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Utility maximization in models of discrete choice
economics letters EI.SEVIER
Economics Letters 49 (1995) 91-94
Utility maximization in models of discrete choice Martin Peitz Wirtschaftstheoretische Abteilung H, Bonn University, Adenauerallee 24-42, 53113 Bonn, Germany Received 7 September 1994; accepted 8 November 1994
Abstract
In this paper I construct a direct utility function that has as its counterpart an indirect utility function that is commonly used in models of product differentiation.
Keywords: Utility maximization; Discrete choice JEL classification: L10
1. Introduction In various c o m m o n models of discrete choice it is assumed that consumers either choose one variant of a good of which they consume one unit and none of the other variants, or they choose the outside option. Articles analyzing this class of models usually commence with an evaluation function, which is called a conditional indirect utility function. A consumer chooses the variant with the highest value: if for some variant the value is positive, a consumer buys one unit of the variant that gives him the maximal value. Otherwise, he chooses the outside option. Duality theory suggests that there is an associated direct utility function. However, the results of duality theory cannot be applied to this problem because continuity is not satisfied. Therefore, there are some doubts as to whether the evaluation function is, indeed, an indirect utility function and whether it is consistent with utility maximization. I will construct a direct utility function such that the underlying preference relation satisfies reflexivity, transitivity, completeness, and local nonsatiation. I will then show that this direct utility function has as its counterpart the indirect utility function I was looking for. Hence, consumer behavior in discrete choice models of the type presented can be derived from utility maximization. 0165-1765/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSD1 0 1 6 5 - 1 7 6 5 ( 9 5 ) 0 0 6 6 5 - 6
M. Peitz / Economics Letters 49 (1995) 91-94
92
2. Notation and results
Suppose there is a m a r k e t with n different variants of a good that is vertically differentiated. Variant i, i = 1 , . . . , n, is described by its quality q~ E [q_q_,~], q ~> 0 and ~ < ~, and its price, Pi > 0. These variants are ordered such that qi < q~+ 1, i = 1 . . . . . n - 1. Each c o n s u m e r has the same reservation price R for the good with quality 0 (this variant may be artificial). It will turn out that R cannot be interpreted as income. The valuation of quality is d e n o t e d by 0 ~ [_0, 0] C [0, ~] and defines a type of consumer. I will give a direct utility presentation for the following indirect utility function:
v(p, y; 0) = max[0, max v,(p, y; 0 ) ] ,
(1)
where vi(p, y; O)= R - p i + Oqi is called the conditional indirect utility function of type 0. This kind of indirect utility function was first p r o p o s e d by Mussa and R o s e n (1978) and has been used in many articles on vertical product differentiation; c o m p a r e , for example, A n d e r s o n et al. (1992). As an exception to other articles in the field, Mussa and R o s e n defined a conditional direct utility function. However, they a priori begin within a f r a m e w o r k of discrete choice and unit d e m a n d whereas I derive this behavior from utility maximization. Note that the results in this paper can be rewritten for linear r a n d o m utility models frequently used in models of horizontal product differentiation. In the class of models u n d e r consideration, consumers buy one unit of up to one variant. For the n variants, I introduce the vector of quantities x ~ No+. In addition there is a n u m e r a i r e good with quantity x 0 1> 0 and normalized price P0 - 1. This means that the price for the variants and income are measured in units of the n u m e r a i r e good. I n c o m e of a c o n s u m e r of type 0 is interpreted as y = R + @, which is the reservation price of a c o n s u m e r of type 0 for a unit of the variant with the highest possible quality. Thus, consumers with a higher income care more about quality. Note that y can be m o r e generally interpreted as a share of income. It is only important that the relevant budget for the c o n s u m e r is greater or equal to R + 0~. This is always satisfied because a reservation price says how m u c h a c o n s u m e r is willing to pay for one unit, given his income. The budget constraint is p - x + x 0 <~y. T h e interpretation of A n d e r s o n et al. (1992) of R as income is inconsistent with utility maximization because this implies that for a p~ ~ [R, R + Oq~] a c o n s u m e r of type 0 prefers variant i over the outside option and hence violates the budget constraint that is implicit in any indirect utility function. D e n o t e e i the n-dimensional vector that has a 1 at the ith position and a 0 elsewhere. Since I am looking for a justification of discrete choice and unit d e m a n d , the direct utility function u(xo, x; O) to be constructed should satisfy the following properties: (P1) U(Xo + A , x ; O ) > u ( X o , X ; O ) , A >O , (P2) U(Xo, ei;O)>~U(Xo, O,;O), i= 1 , . . . ,n, (P3) u(x0, 0,; 0) = u(x o, Aei; 0), i = 1 , . . . ,
n, A E (0, 1),
M. Peitz / Economics Letters 49 (1995) 91-94
93
(P4) u(x o , e ~ ; O ) = u ( x o , ( l + A)e~;O), i = l , . . . , n , A > O , (P5) u(xo, ei;O ) = u ( x o , x ; O ) , x > e i and x / ~ e i , j > i , (P6) T h e r e exists an index i, 1 ~< i ~< n, such that u(x 0, ei; 0) ~> u(x 0, x; 0), x E ,qt+. (P1) says that consumers are locally nonsatiated in the n u m e r a i r e good. By (P2) no c o n s u m e r is worse off by taking a unit of some variant in addition to the c o n s u m p t i o n of the n u m e r a i r e good. According to (P3), consumers do not evaluate a quantity of less than one unit and, according to (P4), they do not evaluate a quantity of m o r e than one unit. (P5) says that, given the quantity of the numeraire good, consumers do not strictly prefer m o r e than one unit of variant i and less than one unit of the other variants over one unit of variant i and zero units of the others. (P6) reads: there exists a best variant i such that a c o n s u m e r weakly prefers e, over any x E .)l+ T o capture c o n s u m e r behavior in discrete choice models with unit d e m a n d (see ( P 2 ) - ( P 6 ) above) I introduce the following indicator functions for i = 1 , . . . , n: n
{~' Xi(x) =
,
,
ifxi~l'xi>~xjf°rjxi otherwise,
f°rj>i,
with the convention that, in the case of indifference, the variant with the smallest index is chosen. This gives a 1 to the variant with the highest quantity a m o n g variants if it is greater or equal to 1; otherwise, it gives a 0. The direct utility function of type 0 is then defined as u(x 0, x; O) = ~'~ (R + Oqi)xi(x ) + x o - R - ~ .
(2)
i
It is easily seen that the underlying preference relation satisfies reflexivity, transitivity and c o m p l e t e n e s s (see, for example, Barten and B 6 h m , 1982). It is also clear that the following result holds.
Result 1. u(xo,x; O) defined in (2) satisfies ( P 1 ) - ( P 6 ) . A n indirect utility function is defined as
v ( p , y;O) = mX(I.X a x {u(x 0 , x ; 0 ) l p . x + x o<~y} . I am n o w in a position to state the relationship between the functions defined by (1) and (2) w h e n y = R + 0~.
Result 2. T h e function defined in (1) is the indirect counterpart of the direct utility function defined in (2).
M. Peitz / Economics Letters 49 (1995) 91-94
94
Proof. v(p, y; O) = mX(I,X a x { u ( x , , , x ; O ) l p .x + xo <~R + Oq} = max[u(R + 0F, 0,,;0), u(R + 0~1 - P l , e l ; O ) , . . .
=max[O,R + O q l - P l . . . . , R + O q , - p , , ] .
, u(R + 0 q - p n , en; 0)]
[]
The discrete choice model presented is thus consistent with utility maximization.
Acknowledgment Research for this paper was done while I was staying at C E P R E M A P , Paris. Financial support from the French G o v e r n m e n t is gratefully acknowledged.
References Anderson, S., A. de Palma and J.-F. Thisse, 1992, Discrete choice theory of product differentiation (MIT Press, Cambridge, MA). Barten, A. and V. B6hm, 1982, Consumer theory, in: K.J. Arrow and M.D. Intriligator, eds., Handbook of mathematical economics, Vol. II (North-Holland, Amsterdam) 381-429. Mussa, M. and S. Rosen, 1978, Monopoly and product quality, Journal of Economic Theory 18, 301-317.
