Journal of Economics and International Finance


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1 Journal of Econoics and International Finance Volue 8 Nuber January, 014 Volue ISSN Nuber 8 August 015 ISSN
2 ABOUT JEIF The Journal of Econoics and International Finance (JEIF) is published onthly (one volue per year) by Acadeic Journals. Journal of Econoics and International Finance (JEIF) is an open access journal that provides rapid publication (onthly) of articles in all areas of the subject such as econoetrics, trade balance, Mercantilis, Perfect copetition etc. The Journal welcoes the subission of anuscripts that eet the general criteria of significance and scientific ecellence. Papers will be published shortly after acceptance. All articles published in JEIF are peer reviewed. Contact Us Editorial Office: Help Desk: Website: Subit anuscript online
3 Editors Prof. Nathalie Jeanne Marie HILMI Professor of Econoics and Finance, International University of Monaco, Hedge Funds Research Institute, Monte Carlo, Principality of, Monaco. France Prof. Osaah M. Al Khazali Professor of Finance, School of Business and Manageent Aerican University of Sharjah, P.O. Bo 6666, United Arab Eirates, Dr. Guneratne B Wickreasinghe School of Accounting Faculty of Business and Law Victoria University Melbourne Victoria, Australia Dr. Melte Gürünlü Departent of Econoics and Finance University of Turin, G.Prato, Italy. Prof. Yongrok Choi Departent of International Trade, Inha university, Incheon, Korea. Prof. Mohaed Osan Ahed Bushara Departent of Agricultural Econoics; FAS; Gezira University; P. O. Bo 0; Wad Medani; Sudan. Prof. Anyanwu John Chukwudi Developent Research Departent African Developent Bank 15 Avenue du Ghana BP 33, 100 Tunis Belvedere Tunis Tunisia Prof. S. E. Neaie Departent of Econoics, Institute of Financial Econoics, Aerican University of Beirut, Beirut, Lebanon. Dr. Adrei Vernikov Banking Departent, Higher School of Econoics P.O. Bo 43, Moscow 15057, Russia. Prof. Keith Pilbea Departent of Econoics, City University, Northapton Square, London EC1V OHB. UK.
4 Editorial Board Dr. Gbadebo Olusegun ODULARU Regional Policies and Markets Analyst, Foru for Agricultural Research in, Africa (FARA), PMB CT 173 Cantonents, Gowa Close, Roan Ridge, Accra, Ghana. Dr ilhan Ozturk Çağ University, Faculty of Econoics and Adinistrative, Sciences, Adana Mersin karayolu, uzeri, 33800, Mersin,Turkey. Professor. Abdelkader BOUDRIGA Professor of finance, School of Econoics and Coerce, Tunis, Tunisia. Dr. Shivakuar Deene Dept. of Coerce and Manageent, Karnataka State Open University, Manasagangotri, Mysore , Karnataka India. Mohaed Oran The Egyptian Echange, 4 (A) El, Sherifein St, Down, Town, Postal Code 11513, P.O. Bo 358 Mohaed Farid, Cairo, Egypt. Dr. Kola Subair Adjunct Professor, Business and, Financial Econoics, Aerican Heritage University, California, USA. Dr. Bora Aktan Assistant Professor of Finance, Yasar University, Faculty of Econoics and, Adinistrative Sciences, Departent of Finance, Selcuk Yasar Capus, Universite Caddesi, No , Bornova, Izir, Turkey. Dr. Davide Furceri Office of the Chief Econoist, Econoics Departent,, Rue André Pascal, Paris Cede 16, France. Dr. ABDUL JALIL Wuhan University, Econoics and Manageent School, Wuhan,Hubei, PR China. Prof. Silvia Ciotti Dept of Social and Environental Sciences, St. John International University, Della Rovere Castle Rey Square, Vinovo (Turin), Italy. Prof. Tessaleno Devezas Advanced Materials and Technological, Forecasting, University of Beira Interior, Covilhã, Portugal. Dr. Nikolay Sukholin Autonoous University, Santo Doingo, Doinican Republic. Prof. Dev Tewari Deputy Dean, Faculty of Manageent Studies Professor, School of Econoics and Finance, Westville Capus, University of KwaZulu Natal Resource Econoics, Durban, South Africa. Dr. Tarek Chebbi Faculty of Law, Econoics and Political Science University of Sousse, Erriadh City 403 Sousse, Tunisia Professor Hichri Walid Gate & Uinversity of Lyon, LAREQUAD Gate, 93 Chein des ouilles, Ecully France. Dr.Sunderasan Srinivasan Navillu Road 7th Cross, Kuvepunagar, Mysore 57003, India. Dr. P. Malyadri Governent degree College,Osania University Tandur ,Rangareddy District India.
