Answer to Question #294270 in Macroeconomics for Jabesa

Question

If a consumer is consuming two commodity X and Y and his utility function U(X,Y)=2xy+6.if the price of the two commodity are 2 and 6 respectively, and consumer has a total income of 80 birr to be spent on the two goods,a)find the utility maximizing Quantity of good X and Y (b) find the MRSxy at equilibrium (c) find the MRSyx at equilibrium

Answer 1

Solution:

a.). Utility maximizing quantity = MRS_XY = $\frac{MUx}{MUy} = \frac{Px}{Py}$

MUx = $\frac{\partial U} {\partial X}$ = 2y

MUy = $\frac{\partial U} {\partial Y}$ = 2x

Px = 2

Py = 6

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

$\frac{2y}{2x} = \frac{2}{6}$

Y = $\frac{x}{3}$

Substitute the figure of X in the budget constraint:

Budget constraint: M = PxX + PyY

80 = 2x + 6y

80 = 2x + 6($\frac{x}{3}$) = 2x + 2x = 4x

80 = 4x

x = 20

y = $\frac{x}{3}$ = $\frac{20}{3}$ = 6.67

The utility maximizing quantity of X and Y = (20, 6.67)

 

b.). MRSxy at equilibrium:

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

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

X = 3y

 

c.). MRSyx at equilibrium:

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

= $\frac{2x}{6} = \frac{2y}{2}$

Y = $\frac{x}{3}$