Generalised Translation of Indirect Utility Functions

Generalised Translation of Indirect Utility Functions DENIS CONNIFFE National University of Ireland Maynooth Abstract This paper considers the derivation of new demand systems from existing ones through replacing an indirect utility function by , where p is a vector of prices and y is income. This is a generalisation of Gorman translation ),( yU p})/(,{ j ypyyU jj   p ,( yU ) jj p    p and will be shown to be effective in terms of producing new demand systems with both good regularity and flexibility properties. JEL Classification: D 11 Keywords: Translation, indirect utility functions, demand equations. Address for correspondence: Denis Conniffe, NIRSA, Economics Department, National University of Ireland, Maynooth, Co. Kildare, Ireland. Tel: 353 1 7086299; Fax: 353 1 7083934; e-mail: [email protected] 2 I INTRODUCTION Gorman (1975) introduced the “translation” device to incorporate extra parameters into utility functions and demand equations. If is the original indirect utility function, where p is a ),( yU p vector of prices and y is income, the translated utility function is ),( jj pyU   p, (1) where the j  are the subsistence quantities, y is assumed jj p   and summation is over n commodities. Gorman showed that if were the original demand equations, the translated ),( yqip equations are ijji pyq      ),(p. (2) For example, if is the simple homothetic utility function y/P, where P is a weighted ),( yU p geometric mean of prices, )log(log jj pP   , the demand equations iii pyq /   are translated to the famous Stone-Geary linear expenditure system (LES) )( jj i i ii py p q     . The idea of this paper is to replace (1) by the more general translation of to ),( yU p                   j y p yyU j j   ,p. (3) This reduces to (1) if all 1 j  . The condition jj py    is replaced by . (4)  j jypy jj    1 which will hold for positive j  if y is not too small. If a i  is negative, i  must also be negative. This paper is particularly concerned with how generalised translation can produce demand systems with both good regularity and flexibility properties. Regularity means that, given appropriate ranges for the parameters, the indirect utility function complies with the constraints implied by rational economic behaviour.1 Ideally, this should be possible for all prices and incomes (global regularity), but 1 That is, a con aximises direct utility under a budget constraint. This implies the indire utility function ),( yU p should be homogeneous of degree zero in income y and prices p, nonsumer m ct 3 should at least hold for all values of these variables relevant to the situation under study. Flexibility is also required in that the corresponding demand system, while satisfying regularity, should be able to model a reasonably comprehensive spectrum of consumer behaviour - so the possible values of income and price elasticities, which are functions of parameter values, should not be seriously restricted2. The influential ‘flexible functional forms’ approach, employing Taylor series approximations to general utilitity (or cost) functions, sought systems embodying flexibility and hoped for regularity, but it seems that often their flexibility depends on their parameters being allowed to take values that contradict regularity3. Such models generally cannot test if observed consumption patterns do or do not accord with economic theory. The approach in this paper will be to start from globally regular systems and to improve their flexibility by generalised translation4. Properties of the general translated utility (3) are examined in section 2 and the corresponding demand system derived. Income and price elasticities are obtained and presented in terms of the elasticities of the parent system and the parameters of the generalised translation. Section 3 illustrates these results by considering a particular case, fairly parsimonious in parameters, that is interesting in its own right. Section 4 applies generalised translation to more parameter rich, though still globally regular, utility functions. In section 5 the possible application of generalised translation to nonglobally regular utilities is discussed. Finally, in section 6, connections between generalised translation and Houthakker’s (1960) indirect addilog system are explored5. II. TRANSLATED DEMAND EQUATIONS AND ELASTICITIES Let decreasing in y, non-increasing in p, and convex or quasi-convex in p. These constraints imply corresponding conditions (aggregation, homogeneity, Slutsky symmetry and negativity) on the demand equations. 