Question

What is his marginal rate of substitution between X =3 and Y=4? If A=1, a = 0.4 and b =
0.5.

Answer 1

Using given values, utility function is

$\text{U = AXaBy}$

$U = X0.4Y0.5$

$\text{MUx }= \dfrac{\partial U}{\partial X }=\dfrac{ (0.4 \times Y0.5)}{ (X0.6)}$

$\text{MUy} = \dfrac{\partial U}{\partial Y} = \dfrac{(0.5 \times X0.4) }{ (Y0.5) }$

$\text{Marginal rate of substitution} = - \dfrac{MUx }{ MUy}$

$= -\dfrac{\dfrac{ (0.4 x Y0.5) }{ (X0.6)}}{ \dfrac{ [(0.5 x X0.4) }{ (Y0.5)}}$

$= - \dfrac{0.4}{0.5}\times \dfrac{Y}{X}$

$= - \dfrac{4}{5} \times \dfrac{4}{3}$

Thus:

$\green{\text{Marginal rate of substitution}= - 1.0667}$