Chapter 1. Consumer Choice under Certainty. 1.1 Budget, Prices, and Demand

Transcription

1 Chapter 1 Consumer Choice under Certainty 1.1 Budget, Prices, and Demand Consider an individual consumer (household). Her problem: Choose a bundle of goods x from a given consumption set X R H under a budget constraint, for given xed prices p 2 R H +. The elements of the decision problem: H is the dimension of the goods space. To x ideas, we consider the basic static model. Price vector p is xed and not manipulable by the consumer. Budget b 2 R + is either given in value ( purchasing power ) or as the value of initial endowments w 2 R H + : b = p w. Note on Notation : For p; q 2 R H ; p q = p T q = HX p i q i (vector product with, matrix multiplication without) Let B = B b;p = fx 2 X; p x bg be the budget set. X is often taken to be equal to R H +. Strictly speaking, this is unrealistic: 1 i=1

2 2 CHAPTER 1. CONSUMER CHOICE UNDER CERTAINTY In any given period, the supply of any good is bounded. Most goods are not arbitrarily divisible. However, for most standard applications these objections are of minor importance. But sometimes it is more appropriate to model consumer behavior as discrete choice problems (on the other hand note: smoothing over several periods). 1 Remark: On discrete choice theory and their application to imperfectly competitive markets see e.g. Anderson, de Palma and Thisse (1992), Discrete Choice Theory, MIT Press. The structure of the choice problem is: prices, budget (+ consumer s tastes)! preferred bundles De nition: (Marshallian) demand correspondence ': x 2 '(p; b) () x is among the most preferred elements in B p;b. Remark: The consumer s choice need not to be unique! notion of correspondence. Two properties which are usually (naturally?) imposed on demand correspondences: (H) Homogeneity: ' is homogenous of degree 0: '(p; b) = '(p; b) for all > 0 (no money illusion ). Warning: see the experiments by Fehr and Tyran (1998). (BI) Budget identity: For every p >> 0 and b > 0: p x = b for all x 2 '(p; b) 1 When combined with utility theory (see Chapter 2), the assumption that X = R H + is problematic for yet another reason: subsistence levels (very low levels of certain goods have the same value as the zero level).

3 1.1. BUDGET, PRICES, AND DEMAND 3 ( no waste ). In most applications one also imposes: (SV) Single-valuedness: ' is a function. Properties (H) and (BI) are relatively simple conditions. Yet, they do not impose much structure on the choice problem. Economists are interested in more structure, for several reasons: individual demand does seem to be somewhat consistent, econometrically, more structure on the derived equations is desirable for testing observed household behavior, theoretically, the predictions derived from (H) and (BI) alone are not satisfactory. Therefore, we will explore two important additional properties of demand. For simplicity, we assume that ' is a function. (WARP) The demand function ' satis es the Weak Axiom of Revealed Preferences: For any two (p; b); (p 0 ; b 0 ): if p '(p 0 ; b 0 ) b and '(p; b) 6= '(p 0 ; b 0 ) then p 0 '(p; b) > b 0 Interpretation: the hypothesis states that the choice in situation (p 0 ; b 0 ) is possible in situation (p; b), but is not made in that situation: '(p; b) is revealed to be preferred to '(p 0 ; b 0 ), the postulate is that the choice in situation (p; b) be not possible in situation (p 0 ; b 0 ) (graphically: the in Figure 1 is not allowed). In other words, if '(p; b) is revealed to be preferred to '(p 0 ; b 0 ) in one situation, it should be preferred whenever this is possible (a ordable).! Samuelson (Economica 1938) Remark:

4 4 CHAPTER 1. CONSUMER CHOICE UNDER CERTAINTY x 2 B p,b B p',b' x 1 Figure 1.1: An illustration of the Weak Axiom The Weak Axiom captures a notion of consistency of choice. There are other versions of the Weak Axiom (see, e.g., the exercises), and other, related axioms. Di erent way of putting the above de nition: De ne a binary relation W on X by: xw y () x 6= y; 9(p; b) 2 R H+1 + such that x = '(p; b) and p y b ( x is revealed preferred to y ) Then (WARP) states that W is asymmetric. The other important property to study is: (LD) A demand function ' satis es the (strict) Law of Demand if it is strictly monotone decreasing in p, i.e. if (p 0 p) ('(p 0 ; b) '(p; b)) < 0 for all b > 0 and p 6= p 0.