2 More formal definitions of flexibility exist, differing in detail. See, for example, Diewert (1974). 3 Caves and Christiansen (1980), Barnett and Lee (1985) and Cooper and McLaren (1992) have discussed the difficulty of reconciling flexibility and regularity for such systems. 4 Of course, there are other approaches to improving regularity properties. Barnett (1983), Barnett and Lee (1985) and Chalfant (1986) reported wider regularity regions resulting from approximations based on Laurent, Muntz-Satz and Fourier expansions rather than Taylor series. Lewbel (1987) and Cooper and McLaren (1996) have also proposed systems. 5 Lewbel (1985) has already used the term generalised translation in a paper on incorporating demographic effects into demand equations, which was a theme of considerable interest to Gorman. But the generalised translation of this paper is so obviously a generalisation of Gorman’s that it seems inappropriate to call it anything else. 4 j y p y   , yz j j         that the translated utility is If s homogeneous of degree zero in income and as it ought to be if a valid utility function, it is clear that is also. ),( zU p. ),( yU piso prices, ),( zU p y z z zU y zU       ),(),( pp                 yyz j jj j j  1 (5)              jj pp zU  ),(p ssuming the original utility is non-decreasing in income, the derivative of with respect to z non negative. Taking the terms within the chain brackets, equation (4) implies the first and second rms combined are positive while the third is also positive because, as previously mentioned, ),( zU pA is i  te and i  are presum o thed to have the same sign. S e translated utility is non-decreasing in income. y z z U p U pi zU i        ),(p                   p UU i ii  (6)     1 i yzpi  utilities. show that above so s will be gular. This corresponds to the situation with Gorman translation. For example, the simple omothetic utility function y/P,  So assuming the original utility is non-increasing in prices, the translated utility is also. Proofs of the convexity of translated utilities are more complicated and depend on the forms of the original They are provided in Appendix 1 and me income level the translated system re )log(log jj pP    h , is globally convex in prices, while ypy jj /)(   , which gives the LES, is convex if . (7) s nds to increase with time. The term ‘effective global regularity’ has been applied to regularity 222 2)( ppy   iijji So although Gorman translation increases flexibility by introducing extra parameters, it may invalidate regularity at very low incomes. This has not been seen as a difficulty, because, at least for analyse with time series data, interest focuses on inferences valid for recent or current time periods and income te 5 everywhere except at low incomes6. Generalisations of (7) for generalised translation can be deduced ater sections. From Roy’s lemma the demand equations are from the results of Appendix 1 and will appear in l    y jij )1(1                      j i p y p z zU p zU yq j i ii i it     ),( / ),( ),( 1 pp p, re denotes the demand equation from the translated utility. Also by Roy’s lemma, it q whe                        j y p yyqzq z zU p zU j jioio i   ,),( ),( / ),( pp pp , where enotes the demand function derived from the original utility function. So io qd                                          j i j y p y p y p yyq yq j jij i ii j jio it      )1(1 , ),( 1 p p , or, more tidily,                   1 , 1i y p zi ii   p, or, in terms of budget shares   ),( q V yq ioit p                     i y p y z zw V yw i iiioit   , 1 ),( pp , (8) where j y Vjj     )1(1 pj        cross-pri rms of and , the income and price elasticities of the original system. They are: and ),( ywit pis the budget share of the translated system as a function of p and y, and ),( zwio pis the budget share of the original system as a function of p and z. The income, own-price and ce ))( zp, io E),( zeiko p elasticities of the translated system are derived in Appendix 2, in te 6 The term seems due to Cooper and McLaren (1996). 6        jjoi it io jjtiio it io it zw yw zw yV z ywzE yw zw E  ),()1( ),( ),( 1),(),( ),( ),( p p p pp p p           yV z yw zw zEzwzeywzee it io ioioiioitiiioiit ),( ),( 1),(),(),(1),()1(),( p p ppppp                ywzee ktkikoikt ),()1(),( pp  yV z yw zw yw yw zEzwze kt ko it kt ioioiko ),( ),( 1 ),( ),( ),(),(),( p p p p ppp he xample of the next section illustrates this. D TRAN Even if the elasticities of the original system are very restricted, these elasticities are much less so. T E III GENERALISE SLATION OF THE CONSTANT BUDGET SHARE MODEL Returning to PyU /, where )log(log jj pP    , the original demand equations, in budget share form, are iio w  , or constant budget shares. Income and own price elasticity are unity and cross-price elasticities are zero. From (8) the translated demand equations are                                         j j y p y p y p w j jj i i ii j ji i      11 1 r, tidily,. o                    i  i iiii y p y z V w  1 (9) is shown in Appendix 1 that this system7 will satisfy effective global regularity if It 22 222 1 22)1(           ii pp ii  (10)           y pz y pz iiiiiiii  , which holds if   iiji     ,1, and .1    It could even hold for an 0 i  if  10 i  , although that may be purely academic. Obviously, (10) reduces to (7) if all j  = 1. If (9) is to have much practical value, it should be more flexible than the LES, which it becomes 7 This system has been examined in greater detail in Conniffe (2002a), but was not then understood as a case of generalised translation. 7 when all j  = 1. The limitations of the LES include the linearity of all its Engel curves8, its inability ffe inc par ratio of income elasticities anslated income elasticity, dropping, for convenience, ripts indicating translated and original, to cater for inferior goods and the tying of price e cts to ome aspects of price changes. In ticular, the ratio of the cross-price elasticities ik e and jk e equals the i Eand j E, and the possibility of complementary goods is excluded. From the formula of the previous section for tr the subsc     i jjjji i iwyVw     iii z wE )1( 1    . ry ne t ranges d i Eis not generally a simple monotonic function of income. A commodity could be a luxu , or a cessity at differen of income, and could even be an inferior goo (for a positive i  and small i  ). However, as y the budget shares (9) tend to constancy and i E to unity, which is not unreasonable and is a property of many other demand systems. That Engel curves can take a large the demand equation and the income elasticity, although as From the formulae of the previous section, the own-price and cross-price elasticities variety of shapes is clear from y ey will approach linearity. th are          wwe iiii  )1(1 yV z wi i ii   1 and             Kk i kik yV w w e  1,   iz  s of cross-price elasticities need not equal ratios of income elasticities. The compensated price so ratio effect y q q p qi k k i      is 8xample, Lau, 1986) is t Eonstant prices, are no One of the few generally agreed findings from empirical studies (see, for e hat ngel curves, the relationships between expenditure and income at c n-linear for at least some goods. 8 )}1()1({)}1(1{ i ikki ppppyVyVy ikkikiki qqy zz qq  kjjjjki w        and this can generally be negative or positive, allowing for complements as well as substitutes. So the extra n parameters contained in (9), relative to the LES, do substantially increase flexibility9. Of ourse, any demand system with just 3n-1 parameters, will fall short of full flexibility if there are more lly if data re scarce, or observed price ranges do not embody sufficient independent variation to estimate a very etailed model of interacting price effects. NIOUS UTILITIES igina . Initially, a util or so10, par s from two utilit on rameters, are oncave in prices. Then (Conniffe, 2002b) the sum, product and the reciprocal of reciprocals (harmonic mean) are also globally regular utility functions. For example, with and c than a few commodities. Sometimes, however, parsimony of parameters is desirable, especia a d IV GENERALISED TRANSLATION OF LESS PARSIMO reater flexibility of price effects can be modelled by generalised translation of a less parsimonious, G but still globally regular, or l utility ity of the form Py/will be considered, where is a function of more than n parameters, but is concave in prices. P It is easy to generate a globally regular homothetic utility function with 2n, ameter y functi s /y and 2 /Py , where 1 P and 2 P, both functions of n pa 1 P c j j pP  = 1 jj pP  = 2, with i  and i  positive, the harmonic mean of the utility functions is    jjj pp y Uj   , which is globally regular with 2n – 1 parameters. It gives the demand system jjj iiji ij j wp+p p+p =      , which still, of course, gives a unitary income elasticity, but price elasticities of ollak (1972) described