Shadow prices are well understood and are widely used in economic
applications. However, there are limits to where shadow prices can be applied
assuming their natural interpretation and the fact that they reflect the first
order optimality conditions (FOC). In this paper, we present a simple ad-hoc
example demonstrating that marginal cost associated with exercising an optimal
control may exceed the respective cost estimated from a ratio of shadow prices.
Moreover, such cost estimation through shadow prices is arbitrary and depends
on a particular (mathematically equivalent) formulation of the optimization
problem. These facts render a ratio of shadow prices irrelevant to estimation
of optimal marginal cost. The provided illustrative optimization problem links
to a similar approach of calculating social cost of carbon (SCC) in the widely
used dynamic integrated model of climate and the economy (DICE)

Khabarov, Nikolay

Obersteiner, Michael

Smirnov, Alexey

English

arXiv

arXiv.org e-Print Archive

Shadow prices and optimal cost in economic applications

Shadow prices are well understood and are widely used in economic applications. However, there are limits to where shadow prices can be applied assuming their natural interpretation and the fact that they reflect the first order optimality conditions (FOC). In this paper, we present a simple ad-hoc example demonstrating that marginal cost associated with exercising an optimal control may exceed the respective cost estimated from a ratio of shadow prices. Moreover, such cost estimation through shadow prices is arbitrary and depends on a particular (mathematically equivalent) formulation of the optimization problem. These facts render a ratio of shadow prices irrelevant to estimation of optimal marginal cost. The provided illustrative optimization problem links to a similar approach of calculating social cost of carbon (SCC) in the widely used dynamic integrated model of climate and the economy (DICE)

Khabarov, N.

Smirnov, A.

Obersteiner, M.

International Institute for Applied Systems Analysis (IIASA)

