Question

1. Assume that Roberts’s utility from consuming good X and good Y is given by the following function:

U = X^0.3Y^0.7

Where X is the quantity of good X while Y is the quantity of good Y.

Assume the price of X (P_X) is £25, the price of Y (P_Y) is £35 and he has a budget of £1000 to spend on the two goods. 

c.     Using the demand functions, calculate the quantities of X and Y Robert should purchase to maximise his utility. Calculate the utility this optimal consumption bundle provides.

Answer 1

Objective: Maximise $X^{0.3}Y^{0.7}$

Subject to:

25X + 35Y = 1000

x≥ 0, y≥0

Px = £25

Py = £35

TC = Px *X + Py*Y

$TC = P_{x} \times X + P_{y} \times Y$

TC = 25X + 35Y = 1000

f(x,y) = $X^{0.3}Y^{0.7}$

g(x,y) = 25X + 35Y = 1000

L(λ, x, y) = X^0.3Y^0.7 − λ(25X + 35Y - 1000)

0.3X^-0.7Y^0.7 = 25λ

0.7X^0.3Y^-0.3 = 35λ

25X + 35Y = 1000

$\frac{0.3}{0.7}X^{-0.7-0.3}Y^{0.7 – - 0.3}$ = $\frac{25λ}{35λ}$

$0.428571X^{-1}Y^{1}$ = 0.714286

Y = 1.666669X

25X + 35Y = 1000

25X + 35(1.666669X) = 1000

25X + 58.33342X = 1000

83.33342X = 1000

X = $\frac{1000}{83.33342}$

X = 12

Y = 20

U = $X^{0.3}Y^{0.7}$

U = $12^{0.3}20^{0.7}$

U = $2.107436 \times 8.141811$

U = 17.15834