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ARTICLE IN PRESS JID EOR m5G December 26 2019 8 3 European Journal of Operational Research xxx xxxx xxx Contents lists available at ScienceDirect European Journal of Operational Research journal homepage Interfaces with Other Disciplines Bounded rationality and thick frontiers in stochastic frontier analysis Mike G Tsionas Lancaster University Management School LA1 4YX UK a r t i c l e i n f o Article history Received 11 June 2019 Accepted 7 December 2019 Available online xxx Keywords Decision support Bounded rationality Technical effi ciency Stochastic Frontier models Bayesian analysis a b s t r a c t Recent research has proposed a statistical test based on the notion that agents have bounded rationality if and only if more attractive states are chosen with larger probability We propose and implement a statistical test for bounded rationality in the context of stochastic cost frontiers Bounded rationality is related to probabilistically cost effi cient distributions The test is based on comparing a discrete set of probabilities with the theoretical distribution under bounded rationality Implementation is shown to be quite easy in a Bayesian framework using the Bayes factor for model comparison between estimated and theoretical probabilities The bounded rationality model introduces only an extra parameter in frontier models and therefore it is quite practical to use in applications as a general semi parametric model for ineffi ciency 2019 Elsevier B V All rights reserved 1 Introduction Ub e Andersson J rnsten Lillest l and Sandal 2017 proposed a statistical test based on the notion that agents have bounded ra tionality if and only if more attractive states are chosen with larger probability see Erlander 2010 When similar agents make discrete choices from a choice set C 1 C we expect an aggregate distribution of the form Probability of selecting choice c e V c C c 1 e V c c 1 C 1 where C is the number of possible choices and V c is expected util ity corresponding to choice c It is important to emphasize that irrespective of the distribution governing the choices the resulting statistical distribution will be the same due to the Fisher Tippet Gnedenko theorem the analogue of central limit theorem applied to the distribution of the maximum For example under a Gum bel distribution in Manski 1977 random utility model it is well known that we obtain 1 Moreover 1 can be derived from max imum entropy considerations see e g Wilson 1967 Anas 1983 and Erlander and Stewart 1990 It is also the solution of the max imum utility problem see Erlander and Stewart 1990 as well as a direct implication of probabilistic cost effi ciency see Erlander 2010 In addition there are many other approaches e g Mattsson and Weibull 2002 and Matejka and McKay 2015 yielding the same model Applications of bounded rationality are many includ ing Barucci and Landi 1996 Hampf 2017 Jiang Fang Fan and Wang 2013 Liu Lv Li and Tang 2017 etc E mail address m tsionas lancaster ac uk A probability distribution p p 1 p C is probabilistically cost effi cient if and only if given that a sample has greater total cost then it is always less probable In turn these implies that the probabilities must satisfy p c e K c C c 1 e K c c 1 C 2 for some 0 where K c is the cost of choice c and Erlander 2010 Chapter 4 Ub e et al 2017 in turn propose a formal chi square statistic for bounded rationality whose null hypothesis is that estimated probabilities follow the pattern in 2 for a cer tain value of If is large alternatives with maximum utility are chosen with probability 1 at the limit If it is small alterna tives are equally probable In our case K c is defi ned in cost units and it is diffi cult to improve it Bogetoft Lee December 26 2019 8 3 a decision maker may fi nd it optimal to settle at lower levels of effi ciency The general idea of rational ineffi ciency is that there are gains from ineffi ciency and therefore costs of improving ef fi ciency Bogetoft for example a loyal pool of highly qualifi ed and content em ployees or to substitute for inputs that are not accounted for for example a higher direct wage bill or higher turn overs in the la bor force op cit p 246 From this point of view improving effi ciency is costly and this should be taken explicitly into account in estimation In addition the thick frontier idea introduced by Berger and Humphrey 1992 relies on the concept that the error terms of say translog cost functions within the lowest and highest cost quartiles refl ect only measurement error and so the differences be tween the lowest and highest cost quartiles can be attributed to ineffi ciencies and perhaps market factors as well Therefore for banks in the lowest cost quartile and for banks in the highest cost quartile different translog cost functions are estimated and a fi nal measure of ineffi ciency that is presumably free from the effect of market conditions see Berger and Humphrey 1992 pp 259 260 The main drawback of