5 Journal of Econoics and International Finance Table of Contents: Volue 7 Nuber 8 August 015 ARTICLES Research A odel for evaluating recreation benefits with reference 167 dependent preference Tadahiro Okuyaa* and Yasuhisa Hayashiyaa Banking sector refors and output growth of anufacturing 183 sector in Nigeria ( ) Olanrewaju, Oluwagbenga Gideon*, Areo, Adeleke Gabriel and Aiyegbusi Oluwole Oladipo
6 Vol. 7(8), pp , August, 015 DOI: /JEIF Article Nuber: 33EDAF ISSN Copyright 015 Author(s) retain the copyright of this article Journal of Econoics and International Finance Full Length Research Paper A odel for evaluating recreation benefits with reference dependent preference Tadahiro Okuyaa 1* and Yasuhisa Hayashiyaa 1 Departent of Regional Policy, Faculty of Econoics, University of Nagasaki, 13 Kawashiocho, SaseboShi, Nagasaki, ; Japan. Faculty of Econoics and Manageent, Tohoku University, , Aobaku, Sendai, Miyagi, Japan. Received 4 May, 015; Accepted 7 July, 015 In environental valuation studies, it is coonly assued that a utility arises fro an absolute aount of environental quality. This criterion, called absolute evaluation, is used in ethods including the travel cost ethod and the contingent valuation ethod. Studies in eperiental econoics, however, have indicated that an individual s criterion depends on reference dependent preference (RDP) a relative evaluation rather than absolute evaluation. This criterion is used ainly in analysis of biases such as fraing effects or brand choice. The purpose of this paper is to construct a odel for evaluating recreational benefit with RDP. The odel focuses ainly on RDP for an environental quality so as not to conflict with the aio of choices, and the travel cost ethod is used as the odel s basis. First, a structure of utility function is discussed and the benefit with RDP is defined and analyzed based on the relation between the level of RDP and the agnitude of the benefit. Second, the calculating forula of the benefit is derived by the integratingback ethod and tests for consistency between the results of static analysis and the nuerical eaple are perfored. Key words: Benefit analysis, environental quality, reference dependent preference, travel cost ethod. INTRODUCTION In environental econoics, it is a coon assuption that a utility arises fro the aounts of consuing a good and of an environental quality (Freean III et al., 014). For eaple of the environental quality, nitrogen or sulfur dioide is used as inde of air quality; biocheical oygen deand or phosphorus as water quality; area or a nuber of spices as forest or wetland qualities. Here, the weak copleentarity assuption that the increent of an environental quality leads to the increent of the aount of deand enables researchers to easure the positive or negative benefit of the environental quality change (Mäler, 1974). Thus, ost environental valuation studies have eployed this assuption for valuation ethods (Bockstael and McConnell, 007; Freean III et al., 014). The travel cost ethod (hereafter TCM) is representative of this *Corresponding author. Eail: Tel./Fa: JEL classification: Q6, Q51. Authors agree that this article reain peranently open access under the ters of the Creative Coons Attribution License 4.0 International License
7 168 J. Econ. Int. Finance tendency because it relies on the assuption that there is a closed relationship between the aount of the recreational deand and the level of recreation site s environental quality (Shrestha et al., 00; Herriges et al. 004; Phaneuf and Siderelis, 003; Whitehead et al., 009). However, soe eperiental studies in econoics and psychology pointed out that an individual s decision aking is influenced by reference dependent preference (RDP), which has three characteristics. First, a visitor s behavior is influenced by facts including the reference point constructed by that visitor s previous purchase, knowledge, or initial endowent of a good. Second, a visitor s utility (value) function ay enter the negative diension (this is called a loss). Therefore, a visitor s preference is nontransitive. Third, a visitor s utility function is conve in the negative diension. This is called loss aversion. The purposes of this paper are 1) to forulate a recreation behavior odel with RDP for environental quality and ) to forulate an application odel for benefit calculation. As for the base odel, the travel cost ethod is eployed. This for of preference was presented in the prospect theory by Kahnean and Tversky (1979) and Tversky and Kahnean (1991). In environental valuation studies, RDP is a plausible cause of biases such as a fraing effect i, a status quo bias ii, and an endowent effect iii. These biases occur ost often when a stated preference ethod (SPM) is used for valuing an environental quality iv. As for the theoretical studies, Munro and Sugden (003) iproved the (prospect) theory of Tversky and Kahnean (1991) and added restrictions to the preference condition to epress an eogenous reference point. Bowan et al. (1999) used a gainloss function to epress RDP and analyzed the effect when the reference point is endogenously deterined. Kőszegi and Rabin (006) analyzed the effect of the reference level of consuption under uncertainty in the case that the reference points are eogenously deterined. In epirical studies, Batean et al. (1997) tested the ipact of reference dependent preference on the echange of private goods by eperiental ethods, and Herne (1998) estiated the property of loss aversion. Peters (01), Zeisberger et al. (01), and Li and Ling (015) did recent epirical or theoretical studies on the RDP. Barberis (013) describes the review on the prospect theory. These studies assued RDP for the goods, discussed the for (preference structure) of individuals utility (value) function, and perfored epirical tests by eperiental ethods. Soe epirical studies on the visitor behavior with RDP in arkets have eained RDP on prices. The reference point of this RDP is called a reference price, and a visitor purchases an ite as the result of coparing the prices of goods with the reference prices. A fundaental discussion on visitor behavior with RDP is offered by Winer (1986). It is called the reference price odel (RPM) and is based on the assiilation contract theory (Sherif, 1963) and the adaptation level theory (Helson, 1964). A visitor gets a utility fro the difference between the actual price and the reference price. A visitor s deand is also influenced by this difference. Winer (1986), Mayhew and Winer (199), Lattin and Bucklin (1989), Greenleaf (1995), Ren et al. (014) and Kuar (014) conducted epirical studies and confired the reference price effect for purchasing a good. Putler (199) considered a visitor behavior theory for the RPM by using Kalan s (1968) utility function, which includes the prices of goods as a RDP variable (reference price) in its function. Putler (199) considered the forulation of reference price effect in utility function and analyzed the substitution and incoe effect of reference price for the Marshallian and Hicksian deand functions. Putler s (199) forulation of RPM has not been applied for bundles of goods. Thus, at least, it can be assued that a visitor s preference satisfies the transitivity for the aount of goods. Mayhew and Winer (199, 6) eplained the forulations of the internal and eternal reference prices; The internal reference price is the prices stored in eory on the basis of perceptions of actual, fair, or other price concepts, thus, people adapt to the level of past stiuli and judge new stiuli in coparison with the adaptation level. The eternal reference point is the one provided by observed stiuli in the purchase environent. For eaple, Point of purchase shelf tags that contain inforation about suggested retail price or the actual or unit price of another product against which a price can be copared. The arguent on the reference point of RPM relates to the discussion on RDP in eperiental studies in the sense that the internal or eternal reference points are fored whether they are deterined eogenously or endogenously. As the structures of a reference price in RPM, Bell and Bucklin (1999) assued the (internal) reference price as a reference price fro a visitor s previous purchase occasion; Eery (1970), Hardie et al. (1993), and Kalyanara and Little (1994) assued that the present reference point is the weighted average of the past prices of the ite and/or the weighted average between the past price and the individual s past reference point. The above arguents can be suarized as follows. The first is that although RDP has been confired in SPM or eperiental studies, few have considered revealed preference ethods (e.g., TCM) even if RDP is observed in RPM. The second is that although the reference price effect has been epirically indicated, few studies have considered the effect of RPD for the qualities of goods in a visitor s behavior v. In environental valuation studies, because a ain focus point is how changes in quality influence benefits, it is