5 1.1. BUDGET, PRICES, AND DEMAND 5 The Weak Law of Demand is de ned accordingly. Interpretation: Consider a price-budget situation (p; b). Let p = p 0 p denote a price change, and x = '(p 0 ; b) '(p; b) the demand change induced by p. Then p x < 0 ( price changes and demand changes point in0opposite 1directions ). 0 : If, e.g., only the price of good h changes, p = B p h C then p x : A 0 p h x h, and p h and x h must have the opposite sign. Mathematical Reminder: If ' is di erentiable and monotone, p '(p; b) is negative semi-de nite for all p; b. p '(p; b) is negative de nite for all p; b, then ' is stictly monotone. Remember: A matrix A 2 R H2 is called negative semi-de nite if v Av 0 for all v 2 R H A is called negative de nite if v Av < 0 for all v 6= 0 2 R H Aside: Further suggested references for math: A. Takayama: Mathematical Economics, 2nd ed., CUP A. Chang: Fundamental Methods of Mathemetical Economics, 3rd ed., McGraw- Hill C. Huang, Ph. Crooke: Mathematics and Mathematica for Economists, Blackwell In particular in the context of our course, an excellent book on mathematics and some of the related economics is A. de la Fuente: Mathematical Methods and Models for Economists, CUP What is the relationship between (WARP) and (LD)?

6 6 CHAPTER 1. CONSUMER CHOICE UNDER CERTAINTY Proposition 1 If ' satis es (H), (BI), and (SV), then (LD)=)(WARP), but not the reverse. Proof. Suppose ' satis es (LD). Take any (p; b), (p 0 ; b 0 ) with '(p; b) 6= '(p 0 ; b 0 ) and p '(p 0 ; b 0 ) b: We know that (p p) ('(p; b) '(p; b)) < 0 for all p 6= p: (1.1) Consider p = b b 0 p 0 : Then using (H) we have Hence, (1.1) is equivalent to '(p; b) = '(p 0 ; b 0 ) (1.2) b b 0 p0 '(p 0 ; b 0 ) p '(p 0 ; b 0 b ) b 0 p0 '(p; b) + b < 0 () b b b 0 p0 '(p; b) < p '(p 0 ; b 0 ) {z } b b ) b b b 0 p0 '(p; b) < 0 () p 0 '(p; b) > b 0 (Trick: Introduce a p which allows to obtain '(p 0 ; b 0 ): scale p 0 appropriately) Concerning the converse of Proposition 1, an example of a ' satisfying (WARP) but not (LD) is given in Figures 1.2 and 1.3. Here, H = 2, p 0 1 < p 1, p 0 2 = p 2, b 0 = b. Then (LD) implies ' 1 (p 0 ; b) > ' 1 (p; b). Yet, (WARP) places no restrictions on '(p 0 ; b). (see gure) General intuition: (WARP) is a relatively weak condition on individual behaviors, (LD) a relatively strong one. 1.2 The Weak Axiom of Revealed Preferences Next, we want to understand (WARP) and (LD) a bit better. First we look at (WARP). An important conceptional tool for understanding (WARP) is the Slutsky wealth compensation.

7 1.2. THE WEAK AXIOM OF REVEALED PREFERENCES 7 x 2 admissible for ϕ (p', b) under (LD) ϕ(p, b) x 1 Figure 1.2: (LD) and... x 2 admissible for ϕ (p', b) under (WARP) ϕ(p, b) x 1 Figure 1.3:...(WARP)

8 8 CHAPTER 1. CONSUMER CHOICE UNDER CERTAINTY Idea: Go from (p; b) to (p 0 ; b 0 ) (new choice experiment) without making the consumer worse o. De nition: The change from (p; b) to (p 0 ; b 0 ) is called Slutsky compensated if b 0 = p 0 '(p; b). Proposition 2 Suppose demand satis es (BI) and (SV). Then ' satis es (WARP) if and only if the compensated law of demand holds, i.e.: (CLD) For any Slutsky compensated change from (p; b) to (p 0 ; b 0 ) (p 0 p) ('(p 0 ; b 0 ) '(p; b)) 0 with strict inequality if and only if '(p; b) 6= '(p 0 ; b 0 ). Remark: The compensated law of demand is di erent from (LD): the budget changes! Proof. (i) (BI), (SV), and (WARP) imply (CLD): Consider any Slutsky compensated price change from (p; b) to (p 0 ; b 0 ), and suppose '(p 0 ; b 0 ) 6= '(p; b). Then (p 0 p) ('(p 0 ; b 0 ) '(p; b)) = p 0 '(p 0 ; b 0 ) p 0 '(p; b) p '(p 0 ; b 0 ) + p '(p; b) {z } {z } {z } {z } (1) (2) (3) (4) Here, '(p 0 ; b 0 ) is not a ordable under (p; b), because of (WARP) and p 0 '(p; b) = b 0 (Slutsky compensation). Hence, in term (3), p '(p 0 ; b 0 ) > b. The rst term, (1), is equal to b 0 by (BI). Term (2) is equal to b 0 because of Slutsky compensation. Term (4) is equal to b by (BI). Hence, (p 0 p) ('(p 0 ; b 0 ) '(p; b)) < 0. (ii) (BI), (SV), and (CLD) imply (WARP): Slightly more complicated, in two steps: 1. If (WARP) is violated (i.e. if 9(p; b); (p 0 ; b 0 ) : '(p; b) 6= '(p 0 ; b 0 ); p '(p 0 ; b 0 ) b; p 0 '(p; b) b 0 ) then it is violated for a Slutsky compensated price change (i.e. for (p; b); (p 0 ; b 0 ) such that one of the two binds)

9 1.2. THE WEAK AXIOM OF REVEALED PREFERENCES 9 2. If there are (p; b), (p 0 ; b 0 ), such that '(p; b) 6= '(p 0 ; b 0 ); p '(p 0 ; b 0 ) = b; p 0 '(p; b) b 0 ; then these pairs violate (CLD). The rst step is a little work, the second is trivial by (BI). Remark: (H) is not needed in either direction of the proof. 2 Suppose now that ' is not only (SV), but even di erentiable. (When assuming ' is di erentiable, we shall henceforth always take for granted that int X 6= ;) Locally, around (p; b) we have d' p ' dp {z} {z} b ' {z} {z} db ( total di erential ) H1 H1 11 HH where (dp; db) is a vector of variations spanning the tangential hyperplane at (p; b). Restrict attention to those variations with Slutsky compensated price changes: db = '(p; b) dp Then d' p 'dp b 'db = (@ p ' b '' T )dp If ' satis es (BI) and (WARP), then Proposition 2 implies for all such variations. dp d' 0 (Proposition 2 holds for nite variations, by taking limits and because of the di erentiability of ', it also holds locally (on the tangential hyperplane), i.e. for in nitesimal variations. Note: Because of the limit, we only get ) Since we can multiply the vector dp by arbitrary constants, we have proved: Proposition 3 Suppose that demand ' satis es (BI), (SV), (WARP), and is di erentiable. Then the Slutsky matrix S(p; b) p '(p; b) b '(p; b)'(p; b) T is negative semi-de nite for all (p; b). 2 By the way: one can show (this is elementary but not entirely trivial) that (WARP) and (BI) imply (H).

10 10 CHAPTER 1. CONSUMER CHOICE UNDER CERTAINTY Another way of understanding the above statement is to consider the compensated demand function q 7! '(q; q '(p; b)): The Slutsky matrix is the derivative of this function at q = p. The generic entry of the Slutsky matrix: s ij j ' j s ij is called the substitution e ect for good i with respect to price j changes. The experiment behind the Slutsky compensation: Raise price p j by p j. Then: good j is more expensive, the consumer is overall poorer. The rst e ect usually implies: switch demand from good j to other goods. The second e ect usually implies: adjust all demands downward. The Slutsky experiment eliminates the second e ect and therefore allows to pinpoint the rst e ect. This is re ected in the matrix j : total change of ' i in response to p j ' j : change of ' i in response to a wealth reduction at ' j Therefore: s ij measures the (pure) substitution e ect, and Proposition 3 says that under the stated conditions this e ect is overall negative. Following Hicks (Value and Capital, OUP, 1936), we usually call goods i and j - complements, if s ij < 0; s ji < 0 - substitutes, if s ij > 0; s ji > 0: Remark: Another way to isolate the substitution e ect is by Hicks compensation : adjust the consumer s budget such that she is as well o as before the price change (instead of being able to buy the same bundle). But