Shadow Prices and Optimal Cost in EconomicApplicationsNikolay Khabarova,∗, Alexey Smirnovb,c, Michael Obersteinera,daExploratory Modeling of Human-Natural Systems Research Group, Advancing SystemsAnalysis Program, International Institute for Applied Systems Analysis (IIASA),Schlossplatz 1, Laxenburg, A-2361, AustriabIntegrated Biosphere Futures Research Group, Biodiversity and Natural ResourcesProgram, International Institute for Applied Systems Analysis (IIASA), Schlossplatz 1,Laxenburg, A-2361, AustriacFaculty of Computational Mathematics and Cybernetics, Lomonosov Moscow StateUniversity, Moscow, 119991, RussiadEnvironmental Change Institute, Oxford University Centre for the Environment, SouthParks Road, Oxford, OX1 3QY, UKAbstractShadow prices are well understood and are widely used in economic ap-plications. However, there are limits to where shadow prices can be appliedassuming their natural interpretation and the fact that they reflect the firstorder optimality conditions (FOC). In this paper, we present a simple ad-hoc example demonstrating that marginal cost associated with exercisingan optimal control may exceed the respective cost estimated from a ratioof shadow prices. Moreover, such cost estimation through shadow prices isarbitrary and depends on a particular (mathematically equivalent) formula-tion of the optimization problem. These facts render a ratio of shadow pricesirrelevant to estimation of optimal marginal cost. The provided illustrativeoptimization problem links to a similar approach of calculating social costof carbon (SCC) in the widely used dynamic integrated model of climate∗Corresponding authorEmail address: khabarov@iiasa.ac.at (Nikolay Khabarov)Preprint submitted to arXiv November 28, 2022arXiv:2211.03591v2  [econ.GN]  25 Nov 2022and the economy (DICE).Keywords: Shadow prices, optimal marginal cost, social cost of carbon(SCC), Dynamic integrated model of climate and the economy (DICE),methodology, optimization, systems analysis, marginal value.1. IntroductionThe concept of shadow prices is widely known and has extensive applicationsin economics. There is a large body of literature devoted to exploration ofproperties and limitations of shadow prices in various settings e.g. in com-petitive general equilibrium models [1], valuation of nonmarket goods [2],industrial pollutants [3], environmental efficiency [4], carbon pricing andabatement costs [5], land use regulation [6], land allocation [7], cost of soilerosion [8], patent and knowledge stock valuation [9], water supply [10],power system reliability assessment [11], portfolio optimization [12], socialcost of carbon (SCC) estimation [13], [14].In economic applications formulated as a constrained optimization prob-lem, a shadow price is the change in the optimal value of the objective func-tion (e.g. utility) per unit of a constraint that is relaxed by an infinitesimalamount p.452 in [15]. In the context of continuous time optimal controlproblems, shadow price depends on time and is referred to as a “costatevariable” [16].In this paper, we focus on a non-linear constrained optimization problemsimilar to DICE [13], [14] and extend the analysis [18] to further explore themethod of estimating optimal cost based on shadow prices. Such methodinvolving a ratio of two shadow prices (or marginal values) is employed inDICE for estimation of the social cost of carbon. A similar approach of2deriving a crop price as a ratio of two shadow prices is suggested in [7]. Thepossibility of such derived price to deviate from an (exogenous) market priceis admitted in that paper, whereas explanation of this deviation is suggestedto be in the area of non-market values.The main driving question for the following analysis is whether the costestimate of an optimal action obtained via shadow prices would be equal tothe actual cost corresponding to an optimal solution. To answer that ques-tion, we (1) construct a simple utility maximization problem, (2) solve thatproblem numerically, (3) check first-order optimality conditions to ensurehigh accuracy of the obtained solution, (4) compare the cost of optimal con-trol versus its cost estimate obtained via shadow prices, and (5) demonstratethat the latter is an underestimate of the former. We further illustrate thatthe cost estimated through shadow prices is an arbitrary number, becauseit depends on a particular (mathematically fully equivalent) formulation ofthe optimization problem and can be lower, equal, or higher than the opti-mal cost. We finally conclude that despite shadow prices are derived fromthe first-order optimality conditions, the cost estimate obtained as a ratioof two shadow prices