Answer to Question #254321 in Microeconomics for Chilu

Question

john has an income of k1000 a month which he spends on two goods only, accommodation and food. his utility function is U(A, F)=10A^.6F^.4 where A is accomodation and F is food. the prices of accommodation and food are k15 and k8 respectively. a)how much of F and A does John buy to maximize utility. b.) suppose now that due to minimum housing standards, John is required to buy not less than 50 units of accommodation, how will his income be spent now? c) what level of utility does John get in b.) above

Answer 1

Solution:

a.). U(_A,F) = 10A^0.6F^0.4

Budget constraint: I = P_AA + P_FF

1000 = 15A + 8F

MU_A = $\frac{\partial U} {\partial A}$ = 6A^-0.4F^0.4

MU_F = $\frac{\partial U} {\partial F}$ = 4A^0.6F^-0.6

 

$\frac{MU_{A} } {P_{A}} = \frac{MU_{F} } {P_{F }}$

$\frac{6A^{-0.4}F^{0.4} }{15} =\frac{4A^{0.6}F^{-0.6} }{8}$

Simplify:

F = $\frac{5A}{4}$

Substitute in the budget constraint to derive A:

1000 = 15A + 8F

1000 = 15A + 8(5A/4)

1000 = 25A

A = 40

 

F = 5A/4 = 5(40)/4 = 50

F = 50

 

b.). New Accommodation = 50 x 15 = 750

1000 – 750 = 250

250/8 = 31.25

He will now spend 750 on accommodation and 250 on food using his available income. That is 50 units of accommodation and 31.25 units of food. Therefore, he will spend more on accommodation than food due to the restriction put on accommodation.

 

c.). U(_A,F) = 10A^0.6F^0.4

U(_A,F) = 10(50)^0.631.25^0.4

U(_A,F) = 414.31