Question

The consumer's budget constraint is $6 = 0.25G + P, where G is packs of gum and P is bags of pretzels. The consumer's utility function is U = P^0.5G. The utility-maximizing bundle consists of _____ packs of gum and _____ bags of pretzels.

Answer 1

From the budget constraint of the consumer, we see that disposable income (M) is $6, price of one pack of Gum is $0.25 and that of one bag of pretzels is $1. Also, the utility function is given by- U= P^{​​​​​​0.25}G. This is a problem of unconstraint optimization.

To solve the problem, we introduce the Lagrange function as:

L= U(P,G)+ a(6 -0.25G -P)........{eq.i} ,where, a is a constant called Lagrange multiplier.

We differentiate eq. (i) w.r.t P, G and a to find the optimum values.

Partially Differentiating eq(i) w.r.t G, we get,

.L/G= P^{​​​​​​0.25} - 0.25a.....{eq.ii}

Partially Differentiating eq(i) w.r.t. P, we get,

L/P= 0.25P^-0.25G - a.....{eq iii}

Again partially Differentiating eq (i) w.r.t a, we get,

L/a= 6 - 0.25 G - P.....{eq iv}

Now according to the First Order Conditions of Optimization, eq(ii), eq(iii) and eq(iv) equals to 0.

So, From eq(ii), we get

P^{​​​​​​0.5} = 0.25a.....(1).......{Since, P^{​​​​​​0.5} - 0.25a=0}

From eq(iii), we get,

(0.5 G)/√P = a....(2).....{Similar to previous reason}

Now dividing (1) by (2), we get,

P = 0.25* 0.5 G = 0.125 G...(3)

Now substituting this value of P in eq(iv), we have,

6- 0.25G- 0.125G = 0

or, G = 6/0.375 = 16

So, P = 16*0.125 = 2.

Hence, the utility maximizing combination combination is 16 packs of gum and 2 bags of pretzels.