8 Okuyaa and Hayashiyaa 169 useful for another analysis to consider recreation behavior with RDP. The ain hypothesis of this study is that a reference point (siilar to RDP) for an environental quality eists and relates closely to recreation behavior. Let us iagine a visitor s decision aking when he or she chooses between recreation sites A and B (e.g., river A and river B), and the sites have siilar qualities as Q a and Q b. Traditional recreation odel assuing a single trip states that if an individual prefers Q a to Q b, he or she will choose to go to site A; his or her utility is defined as uq ( a ). However, visitors can often be heard coplaining 1) This place was not as good as the last one, or, ) I have already been to that place, so let s go to a different one. These situations ean that the visitor does not judge the quality of sites in absolute ters. It is possible to consider that there is a reference point in the visitor s preference structure, e.g., the first case would relate the case that utility arises fro u( Qa Q) where Q is the quality of the previous site, and the second case would relate the case that utility arises fro u( Qa, Qb Q), where Q b is the previously visited site and Q Qb. The organization of this study is as follows. Firstly, a visitor behavior odel with the RDP is considered and analysis focuses on the relationship between RDP and deand, followed by the analysis of the benefits of RDP defined following the concept of welfare easures, and static analyses for the relationship between RDP and benefit. These analyses focus on the relationship between the position of the RDP (gain or loss) and the agnitude of the benefit because it is a fundaental consideration siilar to price or incoe in epirical welfare studies. Also, an estiation odel is considered and the total value is derived by an integratingback approach (Larson 199, Eo and Larson 006, and von Haefen 007), and nuerical eaples are perfored to confir consistency of the estiation odel with the static analysis. Finally, the results and the unresolved issues of this study are discussed. MATERIALS AND METHODS Utility aiization proble with reference dependent preference Forulation of reference dependent preference on environental quality In this section, a visitor s recreation activity with RDP is considered by following the consuer behavior odel forulated by Putler (199). A Main difference is that RDP consists of price in Putler (199) s odel but quality in this odel. First, the forulations of RDP are considered. Let z be the aount of a coposite good and be the nuber of recreation activities for a recreation site. Respectively, the prices of these goods are p and p. Let Q be an environental quality in a site in which a political project is z assued to be ipleented. Finally, a reference point (eactly to say, reference quality) to be copared with Q be considered. In this study, let RQ be the reference quality, and the value gained fro coparing Q and RQ is the relative value. Putler (199), who also odeled the reference price effect vi, broke the visitor s judgent on RDP into three stages. In the first stage, the visitor judges the level of relative value by coparing Q and RQ. In the second stage, the visitor evaluates the level. This eans that the visitor evaluates the degree of relative value before evaluating it as his or her utility. In the third stage, the evaluated relative value is reflected in the visitor s utility function. As for the first stage, let DR, be the doain of relative values and RE be an eleent of DR. RE is a gain when RE DR, and this eans that a visitor judges RE Q RQ 0. RE is a loss when RE DR, and this eans that a visitor judges RE Q RQ 0. Finally, RE is zero when RE Q RQ 0. Let zero be included in gains for the sake of siplicity. In forulations, the notation g eans RE DR, the notation l eans RE DR 0 eans RE 0., and the notation In this study, RQ is assued to be endogenously deterined. For eaple, RQ consists of the average of all (hoogeneous) environental qualities (e.g., quality indicators of a river) that the visitor already knows. Thus, Q RQ 0 eans that the environental quality ( Q ) is judged to be relatively better than the (aggregate) qualities which this visitor has eperienced or knows about. Otherwise, Q RQ 0 eans that the environental quality ( Q ) is judged to be relatively worse than the visitor s eperience or knowledge. Equation (1) represents gain, equation () represents loss, and equation (3) represents a duy function for gain and loss because the visitor cannot eperience gain and loss at the sae tie. g I Q RQ if Q RQ 0 (1) l 1 I Q RQ if Q RQ 0 () 1 I 0 Q RQ Q RQ For the second stage, the evaluation for gain and loss, let E() be an evaluation function. The evaluation functions for gain and loss are represented as equation (4). Eg ( g) Q RQ E( g, l) 0 Q Q El () l Q RQ E ( g) 0,li E ( g) 0, E ( l) 0,li E ( l) 0 (5) g g0 g l l0 l Equation (4) iplies that the evaluation would be different for gain and loss. Equation (5) is the conditions on the liit in which case (3) (4)
9 170 J. Econ. Int. Finance Figure 1. Eaple of the utility function with RDP. differentiations, the notations U and U are also used for siplicity. The differentiated condition is a general one for z,, Q, and G Twice differentiable; first differentiation is positive, and second one is negative or zero i (i.e. U / z U z 0, / 0 ). The notations of differentiations on other U z U zz functions follow these notations. The property of loss ( l ) is considered by following an eaple of utility function that has the property of loss aversion (Figure 1). The dotted line is the utility function (value function) illustrated in Tversky and Kahnean (1991) and the solid line is the siplified case of the dotted line for an epirical analysis (of deand function) discussed above. In Figure 1, the first derivatives for losses in both cases are the sae as the gain ( Ul 0 ). The second derivative for losses is suarized as ull 0. Finally, differential conditions of a reference quality fro the utility function are that U / RQ u 0 and U / RQ u 0 REDR g REDR for the area of gains, and U / RQ u 0 and REDR U / RQ u ll 0 for the area of losses. Figure 1 also REDR illustrates that the increase of RQ correlates with the decrease of RE. l gg Utility aiization proble Fro equation (6), the utility aiization proble is defined as equation (7). The Marshallian deand function for recreation activity is derived as equation (8). Notice that the deand is zero if a visitor s utility is negative when RE is a loss. Thus, the utility is assued not to be negative even in the case of loss and the deand is a positive value i. Figure. Recreation deand and reference dependent preference. the utility function considered below becoes a traditional utility function that includes the absolute value of environental quality only when the relative level is equal to zero vii. Here the recreation deand is. So if a visitor uses the environental quality ties, then he or she acquires the E() aount of relative value. Let G be total gain and L be total loss defined as G E g () and L E l (). Then, a visitor s utility function is defined as equation (6) viii. Equation (6) iplies that a visitor has preferences based on both absolute evaluation ( Q ) and relative evaluation ( G, L ). Thus, the visitor s utility function becoes the traditional one fro equation (5) when GL 0. U u( z,, Q, G, L) (6) Here, n U / n denotes the nties differentiation for variable { z,, Q, G, L, RQ, g, l}. As for the first and second Ma u( z,, Q, G, L) s. t. y pzz p (7) z, ( p, p, y, Q, g, l) z (8) As for the Marshallian deand function, the weak copleentarity defined below is assued to hold for the environental quality ( Q ), RE (even at gains and losses), and the recreation deand. Fro the copleentarity, the Marshallian deand function increases when the environental quality or RE increases. Otherwise the deand decreases when the reference quality increases. To observe the difference between traditional deand functions and this odel, Figure illustrates a relation between the deand and the quality. Let the line fro A to D be the line in which RDP is zero, naely, Q RQ 0 for all points on the line (this deand function is equivalent to the one fro traditional econoic theory since it is equivalent to assue Q RQ 0 with the absolute value of environental quality only fro equation (5)). Net, let C be the point at which Q1 RQ 0, and let B be the other point at which the quality is less than the point C Q Q RQ ). Q0 RQ 0 iplies that RE is at loss. ( 0 1 Thus, the deand at point B is less than at point A because point A is the point at which RE is zero. Siilarly, let E be the point at which the quality is greater than point C ( 1 ). This iplies Q Q
10 Okuyaa and Hayashiyaa 171 that RE is at gain. Thus, the deand at E is greater than the one at point D because point D is the point at which RE is zero. As a result, the deand function with RDP is the line BCE. As for the case in which Q is fied and RQ increases, the change of the deand is syetrical to the case of Figure because the increase of RQ iplies RE decreases (goes to loss) fro the definition. As a result, the first derivatives of the deand function for RE are ( )/ g 0 and ( )/ l 0, and the second derivatives ( )/ g 0 and ( )/ l 0 are supposed (See Appendi A). So, the properties of RDP in the deand function are assued to be equivalent to those of RDP in the utility function. In epirical studies, Winer (1986) and Putler (199) eployed the linear for and Suzuki et al. (001) eployed the logistic for to estiate deand functions. In addition, Suzuki et al. (001) set ( ( )/ g)/( ( )/ l) 1 to test the loss aversion in the deand function ii. Finally, the indirect utility function is derived as V u z ( ), ( ), Q, g, l v( p, p, y, Q, g, l) and the ependiture function is derived as y e( p, p, U, Q, g, l) z. Utility iniization proble and the Slutsky equation Utility iniization proble By a siilar process, the utility iniization proble is defined as equation (9) and the Hicksian deand function is derived as h h ( pz, p, U, Q, g, l) iii. Equation (10) is assued to hold h for (), (), and e(). Finally, Shephard s Lea h ( y / p ( ) ) is assued to hold. z, Min p z p s. t. U u( z,, Q, g, l) (9) z h p, p, Q, g, l, e( ) (10) z Let the first derivative of environental quality for the Marshallian deand function be equation (11), that for the Hicksian deand function be equation (1), and that for the ependiture function be equation (13). In equation (13), e( )/ Q 0 is assued (Mäler, 1974). The Slutsky equation is derived fro equation (10) as equation (14). In equation (14), the first ter is the substitute effect; the second ter is the incoe effect; and the third and fourth ters are the gain/loss effects. In this study, the gain/loss effect is assued to be positive. h I 1 I Q Q g l z (11) h ( ) ( ) e( ) ( ) ( ) e( ) I Q Q y Q g y g ( ) ( ) e( ) 1 I l y l Static analysis on the Slutsky equation (14) Net the derivatives of RE for each function are considered. As for the indirect utility function, the first derivatives are V / g 0, V / l 0 and the second derivatives assue V / g 0, V / l 0. Those lead to the first derivatives of the ependiture function as y/ g 0, y/ l 0 and to the y/ g 0, second derivatives as y/ l 0 (See Appendi B). Therefore, equation (15) holds. Equation (15) indicates that the aount of ependiture at a loss is greater than at zero, and the ependiture at zero is greater than at a gain. e, l e,0 e, g (15) Net, the derivatives of the reference quality are considered. The first derivative of the Marshallian deand function is equation (16); it is negative because ( )/ g 0 and ( )/ l 0. The first derivative of the ependiture function is equation (17); it is positive because e( )/ g 0 and e( )/ l 0. The first. derivative of the Hicksian deand function is equation (18); whether it is positive or negative depends on the gain/loss effect. If the gain/loss effect is assued to be positive, equation (18) is negative. I 1 I RQ g l y I e( ) 1 I e( ) RQ g l h ( ) ( ) e( ) I RQ g y g ( ) ( ) e( ) 1 I l y l (16) (17) (18) Finally, the total effects on the deands and the ependiture fro the increase of environental quality and the reference point are calculated by equation (19) fro equations (11), (13), (14), (16), (17) and (18). Then, it is suarized as equation (0). h h h h I 1 I Q Q g l y e I e 1 I e Q Q g l (1) (13), Q RQ Q y y e, Q RQ Q h h ( ) ( ) e( ) Q RQ Q y Q (19)
11 17 J. Econ. Int. Finance h h y y / Q / RQ / Q / RQ Q RQ ( )/ y (0) Corollary 1. If the increase of both the environental quality and the reference point is equivalent, then the increase of (Hicksian) deands and the ependiture are equal to the increase of both function in which assue only absolute value of Q. That is, the arginal benefit with RDP is equivalent to the arginal benefit without RDP (the traditional benefit). Equation (19) iplies that RDP does not influence the aount of deand and ependiture if the environental quality and the reference quality increase or decrease by the sae degree. Equation (0) suarizes the total effect for the arginal benefit. The values of the first and the second parentheses on the right side are equal to the value without RDP. Thus, the benefit with RDP is equivalent to the benefit without RDP. Finally, the choke price and the weak copleentarity are * considered. The choke price p is defined as equation (1), which iplies that the choke price is the price at which the Hicksian deand is zero. Notice that the Hicksian deand includes a gain and a loss. Net, the weak copleentarity is generally defined as ( ) / * 0. However, the ependiture function of this odel e Q p includes the gain and loss effect as equation (13). The odified version of the weak copleentarity is defined as equation (). p * h in p ( ) 0 (1) e( ) e( ) e( ) I 1 I 0 Q * g * l * p Definition of benefit and static analysis Definition of total value p p () It is necessary to differentiate between situations in which a project is ipleented and those in which no project is ipleented to define the benefit fro the change of an environental quality with RDP iv. Let s wo be the superscript representing the quality level at which a project is ipleented and s w be the one at which no project is ipleented. Using the notation, the utility function (and other variables) are rewritten as U s u z s, s, Q s, G s, L s. Note that a visitor does not eperience a gain and a loss at the sae tie, so RE differs based on whether or not the project is ipleented. For eaple, in one case, RE gains when s wo and it will also gain when s w ; however, in another case, RE decreases when s wo and will be at zero when s w. The benefit fro quality change is defined by equivalent variation (EV) as equation (7) and copensating variation (CV) as equation (9). EV and CV can be decoposed into three kinds of benefit fro incoe change (equations (5) and (9)), and benefit fro quality change (equations (6) and (30)). This study eaines only the benefit fro quality change v. Thus, the total value of environental quality (hereafter TV) is suarized as equation (31) vi. Equation (31) iplies EV when s w and CV when s wo. wo wo w wo wo wo wo wo wo wo wo wo EV e( p, p, U, Q, g, l ) e( p, p, U, Q, g, l ) z z (3) wo wo w wo wo wo w w w wo wo wo e( p, p, U, Q, g, l ) e( p, p, U, Q, g, l ) z z z z (4) w w w w w w wo wo wo wo wo wo e( p, p, U, Q, g, l ) e( p, p, U, Q, g, l ) z z (5) w w w wo wo wo w w w w w w e( p, p, U, Q, g, l ) e( p, p, U, Q, g, l ) (6) w w w w w w w w wo w w w CV e( p, p, U, Q, g, l ) e( p, p, U, Q, g, l ) z z wo wo wo w w w w w wo w w w e( p, p, U, Q, g, l ) e( p, p, U, Q, g, l ) z z w w w w w w wo wo wo wo wo wo e( p, p, U, Q, g, l ) e( p, p, U, Q, g, l ) z z z z (7) (8) (9) wo wo wo wo wo wo wo wo wo w w w e( p, p, U, Q, g, l ) e( p, p, U, Q, g, l ) (30) s s s wo wo wo s s s w w w TotalValue e( pz, p, U, Q, g, l ) e( pz, p, U, Q, g, l ) (31) Equation (31) includes the quality itself ( Q ),the gain ( g ), and the loss ( l ). Equation (31) is thus a coprehensive forulation wo w including the absolute evaluation (the evaluation for Q Q ) and the relative evaluation (the evaluation for wo wo w w ( g, l ) ( g, l ) ). Influence of RDP for total value Let p z, p, y, and Q (the part of absolute evaluation in utility function) be fied when s wo and s w. As for locations of RE, there are three possible areas: gain, zero, and loss. Thus, there are 3 3 patterns to deterine the value of TV (e.g., wo w TV e(, g ) e(, g ), (,0 wo w TV e ) e(, l ) ). Figure 3 shows these cases with a possible ependiture function. Let the origin of the arrow line correspond to the aount of ependiture when the wo wo in equation (31)) and project is not ipleented (i.e., e(, g, l ) the end point of the arrow line correspond to the aount of ependiture when the project is ipleented (i.e., (, w w e g, l ) ). For eaple, the third case indicates the aount of ependiture wo, and the eighth case indicates change fro e(, l ) to e(, l w ) wo w the aount of change fro e(, l ) to e(, g ). In these cases, the second and fourth, third and fifth, sith and eighth, and seventh and ninth cases ean the sae changes of TV because each condition differs only in ters of whether the RE benefits: benefit fro price change (equations (3) and (8)), goes fro low to high or fro high to low. Thus, the first, fourth, fifth, sith, and seventh cases are considered. The case of zero to zero (first case) eans there is no effect on the RE. The case of zero to gain (fourth case) eans the increase of the benefit, and the case of zero to loss (fifth case) eans the decrease of the benefit. The case of loss to zero (sith case) eans the increase of the benefit, and the case of gain to zero (seventh case) eans the decrease of
12 Okuyaa and Hayashiyaa 173 Table 1. Benefit and positions of RE With Without Gain Zero Loss Gain Increase Increase Increase Zero Decrease No Effect Increase Loss Decrease Decrease Decrease s w since the difference of ependiture is sallest (i.e., the ninth case in Figure 3). This case is interpreted to be a situation in which a visitor judges an objective environental quality to be better than other qualities when s wo, and judges it to be worse when s w. One reason that a visitor ight judge the quality to be worse despite environental iproveent could be that if other qualities are also iproved at sae tie and those are ore ipressive to the visitor, the reference quality would increase ore than the objective quality would increase. Another reason for such a judgeent could be a gap between the quality change achieved by the project and the quality change the visitor iagines when s wo. Therefore, the result of the project would have a negative ipact for the visitor. The benefit fro this situation is denoted by TV and defined as equation (34). GL s s s wo wo s s s w w TV e( p, p, U, Q, g ) e( p, p, U, Q, l ) (34) GL z z Figure 3. Nine cases of relative evaluation. the benefit. These results are suarized in Table 1. Let us consider the iplications of these benefits. In the cases in which the benefit increases, TV increases ost in the case in which RE changes fro a loss when s wo to a gain when s w since the difference of ependiture is biggest (that is, the eighth case in Figure 3). This case is interpreted to be a situation in which a visitor judges an objective environental quality to be worse than other qualities (which are coprehensively denoted by RQ ) when s wo, and judges it to be better when s w. In short, the ipact of environental quality change on a visitor s RDP is very big. The benefit is denoted by TV and defined as equation (3). LG s s s wo wo s s s w w TV e( p, p, U, Q, l ) e( p, p, U, Q, g ) (3) LG z z The case in which RE is zero when s wo and s w is interpreted to be a situation in which a visitor judges the quality to be equivalent to other qualities when s wo and s w because the visitor s reference quality ( RQ ) changes to the sae degree as the quality change (Corollary 1). One eaple of such a situation would be of another project being ipleented for other environental quality at the sae tie. The benefit fro this situation is denoted by TV 00 and defined as equation (33). s s s wo s s s w TV00 e( pz, p, U, Q,0) e( pz, p, U, Q,0) (33) In cases in which TV decreases, TV decreases ost in the case in which RE changes fro a gain when s wo to a loss when These results are suarized in equation (35). In addition, in the case of environental deterioration, these inequalities becae TV TV TV ). Equation (35) iplies that 1) reverse (i.e. LG 00 GL the total value that is defined only by the absolute value of quality TV ( 00 ) would be a part of the values and ) there is a possibility that the total value can be negative even if the project ais to iprove quality because there is no restriction on the reference quality change. TV TV TV (35) LG 00 GL Decoposition of use and nonuse values and an interpretation of nonuse value The decoposition of use and nonuse value is perfored to investigate the relation between nonuse value and RDP. Equation (31) decoposes TV into use value (equation (36)) and nonuse value (equation (37)) by using the choke price (Neil, 1988; Larson, 199). s s s wo wo wo s s s w w w TV e( p, p, U, Q, g, l ) e( p, p, U, Q, g, l ) z z s * s w w w s s s w w w e( pz, p, U, Q, g, l ) e( pz, p, U, Q, g, l ) e p p U Q g l e p p U Q g l s * s wo wo wo s s s wo wo wo ( z,,,,, ) ( z,,,,, ) s * s wo wo wo s * s w w w ( z,,,,, ) ( z,,,,, ) (36) e p p U Q g l e p p U Q g l (37) Let us consider the definition of nonuse value. Generally, the nonuse value is defined fro the properties of uniqueness and irreversibility (Krutilla, 1967). Regarding uniqueness, equation (37)
13 174 J. Econ. Int. Finance Table. Paraeters. Q g l p y wo Q w Q includes the reference point so that the environental quality is copared with others. There is a possibility that the uniqueness will not hold. However, Krutilla (1967) stated that uniqueness is not necessary condition for his arguent. One reason is that there is a possibility that a siilar environental quality eists in another arket which is difficult to access. Thus, a visitor can copare the objective environental quality viii of one place with others even if the other places are out of reach. If the eistence value defined only by absolute evaluation (in this study, it is the case in which RE 0 ) is the pure eistence value, equation (37) would interpret the ipure eistence value. Integratingback approach Von Haefen (007) presented three ethods to estiate the total value of environental quality by arket data. In this paper, the integratingback approach is eployed viii. This approach is useful for obtaining the total value fro a deand function. The central idea is to derive the quasiependiture function, which was developed by LaFrance (1985). Recently, Eo and Larson (006) presented an estiation odel based on the integratingback approach. This ethod can calculate the use and nonuse value fro arket data such as the forulations