Question

Assume that the prices of good X, Y and Z are as follows R5,R1 and R4 respectively, and the Judith has an income of R37 to spend. HOW much of each good will judith consume in order to maximise her utility? What will be her total utility and marginal utility of the last rand spent on each good? Show all the calculations

Answer 1

Solution:

The budget constraint function:

M = PxX + PyY + PzZ

Where: M = Income

            Px = Price of Good X

            Py = Price of Good X

            Pz = Price of Good X

37 = 5X + Y + 4Z

$\frac{MU_{x} }{P_{x} } = \frac{MU_{y} }{P_{y} } = \frac{MU_{z} }{P_{z} }$

MU_x  = $\frac{\partial U} {\partial X} = \frac{5}{x}$

MU_y  = $\frac{\partial U} {\partial Y} = \frac{1}{y}$

MU_z  = $\frac{\partial U} {\partial Z} = \frac{4}{z}$

$\frac{\frac{5}{x} }{5} = \frac{\frac{1}{y} }{1} = \frac{\frac{4}{z} }{4}$

$\frac{1}{x} = \frac{1}{y} = \frac{1}{z}$

Therefore,

X = Y = Z

Substitute in the budget constraint function:

37 = 5X + Y + 4Z

37 = 5Y + Y + 4Y

37 = 10Y

Y = 3.7

X = Y = Z = 3.7

Good X = 5X = 5 x 3.7 = 18.5

Good Y = Y = 3.7

Good Z = 4Z = 4 x 3.7 = 14.8

b.). The marginal utility of the last rand spent on each good = 3.7 per good.

 

The total utility of the last rand spent on each good = 3.7 + 3.7 + 3.7 = 11.1