their approach is that it can provide inef fi ciencies across quartiles or asset class sizes but not ineffi ciency estimates at the bank level The novel approach that we propose in this paper can deliver ineffi ciencies at the decision making unit level 2 Bounded rationality in stochastic frontier models 2 1 General model Of course there are many defi nitions of bounded rationality in economics Here we adopt the approach of Ub e et al 2017 as a motivation to introduce their kind of bounded rationality in stochastic frontier modeling and in turn establish that it can be extended to a non parametric modeling of the distribution of the one sided error term ineffi ciency in stochastic frontier mod els In this paper as in Ub e et al 2017 we use the follow ing defi nition of bounded rationality agents are boundedly ratio nal if better states are chosen with larger probability Specifi cally Erlander 2010 uses this defi nition to formulate probabilistic cost effi ciency a statistical framework enabling rigorous statistical test ing Even though we in this paper mainly discuss applications to the newsvendor model the fi rst part of the paper is more general in that it can be applied to any context where bounded rationality is defi ned on a standard probability space Ub e et al 2017 p 251 Specifi cally agents are boundedly rational in the sense that they do not always behave as to minimize costs Under full infor mation and a defi nite and indisputable defi nition of cost fully ra tional behavior can be justifi ed Therefore we fi nd the procedure of Ub e et al 2017 particularly well suited for use in the context of cost or production frontiers This is because more effi cient pro duction is desirable yet it requires the use of real resources and it is therefore costly to achieve We assume a stochastic cost frontier of the form y i x i v i u i i 1 n 3 where x i k is a vector of explanatory variables input prices W outputs Y y i is total cost is a vector of parameters in k v i iid N 0 2 and u i is a non negative random variable rep resenting cost ineffi ciency Extension to production frontiers is straightforward Moreover the parametric assumption can be re laxed as in Michaelides Tsionas Vouldis and Konstantakis 2015 Michaelides Vouldis and Tsionas 2010 Tsionas 2017 Tsionas and Izzeldin 2018 Cost effi ciency is r i e u i with support is S 0 r 1 r 2 r C 1 r C 1 1 Of course r 1 r 2 0 is a parameter which is formally needed to ensure that the posterior is fi nitely integrable Fernandez Osiewalski December 26 2019 8 3 2 2 Model under bounded rationality Under bounded rationality we have p c e u c C c 1 e u c r c C c 1 r c c 1 C 8 where u c ln r c represents support points for technical ineffi ciency and 0 is an unknown parameter Our prior for is p e A 1 9 which corresponds to an exponential distribution where A 0 is prior parameter representing the prior expectation and variance of As A the prior in 9 becomes uniform Under this as sumption the posterior of the model is p D n 1 e q 2 2 C c 1 p 1 c n i 1 C c 1 exp 1 2 2 y i ln r c x i 2 ln r c r c C c 1 r c 10 We can rewrite this expression as follows p D n 1 e q 2 2 A 1 C c 1 r c n C c 1 p 1 c n i 1 C c 1 exp 1 2 2 y i ln r c x i 2 ln r c r c 11 The posteriors in 7 and 11 can be analyzed using the Gibbs sampler see Appendix A Depending on the prior expectation and variance of viz A a formal test of the hypothesis of bounded rational or probabilistic cost effi ciency is the Bayes factor in favor of A 7 and against A 1 Sup pose these models are called 2 and 1 respectively with Model 1 being obviously more general so in fact we want to see whether 8 receives support in the light of the data First we defi ne the marginal likelihood of each model M 1 D p p D d d d p M 2 D p D d d d 12 In turn the Bayes factor in favor of Model 2 bounded rational ity and against model 1 general probabilities is BF 2 1 M 2 D M 1 D 13 If BF 2 1 is greater than about 20 Kass December 26 2019 8 3 Fig 1 Distributions of effi ciency Notes Presented are histograms of cost effi ciency for different numbers of support points C the translog cost function is as follows ln C o N i 1 w i w i M m 1 y m y m t t E ln E 1 2 N i 1 N j 1 ww ij w i w j 1 2 M m 1 M m 1 yy mm y m y m 1 2 t t t 2 1 2 EE ln E 2 N i 1 M m 1 wy im w i y m N i 1 wt w i t N i 1 Et w i ln E M m 1 yt y m t M m 1 yE y m ln E Et t ln E 16 From Table 1 it turns out that the model with C 50 consis tently has the highest Bayes factor close to approximately 129 no matter what the value of A is Therefore this should be taken as our estimate of C The largest Bayes factor corresponds to C 50 and A 1 However we proceed based on the assumption that A for various values of C in Fig 1 where we report poste rior densities of effi ciency We use A so that the prior in 9 is uniform For C 10 effi ciency is close to 90 5 with posterior standard deviation 0 19 with considerable probability at values higher than 0 90 In Fig 2 we report marginal posterior densities correspond ing to C 50 using alternative values of A left panel The posterior densities of range from slightly less than 0 5 to 2 3 with a mean close to 1 3 suggesting that cannot be zero so effi ciency is not chosen randomly which would be the case if 0 This in itself provides direct evidence in favor of rational inef fi ciency and bounded rationality in the Ub e et al 2017 sense In the right panel of Fig 2 reported are plots of 8 for various values of C drawn using the posterior mean estimates of with a uniform prior A 0 In Fig 3 left panel we report posterior densities of technical change TC effi ciency change EC and pro ductivity growth PG EC TC for our best specifi cation C 50 and A 1 Evidently technical change averages zero but has two modes near 1 and 1 and ranges from 2 to 3 Effi ciency change av erages 0 014 and ranges from less than 1 to 2 5 contributing to productivity growth of 2 5 on the average and ranging from 2 to slightly over 6 In Fig 3 right panel we report posterior densities of output cost elasticity defi ned as e cy q j 1 ln C W Y t ln Y j which is the inverse returns to scale measure and q is the num ber of output variables in vector Y The evidence suggests that out put cost elasticity varies between 0 77 and 1 2 with high poste rior probability and averages 0 97 with posterior standard devia tion 0 06 approximately Regarding the thick frontier specifi cation in 14 we can com pute Bayes factors in favor of 4 relative to 14 when A These Bayes factors are presented in Table 2 For C 10 classes the Bayes factor favors the thick frontier in 14 As the number of classes increases the two models become nearly the same and the Bayes factor reaches a maximum of 3 389 when C 50 Larger values of C yield Bayes factors which are close to one The evidence suggests that there is no evidence against 4 unless the number of categories is very small 10 in our case It would be interesting to see how the new model performs in terms of more traditional models like the half normal the truncated normal the exponential and gamma distributions In all these cases effi ciency can be estimated using well known MCMC methods 2 The truncated normal distribution assumes u N 2 u the half normal arises as a special case if we set 0 and the gamma distribution denote u Ga P has den sity f u P P u P 1 e u u 0 0 and P is a positive parameter that we restrict to take values in 1 2 3 this case being known as Erlang distributions When P 1 we obtain as a special case the exponential distribution We take as benchmark the exponential 2 Our priors were p const p u 1 u and p 1 Please cite this article as M G Tsionas Bounded rationality and thick frontiers in stochastic frontier analysis European Journal of Oper ational Research https doi org 10 1016 j ejor 2019 12 010 M G Tsionas European Journal of Operational Research xxx xxxx xxx 5 ARTICLE IN PRESS JID EOR m5G December 26 2019 8 3 00 511 522 5 parameter C 50 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 density 00 10 20 30 40 50 60 70 80 91 efficiency 0 004 0 006 0 008 0 01 0 012 0 014 0 016 0 018 posterior probability C 10 C 25 C 50 C 100 prior Fig 2 Marginal posterior densities of C 50 and posterior probabilities Notes Left panel shows marginal posterior distributions of corresponding to different values of the prior parameter A In the right panel shown are posterior probabilities p c plotted against effi ciency levels r c c 1 C 0 03 0 02 0 0100 010 020 030 040 050 060 07 TC EC and PG 0 50 100 150 density technical change efficiency change productivity growth 0 70 80 911 11 21 3 ecy 0 1 2 3 4 5 6 7 density Fig 3 Posterior densities of technical change TC effi ciency change EC and productivity growth PG Notes TC is defi ned as T C it ln C W it Y it t t where t represents the time trend Effi ciency change is EC it ln r it ln r i t 1 Finally productivity growth is PG it T C it EC it We use C 100 A 0 Moreover output cost elasticity defi ned as e cy q j 1 ln C W Y t ln Y j which is the inverse returns to scale measure and q is the number of output variables in vector Y Table 3 Bayes factors against exponential and in favor of competing specifi cations Exponential Erlang gamma P 1 1 0 0 0 Half normal 5 320 Truncated normal 6 415 Erlang gamma P 2 3 432 Erlang gamma P 3 5 478 Thick Frontier C 50 132 381 distribution P 1 and we provide Bayes factors against the expo nential distribution and in favor of other specifi cations in Table 3 Evidently the Bayes factor in favor of the thick frontier using the optimal C 50 and against the exponential is almost 282 so this model receives a lot of support in the light of the data The Bayes factor in favor of the true thick frontier against the next best model truncated normal is 132 381 6 415 20 6 which indicates strong evidence in favor of the new model and against the trun cated normal For comparison purposes we report estimated effi ciency mea sures for all specifi cations in Fig 4 Clearly the different specifi cations imply different effi ciency distributions From Fig 4 it is evident that most