Question

Helen’s preferences over CDs (C) and sandwiches (S) are given by U(S,C) = SC + 10(S + C),with MUC = S+ 10 and MUS =C+ 10.If the price of a CD is $9 and the price of a sandwich is $3,and Helen can spend a combined total of $30 each day on these goods, find Helen’s optimal consumption basket.

Answer 1

First of all we will say

Budget$30=9C+3S or S=10-3C$

Therefore

$U = S•C+10(S+C) = S•C+10S+10C = C(10-3C)+10(10-3C)+10C$

Simplify to

$U = 10C-3C²+100-30C+10C = 100-10C-3C²$

this means

$C≥0 and from budget: C[MAX]=30/9≈3.33 ; at S=0$

$S≥0 and from budget: S[MAX]=30/3=10.00 ; at C=0$

$3.33≥C≥0$

$0≤S≤10$

Now maximize U(C)→MAX

100-10C-3C²→MAX & 10/3≥C≥0

∂U(C)/∂C= -10-3C < 0

Maximum will be a lower boundary since its downward sloping for all allowed ranges.

Therefore utility-maximizing solution backet is:C=0 & S=10 &U[MAX]=100.