Sums and Products of Indirect Utility Functions (NIRSA) Working Paper Series No. 6

1 NIRSA is a research institute dedicated to the inter-disciplinary and comparative study of the impact of global processes on regional and spatial development. It involves several thirdlevel institutions in Ireland: NUI Maynooth, Galway-Mayo Institute of Technology, Institute of Technology, Sligo, Mary Immaculate College - University of Limerick, Waterford Institute of Technology. © 2002 by Denis Conniffe. All rights reserved. Short sections of the text, not to exceed two paragraphs, may be quoted without the explicit permission of the author provided full credit, including the © notice, is given to the source. DISCLAIMER While NIRSA is happy to facilitate the dissemination of working papers produced by associates of the Institute, the responsibility for content rests solely with the authors. ABSTRACT There are relatively few known demand systems that are theoretically satisfactory and practically implementable. This paper investigates building more complex demand systems from simpler known ones by considering sums and products of basic utility functions, an approach that does not seem to have been exploited previously in the literature. Some of the systems that result are interesting and usefully extend the range of available functions. Even the simpler systems that are not sufficiently flexible for the analysis of real world consumption data may still be useful for applied general equilibrium studies and for theoretical explication. Although some systems, instead of being new, turn out to be rediscoveries of already known ones, the way in which they arise as combinations of simple components is of interest in itself in showing them as sub sets of wider classes. 2 I INTRODUCTION An indirect utility function , where p is a vector of prices and y is income, and the demand equations derived from it through Roy's identity ),( yU p y U p U q i i∂ ∂ ∂ ∂ −= /, (1) satisfy demand theory, or utility maximisation, provided meets stringent criteria. These are that U be homogeneous of degree zero in income and prices (p), non-decreasing in y, non-increasing in p, and convex or quasi-convex in p. Then the demand equations satisfy the required constraints of aggregation, homogeneity, Slutsky symmetry and negativity utilities. These criteria for the validity of indirect utility functions are very restrictive on the choice of functional forms, even with restrictions placed on the parameters occurring in the forms. There are relatively few known functions U that satisfy validity conditions for all, or even for all plausible values of prices and income and some of them are very basic. This paper investigates building more complex demand systems from simple known ones by considering sums and products of basic utility functions. ),( yU p The basic combination devices, which will be described in section II, are quite simple, but at least as far as this author knows, they have not been exploited previously in the literature in order to expand the range of valid demand systems. Some of the simpler systems that result and that will be described in section III, may not be as flexible as might be desired for the analysis of real world survey or time series data on consumer expenditures on commodities. However, they may still be useful for applied general equilibrium studies and for theoretical explication. Some more complex systems, to be derived in section IV, are more flexible and perhaps usefully extend the range of available functions. As might be expected, some systems, instead of being new, turn out to be rediscoveries of already known ones. However, even the way in which they arise as combinations of simple components is of interest in itself in showing them as sub sets of wider classes. 3 II DEMAND EQUATIONS FROM SUMS AND PRODUCTS OF UTILITIES Suppose we have two (indirect) utility functions and satisfying all validity criteria. Then the criteria obviously apply to 1 U2 U 21 UU + (the sum of two quasi-convex functions is quasi-convex or convex) and indeed to 21 )1( UU λ λ + − , where λ is a positive constant, and corresponding demand systems can be derived. Let ),( 11 yww ii p = and be the sets of demand equations, in budget share form, resulting from application of Roy's identity to and respectively. Then by applying (1) to ),( 22 yww ii p= 1 U2 U21 )1( UU λ λ + − and simplifying, the demand equations corresponding to this sum of utilities turn out to be y U y U y U w y U y U y U ww iisi ∂ ∂ + ∂ ∂ − ∂ ∂ + ∂ ∂ + ∂ ∂ − ∂ ∂ − = 21 2 2 21 1 1)1()1( )1( λλ λ λλ λ , (2) or the original individual demand formulae weighted by (apart from constants) the derivatives of utilities with respect to income. The sub-script s denotes the utilities were summed. For the special case of utility functions of the form 1 1P y U= and 2 2P y U=, (3) where and are price indices, the validity of utility functions reduces to the validity of the price indices and it is then evident that the utility function 1 P2 P 21 )1( PP y U λλ +− = is also valid. So we will be interested in the properties of the weighted sum of utility functions 2 21 2 1 21 1 )1()1( )1( U PP P U PP P U λλ λ λλ λ +− + +− − =. Applying Roy's identity to this gives 21 2 2 21 1 1)1()1( )1( PP P w PP P ww iiwsi λλ λ λλ λ +− + +− − =, (4) the individual demand formulae weighted (apart from constants) by the price indices, or the reciprocals of the derivatives of utilities with respect to income. 