Ecological Economics 70 (2011) 1565–1567

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Ecological Economics

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Commentary

Smallholder timber sales along the Transamazon Highway: a comment

Robert Walker a,⁎, Eugenio Arimab,c

a Department of Geography, Michigan State University, East Lansing, MI, United Statesb Department of Geography and the Environment, University of Texas, Austin, TX, United Statesc Instituto do Homem e Meio Ambiente da Amazônia (IMAZON), Belém, Brazil

A R T I C L E I N F O

Article history:Received 1 October 2009Received in revised form 17 November 2010Accepted 21 November 2010Available online 4 March 2011

Keywords:AmazonLoggingForestryDeforestation

1. Introduction

In a recent article published in this journal, Amacher et al.(2009) develop a model of timber resource exploitation by smallholders in the Brazilian Amazon. Such individuals control largestocks of tropical hardwoods, so gaining insight into their forestrypractices is key to effective environmental policy. Amacher et al.(2009) consider a single period optimization of householdwelfare. They provide a formal statement of this problem, andconstrain it with respect to household labor, land, and timberresources. They use their statement to lay the groundwork for astatistical examination of household decisions with respect toextraction of non-timber forest products, agricultural activities,and the sale of wood to loggers. Amacher et al. (2009) claim thattheir study area, central Pará State, possesses incomplete labormarkets, which motivates a “non-separable” formulation. In theabsence of a wage rate, a key theoretical variable is μ, the“opportunity cost of household time” (Amacher et al., 2009,1790). The present comment applauds efforts like those ofAmacher et al. (2009), but takes issue with certain aspect oftheir model statement, as well as claims they make about theempirical world it represents. It offers an alternative formulation,and closes with several remarks about the empirical situation themodel is meant to address.

⁎ Corresponding author.E-mail address: rwalker@msu.edu (R. Walker).

0921-8009/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolecon.2010.11.018

2. Model Issues

Amacher et al. (2009) seek to maximize utility subject toconstraints on labor and available timber. Specifically, their max-imand is

U f S0 A−A� �

−Ss; LN� �

; x;Qc; T−L;Ωh i

;

where variable definitions, parameters and functions are given inTable 1. The first argument of the utility function, non-timber forestproducts (NFTPs), is governed by a production function, f(y, LN), withy=S0(Ā−A)−Ss. This is increasing in forest stock, y, and laborallocated to NFTP extraction, LN; it is decreasing in harvest, Ss. Qc, theamount of agricultural production consumed on site by the house-hold, is governed by a production function Q=Q(A, LA, KA), thesurplus of which, Q−Qc, is marketed. Leisure enters the utilityfunction as T−L, where L=LN+LA. (Amacher et al., 2009, p. 1789).Amacher et al. (2009) constrain harvest, Ss, as 0≤Ss≤S0Ā; income asI=pfSs+R + pA(Q−Qc)+wZ−x−rKA ≥0; and labor as L = T− l.Their Lagrangian is

EU f :ð Þ;x;Qc;T–L;Ωð Þ + λI + μL + γ Ss−S0Ah i

η Ssð Þ sicð Þ

with multipliers λ, μ, and η. The choice variables are LN, LA, Ss, x and Qc(Amacher et al., 2009, p. 1790, footnote 8). Aside from its manifesttypo, we take issuewith this statement for twomain reasons. First, theupper bound to harvest allows the household to remove wood fromthe entire property, although A has already been cleared foragriculture. Thus, a negative forest volume enters the NFTP produc-tion function if harvest attains its upper bound. Second, the laborconstraint is incomplete in the Lagrangian, where the multiplier termis given as μL (Amacher et al., footnote 8). That is, T and l are missing.1

The present comment gives a statement of the optimizationproblem consistent with the empirical circumstances described byAmacher et al. (2009). This does not mean we concur with theirdescription, particularly as regards the labor market, an issuebroached in the conclusions. More than one way can be found toformulate an optimization problem; the approach taken here reflectsour objectives to (1) correctly state the upper bound on harvest, andto (2) provide a labor constraint yielding the shadowprice of time as it

1 Total time endowment, T, and leisure, l, need to appear, whether the constraint isan upper bound or an equality.

Table 1Terms in Amacher et al. (2009).