Economics Letters 49 (1995) 91-94
Utility maximization in models of discrete choice Martin Peitz Wirtschaftstheoretische Abteilung H, Bonn University, Adenauerallee 24-42, 53113 Bonn, Germany Received 7 September 1994; accepted 8 November 1994
Abstract
In this paper I construct a direct utility function that has as its counterpart an indirect utility function that is commonly used in models of product differentiation.
Keywords: Utility maximization; Discrete choice JEL classification: L10
1. Introduction In various c o m m o n models of discrete choice it is assumed that consumers either choose one variant of a good of which they consume one unit and none of the other variants, or they choose the outside option. Articles analyzing this class of models usually commence with an evaluation function, which is called a conditional indirect utility function. A consumer chooses the variant with the highest value: if for some variant the value is positive, a consumer buys one unit of the variant that gives him the maximal value. Otherwise, he chooses the outside option. Duality theory suggests that there is an associated direct utility function. However, the results of duality theory cannot be applied to this problem because continuity is not satisfied. Therefore, there are some doubts as to whether the evaluation function is, indeed, an indirect utility function and whether it is consistent with utility maximization. I will construct a direct utility function such that the underlying preference relation satisfies reflexivity, transitivity, completeness, and local nonsatiation. I will then show that this direct utility function has as its counterpart the indirect utility function I was looking for. Hence, consumer behavior in discrete choice models of the type presented can be derived from utility maximization. 0165-1765/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSD1 0 1 6 5 - 1 7 6 5 ( 9 5 ) 0 0 6 6 5 - 6
M. Peitz / Economics Letters 49 (1995) 91-94
92
2. Notation and results
Suppose there is a m a r k e t with n different variants of a good that is vertically differentiated. Variant i, i = 1 , . . . , n, is described by its quality q~ E [q_q_,~], q ~> 0 and ~ < ~, and its price, Pi > 0. These variants are ordered such that qi < q~+ 1, i = 1 . . . . . n - 1. Each c o n s u m e r has the same reservation price R for the good with quality 0 (this variant may be artificial). It will turn out that R cannot be interpreted as income. The valuation of quality is d e n o t e d by 0 ~ [_0, 0] C [0, ~] and defines a type of consumer. I will give a direct utility presentation for the following indirect utility function:
v(p, y; 0) = max[0, max v,(p, y; 0 ) ] ,
(1)
where vi(p, y; O)= R - p i + Oqi is called the conditional indirect utility function of type 0. This kind of indirect utility function was first p r o p o s e d by Mussa and R o s e n (1978) and has been used in many articles on vertical product differentiation; c o m p a r e , for example, A n d e r s o n et al. (1992). As an exception to other articles in the field, Mussa and R o s e n defined a conditional direct utility function. However, they a priori begin within a f r a m e w o r k of discrete choice and unit d e m a n d whereas I derive this behavior from utility maximization. Note that the results in this paper can be rewritten for linear r a n d o m utility models frequently used in models of horizontal product differentiation. In the class of models u n d e r consideration, consumers buy one unit of up to one variant. For the n variants, I introduce the vector of quantities x ~ No+. In addition there is a n u m e r a i r e good with quantity x 0 1> 0 and normalized price P0 - 1. This means that the price for the variants and income are measured in units of the n u m e r a i r e good. I n c o m e of a c o n s u m e r of type 0 is interpreted as y = R + @, which is the reservation price of a c o n s u m e r of type 0 for a unit of the variant with the highest possible quality. Thus, consumers with a higher income care more about quality. Note that y can be m o r e generally interpreted as a share of income. It is only important that