is, generally speaking, a not optimal arbitrary numberand, as such, is of no use in applications.2. Optimization problem formulationWe consider a hypothetical fruit storage facility operating in two time peri-ods i = {1, 2} and in these periods carrying pi ton of product (e.g. apples).The storage facility manager has to protect apples from fruit flies and indoing so they decide on the pesticide application rate µ ∈ [0, 1]. Apples,not fully treated with pesticides support flies’ reproduction and so virtually3”emit” ei flies, described by the flies emission rate σ per ton of untreatedapples. Flies in quantity ei produce damage to the stored product, whichis described by the share d(ei) of stored apples that are damaged by flies.Pesticides treatment cost is described by the cost function g(µ). The shareof product saved from consumption (selling) in period 1 is described by thesaving ratio s, which is a control variable along with µ. There is no pesti-cides treatment in the period 2 and all product left from period 1 (less theshare damaged by flies) is consumed (sold) in period 2. Flies are completelyremoved after the end of the 1st period and before beginning of the 2ndperiod, so that e1 does not have impact on the 2nd period. The objective ofthe fruit storage facility manager is to maximize utility, which depends onconsumption ci in each of the two time periods. Mathematical formulationof the problem is presented below:maximizeµ,s√c1 +√c2, (1)c1 = [1− d(e1)]p1(1− s)− p1g(µ), (2)e1 = σp1(1− µ), (3)p1 = const, (4)p2 = [1− d(e1)]p1s, (5)c2 = [1− d(e2)]p2, (6)e2 = σp2, (7)0 ≤ µ ≤ 1, 0 ≤ s ≤ 1, (8)µ, s : c1 ≥ 0, c2 ≥ 0. (9)43. Optimal cost of control and its estimate from shadow pricesLet’s consider the problem (1) – (9) for a simple set of functions and con-stants:d(ei) = ei, (10)p1 = 1, (11)g(µ) = µ2, (12)σ = 1. (13)So the problem (1) – (9) can be formulated asmaximizeµ,s√c1 +√c2, (14)c1 = (1− e1)(1− s)− µ2, (15)e1 = 1− µ, (16)c2 = (1− e2)µs, (17)e2 = µs, (18)0 ≤ µ ≤ 1, 0 ≤ s ≤ 1, (19)µ(1− s)− µ2 ≥ 0. (20)The problem (14) – (20) can be rewritten asmaximizeµ,s f(µ, s) (21)f(µ, s) =√µ(1− s)− µ2 +√(1− µs)µs, (22)0 ≤ µ ≤ 1, 0 ≤ s ≤ 1, (23)1− s ≥ µ. (24)The function f(µ, s) is strictly concave in the specified domain (23), (24)as illustrated by the plot of this function in Figure 1. Therefore, there is a5unique solution to the problem (14) – (20) which is an internal point of thedomain (23), (24) or, equally, of the domain (19), (20).00.20.40.60.81 0 0.2 0.40.6 0.8 100.10.20.30.40.50.6msFigure 1: Function f(µ, s) in the maximization problem (21) - (24), here we denotedµ ≡ m.Let’s consider an internal maximum point of the problem (14) – (20)i.e. when the constraints of the type of inequality (i.e. (19) and (20))are strictly satisfied. In this case Lagrange multipliers corresponding toinequality constraints are all zeros, and the first order optimality conditions(FOC) can be expressed through the Lagrange function L employing onlyfour multipliers λ1, λ2, ξ1, and ξ2 as follows:6L = −√c1 −√c2 − λ1[(1− e1)(1− s)− µ2 − c1]− ξ1[1− µ− e1]− λ2[(1− e2)µs− c2]− ξ2[µs− e2], (25)∂L∂µ= 2λ1µ+ ξ1 − λ2(1− e2)s− ξ2s = 0, (26)∂L∂s= λ1(1− e1)− λ2(1− e2)µ− ξ2µ = 0, (27)∂L∂c1= − 12√c1+ λ1 = 0,∂L∂c2= − 12√c2+ λ2 = 0, (28)∂L∂e1= λ1(1− s) + ξ1 = 0,∂L∂e2= λ2µs+ ξ2 = 0. (29)The problem (14) – (20) was solved numerically using GAMS1 and all equa-tions and inequalities (15) – (20) have been checked manually along withthe FOC (26) – (29) and found to be all satisfied with high precision in theorder of 10−8. Inequality constraints (23), (24) and, equally, (19), (20) arestrictly satisfied: µ = 0.5, s = 0.206. This indicates that the solution foundnumerically has high precision.According to the meaning of the Lagrange multipliers [15], λ1 and ξ1 areincrements in the optimal value of the objective function (14) when one unitis added to the right hand side of the equations (15) and (16) respectively.Let’s pose the question on how many units of consumption x (expressed inapples or equally in dollars) would need to be added to the consumptionequation in the first period (15) to compensate one unit of flies added tothe equation (16) so that the resulting value of the objective function (14)1https://www.gams.com7would not change. The answer is provided by the following equation relyingon the meaning of λ1 and ξ1 (where increment of the objective functioncorresponding to λ1 is scaled to x units)xλ1 + ξ1 = 0. (30)This equation holds for ”sufficiently small” λ1 and ξ1 with increasingly highprecision2 and states that the added quantities of one unit of ”flies” and xapples (dollars) lead together to zero increment in the optimal value of