derived below. Let the deand function be equation (38). Here Q Q I g (1 I) l. As for the estiation, visit Q g l nuber ( ), travel cost ( p ), incoe ( y ), and environental quality ( Q ) are observed in the recreation arket. In addition, other environental qualities ust be accounted for to deterine the reference quality. It is necessary to research the data of visitors knowledge about other qualities, or their eperience of sites they have visited. Then the reference quality is constructed as shown above and the data of gain ( g ) or loss ( l ) are calculated i. ln[ ( )] p y Q (38) The quasiependiture function is equation (39), where the constant of integration is Uep Q. The (indirect) utility function is equation (40). Let the price and the incoe be fied for siplicity, and the notation s be oitted in these variables. Then the total value is derived as equation (41), where s s ep Q p y Q. TV consists of the deand s s s ( ( p, y, Q ) ) and the quality ( s Q ). Finally, TV is decoposed into nonuse value (NUV) as in equation (4) and use value (UV) as in equation (43). / ep p Q UepQ 1 e p, U, Q ln (39) U (1/ )ep( y) (1/ )ep( p Q) ep( Q) w wo 1 Q 1 ln Q TV w wo ep Q Q w wo NUV Q Q (40) (41) (4) 1 ln w 1 wo UV TV UV Q Q Paraeters for benefit calculation (43) A project concerning quality iproveent is assued. The siulation is focused only on the total value because nonuse value is defined as erely the difference of quality change i. Paraeters are specified in Table. The paraeters,, Q,, p, and y are the sae as those in Tables 1 and of Eo and Larson (006) (the results of estiation odel for nonuse value). The variables g and l are originally designed so as to satisfy the property of RDP of deand function ( ( / g)/( / l) 1)). For the quality level, Eo and Larson (006) used biocheical oygen deand (BOD) ii for the wo w estiation; Q 14 and Q 17 are designed as these levels. Since positive utility does not arise for values ( Q ) under 10 in this odel, the quality levels and reference qualities are set at values over than 10. Therefore, this foration cannot be used for arbitrary values of paraeters. The siulation is thus perfored for reference quality ( RQ ) ranging fro 10 to 0 in onepoint increases. RESULTS AND DISCUSSION The siulations were perfored by each functions (equations (38) to (41)). The ain focus of the discussion is to eaine the differences between the traditional benefit calculation odel and the odel of this paper by analyzing the relations between RQ and TV. Deand function Figure 4 shows the change of the deand corresponding
14 Okuyaa and Hayashiyaa 175 Deand levels Utility levels Reference point Q=14 Q=17 Figure 4. Reference quality and deand function Reference quality Figure 6. Reference quality and utility function. Ependiture levels Reference quality Figure 5. Reference quality and ependiture function. to each reference point. The gray line is the deand wo function when the quality level is Q and the black line is w the deand function when the quality level is Q. The point 14 for the gray line and the point 17 for the black line are the points at which RE 0 (i.e., the inflection points for each function). The inflection points are not discussed in previous section. However, this forulation is eployed because it is coonly used in RPM odels (for the forulation of deand function) and eperiental studies (for the forulation of value function). In previous section, the relation between the deand and RE was analyzed. The deand decreases when RE becoes negative in Figure. Figure 4 shows both deand levels decrease corresponding to the increase of RQ (the increase of RQ eans that RE becoes negative). This feature is the sae in Figure. Ependiture and utility function The ependiture function (equation (39)) is shown in Figure 5. In previous section, the condition y / RQ 0 is discussed (equation (17)) and it is reflected in Figure 5. Here, the U in the ependiture function is the utility at point 14 in Figure 6. As for the property of RDP, the gradient at the loss is greater than the gradient at the gain. Siilarly, the condition U / RQ 0 for the (indirect) utility function is reflected in Figure 6. Especially, the properties of loss aversion (li g0ug)/(li l0ul) 1 are observed. That is, the properties of loss aversion in the deand function (e.g., g 0.03 and l 0.07 ) are also reflected in the utility function. Benefit calculation Figure 7 shows the change of TV corresponding to each level of reference point. The black line shows the case in which RE in the second ter of equation (31) is fied at zero and RE in the first ter of equation (31) changes (eactly to say, RQ in the first ter of equation (31) changes) iii. As a result, the total value decreases
15 176 J. Econ. Int. Finance Total value Figure 7. Reference quality and total value. Table 3. Eaple for Table 1 fro Figure 5. With Without Gain Zero Loss Gain [8,10] [7,10] [5,10] Zero (1,3) (3,3)[7,7] [5,7] Loss (1,6) (3,6) (6,9) Total Value Reference quality RE=0(without) RE=0(with) 1 3 Cases Figure 8. Relative evaluations and size of total value. following the increase of RQ. This case corresponds to the fourth case in Figure 3. The increase of RQ in the first ter eans that the value (the aount of ependiture) of the first ter of equation (31) decreases, so the difference between the first and second ters of equation (31) is a decrease. The black line also shows the relation between RE and the total value in Table 1. Let (row, colun) be the eleent in the atri corresponding to the row and colun of Table 1. (e.g., (zero, loss) eans decrease in Table 1). In Figure 5, for eaple, the difference between the value at point 3 and that at point 6 (hereafter abbreviated as (3, 6)) corresponds to (zero, loss) if the wo value at 3 is the one when Q Q and the value at 6 is w the one when Q Q. Siilarly, other situations can be considered (e.g., (6, 9) corresponds to (loss, loss)). Other eaples are listed in Table 3. The gray line in Figure 7 shows that RE in the first ter of equation (31) is fied and RE in the second ter of equation (31) changes. As a result, the total value increases following the decrease of RQ. This case corresponds to the sith case in Figure 3. The increase of RQ in the second ter of equation (31) eans that the value of the second ter decreases, so the difference between the first and second ters of equation (31) is an increase (the total value increases). The gray line also shows the relation in Table 1. Let iv [row, colun] be the eleent in the atri corresponding to the row and colun of Table 1 (e.g., [zero, gain] eans increase fro Table 1). In Figure 5, for eaple, the difference between the values at point 7 and point 10 (abbreviated as [7, 10] below) corresponds to [zero, gain] if the value (the aount of ependiture) at wo 7 is the one when Q Q and the value at 10 is the one wo when Q Q. Siilarly, other situations can be considered (e.g., [8, 10] corresponds to [gain, gain]). Other eaples are listed in Table 3. Figure 8 shows the relation in equation (35). The situations (the aount of deand and environental wo w qualities when Q Q and Q Q ) are given in Figure 7. Case 1 indicates the value of TV LG when the first ter of equation (3) is set as point 0 on the gray line and the second ter of equation (3) is set as point 10 on the black line in Figure 7. Siilarly, Case indicates the value of TV 00 when the first ter of equation (33) is set as point 14 on the gray line and the second ter of equation (33) is set as point 17 on the black line. Case 3 indicates the value of TV GL when the first ter of equation (34) is set as point 0 on the gray line and the second ter of equation (34) is set as point 10 on the black line. The result of equation (35) is confired. Figure 9 shows the price change and the total value. The black line is the sae as in Figure 7, the gray line is the line when the price of the black line set 0, and the dotted line is the line when the price of the black line set 40. The total value decreases when the price increases.