Variables and parametersSs wood volume extractedLA agricultural laborLN extractive laborX market goodQ c subsistence consumptionQ agricultural productionl leisureA ̅ property sizeA area in agricultureT time endowmentKA capitalPf wood price (x, numeraire)pA farm good price (x, numeraire)S0 density of woodΩ household attributesI income

ConstraintsSs≤S0(A ̅) Inequality 1 in Amacher0≤Ss Inequality 2 in Amacher0≤pfSs+R+pA(Q−Q c)+wZ−x− rKA= IL=LA+LN=T− l

FunctionsNon-timber forest products: f= f (S0(A ̅−A)−Ss), LN)Farm production: Q=Q(A, LA, KA)Utility: U( f, x, QC, l)

1566 R. Walker, E. Arima / Ecological Economics 70 (2011) 1565–1567

relates to allocation across both labor and leisure activities. Let utilitybe U[n, x, Qc, l], where x and Qc are as above, l is leisure, and the non-forest timber product, NFTP, is n. Rewrite as U[n(Ss, LN), x(LA, Ss, Qc),Qc, l], where n(Ss, LN)≡ f(S0(Ā−A)−Ss, LN) as in Amacher et al.(2009). In a departure from Amacher et al. (2009), embed the incomeconstraint in the purchase of the market good, x=x(LA, Ss, Qc)≡pfSs+pa(Q(LA)−Qc)+R−rKA, where A and KA are suppressed in theproduction function, and wZ is eliminated, under the assumption itreflects wages earned by out-hiring of household labor, expresslyforbidden (Amacher et al., 2009).2 Now, rewrite the utility function asU[n(Ss, LN); x(LA, Ss, Qc); Qc; l]≡g(LN, LA, Ss, Qc, l), and restate themaximization problem:

LN ; LA; Ss;Qc; lMaximize g LN ; LA; SS;Qc; lð Þ;

subject to LN+LA+ l=T; 0≤LN, LA, l; and 0≤Ss≤S0(Ā−A). Thisdiffers from Amacher et al., (2009) in (1) stating the time constraintwith equality; in (2) restricting harvest to uncleared lands; and in (3)embedding income directly in the optimization problem as anequality. Solution by the Karush–Kuhn–Tucker conditions involveseleven unknowns, including the five choice variables, and multipliersfor the time constraint, the bounds on harvest, and non-negativity forlabor allocations and leisure. For illustrative purposes, assumepositive labor and leisure choices, obviating the need for associatednon-negativity constraints; this is reasonable for smallholders in thestudy area. Following conventions in signing to yield positivemultipliers for the remaining inequality constraint, the Lagrangian is(Miller, 1978):

g LN ; LA; Ss;Qc; lð Þ–μ LN + LA + l–Tð Þ + ηSs–γ Ss−S0 A−A� �h i

;

2 Amacher et al. (2009) formulate this as a non-negativity constraint. In the absence ofsavings and investment, it would be an equality, the likely case. Amacher et al. (2009) alsodefine income, I, as earnings plus other revenues (wood and crop sales+R+wZ) minusexpenditures (goods purchased, payments to capital). This includes an undefined term, wZ(Amacher et al., 2009, 1790). All of this shows up as I in the statistical Eq. (7).

and the first order conditions are:

Un fLN−μ = 0 ð1Þ

paUxQ LA−μ = 0 ð2Þ

Ul−μ = 0 ð3Þ

−Un fy + pf Ux + η−γ = 0 ð4Þ

−paUx + UQ c= 0 ð5Þ

LN + LA + l = T ð6Þ

ηSs = 0 ð7Þ

γ Ss−S0 A−A� �h i

= 0 ð8Þ

where Uz = ∂U∂z ; fz =

∂f∂z, and Q z =

∂Q∂z for arbitrary z. This system has

eight unknowns, [LN, LA, Ss, Qc, l, μ, η, and γ], and must be solved byconsidering alternative cases for the inequality multipliers; theseallow for three feasible solutions, [η, γ=0], [ηN0, γ=0], and[η=0, γN0]. Due to space limitations, we consider only the first,which implies an interior solution for Ss (as in Eq. (7) of Amacher et al.,2009). Thus, with positive but less than maximal harvest, Eqs. (1)–(6)(replacing 4 by 4a:−Unfy+pfUx=0) yield six solution values [LN,* LA,*Ss,* Qc,* l,* μ*], which maximize household welfare under appropriateassumptions about utility and production functions.

