Question

1.     Given utility function U(x,y) = X^1/4Y^3/4, Where price of X=8Br and Price of Y = 2 Br and income of the consumer is 480 Br;

A.    Compute the utility maximizing level of X and Y

B.    Calculate the marginal rate of substitution of X to Y at equilibrium and interpret the result

C.    Compute the total utility at equilibrium

Answer 1

A. The utility maximizing level of X and Y;

$$MU_x=0.25x^{-0.75}y^{0.75}$$

$$MU_y=0.75x^{0.25}y^{-0.25}$$

$${\frac{MU_x}{p_x}}=\frac {MU_y}{p_y}$$ $$x \times p_x+ y \times p_y=M$$

$$y=4x$$

$$8 \times x+2 \times y=480$$

$$x=30, y=120.$$

B.  The marginal rate of substitution of X to Y at equilibrium;

$$\frac {\partial U}{\partial x \partial y} =\frac {0.1875\times y^{0.5}} {(x)^{0.5}}$$

$$MRS x.y = \frac {\partial U} {\partial x \partial y} = \frac {0.1875\times 120^{0.5}}{( 30)^{0.5}}=0.375$$

C.    The total utility at equilibrium;

=$30^{0.25}\times120^{0.75}$

=84.85