11 1.2. THE WEAK AXIOM OF REVEALED PREFERENCES 11 for this experiment we need to know how to measure the consumer s wellbeing. See Chapter 2. Proposition 3 implies that s hh 0 8h: Hence, the partial demand function for good h, ' h (:::; p h ; :::), can only be increasing if the wealth e ect for that good is negative and su ciently strong. Reminder: A good i < 0 called inferior (for the given consumer), a good i > 0 normal. Proposition 4 For a normal good, the partial demand function is decreasing. Can we strengthen Proposition 3 to have negative de niteness? Answer: No. The Slutsky matrix is necessarily singular (does not have full rank), and the price vector lies in its null space: p S(p; b) = 0 (see the exercises). Further question: But do we have at least rank S = H 1? Answer: In general, no (see the exercises). But with additional structure (see chapter 2), the answer is yes. Final question: Is there a converse to Proposition 3? (this would clearly be useful for verifying the (WARP) directly...) Proposition 5 If ' satis es (H), (BI), (SV), and is di erentiable, and if S(p; b) is negative de nite on T p := v 2 R H ; p v = 0 (1.3) then ' satis es (WARP). Remark: Because p S(p; b)p = 0; (1.3) is equivalent to the condition w that w S(p; b)w < 0 for all w 6= 0; 6= p. kwk kpk Proof. Kihlström, Mas-Colell, Sonnenschein (Econometrica 1976) Hence, (WARP) and the negative semi-de niteness of S are not quite the same. Because of that, sometimes a weaker Weak Axiom is used, which is indeed equivalent to the negative semi-de niteness of S.

12 12 CHAPTER 1. CONSUMER CHOICE UNDER CERTAINTY 1.3 The Law of Demand De nition: Given a Marshallian demand function ' h, the function e h : R +! R; e h (b) = ' h (p; b) is called the Engel curve for good h at price p.! Remark: In applied work, Engel curves often relate expenditure shares to total expenditure: w h (b) = p h e h (b)=b. A classical empirical Engel curve is the PIGLOG form, identi ed by Muellbauer (Econometrica 1976): w h = A h (p) + B h (p) log b Proposition 6 Let ' be a di erentiable function with Slutsky p ' = S A: (where A b '' T ) A(p; b) is positive semi-de nite, if ' has linear Engel curves (locally, at p). In this case, v Av = 0 () v? '(p; b): Proof. Suppose that '(p; b) = (p)b; : R H +! R H. b '(p; b) = (p); and A(p; b) = (p)(p) T b. Hence, v Av = v T T vb = (v ) 2 b 0 for all v. Equality () v? () v? '(p; b). What about a converse? The general statement is the following: Proposition 7 Let ' be a di erentiable function satisfying (BI). Then A(p; b) is positive semi-de nite for all (p; b) if and only if ' has linear Engel curves. Proof. Step 1: A(p; b) is positive semi-de nite if and only if 9(p; b) 0 b '(p; b) = (p; b)'(p; b). Proof: ( : v Av = v T '' T v = (v ') 2 0 all v. ) : Suppose no such exists, b ' and ' are not collinear. Then (see Figure 1.4) there is a v such that v Av = b ')(v ') < 0 {z } {z } 2R Step 2: A : R H+1! R + b ' = ' exists if and only if 9 : R H! R H 2R

13 1.3. THE LAW OF DEMAND 13 ϕ b ϕ v Figure 1.4: Finding a v and B : R H+1! R b B 0, such that '(p; b) = (p)e B(p;b) all (p; b) Proof: (= : Take b. =) : In each component we have an ordinary di erential equation in b ' h = ' h. Standard ' h = b (ln ' h ) = implies that 9 h = h (p) (constant of integration) : ' h (p; b) = h e B(p;b) b B(p; b) = (p; b). Last step: Suppose A is positive semi-de nite and ' satis es (BI). Then we have shown that '(p; b) = (p)e B(p;b). Now, (BI) implies ' h p (p)e B(p;b) = b () e B(p;b) b = p (p) ) '(p; b) = (p) p (p) b

