1.3 The Indirect Utility Function

Transcription

1 1.2 Utility Maximization Problem (UMP) (MWG 2.D, 2.E; Kreps 2.2) max u (x) s.t. p.x w and x 0 hx Xi For a cts preference relation represented by a cts utility fn, u ( ): 1. The UMP has at least one solution for all strictly positive prices and non-negative levels of income. 2. If x is a solution of the UMP for given p and w, thenx is also a solution for (ap, aw) for any positive scalar a. i.e. x (p, w) x (ap, aw) [Homogeneity of degree 0 of demand.] 3. If in addition we assume preferences are locally non-satiated then x being a solution of the UMP implies P p x = w. 4. If in addition we assume preferences are convex (i.e. u is quasi-concave) then the set of solutions x (p, w) to the UMP is a convex set. 5. If preferences are strictly convex then the solution to the UMP is unique and x (p, w) is a continuous function of p and w The Indirect Utility Function Assume that u ( ) is a cts fn that represents locally non-satiated preferences. The indirect utility fn v (p, w) is:- 1. homogeneous of degree zero in p and w i.e. v (p, w) v (ap, aw) for all a>0 2. strictly increasing in w 3. non-increasing in p 4. quasi-convex in p and w, thatis v (αp +[1 α] p 0,αw+[1 α] w 0 ) max [v (p, w),v(p 0,w 0 )] 2

2 1.4 The Expenditure Minimization Problem (EMP) For a cts preference relation represented by a cts utility function, u ( ): 1. The EMP has at least one solution for all strictly positive prices & u u (0). 2. If x is a solution of the EMP for given p and u, then x is also a solution for (ap, u) for any positive scalar a. i.e. h (p, u) h (ap, u) [Homogeneity of degree 0 in prices.] 3. If in addition we assume preferences are convex (i.e. u is quasi-concave) then the set of solutions h (p, u) to the EMP is a convex set. 4. If preferences are strictly convex then the solution to the EMP is unique and h (p, u) is a continuous function of p and u. 3 Properties of the Expenditure Function Assume that u ( ) is a cts fn that represents locally non-satiated preferences. The expenditure function e (p, u) is:- 1. homogeneous of degree one in p i.e. e (p, u) ae (p, u) for all a>0 2. strictly increasing in u 3. non-decreasing in p 4. concave in p, thatis e (αp +[1 α] p 0,u) αe (p, u)+(1 α) e (p 0,u) 4

3 1.5 UMP & EMP with Derivatives Constrained Optimatization and the Kuhn-Tucker Conditions (reference: MWG appendix M.K; Kreps Appendix A) Problem max f (x) x R N s.t. g m (x) = 0, m =1,...,M h k (x) 0, k =1,...,K Form the Lagrangian function:- L (x, μ, λ) =f (x) μ m g m (x) λ k h k (x) 5 THEOREM (Assuming the constraint qualification is satisfied) For (x,μ,λ ) to be a solution to the above constrained optimization problem, (x,μ,λ ) must satisfy (i) x n L (x,μ,λ )=0for all n =1,...,N (ii) L (x,μ,λ )=f (x ) and λ k 0, forallk =1,...,K 6

4 (i) and (ii) can be re-expressed as the Kuhn-Tucker FONCs for a maximum: (A) x n f (x )= μ m g m (x )+ x n λ k h k (x ), n =1,...,N x n (B) λ kh k (x ) = 0 k =1,...,K & g m (x )=0, m =1,...,M. which implies complementary slackness, i.e., λ k > 0 h k (x )=0and h k (x ) < 0 λ k =0 7 max u (x) x R L For UMP s.t. x 0, =1,...,L ; μ p x w 0, k =1,...,K ; λ Form the Lagrangian fn: Ã L! X L (x, μ, λ) =u (x) μ ( x ) λ p x w 8

