I am having some difficulties understanding the calculations of marginal utility.
On this problem (,)= square root of PM
I know that MRS is equal to MUp/MUm but i do not understand how its possible that :
MUp=1 / square root of 10 and
MUm=2.5/ square root of 10
I have some learning disabilities and any clear and simple answer on how to get this values would be greatly appreciated.
I know that MUp is calculated by finding the partial derivative of U with respect to p but where do i get these values to do these calculations? Thanks!!
You can estimate these partial derivatives from the graph by reading the change in $P$ (respectively, $M$) required to reach the next highest labeled indifference curve. Then $$ \frac{\partial U}{\partial P} \approx \frac{\Delta U}{\Delta P} \text{ and } \frac{\partial U}{\partial M} \approx \frac{\Delta U}{\Delta M} $$
There isn't enough information in the diagram to show that $\frac{\partial U}{\partial P}(A) = \frac{1}{\sqrt{10}}$ and $\frac{\partial U}{\partial M}(A) = \frac{5}{2\sqrt{10}}$, though. Since $U(2,5) = \sqrt{10}$, I am guessing that $U(M,P) = \sqrt{MP}$ as a definition on another slide.
If that is the case, then you can compute the partial derivatives with the power rule. $U(M,P) = P^{1/2} M^{1/2}$, so \begin{align*} \frac{\partial U}{\partial M} &= \frac{1}{2} P^{1/2} M^{-1/2} = \frac{\sqrt{P}}{2\sqrt{M}} \\ \frac{\partial U}{\partial P} &= \frac{1}{2} P^{-1/2} M^{1/2} = \frac{\sqrt{M}}{2\sqrt{P}} \\ \end{align*} So at $A = (2,5)$, we have \begin{align*} \frac{\partial U}{\partial M}(2,5) &= \frac{\sqrt{5}}{2\sqrt{2}} = \frac{\sqrt{5}}{2\sqrt{2}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{5}{2\sqrt{10}} = \frac{2.5}{\sqrt{10}}\\ \frac{\partial U}{\partial P}(2,5) &= \frac{\sqrt{2}}{2\sqrt{5}} = \frac{\sqrt{2}}{2\sqrt{5}}\cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{1}{\sqrt{10}} \\ \end{align*}