1 13. Expenditure minimization Econ 494 Spring 2013

2 Agenda Expenditure minimization Set up problem and solve Relation to utility maximization Graphical illustration Slutsky equation

3 Expenditure minimization Dual of the utility maximization problem Mathematically identical to cost minimization Suppose we assume that consumers minimize the cost of achieving a given level of utility Hold utility constant (rather than income) This problem will give us Hicksian, or compensated, demands Useful for welfare analysis “How much money income can be taken away from an individual to make her as well off after some change (e.g., price decrease) as she was before?” – Compensating variation

4 Expenditure minimization The expenditure function is: HOD(1) in prices (p 1, p 2 ) increasing in (p 1, p 2, u) concave in (p 1, p 2 ) same as cost min

5 FONC and SOSC By IFT, solution to FONC are x i h (p 1, p 2, u) and h (p 1, p 2, u)

6 Compare E-min and U-max Both E-min and U-max require tangency of indifference curve and budget line. Difference is in the constraints From the FONC: m = 1 / h

7 Compare E-min and U-max E-MIN Solution to FONC: x i h (p 1, p 2, u) h (p 1, p 2, u) Find lowest budget line that reaches indifference curve Holds utility constant Demand unobservable Only substitution effect U-MAX Solution to FONC: x i m (p 1, p 2, M) m (p 1, p 2, M) Find highest indifference curve that reaches budget line Holds income constant Demand is observable Both substitution and income effects

8 Apply envelope theorem to expenditure function represents the marginal cost of utility the additional money required to attain a higher level of utility Same idea as Shephard’s Lemma Derivative of indirect expenditure function wrt price yields Hicksian demand function

9 Apply envelope theorem for comparative statics By concavity of E (p 1, p 2, u) in prices: we get refutable hypotheses, but they are unobservable !! All e-min comparative statics are same as cost min.

10 E-min vs. U-max Graphical illustration If initial price p 1 0, then choose A. E-min vs. U-max Graphical illustration Slope = – p 1 1 / p 2 M / p 1 1 C U1U1 U0U0 A M / p 1 0 x1x1 x2x2 M / p 2 B If price falls to p 1 1,then choose B for E-Min, and choose C for U-max. Slope = – p 1 0 / p 2

11 Hicksian demands Hicksian, or compensated demands, are the solution to the FONC of the E-min problem Properties of Hicksian demands: HOD(0) in prices Comparative statics for Hicksian demands produce refutable hypotheses, but are not observable Slutsky equation lets you convert these into something observable so you can test implications

12 Under what conditions will bundle of goods chosen under U-max and E-min be the same? Four identities 1. E (p 1, p 2, V (p 1, p 2, M))  M The minimum expenditure necessary to reach utility V (p 1, p 2, M) is M. 2. V (p 1, p 2, E (p 1, p 2, u))  u The maximum utility from income E (p 1, p 2, u) is u. 3. x i m (p 1, p 2, M)  x i h (p 1, p 2, V (p 1, p 2, M)) Marshallian demand at income M is the same as the Hicksian demand with a utility level V (p 1, p 2, M) that can be reached with income M. 4. x i h (p 1, p 2, u)  x i m (p 1, p 2, E (p 1, p 2, u)) Hicksian demand at utility u is the same as Marshallian demand with an income level E (p 1, p 2, u) that will achieve the same level of utility u.

13 Prove identity 4 Substitute solution into budget constraint: By definition of the expenditure function: Therefore:

14 Prove identity 2 By definition of indirect utility function:

15 Slutsky equation: Modern derivation Note ! Can also be done for cross-prices Unobservable, but a function of observables.

16 Slutsky equation The Slutsky equation decomposes a change in demand induced by a price change into: substitution effects (move along indiff. curve) income effects (move to new indiff. curve) You can do the same thing for cross-price effects.

17 Income and substitution effects for a decrease in p 1 A C B x1x1 x2x2 U1U1 U0U0 x10x10 x1hx1h x1mx1m Subst. effect Income effect

18 Slutsky equation in elasticity form Silb. § 10.6 has other elasticity formulae. Useful for demand estimation

19 Slutsky matrix We said that because the expenditure function is concave in prices, then:

20 Integrability How do I know that the demand functions I estimated came from rational (U-max) decisions? They do if Slutsky matrix is NSD and symmetric Integrating back from demand function to indirect utility function Should be able to go from: Indirect utility fctn.  x i m () x i m ()  Indirect utility fctn.