16 Okuyaa and Hayashiyaa 177 Total value Figure 9. Price change and total value. Total value reference quality TV(without=0,p=9.91) TV(without=0,p=0) Reference quality TV(without=0) Figure 10. Etree case of reference quality. Finally, a proble on calculation is discussed. It is called the etree oving to loss. Figure 10 shows the case (by using the black line fro Figure 7) in which the reference quality changes etreely to loss. In this case, the value goes to negative such as at point 30 despite the project s ai of environental iproveent. This happens because there is no restriction on the value of reference quality for s wo and s w (therefore, this case occurs in the case of the gray line, e.g., if RQ goes below 10). Thus, a boundary condition should be set for the reference quality or preference structures. Estiation of deand function Final section described how to estiate the deand function represented as equation (38). In estiating, an independent variable is the visit nuber ( ); dependent variables are the travel cost ( p ) and household incoe ( y ) observed in the recreation arket (or collected by a survey). The data on environental quality ( Q ) in recreational sites are also used for a dependent variable. Here, the reference quality data ( RQ ) would be collected by asking respondents; For eaple, how uch quality level ( RQ ) of the environent do you need? If RQ Q, the respondents would be categorized as gainrespondents. In the inverse case, lossrespondents. Researchers would arbitrarily decide how to categorize the case of RQ Q ; either include it in gainrespondents or lossrespondents. Whether respondents preferences are the absolute or relative valuation would be eained by coparing the deand functions; ( p, p, y, Q) and ( p, p, y, Q, g, l). z Conclusion Most environental valuation studies have eployed absolute evaluation assuption for valuation ethods. However, soe eperiental studies in econoics and psychology pointed out that an individual s decision aking is influenced by reference dependent preference, naely relative evaluation. The purposes of this paper are 1) to forulate a recreation behavior odel with RDP for environental quality and ) to forulate an application odel for benefit calculation. As for the base odel, the travel cost ethod is eployed as discussed below. First, the odeling of RDP in a visitor s utility function and the properties of RDP on deand and ependiture functions were eained. The analysis for the deand function revealed that the siultaneous changes of the environental quality and the reference quality are equivalent to the condition in which only the absolute evaluation is considered. This iplies that soe effects (e.g., a fraing effect) of RDP for the value of the benefit arise only when the quality and the reference point change in different directions. Second, the definitions of the benefits and static analysis were considered. The finding was that the total value defined only by absolute evaluation is one of the z
17 178 J. Econ. Int. Finance benefits that include RDP. Thus, it is necessary to deterine whether the benefit (or, in ters of this study, visitor behavior) includes RDP or not. If the benefit includes RDP, then the benefit can change depending on the reference points. This paper copare the benefits 1) relative evaluation is loss before ipleenting a project and relative evaluation is gain after ipleenting a project, ) relative evaluation is zero before ipleenting a project and relative evaluation is zero after ipleenting a project, 3) relative evaluation is gain before ipleenting a project and relative evaluation is loss after ipleenting a project. As a result, the benefit of first case is bigger that the second case and the second is bigger than the third case (equation (35)). Third, this property was confired though siulations. Since the coputable forulation reflects the theoretical findings, this odel can be used to estiate the recreation deand function with RDP and to calculate the benefit. Finally, soe probles for epirical study should be entioned. First, the recreation deand for a single site is assued in this paper. However, the structure of reference quality needs the aggregation of other qualities that a visitor has already eperienced or knows about. Thus, it is natural to assue there are ultiple sites for recreation deand. One solution could be to use the ultiplesite trip for travel cost ethod, e.g., the Kuhn Tucker Model. Second, the loss effect cannot be estiated if there is no reservation utility (discussed in theoretical analysis). Since the loss effect eans that the utility is at a negative value, it is possible for a visitor not to select such recreation sites. Third, if the influence of RDP is confired, there is a possibility that the benefit will change over a long period of tie due to the change of the reference quality. Thus it ay be necessary to consider the structures of RDP for a dynaic odel. Notes i. The fraing effect is a phenoenon in which an individual s preference changes depending on how options are presented (fraed) in a questionnaire (Tversky and Kahnean, 1991). ii. The status quo bias is a phenoenon in which an individual tends to prefer to reain at the status quo due to an aversion toward loss (Kahnean et al., 1991). iii. The endowent effect is a phenoenon in which an individual feels that a good has a higher value once he or she has becoe the owner of the good. This effect has been eplained as being equivalent to status quo bias (Kahnean et al., 1991). However, since it is not clear in these studies whether or not the property right is the ain coponent of the status quo, these two biases are eplained separately. iv. These biases are discussed etensively in the proble of the disparity between willingness to pay and willingness to accept. Mitchell and Carson (1989) present cases of this disparity, and RDP is one of the. Haneann (1991) deonstrated the cause theoretically without RDP. However, recent studies indicate RDP as the ain cause of endowent effect (Horowitz and McConnell, 00; Plott and Zeiler, 005; Brow, 005). v. Suzuki et al. (001) estiated the deand function with RDP for services qualities in the airline arket. vi. Putler (199) odeled the reference price effect as ref follows: let p be the price of a good, p be the reference ref price of the good. Then the gain is p p 0, and the ref loss is p p 0. This iplies that a visitor gains a utility if the price is ore inepensive than the reference point. vii. Putler (199) set E( ) 0. However, it is generally assued that Eg ( g) 0 and El ( l) 0 in the utility function with RDP. Thus, those conditions are eployed in such studies as Munro and Sugden (003). viii. In addition, since G and L are directly involved in the utility function, it ay be useful to construct a utility function to assue functions F and F such as G F ( ) E ( g) and L F ( ) E ( l). An eaple of the g g Utility function is, where l l g F E g g and 1/ g( ), g( ) 0.5 1/ Fl( ), El( l) l. i. As for the notation, the first derivatives are denoted as U z u, the second derivatives are denoted as / z U / z uzz U / z uz, and the cross derivatives are denoted as. Note that u gl is not defined.. Foral conditions are based on a odified version of Bowan et al. (1999). The first is that U is strictly increasing in RE. The second is to epress the relation between the arginal utility of a loss and the arginal utility of a gain, defined as U ( RE) U ( RE) U( RE) U( RE) for 0 RE RE. The third is to represent an assuption of diinishing arginal sensitivity defined as RE is strictly concave in RE 0 and RE is strictly conve in RE 0. The fourth is to represent that a person can evaluate losses even when coparing very sall losses to very sall gains, given that there eists a value M s.t. li 0( u ) l RE RE REDR /( ure REDR ) M. In addition, if RE is the linear for, a siple condition to epress loss aversion is li g0ug/lil0ul 1. i. It is natural to think there is a value of RE at which the value of the utility becoes positive if RE eceeds the value ( RE RE U 0 ). This iplies that it is necessary to assue a reservation utility for a recreation activity ( ) if a recreation activity occurs even in the case that RE is at loss. This study assues that the absolute value of Q in the utility function will perfor the role.