14 14 CHAPTER 1. CONSUMER CHOICE UNDER CERTAINTY Proposition 7 shows that for wealth e ects to be well-behaved (i.e. have the positive monotonicity one would intuitively expect and which would be nicely compatible with the negative monotonicity of the Slutsky matrix), Engel curves must be linear. This is quite an assumption as we shall see. Proposition 8 If ' satis es (BI), (SV), (WARP), is di erentiable, has linear Engel curves, and S(p; b) has rank H-1 for all (p; b), then ' satis es (LD). Proof. By proposition 3, S is negative semi-de nite. This and the linear Engel curves imply, by the above, that p 'v = v (S A)v 0 all v Full rank means: v Sv < 0 for all v 6= 0 not collinear with p. Negative de niteness: p 'v = 0 () v Sv = 0 and v Av = 0 () v k p and v? '(p; b) (Proposition 6) This is impossible because p is not orthogonal to '(p; b) (p '(p; b) = b 6= 0). Note that without the full-rank condition on the Slutsky matrix, we still obtain the Weak Law of Demand. 1.4 Applied Demand Analysis References: M. Browning: Children and Household Economic Behavior, JEL 30, 1992, A. Deaton, J. Muellbauer: Economics and Consumer Behavior, CUP 1980, A. Deaton: Understanding Consumption, OUP 1992, A. Deaton: The Analysis of Household Surveys, PUP 1997, W. Hildenbrand: Market Demand, PUP 1994, G. Stigler, The Early History of Empirical Studies of Consumer Behavior, JPE 42, 1954, The Summer 2001 issue of the Journal of Economic Perspectives.

15 1.4. APPLIED DEMAND ANALYSIS Who is the consumer? Smallest decision making unit: the household - a group of people living at the same address having meals prepared together and with common housekeeping (from the British Family Expenditure Survey ). Literature on intra-family decision: G. Becker: A Treatise on the Family, HUP 1981, M. Browning, F. Bourguignon, P.A. Chiappori, V. Lechene: Income and Outcomes: A Structural Model of Intrahousehold Allocation, JPE 102, What are the goods? Typically, in applications individual goods are grouped into di erent categories to facilitate their description (see the next section). Example: Aggregation in the French Enquête Budget de Famille : 1. Food at home 8. Fuel, light and power 2. Food outdoors 9. Durable household goods and dom. services 3. Nonalcoholic drinks 10. Hygiene and health 4. Alcoholic drinks 11. Transport and communication 5. Clothing 12. Tobacco 6. Footware 13. Culture and leisure 7. Housing 14. Other goods What is the household s budget? Decompose short-run income (+existing wealth) in at least two steps: gross (monthly, yearly) income - statutory deductions (income tax, social security,...) = disposable income (y) - savings (+dissavings) = total expenditure (b) If future prices are xed (and known to be xed), short-run demand (demand in period ), ' can be studied in two ways: either ' = '(p ; b ): consumption as a function of current total expenditure. Problem: b = b(p ; y ) is endogenous.

16 16 CHAPTER 1. CONSUMER CHOICE UNDER CERTAINTY or ' = '(p ; y ): consumption as a function of current disposable income. Problem: ' typically does not satisfy the budget identity. Remedy: De ne Slutsky compensated income (at price p 0 ) as y 0 = p 0 ' (p ; y ) + y p ' (p ; y ) (household can a ord old consumption and old saving) and compensated demand as p 0 7! ' (p 0 ; y 0 ) (1.4) Then one can still de ne the usual Slutsky decomposition and show that the Slutsky matrix (the derivative of (1.4) at p 0 = p ) is negative semide nite. Remark: If future prices cannot be taken to remain xed when p changes, then short-run demand is more complex. One usually has to impose timeseparability of ' (see below) in order to obtain tractable models. 1.5 Composition and Separation The theory developed thus far is valid for any number of goods. Taken literally, H is a very large number for any planning period. It becomes even larger if one takes into account that household planning typically involves many periods. If one assumes that future prices are known, the present model easily generalizes to this case: Let b l long-run budget (e.g., expected life-time income), periods = 1; :::; t H 0 l number of goods in the static (per-period) choice problem H = th 0 l number of time-contingent goods p T = (p 11 ; :::; p 1H ; p 21 ; :::; p 2H ; :::; p t1 ; :::p th ) l dynamic price vector Given this enormous dimension of the goods space, applied demand analysis uses the two important concepts of separation and composition. De nition: Let G = fh 1 ; :::; h HG g be a group of commodities. The demand function ' is separable with respect to G if there exists a demand function ' G : R H G+1 +! R H G and an expenditure function b G = b G (p; b) such that ' hi (p; b) = (' G ) i (p h1 ; :::; p hg ; b G (p; b)) for h i 2 G:

17 1.5. COMPOSITION AND SEPARATION 17 In other words: demand for goods in group G depends solely on the prices of the goods in group G and on total expenditure on goods in G. Example 1: Two-stage budgeting: Suppose that x T = x 1 ; :::; x {z H1 } ; x H 1 +1; :::; x H1 +H {z } ; :::; :::x 2 H {z} group G 1 group G 2 group G L If demand is separable with respect to G 1 ; :::; G L, then the household s choice problem can be analyzed in two steps: 1. Choose budgets b 1 ; :::b L for each commodity group 2. Choose quantities x h1 ; :::; x hgl of all goods in each group G l, for given b l. Example 2: Time-separability: There exist t functions ' and b, = 1; :::; t such that ' h (p 1 ; :::; p t ; b) = ' h(p ; b ); (1.5) for all and h, where p is the vector of period- prices and b = b (p; b). Note that in the case H = 1 (a convenient assumption in macroeconomics) assumption (1.5) has no bite. Together with the notion of separability, applied work must typically consider demand for groups of goods instead of that for goods, and studies the budgets or budget shares, b G, respectively bg b, of these groups (bg is a function of (p; b)!). Normally, one goes one step further and assumes that the goods in one group can be considered as one single good, a composite commodity. For example, canned tomatoes, fresh tomatoes, salt, etc. are grouped into one good food. Question: Can one de ne quantity and price of a composite commodity? The answer is yes" if one imposes one of two di erent types of restrictions: Restrictions on relative price changes! Hicks-Leontief Composite Commodity Theorem (see exercises) Restrictions on preferences: A su cient condition for the existence of composite prices, in addition to separability, is the homotheticity of preferences (see Chapter 2).

18 18 CHAPTER 1. CONSUMER CHOICE UNDER CERTAINTY An important problem in applying the theory of this chapter is posed by the question: what are price and budget changes? Strictly speaking such experiments are not existent in the data. The closest substitutes in applied work are: cross-sectional data! estimation of Engel curves time-series data (panel or aggregate)! price variations The problem with the former is that unobservable individual characteristics may play an important role (variations of demand may not be due to variations in income even if observable variables, such as numbers of children, etc., are instrumented for). The problem with the latter are possible changing tastes (e.g., the inconsistencies found by Mossin (Econometrica 1972) may result from those). 1.6 Uncertainty and Labor When trying to t the theoretical model of this chapter to the data, it is important to draw attention to two shortcomings of the theory: the omission of labor and of uncertainty. The rst omission concerns the fact that the household s income y t is typically, at least partially, endogenous, the second concerns the problem that choices of future consumption bundles in the dynamic version of the model must be made without full knowledge of the relevant prices and budgets. Concerning labor: Households have some choice over how much to work (for a married couple the labor force participation can be between 0 and 200 per cent) and how much to earn (lower paid, more pleasant jobs versus higher paid, less pleasant jobs). This can be accomodated by introducing a further good, h = 0, non-work satisfaction (or more vaguely leisure ), having a price p 0. Typically, p 0 is approximated empirically by some type of market wage. Then y = p 0 (L x 0 ), where x 0 2 [0; B], B < L. Among the interesting questions with this approach are whether leisure and the other goods are complements or substitutes (see, e.g., Baxter and Jermann (AER 1999), what impact social security has on the choice of leisure and to what extent the choice of leisure is endogenous or exogenous. An important empirical yardstick for discussing these questions is the welldocumented fact that consumption co-varies with household income, or, at the macro level, with the business cycle.

19 1.6. UNCERTAINTY AND LABOR 19 Concerning uncertainty: A precise treatment comes in Chapter 3. Suppose here, di erent from the framework laid out in Section 1.1, that the prices (p 0 ; :::; p H ); > 1, at dates in the future are unknown at date 1. Then the household can only formulate short-run demand ' 1 (p 1 ; y 1 j! 1 ) together with a savings decision, s 1 (p 1 ; y 1 j! 1 ), where! 1 describes the information the household has about future prices (the savings decision can be a simple number or a more complex decision about a portfolio choice). The key question that must to be addressed then is how the household s expectation about the future,! 1, changes if prices p 1 change. Under strong separability assumptions the analysis of consumer behavior is usually not di cult, but often these assumptions are questionable.