Question

Consider a consumer whose preferences over bundles of non-negative amounts of each of two commodities can be represented by a utility function of the form U (, x2) - 4x +2 20x1 Suppose that this consumer is a price taker who faces a finite constant per-unit price for commodity The consumer is endowed with income of y. Throughout this question you may assume one of pi 0 and a finite constant per-unit price for commodity two of p2 > 0. that the consumer has sufficient income to ensure that he or she wants to consume posi- tive amounts of both commodities. You may also assume that the preferences are non-satiated, appropriate second-order conditions for a budget-constrained utility maxi mum, appropriate second-order conditions for a utility constrained expenditure minimum are satisfied, and appropriate constraint qualifications are satisfied. locally (The utility function in this question was drawn from online notes by Prof. Nicholas J. Sanders of Cornell University. The notes were viewed at the website on 24 March 2017.) 1. Is the consumer's utility function quasi-linear? Justify your answer. (5 marks.) 2. Find the consumer's Marshallian demand functions for commodity modity two. (5 marks.) one and com- 3. Find the consumer's indirect utility function. (5 marks.) 4. Find the consumer's expenditure function. (5 marks.) 5. Find the consumer's Hicksian demand functions for commodity one and commodity two. (5 marks.) Question 4 (25 Marks) Consider once again the consumer that was described in Question 3 1. Suppose that the initial price and income vector facing this consumer is (p^, p2, y0) an exogenous shock, the final price and income vector facing this con- (pi, p2,y1. Calculate both the equivalent variation and the compensating Following sumer is variation measures of the welfare impact of this shock. (4 marks.) 2. Suppose final price and income vector is (pi,p2,y°). (Let pj > pf.) How does this simplify the general formulae that you obtained for the equivalent variation and the compen- sating variation measures of the welfare impact of this shock. Calculate the change now that the shock only affects the price of commodity one, so that the in Marshallian consumer surplus measure of the welfare impact of this shock and compare it to the equivalent variation and the compensating variation measures. What do you notice? (8 marks.) 3. Suppose final price and income vector is (pj,ph, y°). (Let pj > p2.) How does this simplify the general formulae that you obtained for the equivalent variation and the compen- sating variation measures of the welfare impact of this shock. Calculate the change in Marshallian consumer surplus measure of the welfare impact of this shock and compare it to the equivalent variation and the compensating variation measures What do you notice? (8 marks.) now that the shock only affects the price of commodity two, so that the

Answer 1

Giuen, 202-41, + X2 1 Pr 2 mcam Comsumer Jham ue nurite Budget consheunl can M Greneral form a Ruasi-linea uhiuty funchen i,Lin cau good) of two ohere strictly Coniane funchion Whiuity funuhion Our u-nr t20X, -41, O,) Lic Sohse the on)- 20 1-41, 2. heru,

Nous aue aru 9reaured to check hat concana then ucX2). Litethte uether 9t it is Stricty is Quasiliniar Ehrtly Concaue 2ox,-4n2 then, delx) 20-8n e,P Shrictly く0 ) oconcaue uhuty funuh'onis Quani Linaa ttenu Our Coniumer ill uant to manimice utility t 201,-4,2 Aubjet Lagrange t20x,- 4%tAM-P, - P Set up the First order Londitons are ancl he

20-8,-AP, -0 20-8n,-AP tke 1 - AR-0 APR Y Substitut YP get, in 26-821 1 8M,20- P 8 Pr marshallian demand Lor n Do P-P the yauu af nin budyet constrauint, Put M:P20P-P

W marshallian Demard for X2 2 8P,M20p,P+P 3 Subs tituti the value ot and n uhlity funihon to get indisect uiuty funihon, vim, p) V +20n,- 4x, 202-P + 20/20P,-P V8RM-20P,Pt 400P,-20P, 4/400 +40PP 64P2 = 64Rm-160PP t&h+ 32008- 160P,P-1600P41 64 P0 ndisact 64 Rm +4P21600P-160 P,P uhlity funchien

Aemchen is funch en o indiruct uhlty (v) and 'pmes harue 4Erpenditue Exom auubon V 64m t42+1600-160P,P 422t 1600p-160RP 6MP うm W091-d0091+,h-

4P21600-160P, P at eauilibnum, expenali turu (e) is Anol to incame Lm) eaual Hane 64PZ tsExperditure fumchan differen funihon Luith get Hucksian demanol erpendi ture Juspert fo hnu lue When ue 5: h'at Hicksian damand tor So et Hicksian demanel 0Y M 16o P2 de &P - aP 16oP-&P 20 P2-P &P

de (64)2 P13200 P-160p)-4p-1600p+ 160, P V- o-160P,-4P-160 0 +16Pe 3200 16 R2 g00 PP-400 P 400P-P - V- Hicksian damand 16 p