a class of demand equations of the form ),,/( Wypfq iii  9 P where W is a homogenous function of all prices and income, as exhibiting “gene all ralised separability”. Because prices, except , take effect through W, there are implied symmetries in how commodity demands are affected by prices of other goods. The LES and the Indirect Addilog System (IAD) of Houthakker (1960) are of this class. But (9) is not, provided the i p j  are non zero. 10 Constraints such as 0  j  reduce parameters by one. 9 )( jjji iiiii ppw j    )1( j we   pj   and j pj   )( jjji kiikik ppw j    we   . system with 4n parameters, given by equation (8), with elasticities given by substituting the above into the formulae of section II. Many other systems are of p ction Combining simple utility functions is not the only way extra parameters could have been obtained. riginal homothetic utility functions with more than 2n parameters are quite possible. For example, Generalised translation then produces a -1 obtainable by other combinations airs of utility fun s. O   2 1 2 1 kjjk pp y U  , could have been taken where the jk  are positive and jk  = kj  11. This has n(n+1)/2 parameters and gives the demand system 2 1 2 1 2 1 2 1 2kjjk jiji i pp pp w     , with price elasticities 2 2 1 2 1 )(4 2 1 kjjki iii iii ppw p we     and 2 2 1 2 1 2 1 2 1 )(4 kjjki ikki kik ppw pp we     . spite of a relative profusion of parameters this system still embodies restrictions on price ffects as In e y q p i k i   = qq k  2 2 1 2 1 ki 22 )(4 kj ik pp pp    sitive if 11  jk is po ik  is, so ruling out complimentarity. So generalised translation, which gives a 11 This is actually the homothetic case of the generalised Leontief utility function. 16 iii p P Pp P P pz U z U Pz U              log11 - and 0, 1 2 2 2 2 . urthermore F                                           ji 22 2 i 2 2 2 2 p Plog p Plog P z log p Plog P z log jiji ii pp P P z pp U and p P P z p U o (A1) and (A2) become and s                                                2 2 ii 2 i 2 2 p z p Plog 2 p Plog log1 ii p z z p P z P and   P                           ijjiji 2 p z p Plog p z p Plog p Plog p Plog log1 z pp P z ji . be rewritten as These can                                          2 2 2 2 2 211loglog1 i i ii ip z z p z p z z p P z p P z P and                                   jijjiiji ppzp z pp z p z pp z             zzzP z zPP P 11log1loglog . atri the Hessian matrix o with respect to prices, which is positive efinite17, plus which, is positive semi-definite, where G is the vector with ith term 2 1 o the Hessian m x (with respect to prices ) of the translated utility is (apart from the S fPlogP/1 multiplier) z by minus 'GG , d ii p z z p P z    1log , lus a diagonal matrix with ith term p 1 2 i  2 )1(             y p p p zi i iii i  minus , where H is the vector with ith term zHH /' 17 If P is concave, is concave. Plog 17 1           i y p p zi ii i   . While 'H H  is negative semi-definite, the addition to it of the diagonal matrix with ith term 2 22 2   i p z  22  i gives a nonnegative definite matrix (diagonals positive and principal minors of higher order zero). ed utility is convex with respect to prices if the n ultiplied by z minus the diagonal matrix with elements 2          yp ii i  So the translat egative of the Hessian of log P m z y p p yiiiii    pii i i i 22 22 1 2)1(                  (A3) ative definite. onvexity for the constant budget share model is nonneg C )log(log jj pP   Now let . The Hessian of log P with respect to prices is So U is onvex with respect to prices if diagonal. c z y p p y p p z ii i ii i i iii i i 22 22 1 2 2)1(                        is positive, or 22 2222 2)1(        ii p p p zi iii i iiii  , (A4) 1            y z y pi  hich is equation (10) of section 3. If all th ej  are positive, the left hand side is of orde , while e right hand side terms are of orde and , so since r 2 y w ri y  2i y  22 i  this positive the condition holds rovided y (and hence z) is not small. The first right hand side term is negative if 0 < i  p < 1, so that a ero i  might even be compatible with convexity in that situation. For at least one of the j  znegative, be the smallest (most negative). The greatest power of y in z is then with coefficient s y  1 let s  18 s ss s p   (  is negative if s  is). So the left hand side rder , while the right hand rms are of order and and it is clear the critical term is i= s. Comparing coefficients f, the condition is s y  22 is of o si y  2i y  22 te  22greater than . This will be true if 0 > 2 s  s  s yss    > -1, even if s  ois small (or even zero). So effective global regularity holds18 if     iiji     ,1, and  .1  ain convex with respect to prices if the negative of the Hessian of log P ultiplied by z minus (A3) is nonnegative definite. Let the smallest eigenvalue of minus the Hessian atrix of log P, that is, the matrix with i,k th term Convexity with Pconcave in prices The translated utility is ag m m ki pp P   log 2  . If log P is strictly concave minus the Hessian is positive definite and can be written equal to be Q+  I, wh dere I is the identity matrix, Q is positive semi-definite an  is positive. Then the condition for regularity is 22 222 )1               p z p zi ii iii    . 1 2 (  ii yypi i    (A5) ms the same as (A4) with replacing i  /2 i p  This see , but it is actually a much stricter condition. Its equivalent, where s  denotes the smallest i  would have been 2 2 22 1 22)1(            ip pz p pz ii   2          i yy iiiiiiis   , r all i and not just i=s. So (A5) probably exaggerates the income required for effective global gularity, although as the left hand side is of orde (assuming fo r 2 y)  i  re , while the right hand side rms are of order and , there is no doubt it will hold as y increases. Less demanding ved for sp ific ases of P. Again, if i y  2i y  22 te  conditions could be deri ec c =0 for some observations (log P not 18 The conditions on the j  and j  correspond to those for the regularity of the utility function of the indirect addilog system. The validity conditions of the IAD have been debated in the literature more than once, as the exchanges between Gamelatos (1973, 1974) and Somermayer (1974), and between Akin and Stewart (1979) and Murty (1982) testify. 19 i  strictly concave) effective global regularity could still hold if 0 < < 1, although the case is ve  probably not of practical interest. For negati the same argument as before gives the condition nd again this will be true if 0 > ss p   2>2 s  as  > -1. rice index. Hessian A(3) to f the He Convexity with Pa translog p As before, convexity requires the negative of the of log P multiplied by z minus be nonnegative definite. The negative o ssian of Log P is    LL , i p/1 ,  is the matrix of coefficients jk  where L is the diagonal matrix with ith element and is the diagonal matrix with ith elemen If, following estimation of the  t ./ 2 ii pS jk  ,  is found to be concave the convexity requirement becomes 22 22 1 22)1(            ii z p pzS i   . 2       y p p y i iiiiiiii  his gives the condition of section V. The condition is the same as (A4) with replacing i S i  T and e same arguments apply as regards the required ranges of i  and i  . But while the i  thwere onstants, the are function of prices and the validity of the argument depends on their being exity with a Generalised Leontief utility return to (A1) and (A2) to take account of terms that no longer vanish. However, uch of the previous approach will still apply. The (translated) utility function is i Sc ositive. p Conv It is now necessary to m 1 2 1 2 1 2 1 2                   U                 z p z p z pj i ij j j  nd could be written in the form , where */ Pz a 2 1 2 1 2 1 2 1 2* jiijiij pppzP   , 20 although is now a function of income, as well as prices. The convexity of the original utility with respect to prices must imply that that is concave in prices and therefore is also. *P *P *logP 2 i * *2 2 *2 2 p logz *log                P Pp P P z p U ii nd a  iii pz PzPPU                *logloglogz 2**2 P P p P P pz           pz *log1 * i 2 * * i i C p P P    *log1 * , say. (A6) Although the first term of (A6), which is easily shown to be 2 1  i  2* )(2        i p z P z/1 beis of order , the term , which may be shown to i C 2 1 2 1 2 jiji pp pz       2 1 3* 1 )( jiij i pP . This will be important later. Returning to (A1), two of its terms z/1 is of order ii ip z pz U p U       2 2 22 now give i i i ippzP PpP          * *2* i z C z p z z P z                 2 11 p log1 *22 i * . he rest of (A1) is Pz log 2 T 2 2 2 2 2 i ip z z U p z z U                 . (A7) arly, terms from (A2) Simil ijjiji p z pz U p z pz U pp U            222 give 21 jijiji p z p z zPp z z P z p z z P z Pz      log1 *log ** P pp P                        * 11 p log1 p **2 us  ji pl i ji p C p C j zz     nd, from the remaining term of (A2), a ji p zzU  2 2 . (A8) p z  ome terms are almost identical to those occurring in the case, with Pz/* P Sinstead of , and, llowing