Conditional Demand Functions and the Implications of Separable Utility

Robert A. Pollak

Southern Economic Journal, Vol. 37, No. 4. (Apr., 1971), pp. 423-433.

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CONDITIONAL DEMAND FUNCTIONS AND T H E IMPLICATIONS OF SEPARABLE UTILITY

ROBERT A. POLLAK*

University of Pennsylvania

Most contributions to demand theory can be divided into two distinct categories. One consists of studies of the foundations of the theory; these are usually concerned with weakening the fundamental axioms to obtain greater generality. The other con- sists of studies which impose strong restric- tions on either preferences or demand func- tions and investigate the implied restric-tions on the other; these sacrifice generality to obtain specific results. In the former category are investigations of the conditions under which preferences can be represented by real valued, continuous utility functions and of the conditions under which demand functions can be defined in the absence of such representability. Examples include Chipman [I], Rader [14], Debreu [2], Son- nenschein [17] and Richter [15]. The latter category includes investigations of the demand functions arising from specific utility functions (e.g., Wegge, 22), of self-dual preference orderings [Houthakker, 71 and, perhaps most prominently, of separable utility functions [Strotz, 19; 20; Gorman, 5; Frisch, 3; Houthakker, 6; Sono, 18; Pearce, 10; 11;Goldman and Uzawa, 41.

The purpose of this paper is to derive and explain the implications for consumer be-havior of the three principal separability hypotheses: the utility tree, block additivity and additivity.' Although most of the results

*Much of the material in this paper is con-tained in my Ph.D. dissertation [Pollak, 121. I am indebted to Franklin M. Fisher, Paul A. Samuel-son, Tony E. Smith, Robert M. Solow, Robert Summers, Henri Theil and an anonymous referee for h e l p f i comments and to the ~ a i i o n a l Science Foundation, the Ford Foundation and the Wood- row Wilson Foundation for financial support. I alone am responsible for the views expressed and the errors which remain.

1 We deal only with separability of the "direct" utility function.

are not new, this paper does more than sum- marize known results and collect them in one place. In the literature on separability, primary if not exclusive emphasis has been placed on the implications of separability for the partial derivatives of the demand functions. In this paper primary emphasis is placed on the implications of separability for the demand functions themselves. The effects of finite price and income changes- which have been neglected in the literature -are also considered. All derivations are based on the use of a new tool-"conditional demand functionso-and the derivations are much more simple than those which have ap- peared in the literature. No mathematics be- yond the rule for taking the derivative of functions of a function is involved, and the mathematical arguments have a straight-forward economic interpretation.

In Section I we define conditional demand functions and consider the relation between conditional demand functions and ordinary demand functions. In Section I1 we define the utility tree and use conditional demand functions to derive its implications for consumer behavior. In Sections I11 and IV we define block additivity and additivity and derive their implications for consumer behavior. Section V examines the implica- tions of the three separability hypotheses for the partial derivatives of the compen- sated demand functions. Finally, the impli- cations of separability are summarized in Section VI.

I. CONDITIONAL DEMAND FUNCTIONS

Consider an individual whose preferences Can be represented by a utility function, U(x, , . . , xn), where xi denotes his con- sumption of the i-th good.2 The individual

To avoid various technical problems, we shall

424 ROBERT A.

is supposed to regard the prices of all goods, (pl , .. . , p,} as given, and to behave as if he were maximizing U(xl , . . . , 2,) sub-ject to the budget constraint

where I denotes total expenditure, hereafter referred to as "income."

In every price-income situation there is a unique collection of goods which yields greater utility than any other collection satisfying the budget constraint. The quan- tities of the various goods in this utility maximizing collection depend on all prices and income; that is

where P denotes the vector of all prices. The functions (h', . .. ,hnj are the "ordinary demand functions."

If the individual's consumption of one good has been determined before he enters the market, we say that the good has been "preallocated." We assume that the indi- vidual is not allowed to sell any of his allot- ment of a preallocated good, and that he cannot buy more of it. For definiteness, suppose that the n-th good is preallocated. We assume that the first n - 1 goods are available on the market a t prices { p l , . . ,p,-lj over which the individual has no

control, and that his total expenditure on these goods, A , is also predetermined. The individual is supposed to choose quantities of the first n - 1 goods so as to maximize

consider only utility functions which satisfy regularity conditions. Let X denote the vector (XI, . . . ,x,). We require

(1) the set of all X for which U(X) 2 0 is strictly convex for all U,

(2) U has strictly positive first order partial derivatives everywhere,

(3) U has continuous second order partial deriv- atives everywhere.

