Question

5. (4 pts) An individual purchases quantities a, b, and, c of three different commodities whose prices are p, q, and r, respectively. The consumer spends m dollars, where m > 2p, and the utility function is given by U(a, b, c) = a + ln(bc) Show that when utility is maximized, the expenditure on each of the second and the third good is equal to p. Find a, b, and c when utility is maximized. (assume Lagrange multipliers gives the max solution)

Answer 1

Page No.- Dale Commodities & a, b, c price pri respectively ท น ๆ 8 : m seed uz at Arlbe) Inlayla - Uz ut libt line t dnxt day I Budget constraint & Capt bq + cr=m.] setting up the hagrangean 2= ut d[m-ap-broer] 2 = 4 at lnb tlne dlm-ap-bq-cr] | First order Conditions in de I-dpzo -@_. de todo - obeto adrzo - © 2h = m-ap-bq - crzo - taking ratio of 6 4 , ® 4 & - 16 = 1 cabove

) substituting value of ( b dc in apt bq tcr = m. - ap + ( 279 + (L) rem apt pt p-m. Ta = m -21 1 do, bt uhlily maximiz silicone Lo= malo, bezeri.com Expenditure on good b= bog i Expenditure an good ea cir Por = P