Answer to Question #221188 in Microeconomics for Vickie

Question

Given the following Cobb Douglas utility function U(X1,X2)=(Xa1,x1-a 2) derive the marginal utilities for good X1 and good x to derive the marginal rate of substitution

Answer 1

Cobb Douglass Utility Function: $U=X_1^{\frac{1}{2}}X_2^{\frac{1}{2}}$

Budget Constraint: $M=P_1X_1+P_2X_2$

Marginal Utility is the utility derived by an additional consumption of a unit of a good.

 

Marginal Utility of X₁:

$MU_{X_1}=\frac{\delta U}{\delta X_1}\\ MU_{X_1}=\frac{dU}{dX_1}\\ U=X_1^{\frac{1}{2}}X_2^{\frac{1}{2}}\\ \frac{dU}{dX_1}=\frac{1}{2}X_1^{\frac{-1}{2}}X_2^{\frac{1}{2}}\\ \frac{dU}{dX_1}=\frac{1}{2}(\frac{X_2}{X_1})^{\frac{1}{2}}$

So, the Marginal Utility of X1 is $\frac{1}{2}(\frac{X_2}{X_1})^{\frac{1}{2}}$

 

Similarly, we can find the marginal utility of X₂:

$MU_{X2}=\frac{\delta U}{\delta X2}\\ MU_{X2}=\frac{dU}{dX2}\\ U=X1^{\frac{1}{2}}X2^{\frac{1}{2}}\\ \frac{dU}{dX2}=\frac{1}{2}X1^{\frac{1}{2}}X2^{\frac{-1}{2}}\\ \frac{dU}{dX2}=\frac{1}{2}(\frac{X1}{X2})^{\frac{1}{2}}$

So, Marginal Utility of X1 is $\frac{1}{2}(\frac{X1}{X2})^{\frac{1}{2}}$

For quantity demand, we need the following condition:

$\frac{MU_{X_1}}{MU_{X_2}}=\frac{P_1}{P_2}\\ \frac{\frac{1}{2}(\frac{X_2}{X_1})^{\frac{1}{2}}}{ \frac{1}{2}(\frac{X_1}{X_2})^\frac{1}{2}}= \frac{P_1}{P_2}$

Simplifying the above expression, we get

$\frac{X_2}{X_1}=\frac{P_1}{P_2}\\ P_1X_1=P_2X_2$

Putting this in the budget constraint

Budget Constraint: $M=P_1X_1+P_2X_2$

$P_1X_1+P_2X_2=M\\ Since\space P_1X_1=P_2X_2\\ P_1X_1+P_1X_1=M\\ 2P_1X_1=M\\ X_1=\frac{M}{2P_1}$

So, the demand for X₁ is$X_1=\frac{M}{2P_1}$

 

We will substitute the value of X₁ in $P_1X_1=P_2X_2$ to find out the demand for X₂

$P_1X_1=P_2X_2\\ X_2=(\frac{P_1}{P_2})\times X_1$

Putting the value of $X_1=\frac{M}{2P_1}$

$X_2=(\frac{P_1}{P_2})\times(\frac{M}{2P_1})\\ X_2=\frac{M}{2P_2}$

So, the demand for X₂ is $X_2=\frac{M}{2P_2}$

Quantity demanded of good $1+2: X_1+X_2$

$X_1+X_2=(\frac{M}{2P_1})+(\frac{M}{2P_2})$