from m of a positiv definite matrix (minu y the Hessian of ) plus two positive semi-definite matrices minus two diagonal matrices ith diagonal terms P as before, can be expressed as fo the su e s * /Pz * logPb w 2  2 * 2       i p z zP and 2 i p z z U     . respectively of orde r i  22/1  and i   2/1 It is worth noting that these terms are in income. , those in and can be seen as following from i Cj C Turning to the extra terms involved ' * *1 ''                  pp zz zCPLL , (A9)  zP C where Lis the vector with elements i ip z CzP zP    *1, * nts . The first matrix in (A9) is positive semi-definite. The second l matrix with diagonal terms e diagonal matrix with diagonal terms i C and Cis the vector with eleme is negative semi-definite, but becomes positive semi-definite if the diagona 2* 2i zCP is added to it. The same applies to the third term if th 2 * 2          i p z zP is added. These diagonal matrices have to be subtracted again, of course. The first term in (A7) and the (A8) term arise from the matrix 22 ' 2 2                       pp zz z U and since         2 1 2 1 2 1 2 1 2 2 3 jiijjj pp z p P p U   (A10) is negative the matrix is negative semi-definite. Adding the diagonal matrix with diagonal terms     3 * 2 jj z 2 2 2      zU 2   i p z produces a positive semi-definite matrix. Note that (A10) is of order 2 3  z. t of the final subtracted diagonal matrix is So the ith elemen 2 2 2 * 4    p z                i z U zP 2* 2 22i i zCP p z z U      , (A11) where, of course, 1           i y p p zi ii i   and 1 2 2)1(             i y p p p zi i iii i   . ei  From earlier comments it is clear the terms of (A11), assuming th positive, are respectively of rder o i y  2 1 2, i y   1 2 1  y. 2 and As already said, minus the Hessian matrix of , multiplied by , is positive definite in rices and it is easily seen to be of order * logP * /Pz z or y. p So the same arguments as previously will of income and own price elasticities of translated systems nvenience of n , the dependence of quantities on prices and income will not be indicated xplicitly except for z, the translated income. The income elasticity is emonstrate effective global regularity. d Appendix 2: Derivation For co otation e 23 =yq y it                               1 1i y p zq V i iiio   . y q q y Eit it it   = V 1                         1 )1( i y p yy z z zq iiiiio   y V V qit   . y qit   that Noting and j y p yy Vj jjj             )1( 1 V y z   this is  ii y p yV q y p yVz zq  i ijjit iiiiio                         )1( 1 )1( 1 . o S  it E  ii pq iiiiio     1 )1( 1 y p VyVqz z q yi ijj itit                   )1( . Using  )(( 1 zw y z VwzqVq y p y p y p y i ii       p ioitioit ii ii i ii                , e the and re the budget shares, and that it w)(zwio awher z zq zq z zEio )( io io  )( )( becomes it E        jjoi it io jjtiio it io zw yw zw yV z wzE w zw  )()1( )( )( 1)( )( . The o ricwn-p e elasticity is = iit i pq p                  i it it i iit p q q p   e              1 1i y p zq V i iiio   . V 1                           2 )1()( i y p yp z z zq p zq iiii i io i io   i it p V V q   . i it p q  = Noting that 1 )1( 1           i i iii iy p yp V   , 1          pi i y p zi ii   and 24 this is  yV zq qq V zq q pV zq q z zq p zq V io itiit io it i iio it io i io } )( ){1( } )( { )1( } )( { )( 1                . o equals  iit e S               ze zqio )( )( V zq q y p q zE zq zqp Vq io it ii it i io it ioi iio it )()1()1( )( )(  , or          yV z w zw zEzwzewzee it io ioioiioitiiioiit )( 1)()()()1)(1()(  . Th e cross-price elasticity is                k io k io it k p z z zq p zq Vq p)( k k p V V p        k it it k ikt p q q p e              zE zq ze Vq io it iko it )()( Vq zq y qpzqqpzq kt koktk k ioktkio )( 1)1( )()(  = or          yVww zEzwzewzee kt ko it kt ioioikoktkikoikt 1)()()()1()(  .  z zww )(   REFERENCES Akin, J. S. and Stewart, J. F., 1979. Theoretical Restrictions on the Parameters of Indirect oney Supply and the Flexible Laurent Demand System. he Minflex Laurent , n, D., 1972. The S-branch Utility Tree: A Generalisation of the Linear and Christensen, L. R., 1980. Global properties of flexible functional forms. Business ogarithmic 002. A New System of Demand Equations, Working paper, Economics Department 2. Sums and Products of Utility Functions, Economic and Social Review, 33, 285ically Oriented Demand System with R., 1996. A System of Demand Equations satisfying Effectively Global deal Demand System. 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