Answer to Question #285578 in Macroeconomics for minnie

Question

Consider a consumer who consumes only two goods, x and y. His utility over these two goods is given by U(x,y) = xy. The budget constraint of the consumer is given by 3x + 9y = 216, where 3 is the price of good x, 9 is the price of good y and 216 is the total income of the consumer.

(a) Find the optimal quantities of good x and y that the consumer is going to consume. Show the solution in a graph. What level of utility is the consumer going to achieve with this bundle?

(b) Now assume that the price of good x increases to 6. Find the new optimal consumption bundle and show it in a graph

Answer 1

Solution:

a.). Optimal quantities of good x and y:

$\frac{MUx}{MUy} = \frac{Px}{Py}$

MUx = $\frac{\partial U} {\partial x} = y$

MUy = $\frac{\partial U} {\partial y} = x$

 

$\frac{y}{x} = \frac{3}{9}$

x = 3y

Substitute in the budget constraint:

216 = 3x + 9y

216 = 3(3y) + 9y = 9y + 9y = 18y

216 = 18y

Y = 12

X = 3y = 3 $\times$ 12 = 36

Optimal quantities of x and y = 36, 12

Level of utility = xy = 36 x 12 = 432

The graph is as below:

 

b.). $\frac{y}{x} = \frac{6}{9}$

x = 1.5y

Substitute in the budget constraint:

216 = 6x + 9y

216 = 6(1.5y) + 9y = 9y + 9y = 18y

216 = 18y

Y = 12

X = 1.5y = 1.5 $\times$ 12 = 18

Optimal quantities of x and y = 18, 12

Level of utility = xy = 18 $\times$ 12 = 216

The graph is as below: