Microeconomics II Lecture 4. Marshallian and Hicksian demands for goods with an endowment (Labour supply)

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1 Leonardo Felli 30 October, 2002 Microeconomics II Lecture 4 Marshallian and Hicksian demands for goods with an endowment (Labour supply) Define M = m + p ω to be the endowment of the consumer. The Marshallian demand will be: x = x(p, m + p ω) Differentiation gives: [ ] xl = p l m [ ] xl p l M + x l M ω l Standard Slutsky decomposition gives: [ ] xl = h l x l p l p l M x l. M 1

2 Microeconomics II 2 Substituting one in the other we get: [ ] xl = h l + x l p l p l m [ω l x l ]. m If as in the labour supply case [ω l x l ] 0 then we can get backward bending labour supply (backward bending leisure demand) with leisure a normal good. Cross-price effects: Commodity i and j are net substitutes iff h j p i = h i p j > 0 Commodity i and j are net complements iff h j p i = h i p j < 0 (intuition in terms of own price effect.)

3 Microeconomics II 3 Commodity i is a gross substitute of commodity j iff (notice the wording) x i p j > 0 Commodity i is a gross complement of commodity j iff x i p j < 0 The Slutsky decomposition can be used to state a new property of the Marshallian demand.

4 Microeconomics II 4 Indeed, the substitution matrix can be written in terms of Marshallian demand: S = x 1 x p 1 + x 1 1 m x 1 x p + x 1 L L m..... x L x p 1 + x L 1 m x L x p + x L L L m such a matrix is symmetric and negative semi-definite. We now provide the answer to a question known as the integrability problem. Question: Given a set of observed (Marshallian of course) demands x(p, m) under which conditions we are sure that there exists a utility function of the consumer from which these demands are derived?

5 Microeconomics II 5 Answer: The answer is positive provided that x(p, m) satisfy: 1. adding up; 2. homogeneity of degree 0 in (p, m); 3. the Slutsky (substitution) matrix is symmetric and negative semi-definite.

6 Revealed Preferences Microeconomics II 6 The integrability conditions are stated in terms of observed demand functions, however what we actually observe is rather than the entire demand of a consumer the choices of the consumer for few values of the parameters. Question: Given a finite set of demand data: (p 1, m 1 ),..., (p n, m n ) are the consumer choices we observe x 1,..., x n consistent with the standard model of max of utility subject to a budget constraint?

7 Microeconomics II 7 To be able to build the answer we need to define the revealed preference binary relationship: 1. If the bundle x is chosen and px m then x revealed preferred to x In the case in which the preferences we are trying to recover satisfy a local non-satiation assumption then 1. If the bundle x is chosen and px < m then x strictly revealed preferred to x Therefore if for a two points observation of the type: for (p, m), x chosen and px < m or x strictly revealed preferred to x ; while for (p, m ), x chosen and p x < m or x strictly revealed preferred to x

8 Microeconomics II 8 we have to conclude that the data observed are not consistent with the consumer maximizing his preferences (satisf. local non-satiation) subject to budget constraint. x 1 (p, m) x x (p, m ) x 2

9 Microeconomics II 9 In the event, however, that the following relationship holds: px > m and p x > m then the information available will not allow us to test the utility maximization theory. We now have the elements to introduce the Weak Axiom of Revealed Preferences: A demand function x(p, m) satisfies the weak axiom of revealed preferences if the following property holds for any two price-income situations (p, m) and (p, m ): if px(p, m ) m and x(p, m ) x(p, m) then p x(p, m) > m in other words if x(p, m) is weakly revealed preferred to x(p, m ) and they are different consumption bundles then x(p, m ) cannot be weakly revealed preferred to x(p, m).

10 Microeconomics II 10 The weak axiom has significant implications on the effect of price changes on demand. We need to look at a special kind of price changes. Imagine a situation in which each change in price from p to p is accompanied by an associated change in income that makes the consumer s initial consumption bundle just affordable at the new prices: consumer s income m is then such that m = p x(p, m) alternatively income is changed so that: m = px(p, m) where p = (p p). This is known as Slutsky income compensation and Slutsky income compensated price changes.

11 Microeconomics II 11 Result. Suppose that the demand function x(p, m) satisfies: homogeneity of degree zero underlying preferences are monotonic (locally non-satiated) Then x(p, m) satisfies the weak axiom of revealed preferences if and only if for any compensated price change from (p, m) to (p, p x(p, m)) we have: (p p)[x(p m ) x(p, m)] 0 with strict inequality whenever x(p, m) x(p, m ).

12 Microeconomics II 12 Proof: Assume WA holds. Consider the strict inequality result. We can rewrite the condition as: (p p)[x(p m ) x(p, m)] = p [x(p m ) x(p, m)] p[x(p m ) x(p, m)] Consider the first term, we know p x(p, m) = m and by monotonicity we get p x(p, m ) = m therefore p [x(p m ) x(p, m)] = 0. Consider the second term. By construction x(p, m) is affordable under p, the WA therefore implies: px(p, m ) > m since px(p, m) = m we conclude: p[x(p m ) x(p, m)] > 0

13 Microeconomics II 13 The opposite implication follows from the observation that the WA holds if it holds for every compensated price change. Therefore if the weak axiom does not hold it means that there exists a compensated price change from (p, m ) to (p, m), where px(p, m ) = m, such that WA does not hold: x(p, m) x(p, m ) and p x(p, m) m. However, by monotonicity: p[x(p m ) x(p, m)] = 0 and p [x(p m ) x(p, m)] 0 Hence (p p)[x(p m ) x(p, m)] 0 which is a contradiction.

