Question

Given utility function U=X0.5Y0.5 where PX = 8 Birr, Birr, PY = 2 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 (MRSX,Y) at equilibrium and interpret your result.

Answer 1

a. Finding the utility maximizing combinations of X and Y

• Utility function

$U=X^{0.5}Y^{0.5}$

• cost function

$240=8X+2Y$

• Marginal Utility of X

$MU_{x}=\frac{dU}{dX}$

$MU_{x}=0.5X^{-0.5}Y^{0.5}$

• Marginal Utility of Y

$MU_{y}=\frac{dU}{dY}$

$MU_{y}=0.5X^{0.5}Y^{-0.5}$

• At equilibrium:

$MU_{x}=MU_{y}$

$0.5X^{-0.5}Y^{0.5}=0.5X^{0.5}Y^{-0.5}$

$0.5\frac{Y^{0.5}}{X^{0.5}}=0.5\frac{X^{0.5}}{Y^{0.5}}$

• We may divide each side by 0.5 and then cross-multiply the equation

${X^{0.5}}{X^{0.5}}={Y^{0.5}}{Y^{0.5}}$

$X=Y$

• To solve the problem, let us substitute Y for X in the cost function

$240=8X+2Y$

$240=8X+2X$

$240=10X$

$X=24$

• If $X=Y$ then

$Y=24$

b. Calculating marginal rate of substitution of X for Y (MRSX,Y) at equilibrium and interpreting the result

$MRS_{xy}=\frac{MU_{x}}{MU_{y}}$

$MRS_{xy}=\frac{0.5X^{-0.5}Y^{0.5}}{0.5X^{0.5}Y^{-0.5}}$ =$\frac{X^{-0.5}Y^{0.5}}{X^{0.5}Y^{-0.5}}$ =$\frac{Y^{0.5}Y^{0.5}}{X^{0.5}X^{0.5}}$

$|MRS_{xy}|=\frac{Y}{X}$=$\frac{24}{24}$ =1

• Interpretation:

The consumer is willing to give up one (1) unit of X to get an extra unit of Y and remain with a combination of goods that is equally satisfying