theobjective function (utility). So, x dollars compensate one unit of ”flies”,i.e. x can be seen as the price compensating one ton of more flies and isexpressed asx = − ξ1λ1= 0.794 ($/ton). (31)Despite this kind of ”cost estimate” x (cost incurred by one ton of flies)is calculated using optimal values λ1 and ξ1, it is not the optimal cost adecision maker would pay at the optimum for reducing one ton of flies,which is expressed as− dg(µ)dµ/de1(µ)dµ∣∣∣∣µ=0.5= 1.000 ($/ton). (32)The minus sign in front of the equation serves the purpose of having the costexpressed as a positive number since de1(µ)/dµ < 0. From (31) and (32) itcan be concluded that equation (30) along with the shadow prices λ1 andξ1 do not define the optimal pesticide treatment cost (expressed in dollarsper ton of flies reduction).2In mathematical terms, that means that the absolute deviation of the left-hand side ofthe equation from zero is constrained as |xλ1 + ξ1| ≤ γ(λ21 + ξ21) for some constant γ > 0.84. Arbitrariness of cost estimate from shadow pricesIt turns out that the value of the cost estimate from shadow prices dependson which of the fully equivalent definitions of the problem (14) – (20) isused for such estimate. Table 1 presents the respective values obtained fromequivalent optimization problems where one of the equations (17) or (18)(or both) were substituted with their respective equivalents:c2 = (1− e2)(1− e1)s, (17.a)e2 = (1− e1)s. (18.a)As evidenced by these results, depending on a particular way of specifyingthe same problem, there are cases when the cost estimate is less than, greaterthan, or equal to the optimal cost (32). This fact is easily explained by themeaning of Lagrange multipliers (marginal values to equations), which arelinked to a perturbed problem that substantially depends on a particularequivalent formulation of the original problem.Table 1: Optimal cost estimates obtained as a ratio of shadow prices from four equivalentformulations of the original problem (14) – (20) where alternative equation formulations(17.a) and (18.a) were optionally replacing the original formulations (17) and (18).Equations formulation Cost estimate −ξ1/λ1(17), (18) 0.794(17), (18.a) 0.767(17.a), (18) 1.027(17.a), (18.a) 1.00095. Discussion and conclusionThe example optimization problem analysed in this paper is rather simple ascompared to e.g. DICE model [13], [14]. However, even that simple problemappears to be analytically intractable. A numerical solution of that problemcan be easily found with sufficiently high precision and FOC can be checkednumerically to ensure the validity of the obtained solution. The marginalcost of the optimal control can be analytically expressed and calculated. The”cost estimate” x obtained from the concept of matching marginal quantitiesadded to problems’ equations and preserving the optimal utility, that is, anoptimal cost estimation method employing a ratio of two marginal values(Lagrange multipliers) leads to an estimate clearly deviating from the truecost of optimal control. Moreover, such an estimate is ill-defined, becauseit’s arbitrary (non-unique) and depends on a particular way of expressingexactly the same problem i.e. a formulation that for all controls (inputs)delivers the same trajectories (outputs).While the considered example is rather hypothetical i.e. the overallmodel and particular functions employed to describe damage to fruit, fliesreproduction, and economic cost of pesticides application were not validated,and the measurement units (e.g. tons of flies) might sound awkward, thepurpose of this theoretical ad-hoc construction is to demonstrate a settingin which shadow prices are being used in practice.The semi-analytical example provided here substantially limits the ap-plication of marginal values (shadow prices) for optimal cost estimate ap-proaches similar to (30) including the SCC calculation approach imple-mented in the DICE model [18].10AcknowledgmentsThe authors acknowledge early discussions around the DICE model withtheir IIASA colleagues Elena Rovenskaya, Artem Baklanov, Fabian Wag-ner, Thomas Gasser, and Petr Havlik. These discussions have spurred theinterest to the application of shadow prices in DICE that ultimately re-sulted in the presented analysis. The authors are grateful to their IIASAcolleagues Johannes Bednar, Michael Kuhn, Stefan Wrzaczek, and MichaelFreiberger for useful discussions and feedback. The authors acknowledgeWilliam Nordhaus for making the DICE source code and documentationopenly available as well as the clean and transparent model structure thatall made the present research possible. The authors acknowledge arXiv forhosting the earlier work on this topic [17].FundingAustrian Science Fund (FWF): P31796-N29/“Medium Complexity EarthSystem Risk Management” (ERM).Author contributionsNK has conceptualized the problem and carried out the investigation; MOhas contributed to funding acquisition; NK, AS, and MO have discussedthe results in the process of investigation; NK has drafted the paper; allco-authors have contributed to writing the manuscript.Competing interestsAuthors declare no competing interests.11References[1] Alasdair Smith. “Shadow Price Calculations in Distorted Economies.”The Scandinavian Journal of Economics, vol. 89, no. 3, 1987, pp.287–302. JSTOR, https://doi.org/10.2307/3440199. Accessed 2 Sep.2022.