the relevant budget for the c o n s u m e r is greater or equal to R + 0~. This is always satisfied because a reservation price says how m u c h a c o n s u m e r is willing to pay for one unit, given his income. The budget constraint is p - x + x 0 <~y. T h e interpretation of A n d e r s o n et al. (1992) of R as income is inconsistent with utility maximization because this implies that for a p~ ~ [R, R + Oq~] a c o n s u m e r of type 0 prefers variant i over the outside option and hence violates the budget constraint that is implicit in any indirect utility function. D e n o t e e i the n-dimensional vector that has a 1 at the ith position and a 0 elsewhere. Since I am looking for a justification of discrete choice and unit d e m a n d , the direct utility function u(xo, x; O) to be constructed should satisfy the following properties: (P1) U(Xo + A , x ; O ) > u ( X o , X ; O ) , A >O , (P2) U(Xo, ei;O)>~U(Xo, O,;O), i= 1 , . . . ,n, (P3) u(x0, 0,; 0) = u(x o, Aei; 0), i = 1 , . . . ,
n, A E (0, 1),
M. Peitz / Economics Letters 49 (1995) 91-94
93
(P4) u(x o , e ~ ; O ) = u ( x o , ( l + A)e~;O), i = l , . . . , n , A > O , (P5) u(xo, ei;O ) = u ( x o , x ; O ) , x > e i and x / ~ e i , j > i , (P6) T h e r e exists an index i, 1 ~< i ~< n, such that u(x 0, ei; 0) ~> u(x 0, x; 0), x E ,qt+. (P1) says that consumers are locally nonsatiated in the n u m e r a i r e good. By (P2) no c o n s u m e r is worse off by taking a unit of some variant in addition to the c o n s u m p t i o n of the n u m e r a i r e good. According to (P3), consumers do not evaluate a quantity of less than one unit and, according to (P4), they do not evaluate a quantity of m o r e than one unit. (P5) says that, given the quantity of the numeraire good, consumers do not strictly prefer m o r e than one unit of variant i and less than one unit of the other variants over one unit of variant i and zero units of the others. (P6) reads: there exists a best variant i such that a c o n s u m e r weakly prefers e, over any x E .)l+ T o capture c o n s u m e r behavior in discrete choice models with unit d e m a n d (see ( P 2 ) - ( P 6 ) above) I introduce the following indicator functions for i = 1 , . . . , n: n
{~' Xi(x) =
,
,
ifxi~l'xi>~xjf°rj
f°rj>i,
with the convention that, in the case of indifference, the variant with the smallest index is chosen. This gives a 1 to the variant with the highest quantity a m o n g variants if it is greater or equal to 1; otherwise, it gives a 0. The direct utility function of type 0 is then defined as u(x 0, x; O) = ~'~ (R + Oqi)xi(x ) + x o - R - ~ .
(2)
i
It is easily seen that the underlying preference relation satisfies reflexivity, transitivity and c o m p l e t e n e s s (see, for example, Barten and B 6 h m , 1982). It is also clear that the following result holds.
Result 1. u(xo,x; O) defined in (2) satisfies ( P 1 ) - ( P 6 ) . A n indirect utility function is defined as
v ( p , y;O) = mX(I.X a x {u(x 0 , x ; 0 ) l p . x + x o<~y} . I am n o w in a position to state the relationship between the functions defined by (1) and (2) w h e n y = R + 0~.
Result 2. T h e function defined in (1) is the indirect counterpart of the direct utility function defined in (2).
M. Peitz / Economics Letters 49 (1995) 91-94
94
Proof. v(p, y; O) = mX(I,X a x { u ( x , , , x ; O ) l p .x + xo <~R + Oq} = max[u(R + 0F, 0,,;0), u(R + 0~1 - P l , e l ; O ) , . . .
=max[O,R + O q l - P l . . . . , R + O q , - p , , ] .
, u(R + 0 q - p n , en; 0)]
[]
The discrete choice model presented is thus consistent with utility maximization.
Acknowledgment Research for this paper was done while I was staying at C E P R E M A P , Paris. Financial support from the French G o v e r n m e n t is gratefully acknowledged.
References Anderson, S., A. de Palma and J.-F. Thisse, 1992, Discrete choice theory of product differentiation (MIT Press, Cambridge, MA). Barten, A. and V. B6hm, 1982, Consumer theory, in: K.J. Arrow and M.D. Intriligator, eds., Handbook of mathematical economics, Vol. II (North-Holland, Amsterdam) 381-429. Mussa, M. and S. Rosen, 1978, Monopoly and product quality, Journal of Economic Theory 18, 301-317.