Econ 5150: Applied Econometrics Empirical Analysis Sung Y. Park CUHK
Marshallian demand Under some mild regularity conditions on preferences the preference relation x ર z ( the bundle x us weakly preferred to the bundle z ) can be rationalized by a utility function u( ): u : R n R: Primal Problem: x ર z u(x) u(z). max u(x) s.t p x y (1) Solution: Marshallian demand x = g(p, y) One can obtain the indirect utility function: v(p, y) u(g(p, y))
Dual Problem: Solution: Hicksian demand min p x s.t u(x) = u 0 (2) x = h(p, u) It represents optimal consumption behavior as a function of the utility level. One can obtain the cost or expenditure function: c(p, u) p h(p, u) Cost of achieving utility u at prices p.
Frisch s: FOC of the primal problem: u(x) x i = λp i, i = 1, 2,, n, (3) where λ denotes the marginal utility of income. Interestingly, when utility is additive, u(x) = n i=1 u i(x i ), (3) becomes u i (x i ) = p i /r, where r = 1/λ can be interpreted as the marginal cost of utility at given prices. Then we have (u(x) should be monotonic [nonsatiation]) x i = f (p i /r) Thus demand depends only on own price and some (mysterious) notion of marginal value of income, i.e., λ(p, y).
Stone-Gorman Memorial Example (Linear Expenditure System): u(x) = n β i log(x i α i ) i=1 with β i = 1. [CD utility case: α i = 0] Marshallian demands: u i (x i ) = β i x i α i = λ i p i 1 = β i = λ p i (x i α i ) x i = α i + β i p i [y p i α i ]
p i x i = p i α i + β i [y p i α i ] α i : committed quantity of good i β i : marginal budget share Indirect utility function: Taking exponentials Solving for y v(p, y) = β i log( β i p i (y p i α i ) v(p, y) = exp v(p,y) = (y p i α i ) c(p, u) = u n (β i /p i ) β i. i=1 n (p i /β i ) β i + p i α i i=1
Expenditure function: Solving for y c(p, u) = u n (p i /β i ) β i + p i α i i=1 What are interesting aspects? Welfare analysis... easy to compute true cost of living indices... exact compensation variations required to change prices from p 0 to p 1. Notes: Problem: α i NLS p i x i = p i α }{{} i +β i [y p i α i ] }{{} ex var other var
Theorem If u(x) is (i) continuous; (ii) strictly quasi-concave; and (iii) non-satiated, then expenditure function c(p, u) is homogenous of degree 1 in p, concave, strictly increasing in u, and has partial derivatives which are Hicksian demand functions.
Marshallian demand Hicksian demand x = h(p, u) = h(p, v(p, y)) = g(p, y) x = g(p, y) = g(p, c(p, u)) = h(p, u) Indirect utility Marshallian demand v(c(p, u), p) = u v c + v = 0 y p i p i x i = c p i = v/ p i v/ y = g i(p, y).
Theorem Hicksian and Marshallian demands satisfy the following conditions: i Adding-up: p h(p, u) = p g(p, y) = y ii Homogeneity: h(θp, u) = h(p, u) and g(θp, θy) = g(p, y) iii Symmetry: h i (p, u)/ p j = h j (p, u)/ p i iv Negative semi definiteness: ξ i ξ j h i (p, u)/ p j 0 for any ξ R n and p i h j (p, u)/ p i = 0.
Slutsky Matrix: ( ) hi (p, u) S = = p h(p, u) = 2 p pc(p, u). j the demand response to changes in price holding utility constant. In order to estimate S having a expression in terms of Marshallian demands is useful. g i (p, c(p, u)) = h i (p, u), g i y Classical Slutsky decomposition. i = 1, 2,, n c p j + g i p j = h i p j = s ij s ij = g i y x j + g i p j
In elasticity form η h ij = g i y y p j x j x i y + g i p j p j x i = η iy θ j + η g ij where η h ij and η g ij are the Hicksian and Marshallian price elasticities, respectively, and θ j denote the expenditure share of commodity j. Notes: s ij = s ji does not imply that the corresponding elasticities are symmetric.
Consider g i g i = s ii x i p i y Here the own Slutsky effect, s ii, is necessarily negative. g i / p i can be positive if the last term is sufficiently negative Giffen effect [Koenker (1977, REStat) and Jensen and Miller (2008, AER). It only happens when x i is inferior good g i / y < 0. h i / p j > 0 x i and x j are substitutes h i / p j < 0 x i and x j are complements g i / p j > 0 x i and x j are gross substitutes g i / p j < 0 x i and x j are gross complements