To avoid the problems associated with the fact that every good is not consumed in every price- income situation, we restrict our attention to a region of the price-income space in which the set of goods consumed remains unchanged. Goods not consumed in this region are ignored.

POLLAK

U(xl , . . , x,) subject to the "budget constraint"

and the additional constraint

where 2, denotes his allotment of the n-th good. Hence, his demand for the goods available on the market depends on the prices of these goods, total expenditure on them, and his allotment of the n-th good. That is

We call the function g"n the "conditional demand function" for the i-th good. The -second superscript, n, indicates that the n-th good is preallocated, and the terminol- ogy is suggested by the analogy with con- ditional probability.

Conditional demand functions can also be defined when more t'han one good is preallo- cated. In general, a conditional demand function expresses the demand for a good available on the market, as a function of (1),. the prices of all goods available on the mar- ket, (2) total expenditure on these goods and (3) the quantities of the preallocated goods.

Formally, we partition the set of all commodities into two subsets, 0 and 8.3 We assume that the goods in 8 are available on the market while those in 8 are preallo- cated; thus, if k E 0, then x k is available on the market, while if Ic E B, then x k is pre- allocated. We denote total expenditure on the goods available on the market by A'. The individual is supposed to maximize

Strictly speaking, it is the set of indices-t,he integers from 1 to n-by which the commodities are identified which are partitioned.

A collection of subsets {SI, . . . ,S,}, of a set S , is said to be a partition of S if (a) each element of S lies in a t least one subset of the collection and (b) no element of S lies in more than one sub- set of the collection. That is, the subsets must be exhaustive (UL Si = S) and mutually exclusive (si n si= 0, # j ) .

425 IMPLICATIONS OF

U(xl , , x,) subject to the "budget constraint''

and the additional constraints

xk = zk k E 8. (1.7)

The demand for a good available on the market depends on the prices of the goods available on the market, total expenditure on them, and the quantities of the preallo- cated goods. Thus

where POdenotes the vector of prices of the goods available on the market and Xs denotes the vector of preallocated goods. The function gi" is the conditional demand function for the i-th good; the second superscript, 8, indicates that the goods in 8 are preallocated.

We now examine the relation between conditional demand functions and ordinary demand functions. Suppose that only the n-th good is preallocated and that the indi- vidual's allotment of the n-th good is pre- cisely equal to the amount he would have purchased when facing prices P with income I . That is, x, = hn(P, I ) . Suppose further that the amount he has to spend on the first n - 1 goods is precisely equal to the amount he would spend on these goods when facing prices P with income I: I -p,hn(P, I ) . In this situation, the individual will purchase the same quantities of each of the goods available on the market as he would purchase if he faced prices P with income I and the n-th good were not pre- allocated. That is

hi(p, I) = gi'n[p~, . . . ,p,-I , (1.9)

a ( P , I ) , hn(P , I ) ] i # n

where

a ( P , I ) = I - pnhn(P, I ) . (1.10)

This identity follows directly from the defi- nitions of ordinary demand functions and

SEPARABLE UTILITY

conditional demand functions as solutions to constrained maximization problems.

A similar result holds when_ more than one good is preallocated. Let H'(P , I ) denote the vector of ordinary demand functions for the preallocated goods. For example, if the last two goods are preallocated, we have H'(P, I ) = [hn-'(P, I ) , hn(P, I ) ] . If all goods were available on the market, total expenditure on the goods in 8 would be CkGap,hk(p,I ) and total expenditure on the goods in 0 would be

By the argument used to establish (1.9) i t follows that

hi(p, I ) = gi 'B[~s,a O ( p , I ) , (1.12)

H ~ P ,I ) ) i E 6

where a' is defined by (1.11). Conditional demand functions are useful

in a number of areas in the theory of con-sumer behavior. In this paper we use them to analyze the implications of separability, but they are also useful in the theory of rationing and provide convenient proofs of the Hicksian aggregation theorem and Samuelson's Le Chatelier principle. In addi- tion, they can be used as the basis of a use- ful decomposition of the cross price deriva- tives of the demand function^.^

11. THE TREE CASE

In this section we consider the implica- tions for consumer behavior of the utility tree hyp~thes is .~ A utility function, U(xl ,

. , x,) is said to be a tree if there exists a partition of the n commodities into m sub-sets; m functions, VT(X,) ; and a function V such that

where m 2 2 and X , is the vector of com- modities in the r-th subset. A word about

4 For a full discussion of conditional demand functions and their uses see Pollak [13].

The utility tree hypothesis is also called "weak separability."

426 ROBERT A.

notation: It is convenient to use double subscripts to denote goods. The first sub- script indicates the subset or "branch" to which a good belongs, and the second indi- cates the particular good within the branch. Thus, xTh denotes the k-th good in the r-th branch. We denote the number of goods in the r-th branch by n, . Thus, X , = (x,, , . . . , x,,,) and

A utility function is a tree if and only if the goods can be partitioned into subsets in such a way that every marginal rate of substitution involving two goods from the same subset depends only on the goods in that ~ u b s e t . ~

The utility tree hypothesis has strong intuitive appeal. If we think of such sub- sets as "food," "clothing" and "recreation," it is tempting to assume that these subsets constitute branches of a utility tree. Simi- larly, if U(xl , . . , x,) is an intertemporal utility function, where X t denotes the vector of goods consumed in period t, i t is again tempting to assume that the utility function has a tree structure, reflecting a type of intertemporal "independence." The crucial assumption in both of these cases is that the individual's preference between two collections of goods which differ only in the components of one subset, say "food," is independent of the (identical) non-food components of the two baskets.

We now examine the conditional demand functions corresponding to a utility tree. We assume that the goods in one branch are available on the market, while all other goods are preallocated. For definiteness, suppose that the goods available on the market are in branch r, and that branch r is "food." Formally, tke0 if t = r and tkc8 if t f r. In accordance with our previous notation, A' denotes total expenditure on food and POthe vector of food prices.

0 If the utility function is a tree, i t is easy to verify that this condition is satisfied. Sufficiency is a much deeper question. See Leontief [B; 91.

POLLAK

The conditional demand functions are determined by maximizing (2.1) subject to the budget constraint

and the additional constraints

If we absorb the constraints (2.3) into (2.1) we obtain

Clearly, the utility maximizing values of (GI , , x,,,) are independent of the preallocated goods. For regardless of the levels of the preallocated goods, the indi- vidual has only to maximize VT(X,) subject to (2.2). Hence, the conditional demand function for the goods in 0 are of the form

I t follows from the general relation between ordinary demand functions and conditional demand functions (1.12) that

hTi(p, I ) = gTi'i [Po, ole(p, I ) ] (2.6)

where a' is defined by (1.11)

This result is extremely important. I t means that the demand for a good in the food branch (say, Swiss cheese) can be expressed as a function of food prices and total expenditure on food. I t would be nicer if the demand for Swiss cheese depended only on food prices and income, but this is not what separability implies. Instead, separability implies that income and the prices of goods outside the food category enter the demand functions for food only through their effect on total expenditure on

7 If an individual's preferences can be repre- sented by a well behaved utility function, and if his demand functions satisfy (2 .6 ) ,then his utility function is a tree. See footnote 21.

_ _ - -

- -

-----

427 IMPLICATIONSOF SEPARABLE UTILITY

food. Thus, if we work with expenditure on food rather than total expenditure on all goods, we can ignore the prices of goods outside the food category?

Our characterization of the demand func- tions implied by a utility tree leads directly to several important conclusions about the effects of finite price and income changes. Consider the effect on the consumption of Swiss cheese of a change in the price of some non-food item, say shoes; we do not require that the change in the price of shoes be small. The change in the price of shoes will cause the individual to change his On food (A') and the change in total expenditure on food will cause a change in Swiss cheese consumption. Now consider a change in the price of another non-food item, say, tennis balls. Suppose that the effect on A' of this price change is the same as the effect on A' of the change h the price of shoes. Then the change in the price of tennis balls and the change in the price of shoes will have the same effects on the consumption of Swiss cheese. Similarly, if a change in income which has the same effect on A' as the change in the price of shoes, then i t will also have the same effect on Swiss cheese consumption. A similar result holds for simultaneous changes in income and in the prices of several non-food items.

The literature on separable utility has been more concerned with the ~ a r t i a l derivatives of the demand functions than with the demand functions themselves. The

8 From an econometric point of view, this is less useful than might a t first appear. (1) The assumption that the utility function is a tree does not, in the absence of additional assumptions, yield a sufficiently narrow class of demand equa- tions to permit estimation. (2) Estimation of de- mand functions requires not only specification of their functional form but also specification of their error structure. Although discussion of error structures is beyond the scope of this paper, i t should be remarked that total expenditure on food cannot be taken as exogenous. A similar problem arises if we treat total expenditure on all goods (rather than income) as exogenous. See Summers [21].

implications of the utility tree hypothesis for the partial derivatives of the demand functions follow immediately from the re-sults we have already established. Differ- entiating (2.6) with respect to p8j and I we obtain

That is, the change in the consumption of Swiss cheese caused by a change in the price of shoes is proportional to the change in total expenditure on food caused by the change in the price of shoes (in- come). We may express (2.7) in ratio form a3

ahTi acre -a ~ s j- a ~ s j # r. (2.9)ahri acre's # r, t

aptk aptk

If- (aae/al) + 0, we can eliminate ag" 'e /d~e between (2.7) and (2.8) and obtain

aae ahr' - Kjahr' - r j ah" apaj acre a 1 pp -a1 s # r (2.10)

a1 where p," is defined by

That is, the change in the consumption of Swiss cheese induced by a change in the price of shoes is proportional to the change in the consumption of Swiss cheese induced by a change in income. The factor of pro- portionality is the same for all food items (Swiss cheese, roast beef) but i t does depend

9 We remark that the factor of proportionality, pr*i, is a function of all prices and income as are the analogous factors of proportionality in the block additive and additive cases.

- -

428 ROBERT A.

on the good whose price has changed. We can express (2.10) in ratio form as

-dpsj = -d l s # r . (2.12)h'"-d -hrk

apSi a I

111. THE BLOCK ADDITIVE CASE

In this section we consider the implica- tions of "block additivity." lo A utility function, U(xl , . - ,x,), is said to be block additive if there exists a partition of the commodities into m subsets; m functions, Vr(xr); and a function F, F' > 0, such that

where m 2 2 and X, is the vector of com- modities in the r-th subset." In the block additive case we call the subsets "blocks."

If m 2 3, a utility function is block additive if and only if the goods can be partitioned into subsets in such a way that every marginal rate of substitution involving two goods from different subsets depends only on the goods in those two subsets.12 This implies that every marginal rate of substitution involving two goods from the same subset depends only on the goods in that subset. Thus, if a utility function is block additive with m blocks, i t is also a tree with m branches. This may be seen directly from (3.1), since

where F-' denotes the inverse of the func- tion F, is clearly a utility tree.

The fact that a block additive utility function with m blocks is a tree with m

10 Block additivity is also called "strong sep-arability."

11 To insure thatzy-1 Vr(X,) is a well behaved utility function, we require F to have a continu- ous second derivative.

1 2 I t is easy to verify that this condition is satisfied if the utility function is block additive. Sufficiency is well beyond the scope of this paper. See Leontief [8;91.

POLLAK

branches means that the results of Section I1 apply directly to the block additive case. If are identify the blocks as "food," "clothing," "recreation," etc., we can im- mediately conclude that the demand for Swiss cheese can be written as a function of food prices and total expenditure on food. The implications of this basic result for finite price and income changes and for the partial derivatives of the demand functions have already been discussed. In this section we examine the additional implications of block additivity.

If a utility function is a tree with m branches, in general, we cannot combine two branches into a single branch. For ex- ample, if "food" and "recreation" are two branches of a tree, it is not in general true that the demand for Swiss cheese can be written as a function of food prices, recrea- tion prices, and total expenditure on food and recreation. But if the utility function is block additive, it is always permissible to treat the goods in two (or more) blocks as a single block. This may be seen directly from (3.1), since

is clearly block additive. Hence, if the utility function is block additive

V ~ P ,I) = s'"."[~,, Pt , ae (p , I ) ] (3.4)

where 0 denotes the set of all goods in blocks r and t and B the goods in the remaining blocks.

