Generating Globally Regular Indirect Utility functions

GENERATING GLOBALLY REGULAR INDIRECT UTILITY FUNCTIONS Denis Conniffe Economics Department, National University of Ireland Maynooth Abstract Despite their scarcity in the literature, an abundance of globally regular indirect utility functions, involving as many parameters as desired, exists. They are easily constructed as a function of simple homothetic component utilities. JEL Classification: D 11 Keywords: Global regularity, indirect utility functions. Address for correspondence: Denis Conniffe, Economics Department, National University of Ireland, Maynooth, Co. Kildare, Ireland. Tel: 353 1 7086299; Fax: 353 1 7083934; e-mail: [email protected] 2 1. INTRODUCTION Regularity means that an indirect utility function complies with the constraints implied by rational economic behaviour1. Regularity is global if it holds for all (positive) prices and incomes, given appropriate ranges for the values of parameters occurring in the function. This short paper is concerned with the generation of such globally regular functions. 2. COMBINING HOMOTHETIC COMPONRNTS Consider a set of homothetic utility functions, each of the form (1) ,/),( kk PyyU k  p where k  is positive and is increasing in prices, homogeneous of degree k Pk  in prices and is a concave function of prices with negative definite or semi-definite Hessian. Then the reciprocal of is k P convex in prices, because 2 2 2 2 32 12 12 i k k i k ki k p P P p P Pp P                 and ji k k j k i k k ji k pp P P p P p P P pp P                          2 23 12 12 , so the Hessian of is 1 k P ' 3 2                    pp kk k PP P, which is positive semi-definite, minus the Hessian of , which also gives a nonnegative definite k P matrix. So is convex in p, non-decreasing in y, non-increasing in p and as well as ),( yUkp homogeneous of degree zero in income y and prices. So it is globally regular. A few properties of k Uare worth examining. As is already obvious, is a concave function of prices. For positive 1 k U 1  , , being an increasing concave function of a concave function, is concave in     kk UU 1 1 That is, a consumer maximises direct utility under a budget constraint. This implies the indirect utility function should be homogeneous of degree zero in income y and prices p, nondecreasing 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. ),( yU p 3 prices. For 1  , is an increasing convex function of a convex function and so is convex in   k U prices. For 1  , it is simple to verify that    k Phas a negative definite or semi-definite Hessian and therefore its reciprocal is convex in prices and so is     kk UU    k Py k/. So is   k U convex in prices for all positive  . Also, is convex in prices as it equals k Ulog Py kloglog   and since is an increasing concave function of a concave function, it is concave and minus it is Plog convex. Now consider the function      1    kkUU , (2) where the k  are non-negative and sum to unity2. Commence with positive 1   . is concave   k U in prices and since sums of concave functions are concave,     kkU is concave. Then     1  kkU is a decreasing function of a quasi-concave function and so is quasi-convex. Since             kkkkk UU yy U1 1 1 and  i k k kkkk ip P P UU p U        1 1 1    Uis increasing in income and decreasing in prices. It is obviously homogeneous of degree zero in income and prices since each is. So U is globally regular for k U1   . Now take  negative and put    . Then (2) becomes      1  kkUU . As already shown is convex in prices, so  k U    kkUis convex and since an increasing function of a convex function is quasi-convex, Uis quasi-convex in prices. It is clear that is again increasing U in income, decreasing in prices and homogeneous of degree zero in income and prices. Finally, for 2 The function (2) is of the form of the reciprocal of a CES (constant elasticity of substitution) price index, although here it is an index of component utilities not prices. 4 0  , the familiar limiting argument as 0  (for example, Diewert, 1993) gives or  k k UU   kk UU loglog  . Since each is convex in prices, the sum is, and since antilog is an increasing convex function, k Ulog Uis. So (2) gives a globally regular utility function for all 1   . For utilities of the form , which are special cases of (1), the sums, products and harmonic k Py/ means, which are special cases of (2), have been examined in Conniffe (2002). However, they are of limited interest as the resulting utilities must all be homothetic. 