5 (A) K-T FONC x u (x )= μ + λ p, (B) μ > 0 x =0& x > 0 μ =0 λ > 0 p x = w & p x <w λ =0 v (p, w) = L (x,μ,λ ) Ã L! X = u (x )+ μ x λ p x w = u (x ) 9 For EMP max X L p x x R L s.t. x 0, =1,...,L ; μ u u (x) 0, k =1,...,K ; γ Form the Lagrangian fn: Z (x, μ, γ) = p x μ ( x ) γ (u u (x)) 10

6 K-T FONC (A) p = γ x u (x )+μ, (B) μ > 0 x =0& x > 0 μ =0 γ > 0 u (x )=u & u (x ) >u γ =0 e (p, u) = Z (x,μ,γ ) = p x μ x + γ (u u (x )) = p x 11 The Envelope Theorem THEOREM For the problem max hxi f (x; q) s.t. g m (x; q) =0,form =1,...,M h k (x; q) 0, fork =1,...,K, Let x n 0, forn =1,...,N L (x, μ, λ; q) =f (x; q) μ m g m (x; q) λ k h k (x; q). And let (x,μ,λ ) be a solution to the K-T FONCs, so that v (q) =f (x,q). 12

7 The Envelope Theorem Then dv (q) dq = L (x,μ,λ ; q) = f (x ; q) m λ k h k (x ; q). 13 Proof of Envelope Theorem By direct differentiation:- dv (q) NX f (x ; q) dx n = dq x n=1 n dq + f (x ; q) But from K-T FONCs (A) f (x ; q) = m + λ h k (x ; q) k (1) x n x n x n unless x n 0 constraint binds in which case x n =0& dx n/dq =0. So multiplying (1) by dx n/dq and summing over n leads to: NX f (x ; q) dx N " n x n=1 n dq = X M # X m + λ h k (x ; q) k x n=1 n x n dx n dq (2) 14

8 Now from K-T FONCs (B) we have: μ mg m (x ; q)+ λ kh k (x ; q) 0 (3) Differentiating (3) wrt q: + μ m g m (x ; q)+ λ k h k (x ; q)+ m + λ h k (x ; q) k + NX n=1 NX n=1 λ k μ m g m (x ; q) dx n x n dq h k (x ; q) dx n x n dq = 0 (4) 15 The first term of the LHS is zero as g m (x ; q) =0,andthefourthterm is also zero as recall by the complementary slackness conditions either h k binds in which case h k (x ; q) =0, or it is slack in which case λ k =0and λ k/ =0. Hence from (4) m + λ h k (x ; q) k = NX n=1 μ m g m (x ; q) dx N n x n dq X n=1 λ k h k (x ; q) dx n x n dq (5) 16

9 So combining (2) and (5) we obtain: NX n=1 f (x ; q) dx M n x n dq = X m and hence the desired result: λ h k (x ; q) k dv (q) dq = f (x ; q) m λ k h k (x ; q). 17 Applications of the Envelope Theorem. (a) Roy s Identity: x (p, w) = v (p, w) / p v (p, w) / w Proof:Bytheenvelopetheorem & v (p, w) = L (x,μ,λ ; p, w) p p v (p, w) w = p " u (x )+ Ã L!# X μ x λ p x w = λ x = w L (x,μ,λ ; p, w) " Ã = L!# X u (x )+ μ w x λ p x w = λ 18

10 (b) Shephard s Lemma h (p, u) = e(p, u) p Proof:Bytheenvelopetheorem e(p, u) = Z (x,μ,γ ; p, u) p p = p = x " X L p x # μ x + γ (u u (x )) 19 (c) Slutsky Equation: Obtained by differentiating w.r.t p k the identity h (p, u) x (p, e (p, u)) h (p, u) = x (p, w) + x (p, w) e(p, u) p k p k w p k = x (p, w) + x (p, w) x k (p, w),wherew = e (p, u). p k w Or in Matrix notation D p h (p, u) {z } L L = D p x (p, w) {z } L L +[D w x (p, w)] T x (p, w) T {z } {z } L 1 1 L 20