Question

Exercise 2: Expenditure minimization We assume an individual whose preferences can be represented by the utility functions | Ưới a) = 8 * @a An expenditure-minimizing consumer would try to minimize the amount they spend on both and rach that their utility is at least as high as some set level of utility U. Mathematically, we thus have minha + P such that Ul. 22) 20 1. Please write the Lagrangan formula corresponding to this particular optimization set up oynundo tieomtogel foto it is to balto listo od boro sabilitado 2. Please take the First Order Conditions (F.O.Cs) of the Lagranian formula you wrote in your previous answer. 3. Please derive the compensated/Hiclesian demand functions for Q1 (H(P1, P2, U)) and Q2 (H2(P1, P2,0)) by combining the F.O.Cs you derived above.

Answer 1

Exercise 2's Enbuditime minimizations We assue an individual whose Pretmences can be represented by the Mility function: - U(9,92) = 9,22 Now Price of God 1 = P1 Pice of groda = P2 and income level = I E Hace the expeditie function I = Pizit P2q2 Ľ Now the Lagrangian formula for casudique : minimisation problem can be wrillmas: L = Piai + P2 Q2 +2 [o-9122/ where X= lagrange multiplis. foes for expenditure minimisation Now 2 22 will be 22 2 2 = P1 - 12. o -2 Man -- () . and 22 = P2 - 12;=0 : - REA - -(i) Now equating both values of i from equation (1) and 24. CS Scania i wo get a CamScanner

2 = 1 .-) P2. Piq Par = ER : P2q2 = Piq, u qiga = Now ex ū - 91.92 = 0 substituting q2 = Piay we get! .: T- 21/0721 ) = 0 : - J. Pia,t = 0 as . J P2 = Plan aj = 382024 )'z Now as q2 = Piar = ( p. ) o ( 2 ) PA'*) -) Ihl)-12 2 ( P2 - 2((P,P.,0)) 32 Hill Hicksin demand for ctions for 2,- O pa) z = 21CH,CM,AT) and q= J IP2) 4 now subst fruiting valus of q, and he in the as empenditure equation we get the enforudi fare fiction Scanned with CamScanner

ruce E- Ez Pigit Po 22 - :>> E= Rolls (ht) H4 R D y length E = P, /2 pz²Jkt Pik Pah Th =) E= 2P² P2 h 0 h = E(P., PT ) This is the enfundique function. 5 Now the expenditure function ist E = 2.8, 1/2 P2 % ū % Now taking partial derivatives of the exproditure function with respect to p, and P2 we gets 2 t = (2x ²) Pi 22 p2 J% z lūk qi(HiCP, 8., )) . and 2 - 2 2 = (2x2) Pinpuh = (2+)-The an (H2CP1, P2,)) .. Hace Hicksian demand for chön of good 1 and 2 Can also be derived by taking partial derivatives of the enfort enfundi fare for ction with respect (cs don puce levus pi and P2 api 11 CamScanner