[2] David A. Starrett. “Shadow Pricing in Economics.” Ecosystems, vol. 3,no. 1, 2000, pp. 16–20. JSTOR, http://www.jstor.org/stable/3658662.Accessed 2 Sep. 2022.[3] Dan Wu, Shuwei Li, Li Liu, Jiyao Lin, Shiqiu Zhang, Dynam-ics of pollutants’ shadow price and its driving forces: An analy-sis on China’s two major pollutants at provincial level, Journal ofCleaner Production, Volume 283, 2021, 124625, ISSN 0959-6526,https://doi.org/10.1016/j.jclepro.2020.124625.[4] Elvira Silva, Manuela Magalhães, Environmental Efficiency,Irreversibility and the Shadow Price of Emissions, Euro-pean Journal of Operational Research, 2022, ISSN 0377-2217,https://doi.org/10.1016/j.ejor.2022.08.011.[5] Zhaohua Wang, Yanwu Song, Zhiyang Shen, Global sustainability ofcarbon shadow pricing: The distance between observed and optimalabatement costs, Energy Economics, Volume 110, 2022, 106038, ISSN0140-9883, https://doi.org/10.1016/j.eneco.2022.106038.[6] Junfu Zhang, JUE Insight: Measuring the Stringency ofLand Use Regulation Using a Shadow Price Approach, Jour-12nal of Urban Economics, 2022, 103461, ISSN 0094-1190,https://doi.org/10.1016/j.jue.2022.103461.[7] Aslıhan Arslan, Shadow vs. market prices in explaining land allo-cation: Subsistence maize cultivation in rural Mexico, Food Pol-icy, Volume 36, Issue 5, 2011, Pages 606-614, ISSN 0306-9192,https://doi.org/10.1016/j.foodpol.2011.05.004.[8] Xiaojie Wen, Shunbo Yao, Johannes Sauer, Shadow prices and abate-ment cost of soil erosion in Shaanxi Province, China: Convex expectileregression approach, Ecological Economics, Volume 201, 2022, 107569,ISSN 0921-8009, https://doi.org/10.1016/j.ecolecon.2022.107569.[9] Michiyuki Yagi, Shunsuke Managi, Shadow price of patent stock asknowledge stock: Time and country heterogeneity, Economic Anal-ysis and Policy, Volume 60, 2018, Pages 43-61, ISSN 0313-5926,https://doi.org/10.1016/j.eap.2018.09.001.[10] Alexandros Maziotis, Andres Villegas, Maŕıa Molinos-Senante,The cost of reducing unplanned water supply interruptions:A parametric shadow price approach, Science of The To-tal Environment, Volume 719, 2020, 137487, ISSN 0048-9697,https://doi.org/10.1016/j.scitotenv.2020.137487.[11] Kai Hou, Puting Tang, Zeyu Liu, Hongjie Jia, Kai Yuan, ChongboSun, Yi Song, A fast optimal load shedding method for power sys-tem reliability assessment based on shadow price theory, Energy Re-ports, Volume 8, Supplement 1, 2022, Pages 352-360, ISSN 2352-4847,https://doi.org/10.1016/j.egyr.2021.11.104.13[12] Francesca Mariani, Maria Cristina Recchioni, Mariateresa Ciommi,Merton’s portfolio problem including market frictions: A closed-formformula supporting the shadow price approach, European Journal ofOperational Research, Volume 275, Issue 3, 2019, Pages 1178-1189,ISSN 0377-2217, https://doi.org/10.1016/j.ejor.2018.12.022.[13] William Nordhaus, Climate Change: The Ultimate Challenge forEconomics, American Economic Review 2019, 109(6), 1991–2014,DOI:10.1257/aer.109.6.1991[14] Documentation and source code for the DICE model are available onlineat https://williamnordhaus.com/dicerice-models where the direct linkto the DICE User’s Manual is:http://www.econ.yale.edu/ñordhaus/homepage/homepage/documents/DICE Manual 100413r1.pdf[15] C. Simon and L. Blume, Mathematics for Economists, W. W. Norton& Company, New York, 1994.[16] Dieter Grass, ed. Optimal Control of Nonlinear Processes: With Appli-cations in Drugs, Corruption, and Terror. Berlin: Springer, 2008. ISBN:978-3-540-77646-8.[17] Nikolay Khabarov, Alexey Smirnov, Michael Obersteiner, Social Costof Carbon: What Do the Numbers Really Mean? ArXiv; DOI:10.48550/arXiv.2001.08935[18] Nikolay Khabarov, Alexey Smirnov and Michael Obersteiner (2022),Social cost of carbon: A revisit from a systems analysis perspective.Front. Environ. Sci. 10:923631. DOI:10.3389/fenvs.2022.92363114

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Shadow prices and optimal cost in economic applications

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