Question

Solve for this persons utility level if they are given $500 in cash and $500 in vouchers to purchase good y

U=x^.7 y^.3 Px=10 Py=10

what level of cash would make this person indifferent between the 500 voucher for y and the cash

Answer 1

Let, cash = x; voucher = y

Given,

U = x^0.7 y^0.3

Now, MUx = Derivative of U with respect to x

                        = {(1 × 0.7)x^(0.7 – 1)}y^0.3

                        = {0.7x^(-0.3)}y^0.3

                        = 0.7y^0.3 / x^0.3

Now, MUy = Derivative of U with respect to y

                        = {(1 × 0.3)y^(0.3 – 1)}x^0.7

                        = {0.3y^(-0.7)}x^0.7

                        = 0.3x^0.7 / y^0.7

Now, by equi-marginal utility,

(MUx / Px) = (MUy / Py)

{(0.7y^0.3 / x^0.3) / 10} = {(0.3x^0.7 / y^0.7) / 10}

(0.7y^0.3 / 10x^0.3) = (0.3x^0.7 / 10y^0.7)

Now, by cross multiplication,

(0.7y^0.3 × 10y^0.7) = (0.3x^0.7 × 10x^0.3)

{0.7 × 10 × y^(0.3 + 0.7)} = {0.3 × 10 × x^(0.7 + 0.3)}

7y^1 = 3x^1

7y = 3x

Now, given (y = 500); what will be x?

7y = 3x

7 × 500 = 3x

3x = 3,500

x = 3,500 / 3 = 1,166.67 rounded to two decimal places

Answer: such level of cash is $1,166.67.