specifi cations im ply average effi ciency around 85 approximately The thick frontier has a very different sample distribution and ranges from approx imately 70 100 with a distinct mode around 100 This result which is also shown in Fig 1 bottom left panel is quite remark able in the sense that after crediting banks for costly ineffi ciency average effi ciency is close to 90 and tend to cluster around 100 The results may strike the reader as unreasonable since un der the bounded rationality thick frontier ineffi ciency appears much lower compared to other standard specifi cations As a mat ter of fact effi ciency averages close to 85 are often reported in the literature If one takes the results from standard models as X ineffi ciency meaning that they are not optimal in some sense Please cite this article as M G Tsionas Bounded rationality and thick frontiers in stochastic frontier analysis European Journal of Oper ational Research https doi org 10 1016 j ejor 2019 12 010 6 M G Tsionas European Journal of Operational Research xxx xxxx xxx ARTICLE IN PRESS JID EOR m5G December 26 2019 8 3 0 40 50 60 70 80 91 efficiency 0 5 10 15 density Thick frontier Exponential Half Normal Truncated Normal P 2 P 3 Fig 4 Effi ciency estimates from different specifi cations then by crediting banks for bounded rationality i e absence of full information etc ineffi ciency is much lower as implicitly a large part of what appears to be waste in X ineffi ciency is in fact used to produce unobserved outputs such as quality of service clien tele etc as we argued in the Introductory sector Bounded ratio nality as in Ub e et al 2017 accounts for these facts implicitly of course through parameter which is actually estimated in our application A similar result was obtained by Atkinson and Dorf man 2005 who found that when electric utilities are credited for their reduction of pollution which is costly this actually results in improvement of performance by a substantial amount In the same sense crediting banks for being boundedly rational and rec ognizing that performance improvement requires the costly use of scarce resources makes X ineffi ciency considerably lower Concluding remarks In this paper we proposed a simple Bayesian treatment of bounded rationality in cost stochastic frontier models under prob abilistically cost effi cient distributions The comparison of proba bilistically cost effi cient distributions with a general set of prob abilities of cost effi ciencies reduces to model comparison using Bayes factors and a Gibbs sampler to perform the computations We do not need to assume a specifi c distribution for cost inef fi ciency as its support is discretized and the number of points can be determined using formal Bayesian model comparison The bounded rationality frontier model introduces only one additional parameter and therefore its use in applications is quite practical First we consider the posterior in 7 whose augmented version is p p r J D n 1 exp 1 2 2 q n i 1 y i ln r i x i 2 ln r i n i 1 p J i r J i r i A 1 where r r 1 r n r J i r i 1 if r i r J i r r 1 r n J J 1 J n Here J i c if and only if r i r c for all i 1 n The augmentation is with respect to the latent variables r and J Sup pose the data for the dependent variable is in the n 1 vector y and the data for the explanatory variables in the n k matrix X Conditionally on r i J i n i 1 the posterior of is normal and that of 2 follows a chi square p r J D N k X X 1 X y ln r 2 X X 1 A 2 q y ln r X y ln r X 2 p r J D 2 n A 3 The conditional posterior distribution of r i is p r i r c p J D exp 1 2 2 n i 1 y i ln r c x i 2 ln r c p c i 1 n c 1 C A 4 The conditional posterior distribution of probabilities is p p r J D C c 1 p N c 1 c A 5 where N c is the number of times in the sample where r i r c This corresponds to a multinomial distribution defi ned over S p C C c 1 p c 1 Finally we have p J i c p r D exp 1 2 2 n i 1 y i ln u c x i 2 ln r c p c c 1 C i 1 n A 6 A standard Gibbs sampler can then be used to produce draws from the posterior in A 1 The Gibbs sampler consists in draw ing a large sample s s p s J s r s s 1 S of size S from the posterior conditional distribution In our application we use 50 0 0 0 0 iterations the fi rst 10 0 0 0 0 of which are discarded to mitigate possible start up effects Under bounded rationality the augmented posterior is p r J D n 1 exp 1 2 2 q n i 1 y i ln r i x i 2 ln r i n i 1 p J i r J i r i A 7 where p J i r J i C c 1 r c c 1 C Relative to the previous Gibbs sampler we have to change the following conditional posterior distributions First p is no longer a parameter Moreover we have p J i c r D exp 1 2 2 n i 1 y i ln r c x i 2 ln r c r c C c 1 r c c 1 C A 8 p r J D C c 1 r c n e A C c 1 ln r c A 9 As this distribution is non standard we use a Metropolis Hastings step to draw random numbers from A 9 Under 14 all conditional posterior distributions remain the same In particular A 8 and A 9 are still valid The posterio

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