4 For functions of the form (3), the product of utilities is a valid utility function λλ 2 1 1UU −1. Applying (1) to this gives y U y U y U w y U y U y U ww iimi log log log log )1( log log log log log log )1( log log )1( 21 2 2 21 1 1 ∂ ∂ + ∂ ∂ − ∂ ∂ + ∂ ∂ + ∂ ∂ − ∂ ∂ − = λλ λ λλ λ , (5) the individual demand formulae weighted by the elasticities of utilities with respect to income. The sub-script m denotes the utilities were multiplied. 1 If the logs of utility functions were utility functions, then the fact that the sum of utilities gives a valid utility would suffice for the product of utilities. Convexity is the crucial property. For functions of the form (3), . Since P is a valid price index, it is concave in prices. The log function is concave and increasing, so is concave and therefore is convex. PyU logloglog −= Plog Ulog 5 III SIMPLE HOMOTHETIC COMPONENT UTILITY FUNCTIONS Three simple utility functions can be generated by dividing income by price indices corresponding to (weighted) arithmetic, geometric and harmonic means. They are ∑ = jj ap y U γ , (6) ∏ =j j gp y U α (7) and ∑ = j j hp yU δ (8) respectively. All satisfy validity conditions provided the ss ',' α γ and s' δ are positive with .1=Σ j α The corresponding demand equation systems are obtained by applying (1) and for (6) this gives jj ii ai p p w γ γ Σ = or jj i ai p y q γ γ Σ =, (9) where q denotes quantity. These are Leontief demands in that the ratios of quantities of commodities are always in fixed proportions, irrespective of prices or income. For the ith commodity the own-price elasticity is ai w − and the cross-price elasticity with respect to price k is . Note there is over-parameterisation in (9) as any one ak w− γ could be eliminated by dividing it into numerator and denominator. But the convention 1=Σ j γ is more compatible with the average price interpretation. As is well known, application of (1) to (7) leads to the Bergson, or constant budget share, demands, igi w α = , with own-price elasticity equalling minus one and cross-price elasticity zero. Applying (1) to (8) gives ∑ = j j i i hi p p w δ δ or ∑ = j j i i hi p p y q δ δ 2, (10) with own-price elasticity –2 + and cross-price elasticity . As for (9), there is parameter redundancy in (10) and a hi whk w 1=Σ j δ convention matches with a harmonic mean price index. Other simple utility functions are easily written down, for example, 6 ∑ =2 jj rp y U φ , (11) which is valid if the s' φ are positive and which gives the demand equations 2 2 jj ii ri p p w φ φ Σ = or 2 jj ii ri p yp q φ φ Σ =, (12) with own-price elasticity 1-2 and cross-price elasticity ri wrk w − . Again a redundant parameter can be accounted for by imposing 1 = Σ j φ , which also permits interpretation of the denominator of (12) as a price index. Even with the four utility functions (6), (7), (8) and (11), there are quite a few potential demand systems. Taking the utility functions two at a time, there are six possibilities and the three combination methods via (2), (4) and (5) makes eighteen demand systems. But how much more flexibility do they give? With income appearing as simply as it does in the four starting point utility functions, it is evident that not only have their demand systems unitary income elasticities2, but so will the combination systems because the weights in (2), (4) and (5) are functions of prices and not income. So we are only considering greater flexibility in response to price changes. Taking as a first example the combination of (6) and (7) by (4), gives the demand system ∏ ∏ Σ+− +− = jjj iiji wsi pp pp wj j γλλ λγαλ α α )1( )1( , (13) which can be written as ag iigi wsi pp pp w λλ λ γ α λ +− + − =)1( )1( (14) where the sp'denote means of prices (which are functions of parameters) with the subscript denoting the type of mean. By dividing numerator and denominator of (13) by λ −1and writing ii γγ λ λ = −1 it is possible to write (13) as ∏ ∏ Σ+ + = jjj iiji wsi pp pp wj j γ γα α α . (15) 7 This gets rid of λ and removes the need for any constraint on the s' γ , but although neater, it destroys the interpretation of jj p γ Σ as an arithmetic mean price index. However, the device will be used in section IV. For the demand system (9), corresponding to (6), all goods had to be price inelastic and for the constant budget share model the elasticity had to be –1. It is easily verified that for (14) the price elasticity is ])1[( )1( )1( agwsi g iiwsi ppw p w λλ λ αα +− − −−− , so that price elastic goods are