More generally, if a block additive utility function has m blocks, and if some of these blocks are combined to form m* superblocks, m* < m, then the utility function is block additive in the superblocks. Hence, a block additive utility function with m blocks is a utility tree with m* branches. The additional implications of block additivity can now easily be established by combining blocks into superblocks, treating the resulting block

429 IMPLICATIONS OF

additive utility function as a tree, and ap- plying the results of Section ll.la

In the case of block additivity, i t is par- ticularly interesting to consider the case in which 8 denotes the goods in one block, and 0 the goods in the remaining m - 1blocks. For definiteness, suppose that the goods in block s are in 8 and the remaining goods are in 0. Then

If we take block s to be "clothing," then (3.5) implies that the demand for Swiss cheese can be expressed as a function of the prices of all non-clothing goods and total expenditure on all goods other than cloth- ing. It is not clear that i t is more useful to write the demand for Swiss cheese this way than as a function of all food prices and total expenditure on food. But if the utility function is block additive, we have the op- tion of writing the demand function in either form; the choice between them must depend on the problem a t hand. If we are primarily interested in the effect of various price and income changes on Swiss cheese consumption, i t is probably more conven-ient to write the demand for Swiss cheese as a function of all food prices and total expenditure on food. But if we are interested in the effect of a change in the price of shoes on the consumption of a variety of non-clothing goods (Swiss cheese, tennis balls, etc.) i t is more convenient to write the demand for these goods as functions of all non-clothing prices and total expendi- ture on goods outside the clothing category.

The implications of (3.5) for finite price and income changes should be clear from the discussion in Section 11. To derive the implications of block additivity for the par-

l3 If m = 2, this technique does not yield any implications of block additivity beyond those already obtained for the tree case.

14 These results can also be derived by means of conditional demand functions in the same way that the results for the tree case were derived in Section 11.

SEPARABLE UTILITY

tial derivatives of the demand functions, we differentiate (3.5) with respect to p,j and I:

If (aae/al) + 0, we can eliminate agr"O/d~e between (3.6) and (3.7) and obtain

where is defined by

dae

As in the tree case, the factor of propor- tionality depends on the good whose price has changed (sj); but unlike the tree case, i t is independent of the good whose quantity response we are considering. We can express (3.8) in ratio form as

ah" ahr'

IV. THE ADDITIVE CASE

The third separability hypothesis which we consider is additivity. A utility function, U(xl , , x,), is said to be additive if there exists a differentiable function, F, F' > 0, and n functions, ui(x;), such that

If n 2 3, a utility function is additive if and only if the marginal rates of substitu- tion involving any pair of goods depend only on those two goods.i6 It is more

We also require that F have a continuous second derivative.

la Again, necessity is easy and sufficiency is hard. See Samuelson [16, 174-91.

430 ROBERT A. POLLAK

plausible to interpret the x's in an additive hT(P, I ) = gr"be , ae(p , I ) ] r # s (4.2) utility function as composite commodities, such as ''food" and "clothing," than as specsc commodities such as Swiss cheese, roast beef and shoes. If an individual's utility function is block additive with blocks corresponding to "food," "clothing," etc., and if we form the Hicksian composite commodities corresponding to each of these blocks, then the utility function defined in terms of these composite commodities is additive. In describing the implications of additivity, we shall refer to the commodities as "food," "clothing," etc.

If a utility function is additive, i t is also block additive; for regardless of how the x's are partitioned into subsets, they will satisfy (3.1). This means that the results of the last two sections can be used to deduce the implications of additivity." Let 8 and 8 be any partition of the x's into two subsets. Then the demand for the goods in 8 can be written as a function of the prices of the goods in 8 and total expenditure on these goods. This result holds for all possible partitions of the commodities into subsets, and thus, when the utility function is additive, we may use any partition of the goods which is convenient. For example, we can write the demand for food as a function of the price of food, the price of recreation, and total expenditure on food and recrea- tion :

hT(P,I ) = gr"br , pt , ae(p ,111 where @ = {r, t ) and all other goods are in 8.18Or, if we prefer, we can write the demand for food as a function of the prices of all goods except clothing and total expenditure on all goods other than clothing:

l7 If there are only two goods, our technique breaks down. Oddly enough, additivity is more d8icult to deal with when there are exactly two goods than when there are n goods, n 2 3. Simi-larly, block additivity is more difficult when there are only two blocks than when there are three or more. For the implications of additivity when there are only two goods, see Samuelson [16, 174-91.

18 This condition is mentioned by Samuelson 116, 174-91.

where 8 = 11, , s - 1, s + 1, , n)and 8 = {s). The implications of these results for finite price and income changes are obvious, and we will not discuss them.

To examine the partial derivatives of the demand functions we differentiate (4.2) with respect to p, and I:

That is, the change in the consumption of food induced by a change in the price of clothing (income) is proportional to the change in expenditure on all goods other than clothing. We remark that the sign of aae/ap, depends on the own price elasticity of clothing.

If_ (aae/al) t. 0, we can eliminate agT" /a~ ' between (4.3) and (4.4) and obtain

affe -ahr -- ap,--ah' - p - r f s- ,ah'

(4.5)ap, aae-a I a1 ar

where the factor of proportionality p-s defined by

aae-

That is, the change in the consumption of food induced by a change in the price of clothing is proportional to the change in the consumption of food induced by a change in income. The factor of proportionality de-pends on the good whose price has changed, but not on the good whose quantity response we are considering. We can express (4.5) in ratio form as

IMPLICATIONS OF SEPARABLE UTILITY 431

V. COMPENSATED DEMAND FUNCTIONS

In this section we examine the implica- tions of separability for the partial deriva- tives of the compensated demand functions.

Corresponding to every level of utility and set of prices is a unique collection of goods which allows the individual to attain that level of utility a t minimum cost. The quan- tities of the various goods in this cost mini-mizing collection depend on all prices and the level of utility to be attained (real income); that is

The functions if1, , f"} are the "com- pensated demand functions." It is well known [Samuelson, 16, 1031, that

where

U = u[hl(p, I ) , ... ,hn(P, I ) ]

and that

A . The Tree Case

If the utility function is a tree, substitut- ing from (2.10) into (5.2) we obtain

where the factor of proportionality a,"' is defined by

ar 8 1 = pr8a + h" 19 (5.5)

That is, the change in the consumption of Swiss cheese induced by a compensated

19 ~ , * jlike and h*i is a function of all prices and income, as are the analogous factors of pro- portionality in the block additive and additive cases.

change in the price of shoes is proportional to the change in the consumption of Swiss cheese induced by a change in income. The factor of proportionality is the same for all food items, but depends on the good whose price has changed. This result can easily be expressed in ratio form.

If we reverse the roles of r i and sj and consider the effect on the consumption of shoes of a compensated change in the price of Swiss cheese we obtain

~ i v i & n ~(5.4) and (5.6) by (ahTi/aI) (ahG/aI), and observing that the two re-sulting expressions are equal by (5.3) we obtain

8j-'Jr ahna-a 1

The value of this expression is clearly independent of both i and j , although it does depend on both r and s. We denote the common values of these ratios by I',, = I',,.20 Thus, if the utility function is a tree, there exist r 's such that

We may express the 'J'S and p's in terms of the r 's as

20 r,, is a function of all prices and income, as are the analogous factors of proportionality in the block additive cases.

21 This condition is both necessary and s&i- cient. See Goldman and Uzawa [4]. Hence ,(2.6) together with the symmetry condition (5.3) is necessary and s a c i e n t .

--

432 ROBERT A.

The intuitive significance of (5.7), (5.8), (5.9) and (5.10) is not obvious. These resdts are "consistency" conditions; they follow from the assumption that the indi- vidual's preferences can be represented by a well behaved utility function and are direct consequences of the symmetry conditions (5.3).

B. The Block Additive Case

If the utility function is block additive, the argument used in the tree case can be used to show

where a'' is defined by

Proceeding as in the tree case, it can be shorn that there exists a function r such that

af

a1 a1 and

gri = r ahri (5.14)

C. The Additive Case

If the utility function is additive, then

where

a" pS + h8. (5.17)

Hence, there exists a function r such that

POLLAK

and

r ah' a = r - ar

VI. CONCLUSION

The implications for the demand functions of the three separability hypotheses can be summarized briefly.

1. If the utility function is a tree, then the demand for a good can be written as a function of the prices of all goods in its branch and total expenditure on these goods. Since total expenditure on these goods depends on the prices of the goods in all branches and income, demand for a good is not independent of the prices of goods in other branches.

2. If the utility function is block additive with m blocks, it is also a tree with m branches, so the tree results hold without modification. In addition, blocks can be combined to form superblocks (e.g., food and recreation can be treated as a single block) and the demand for any good in a superblock can be written as a function of the prices of the goods in that superblock and total expenditure on these goods. As in the tree case, total expenditure on the goods in a superblock depends on all prices and income.

3. From a technical viewpoint, additivity is an extreme form of separability in which any subset of goods constitutes a branch. Thus, if we consider any subset of goods, the demand for a good in that subset can be written as a function of the prices of the goods in the subset and total expenditure on them. The prices of the goods outside the subset are relevant because, and only because, they affect total expenditure on the goods in the subset. As a matter of interpretation, the additivity hypothesis is more reasonable when applied to broad cate- gories of goods than to individual goods.

The advantages of using conditional de-

IMPLICATIONS OF SEPARABLE UTILITY 433

mand functions to analyze the implications of direct separability are two-fold. First, they focus attention on the demand functions themselves, rather than on their partial derivatives. The traditional calculus ap-proach, on the other hand, tends to focus attention on partial derivatives of demand functions. The emphasis on the demand functions is inherently preferable for both theoretical and empirical work. The impli- cations of the three separability hypotheses for the effects of finite price and income changes and for the partial derivatives follow directly from these demand function results.

Second, conditional demand functions lead to a decomposition of the allocation process which is particularly useful for analyzing the implications of direct separa- bility. More specifically, they enable us to separate the allocation of income among categories from the allocation of expenditure within categories. This focuses attention on the fundamental implication of direct separability: the "within category" alloca-tion depends only on the prices of the goods in a category and total expenditure on them. Hence, consumption of a good in one cate- gory is influenced by the prices of goods in other categories only through their effect on category expenditure. Similarly, income in- fluences consumption of goods in a category only through its effect on category expen- diture.

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Conditional Demand Functions and the Implications of Separable UtilityRobert A. PollakSouthern Economic Journal, Vol. 37, No. 4. (Apr., 1971), pp. 423-433.Stable URL:

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[Footnotes]

4 Conditional Demand Functions and Consumption TheoryRobert A. PollakThe Quarterly Journal of Economics, Vol. 83, No. 1. (Feb., 1969), pp. 60-78.Stable URL:

http://links.jstor.org/sici?sici=0033-5533%28196902%2983%3A1%3C60%3ACDFACT%3E2.0.CO%3B2-5

References

2 Continuity Properties of Paretian UtilityGerard DebreuInternational Economic Review, Vol. 5, No. 3. (Sep., 1964), pp. 285-293.Stable URL:

http://links.jstor.org/sici?sici=0020-6598%28196409%295%3A3%3C285%3ACPOPU%3E2.0.CO%3B2-X

13 Conditional Demand Functions and Consumption TheoryRobert A. PollakThe Quarterly Journal of Economics, Vol. 83, No. 1. (Feb., 1969), pp. 60-78.Stable URL:

http://links.jstor.org/sici?sici=0033-5533%28196902%2983%3A1%3C60%3ACDFACT%3E2.0.CO%3B2-5

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14 The Existence of a Utility Function to Represent PreferencesTrout RaderThe Review of Economic Studies, Vol. 30, No. 3. (Oct., 1963), pp. 229-232.Stable URL:

http://links.jstor.org/sici?sici=0034-6527%28196310%2930%3A3%3C229%3ATEOAUF%3E2.0.CO%3B2-1

18 The Effect of Price Changes on the Demand and Supply of Separable GoodsMasazo SonoInternational Economic Review, Vol. 2, No. 3. (Sep., 1961), pp. 239-271.Stable URL:

http://links.jstor.org/sici?sici=0020-6598%28196109%292%3A3%3C239%3ATEOPCO%3E2.0.CO%3B2-M

22 The Demand Curves from a Quadratic Utility IndicatorL. L. WeggeThe Review of Economic Studies, Vol. 35, No. 2. (Apr., 1968), pp. 209-224.Stable URL:

http://links.jstor.org/sici?sici=0034-6527%28196804%2935%3A2%3C209%3ATDCFAQ%3E2.0.CO%3B2-1

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