14 Microeconomics II 14 The inequality we have obtained can be written as: p x 0 This is known as the compensated law of demand. When x(p, m) is differentiable the compensated law of demand becomes: dp dx 0 where dm = dpx(p, m). We then obtain: or dx = D p x(p, m)dp T + D m x(p, m)dm dx = D p x(p, m)dp T + D m x(p, m)dpx(p, m)

15 Microeconomics II 15 or dx = [ D p x(p, m) + D m x(p, m)x(p, m) T] dp T which gives us: dp [ D p x(p, m) + D m x(p, m)x(p, m) T] dp T 0 or dps(p, m)dp T 0 which yields that WA holds iff the substitution matrix is negative semi-definite. Problem is symmetry.

16 Microeconomics II 16 Question: what is a set of necessary and sufficient conditions that rationalize demand behaviour as derived from a consumer max utility subject to budget constraint. Answer: Strong Axiom of revealed preferences The demand x(p, m) satisfies the SA iff for any list (p 1, m 1 ),..., (p N, m N ) with x(p n+1, m n+1 ) x(p n, m n ) for all n N 1 we have: p N x(p 1, m 1 ) > m N whenever p n x(p n+1, m n+1 ) m n for any n N 1.

17 Microeconomics II 17 In words, if x(p 1, m 1 ) is directly or indirectly revealed preferred to x(p N, m N ) then x(p N, m N ) cannot be directly or indirectly revealed preferred to x(p 1, m 1 ). Essentially SA implies that given any finite set of demand data, it is not possible to construct a cycle of the type: x n 1 r.p. x n 2 r.p.... r.p. x n 1 where r.p. is strict in at least one case. Theorem: SA is both a necessary and sufficient condition for the existence of a utility function that rationalize the observed behaviour as a utility maximization one.

18 Microeconomics II 18 The hard part of the proof is sufficiency. The SA is therefore equivalent for an homogeneous of degree zero Marshallian demand that satisfies adding up conditions to the symmetry and negative semidefiniteness of the substitution matrix. Example of how to use GARP. Consider the following data: x 1 = x2 = x3 = p 1 = (10, 10, 10) p 2 = (10, 1, 2) p 3 = (1, 1, 10) where m 1 = 300, m 2 = 130 and m 3 =

19 Microeconomics II 19 Notice that: In period t = 1 the price was p 1, x 1 was chosen but x 3 was affordable: x 1 s.r.p. x 3. In period t = 2 the price was p 2, x 2 was chosen but x 1 was affordable: x 2 w.r.p. x 1. In period t = 3 the price was p 3, x 3 was chosen but x 2 was affordable: x 3 s.r.p. x 2. Hence x 2 w.r.p. x 1 s.r.p. x 3 s.r.p. x 2 which violates SA.

20 Microeconomics II 20 Consumer Surplus Assume that p changes from p 0 to p 1 (price decrease p 1 p 0 or p 1 l p 0 l l). We cannot measure the consumer s gain in terms of utility (utility is not cardinal) however we can ask either of these alternative questions: 1. At the new price level p 1 what change in income would restore the original level of utility for the consumer? This change in income is known as compensating variation CV such that: V (p 0, m) = V (p 1, m CV ) Notice that is p 0 p 1 then CV > 0.

21 Microeconomics II At the old price level p 0 what change in income would induce the new level of utility for the consumer? This change is known as equivalent variation EV such that: V (p 1, m) = V (p 0, m + EV ) Notice that is p 0 p 1 then EV > 0. Both CV and EV can be defined by means of the expenditure function (graph): CV = e(p 0, u 0 ) e(p 1, u 0 ) = L p 0 l e(p, u 0 ) = dp l l=1 p 1 p l l L p 0 l = h l (p, u 0 )dp l l=1 p 1 l by Shephard s lemma and where u 0 is the level of utility achieved when p = p 0.

22 Microeconomics II 22 EV = e(p 0, u 1 ) e(p 1, u 1 ) = L p 0 l = h l (p, u 1 )dp l l=1 where u 1 is the level of utility when p = p 1. p 1 l p h(p, u 0 ) h(p, u 1 ) p 0... p 1... x(p, m).. q 0 q 1 q

23 Microeconomics II 23 These two measures refer to different situations: CV suitable to compensate individuals once a project has gone ahead; EV useful to compare in advance the effect of different projects. As an example assume that only the price of one commodity p l changes from p 0 l to p 1 l p 0 l. If commodity l is normal the Marshallian demand is more steep than the Hicksian demand (by Slutsky) x l p l h l p l = x l m x l < 0

24 Microeconomics II 24 Define the consumer surplus for commodity l when the price is p l to be: CS = For a normal good: pl p l x l (p, m)dp l CV < CS < EV For an inferior good: CV > CS > EV When the income effect is zero: CV = CS = EV this is the case for quasi-linear utility function u(x 1, x 2 ) = u(x 1 ) + x 2 where x 1 m = 0