i≠j and satisfies gi∗(θi,θ−i)−gj∗(θi,θ−i)>0∀θi<θih , then ψi(.,θ−i) .

The proof of the lemma is provided below.

Indeed θi means that

gi(.,ti) and θi for gi∗(.,θ−i) . Using the condition eq. (14) and the preceding lemma, we deduce that gi∗(θil,θ−i)>0 . Since (iii) is decreasing in θih>θil∈△i(θ−i) we also deduce that gi∗(θih,θ−i)<max{0,maxj≠igj∗(θih,θ−i)} is decreasing in qi∗(θih,θ−i)=0 . Therefore −∂2ωi∗(θih,θi,θ−i)qi∗(θih,θ−i)=0≤−∂2ωi∗(θil,θi,θ−i)qi∗(θil,θ−i) is non increasing as the maximum of decreasing functions, thus gi∗(θih,θ−i)>max{0,maxj≠igj∗(θih,θ−i)} .

(ii) Fix gi∗(θˆi,θ−i)>max{0,maxj≠igj∗(θˆi,θ−i)},∀θˆi≤θih , if gi∗(θˆi,θ−i)>0⇒ψi(θˆi,θ−i)X(Ti∗(θˆi,θ−i))>KerTi∗(θˆi,θ−i)⇒ψi(θˆi,θ−i)>0. , then

λ(Ti∗(θˆi,θ−i))>0 and ∂2ωi∗(θih,θi,θ−i)qi∗(θih,θ−i)−∂2ωi∗(θil,θi,θ−i)qi∗(θil,θ−i)=∂2ωi∗(θih,θi,θ−i)−∂2ωi∗(θil,θi,θ−i)=∂∂θˆi[∂2ωi∗(θˆi,θi,θ−i)]∣θˆi=θio(θih−θil),θio∈(θil,θih)=∂iTi∗(θio,θ−i)∂iCi(θ)λ(Ti∗(θio,θ−i))erTi∗(θio,θ−i)(erTi∗(θio,θ−i)−1)2>0. . Now suppose that θ_<θo<θh<θˉ .

v(θo)≤0 implies that v(θh)>0 .

v(θo)=0

It follows from eq. (28) that: v(θo)<0 . Using the mean value theorem we may write the following equalities:

v(θh)>0

Proof of Lemma 3

Assume that θoˆ∈(θo,θh) , v(θoˆ)=0 and θ1=Sup{θ∈[θo,θh]:v(θ)=0} . We can assume without loss of generality that v . Indeed, if v(θ1)=0 then since v′(θ1)<0 , the intermediate value theorem implies that there exists v such that θ1 . Let θ2∈(θ1,θh):v(θ2)<0=v(θ1) . Because v(θ3)=0 is continuous we have θ3∈(θ2,θh) and therefore θ1 . This means that θ3>θ1 is locally decreasing around θ1 : there exists θ2∈(θ1,θh):v(θ2)<0=v(θ1) . Again the intermediate value theorem implies that v(θ3)=0 for some θ3∈(θ2,θh) . This contradicts the fact that θ1 is the supremum since θ3>θ1 .

References

Baron, D.P. 1989. “Design of Regulatory Mechanisms and Institutions.” In Handbook of Industrial Organisation, edited by R. Schmalensee and R. D. Willing, 1347–1447. New York: North Holland.10.1016/S1573-448X(89)02012-1Search in Google Scholar

Branco, F. 1996. “Multiple Unit Auctions of an Indivisible Good.” Economic Theory 8:77–101.10.1007/s001990050078Search in Google Scholar

Chen, Y., and S. Xiong. 2015. “The Incentive Compatibility Critique.” mimeo, University of Bristol and National University of Singapore.Search in Google Scholar

Crémer, J., and R.P. McLean. 1985. “Optimal Selling Strategies under Uncertainty for a Discriminating Monopolist when Demands are Interdependent.” Econometrica 53:345–361.10.2307/1911240Search in Google Scholar

Crémer, J., and R.P. McLean. 1988. “Full Extraction of the Surplus in Bayesian and Dominant Strategy Auctions.” Econometrica 56:1247–1257.10.2307/1913096Search in Google Scholar

Faustmann, M. 1849. “On the Determination of the Value which Forest Land and Immature Stands Possess for Forestry, Reprinted in: M. Gane, ed.” Martin Faustmann and the Evolution of Discounted Cash Flow(in En & h, 1968), Oxford Institute Paper 42.Search in Google Scholar

Laffont, J.J., and J. Tirole. 1993. A Theory of Incentives in Procurement and Regulation. Cambridge: MIT Press.Search in Google Scholar

McAfee, R.P., J. McMillan, and P.J. Reny. 1989. “Extracting the Surplus in the Common-value Auction.” Econometrica 57:1451–1459.10.2307/1913717Search in Google Scholar

Myerson, R.B. 1981. “Optimal Auction Design.” Mathematics of Operations Research 6:619–632.10.1287/moor.6.1.58Search in Google Scholar

Paarsch, H.J. 1997. “Deriving an Estimate of the Optimal Reserve Price: An Application to British Columbian Timber Sales.” Journal of Econometrics 78 (1):333–357.10.1016/S0304-4076(97)00017-1Search in Google Scholar

Pavan A., I. Segal, and J. Toikka. 2014. “Dynamic Mechanism Design: A Myersonian Approach.” Econometrica 82 (2):601–653.10.3982/ECTA10269Search in Google Scholar

Payandeh, B. 1973. “Plonski’s Yield Tables Formulated,” in Ottawa: Departement of Environment, Publication No.1318: Canadian Forestry Service.Search in Google Scholar

Riley, J.R., and W.F. Samuelson. 1981. “Optimal Auctions.” American Economic Review 71:381–392.Search in Google Scholar

Tatoutchoup, D., and G. Gaudet. 2011.“The Impact of Recycling on the Long-run Forestry.” Canadian Journal of Economics 44:804–813.10.1111/j.1540-5982.2011.01655.xSearch in Google Scholar

Tatoutchoup, F.D. 2015. “Optimal Forestry Contract under Asymmetry of Information." Scandinavian Journal of Economics 117 (1):84–107.10.1111/sjoe.12083Search in Google Scholar

Tatoutchoup, F.D. 2017. “Forestry Auctions with Interdependent Values: Evidence from Timber Auctions.” Forest Policy and Economics 80:107–115.10.1016/j.forpol.2017.02.004Search in Google Scholar

Published Online: 2018-06-09

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/bejte-2016-0058

Keywords for this article

auctions; faustmann rule; forest rotation; forestry contract; interdependent costs; mechanism design