possible. The cross-price elasticity with respect to price k is ])1[( )1( agwsi g kiwsk ppw p w λλ λ αα +− − −− , so that cross-price elasticities with respect to k are not constant over commodities, unlike the situation for (9) where they all equalled minus the budget share of good k. So the weighted sum of (6) and (7) does give a system with scope to represent a greater range of economic behaviour. Similar remarks apply to the sum and product combinations via (2) and (5), which are ag a g iiai si pp p p pp w)1( )1( λλ λγαλ −+ +− = and a ii imi p p w γ λαλ +−= )1( respectively. For these two systems to become the same and also equal to (14) would require ag pp =. But as is well known, a geometric mean is always less than an arithmetic mean unless all commodities have the same price. Although the systems are distinct, they have an evident similiarity – own price appears explicitly and linearly in all, while the other prices (and own price) occur implicitly through the price indices. The corresponding demand systems for combinations of (7) and (8) are hg h i i gi wsi pp p p p w λλ δ λαλ +− +− =)1( )1( 2 , jj qlog 2 A well known related characteristic of these basic demand systems is that they could have been derived from additive direct utility functions, for example, α Σ in the case of constant budget share demands, or Σ in the case of (12). jj q φ / 2 8 hg gh i i hi si pp pp p p w)1( )1( λλ δ λαλ −+ +− = and i hi imi p p w δ λαλ +−= )1( respectively, where the harmonic mean is ∑ = j j h p p δ 1. Again, equality of the systems requires hg pp = , but a harmonic mean is always less than a geometric mean unless prices are equal. So again the systems are distinct, with the similarity that the reciprocal of own price appears explicitly and linearly in all three, while the other prices feature only through the price indices. As might be expected, the combinations display more flexibility in price elasticities than their components did. For example, for the demand system (10), corresponding to (8), all goods had to be price elastic, but the combinations relax this. The demand systems for combinations of (7) and (8) can be obtained too and similar comments apply. The equations for a commodity are found to explicitly feature both own price and its reciprocal, which has benefits for own-price flexibility, but other prices again feature only through price indices. Now consider the combination of utility functions (6) and (11), or demand systems (9) and (12), via (4). This leads to rqa rq ii ii wsi pp p p p w λλ φ λγλ +− +− =)1( )1( 2 , (16) where rq p denotes the root quadratic price index . 2 ∑jj p φ As already mentioned, all the demand systems derived in this section, have unitary income elasticity and this could be seen as a serious inflexibility. So it is if we are trying to model observed consumer demand. However, it is often considered a desirable property in applied general equilibrium studies, sometimes along with extreme parsimony in parameters. For 9 example, if (16) is further simplified by taking n iii /1 = = φ γ , where n is the number of commodities, we get (in quantity rather than budget share form) the single parameter system ])1[( )1( rqa rq i wsi ppn p p y q λλ λλ +− +− =, (17) with a p and rq p now the simplest price mean and the square root of the simple mean of squared prices. But (17) is the new class of demand equations proposed by Datta and Dixon (2000)3 for general equilibrium models, which they believe will also be useful in a variety of other applications. From the development here it is evident theirs is a sub-class of a much wider one. They see the 'linearity' in explicit own price4 in (17) as a particular virtue, but that is not unique to their case. The systems resulting from combinations of (6) and (11) via (2) and (5), with the same imposition of n iii /1 = = φ γ , ])1[( ])1[( 2 arq i rg a a rq si ppn p p p p p y q λλ λλ +− +− = and a i rg a mi pn p p p y q ])1[( 2 λλ +− = share the property. Nor are these the only ones. The formulae given earlier for the demand systems from combinations of (7) and (8), if written as equations for quantities rather than budget shares, show that all three have the property. Presumably too, there will be occasions when more than a single parameter is desired, so that the more general formulae are applicable. Perhaps there may even be be situations where 'linearity' in the reciprocal of own price may be desirable instead of, or as well as, 'linearity' in own price. It is not implausible to suspect there are other useful systems for use in general equilibrium modelling besides (17) to be obtained from combinations of simple components systems. 3 They use λ −instead of λ , defining it to be negative. Of course, they prove the validity conditions directly for their system rather than deducing them from the properties of components. 4 They argue, citing Dixit and Stiglitz (1977) that sometimes the non-linearity due to implicit in price indices is unimportant. i p