3. EXAMPLES Consider   jj p y U  1 and   jiij pp y U  2. Both are easily shown to be convex in prices, the former strictly so, provided the j  and ij  are positive and ij  = ji  . Take 3/1,3/2 2     iand 1   , so that (2) is a weighted harmonic mean of and . It is 1 U2 U 1 2 1 2 1 2 1 23                                    y p y p y pj i ij j j  , (3) Diewert’s (1974) generalised Leontief utility function. Diewert proved its global regularity, but that is immediately evident from the derivation here. More importantly, (3) is just one of many possibilities, all globally regular. Taking 1  , with the same  values as before, gives a weighted arithmetic mean of and 1 U2 U 1 2 1 2 1 1 2 1 3 1 3 2                                              y p y p y pj i ij j j  . (4) The demand systems following from (3) and (4) are not the same and show differences that are possibly important in practice. For example, the income elasticities of the generalised Leontief demand system are 5                                                                2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 y p y p y p y p y p y p w E j i ij j j j i ij i i i i    , which must be greater than a half. The income elasticities corresponding to (4) are                                                2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 2 1 /2 / 2 3 jiijjj j j jiijjj i i i i ppp y p ppp y p w E     ’ which must be less than 3/2, which might be more acceptable than constraining them to exceed a half. Regularity need not imply the capability to model a wide range of consumer behaviour. The generalised Leontief’s deficiencies in this regard have been mentioned by Caves and Christensen (1980) and Diewert and Wales (1987) and (4) has its own inflexibilities. Hoewever,  , or a  , could be treated as an unknown parameter to gain more flexibility. The case of 0  , giving products of utilities is particularily inflexible since the resulting function is still homothetic. Many other choices of and are obviously possible; for example, could be taken to be 1 U2 U1 U j j p y   1, where 1    j. Using (2) to combine this with the previous with 2 U  =1 and 2/1 21    gives 1 2 1 2 1 2                                     y p y p y pj i ij j j   , and the corresponding demand equations are                                                        2 1 2 1 1 2 1 2 1 y p y p y p y p y p y p w j i ij j j ij i j i i j j     . 6 Clearly, three or more component utility functions could be employed in (2) to produce even more heavily parameterised, but still globally regular, utility functions. While the flexibility of the resulting demand systems would be increased, especially if  , or the  were also parameterised, the data requirements for the estimation of so many parameters could be a serious practical difficulty. 4. DISCUSSION This paper has shown how to generate highly parameterised globally regular utility functions from (2) using the globally regular components (1). Familiar and more parsimonious utility functions could also be seen as following from (2). Houthakker’s (1960) indirect addilog function j j jp y          , where i  and i  have the same sign and i  >-1, is globally regular and is obviously (2) applied to j py/to the power of j  with 1  and jj    . The components employed in section 3 could be considered as resulting from application of (2) to still simpler components. For example, let kk pyU /, a utility function corresponding to expenditure of the whole budget on good k. Using (2) with 1  and jj    gives the of section 3. 1 Ukiik ppyU /, a utility function corresponding to expenditure of half the budget on good i and half on good k, is (2) applied to and i U k Uwith 0  and 2/1 ki   . Then applying (2) to with ik U1   and  for  gives 2 Uof section 3. Of course, many utility functions have appeared in the literature that are not of the class generated by (2) from components of the form (1). But they are not globally regular. REFERENCES Caves, D. and L. R. Christensen (1980): “Global properties of flexible functional forms”, American Economic Review, 70, 422-432. Conniffe, D. (2002): “Sums and Products of Utility Functions”, Economic and Social Review, 33, 285-295. Diewert, W. E. (1974): “Applications of duality theory”, in Frontiers of Quantitative Economics Vol. II, ed. By M. D Intriligator, and D. A.Kendrick, Amsterdam: North-Holland, 106-171. Diewert, W. E.and T. J. Wales (1987): “Flexible functional forms and global curvature Conditions”, Econometrica, 55, 43-68. Houthakker, H. S. (1960), “Additive Preferences”, Econometrica, 28, 244-256.