3. Discussion

The reformulated problem presented here allows a corner solutionat the upper bound of harvest that does not exceed the supply of rawmaterials. Further, the labor constraint provides a multiplier, μ,interpretable as the shadow-price of time, namely the marginal utilitygain from a unit increase in household time endowment (see Varian,1978, p. 269). An unresolved issue in the statistical estimation is thatAmacher et al. (2009) use for their dependent variable “total quantityof wood sold,” which includes, presumably, sales from initialoccupation of the property (See their Footnote 4). This is inconsistentwith the modeled choice variable, Ss, the harvest off lands remainingin the “reserve,” after agricultural deforestation, A.3 A final issueinvolves the labor market assumption; specifically, markets exist forgoods (i.e., x and Q−Qc) and logs, but not labor. This contradicts ourobservation of a labor market in the study area, central Pará State. Infact, by 1996, many smallholders hired workers in the region, and arural wage rate existed (10 reais per day, the diario; see Walker, 2003,p. 392; Caldas et al., 2007, p. 97). By 2003, when Amacher et al. (2009)conducted their field work, power lines delivered electricity to muchof the eastern Transamazon betweenMarabá and Itaituba. Many smalltowns had bank branches, and carros de linha brought transportationservices to the settlement roads, or travessões, leading off theTransamazon Highway. We by no means suggest that all smallholdersin the study area are labor market “active,” and have encounteredthose that fit a “non-separable” description. But if loggers can drive tosmallholdings and pay for logs, workers can hitch a ride and hire outto farmers flush with earnings from wood sales. In 2007, nearly800,000 people lived in the counties of Pará State traversed by theTransamazon Highway (Walker et al., 2011). Presumably, many ofthem were rural workers.

3 A secondary point is that the right-hand-side of their Eq. (7) possesses anargument, I, that is zero if the budget constraint is functioning. The parts of I that arefixed household attributes (e.g., R) should enter the right-hand-side of (7); revenuesfor wood sale, pfSs, are endogenous.

1567R. Walker, E. Arima / Ecological Economics 70 (2011) 1565–1567

Acknowledgements

The authors would like to acknowledge support from the NationalScience Foundation under BCS-0822597: Territorializing ExploitationSpace and the Fragmentation of the Amazon Forest. The viewsexpressed belong to the authors and do not necessarily reflect those ofthe National Science Foundation.

References

Amacher, G.S., Merry, F.D., Bowman, M.S., 2009. Smallholder timber sale decisions onthe Amazon frontier. Ecological Economics 68, 1787–1796.

Caldas, M., Walker, R., Perz, S., Arima, E., Aldrich, S., Simmons, C., 2007. Theorizing landcover and land use change: the peasant economy of colonization in the AmazonBasin. Annals of the Association of American Geographers 97 (1), 86–110.

Miller, R.E., 1978. Modern Mathematical Methods for Economics and Business. Kreiger,Huntington, New York.

Varian, H.R., 1978. Microeconomic Analysis. Norton, New York.Walker, R., 2003. Mapping process to pattern in the landscape change of the Amazonian

frontier. Annals of the Association of American Geographers 93 (2), 376–398.Walker, R., Perz, S., Arima., E., Simmons, C.S., 2011. The Transamazon Highway:

Past, Present, and Future. In: Brunn, S. (Ed.), In Engineering Earth: The Impactsof Megaengineering Projects. Kluwer Academic Publishers, The Netherlands,pp. 569–600.