18 Okuyaa and Hayashiyaa 179 Thus, u(, Q, L) 0 for RE RE is assued. ii. Let be each paraeter. For eaples, a linear for is p p y y QQ gg ll and a logistic for is 1/{1 ep( p y Q g l)} p y Q g l in the notation of this paper. The condition to epress the loss aversion as / 1 for the linear for and g l ( ) ( ) / 1 for the logistic for. This g l g l iplies that these deand functions ust reflect the properties of utility function (although those would not be reflected eactly). iii. In addition, the ependiture function is defined as h h y pzz p by using the Hicksian deand function. iv. Whether the project is for environental iproveent or for environental deterioration does not atter. The difference corresponds to the definition of willingness to pay and willingness to accept. However, the project is ainly assued to be an environental iproveent project in this study, as discussed below. v. An additional decoposition is shown in Appendi C. vi. In epirical studies, equation (31) would be rewritten wo Q astv e( ) / Qd w. It would be necessary to Q consider the kink at RE 0 if the doain of RE changes between s wo and s w. Thus, equation (31) is the definition when e() is differentiable at RE 0. vii. He also cannot access to the objective environental quality in the definition of equation (37). The reason why he cannot access is due to eploying the choke price to define the equation. See equation (1). The choke price is defined as the price at which (Hicksian) deand is zero. This interpreted as being a situation in which a visitor cannot visit the recreation site because of high travel costs. viii. The related papers concerning this ethod are Neil (1988) and Bullock and Minot (006). The probles are 1) there is no guarantee to derive the choke price fro arbitrary functional for to deterine the use value and nonuse value, and ) it is difficult to apply the ethod to the case of ultisite trips. i. For eaple, assue that the reference quality coes fro the average of the qualities that a visitor has known. If visitor A who plans to go to site 1, which has quality= knows site s quality=3, site 3 s quality=1, and site 4 s quality=5, then the visitor s reference quality is 11/4=.75 and RE =.75= Thus, visitor A s RDP is at loss. If visitor B, who plans to go to site 5, has RE =5.75=.5, the RDP is at gain. For a data set, it is siple to set if a visitor s RE is at gain, then his or her data of loss is zero (Suzuki et al., 001).. Eo and Larson (006) assued the constant of Uep Q. is used to estiate the integration as eistence of nonuse value. Thus, the odel in this study assues 1 and adds RE to their odel. i. In addition, the result of analysis for eistence value are siilar one for the total value since the difference of both definition (total value and the nonuse value) are only the definition of price and the prices are fied in each definition (See equation (31) and equation (37)). ii. 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20 Okuyaa and Hayashiyaa 181 Appendi A. Deand function Let be the undeterined ultiplier and the Lagrange equation be defined by equation (A.1). The first derivatives of equation (A.1) are suarized as equation (A.). Let dp, dp, dy, and dq be zero, and p be equal to 1. Then, the z total derivative of the deand function is derived as equation (A.3) by using Craer s rule. Here, A is the deterinant of the atri (the first parenthesis of the left side of equation (A.)). u( z,, Q, g, l) ( pzz p y) (A.1) uzz uz pz dz dpz ( uzqdq uzgdg uzldl) uz u u Q d dp ( uqdq ugdg uldl) pz p 0 d dy ( zdpz dp ) (A.) 1 d {( ugdg uldl ) uq ( uzgdg uzldl )} A (A.3) The first derivatives are equation (A.4) fro equation (A.3). It is necessary to hold u u u 0 for gains and u z g Q zg u u 0 for losses to satisfy the first differential condition discussed above. The second derivatives are equations l Q zl (A.5) and (A.6). It is necessary to hold equation (A.5) as nonpositive for gains and equation (A.6) as nonnegative for losses to satisfy the second differential condition. These conditions are suarized as equation (A.7). d u u u d u u u, g dg A l dl A g Q zg l Q zl { ( ug uquzg ) / g} A ( ug uquzg )( A / g) g A { ul uquzl / l} A ( ul uquzl )( A / l) l A { ( u g u Q u zg )/ g} ( u )( / ), l u Q u zl A l A A ( u u u )( A / g) { ( u u u )/ l} g Q zg l Q zl (A.4) (A.5) (A.6) (A.7) Finally, an eaple of deand function and the derivatives are illustrated by using the eaple of utility function in 1/ 1/ footnote 8; U z ( Q 0.5 g l). Fro the first derivatives of the Lagrange equation, the deand function is derived as equation (A.8), and the first and second derivatives for gains and losses are equations (A.9) and (A.10), where B p C p 0 and C { Q 0.5 g l} 0. C { Q l} 0 is assued to ensure the utility is positive even if RE is at loss and the deand would be positive(see footnote 10). The first derivatives are satisfied in both gains 1 and losses; however, the second derivatives need the conditions p B C for gains and 4p B C 3 0 for losses. This iplies that the deand function satisfies the conditions when 1 3/ 4 p B C 9/ 4 holds. y p ( Q 0.5 g l) p g l p yb C 0 3 4p yb C 0,, 0.15 p y p B C 1.5B C g 1 4 8p y 4 p B C 3A B l 1 4 (A.8) (A.9) (A.10)
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