Question

given utility function u=x0.75 y0.25 where px = 2 birr, birr, py = 4 birr and the income of the consumer is, m= 240 birr. a. find the utility maximizing combinations of x and y. b. calculate marginal rate of substitution of x for y at equilibrium and interpret your result.

Answer 1

Solution:

Derive the budget constraint:

I = PxX + PyY

240 = 2X + 4Y

The utility maximizing rule is where $\frac{MU_{x} }{MU_{y}} = \frac{P_{x} }{P_{y}}$:

U(x,y) = x^0.75y^0.25

MUx = $\frac{\partial U} {\partial x} = 0.75x^{0.75-1} y^{0.25} = 0.75x^{-0.25} y^{0.25}$

 

MUy = $\frac{\partial U} {\partial y} = 0.25x^{0.75} y^{0.25-1} = 0.25x^{0.75} y^{-0.75}$

$\frac{P_{x} }{P_{y}} = \frac{2 }{4} = \frac{1 }{2}$

$\frac{0.75x^{-0.25} y^{0.25}} {0.25x^{0.75} y^{-0.75}} =$ $\frac{1 }{2}$

$\frac{3y }{x} = \frac{1 }{2}$

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

Substitute in the budget constraint:

240 = 2X + 4Y

240 = 2X + 4($\frac{x}{6}$)

Multiply both sides by 6:

1440 = 12X + 4X

1440 = 16X

X = 90

Y = $\frac{x}{6}$ = $\frac{90}{6}$ = 15

U(x,y) = (90,15)

The utility-maximizing combinations of X and Y that the consumer will consume = 90 and 15

MRTSxy = $\frac{MU_{x} }{MU_{y}}$

MUx = 0.75x^-0.25y^0.25

MUy = 0.25x^0.75y^-0.75

MRTSxy = $\frac{0.75x^{-0.25} y^{0.25}} {0.25x^{0.75} y^{-0.75}} = \frac{3y}{x} = \frac{3(15)}{6} = \frac{45}{90} = \frac{1}{2}$

MRTSxy = $\frac{1}{2}$