Question

Question 2 (20 points) A consumer purchases two goods x ano y. The consumer's income is I. His utility 18 * and y. Px is the price of x. Py is the price of Is 1. His utility is given by U(x,y) = xy a) Calculate consumer's optim uncompensated demand) s optimal choice of x and y under his budget. hinc b) Derive the indirect utility function. c) Are these two goods normal goods? Why! d) Derive the expenditure function. e) Calculate the compensated demand. + Pu Y = I , uttry: losy)= vey Mait : Px X

Answer 1

2 U(1,4) - 42% - Budget constraint :- - Prox + ly y = I. a betting up a legrange. 0 - 3 + (-- Q3 - 8 ) - It order conditions - 0 from 6 and in hiy x le

bubstituting it in epation line - Pax - Bul. a z 2 - Bell + bcx 22 Amcompensated demand of a Uncompersated demand of 'y' b) A Sudisect whility function (10(f.z)) for indirect utility funcion me just need to substitute the site and you in the original utility function ... ie. 0(2.2) = u(342.7), y(e, 2) So using the substitution :

As, u = say So, 0412) - que) - inte de Indirect lilility sfunction . c) not normal good is a good whose demand increases as the Income of the consumer increases . As, we can see x* = 1 and yo = 3 . - As, Both at and you are directly proportional to I (Income) - Hence, with rise in 'I', at and you rises - so both the goods are normal goods. d) Expenditure function (ecru) by suality ince know . elring - e (P, 140 ,2)) = I 1. So, using the indirect utility function and afflying duality.

So, Eubstitute and off, I) with i'. I with elp,4) po do, w403) - EN U = Felpin) = 24 JERK – Reaperehituse function e) compensated demand or Miksian demand Chmus) mots fer Shepherd's lawma. hai (ru) = selfons for its good der har lang z below) = g zu J Pe Py әа = fue detenu de

4, The One) - u.Ja - domand labely hey lov) - Because - J zu ilaly - Ju IPM qu 2 sbely hy (pin) lui Pc 1 - Compensaled demand