Answer to Question #121421 in Microeconomics for RMJ Halloum

Question

A student has income of 250 available to the apartment and other services during one
year. Housing services are measured in units x. Higher values ​​of x mean a larger apartment with
better standard (think 'square meters').Consumption of "other goods" is measured
in units y.

Let the rental price per unit of residential consumption be Px= 10. Let the price of one unit
other benefits are Py= 1.

We assume that the individual has preferences that can be represented by the utility function

U(x.y) =x + √y

(g) (5 points) Find the consumer's optimal consumption of housing services (x) and consumption of other goods (y).

(h) (5 points) Let's say that it is a general inflation that makes both the consumer's
income double and at the same time prices double. Will this affect the consumer's choice?
(Explain)

Hint: Show that the budget line is unchanged.

Answer 1

g) Consumer's optimal consumption 

Objective: Maximize XY^0.5

Subject to:

10X + Y = 250

x≥ 0, y≥0

f(x,y) = XY^0.5

g(x,y) = 10X + Y = 250

L(λ, x, y) = XY^0.5− λ(10X + Y - 250)

∂L/∂X = Y^0.5 – λ(10) = 0

∂L/∂Y = 0.5XY^-0.5- λ(1) = 0

∂L/ λ = -1(10X + Y – 250)

Getting the Ratios of the FOC

Y^0.5 / 0.5XY^-0.5 = 10λ/λ

2Y^{0.5- - 0.5} X^-1  = 10

YX^-1  = 10 /2

Y / X = 5

Y = 5X

Y = 5X

10X + Y = 250

10X + 5X = 250

15X= 250

X= 250/15 = 16.67

Y = 5(16.67)

Y = 83.33

h) Effect on consumers choice.

Utility function: XY^0.5

Consumer’s income double: 250*2 = 500

Prices double: 10X*2 + Y*2 = 500

Prices : 20X + 2Y = 500

L(λ, x, y) = XY^0.5− λ(20X + 2Y = 500)

∂L/∂X = Y^0.5 – λ(10) = 0

∂L/∂Y = 0.5XY^-0.5- λ(1) = 0

∂L/ λ = -1(10X + Y – 250)

Getting the Ratios of the FOC

Y^0.5 / 0.5XY^-0.5 = 10λ/λ

2Y^{0.5- - 0.5} X^-1  = 10

YX^-1  = 10 /2

Y / X = 5

Y = 5X

Y = 5X

20X + 2Y = 500

20X + 2(5X) = 500

20X + 10X = 500

30X= 500

X= 500/30 = 16.67

Y = 5(16.67)

Y = 83.33

The values of X and Y are unchanged even when the income doubles and thus the budget line is unchanged even with the income and prices doubling as shown in the diagram below: