Answer to Question #291739 in Microeconomics for jay

Question

One most commonly used utility function is the Cobb-Douglas utility function which of the form 𝑼(𝑿, 𝒀) = 𝑿 𝜶𝒀 𝜷 where α and β are positive constants. a. Show that this function exhibits diminishing marginal utility for both goods X and Y (4mks) b. Show that the indifference curves of this utility function are convex (i.e show that is there is diminishing marginal rate of substitution between X and Y) 

Answer 1

If the slope of marginal utility ( derivative of marginal utility, is negative, consuming more units of a good increases utility by smaller and smaller increment shows the case of diminishing marginal utility.

$U=(X^{\alpha}Y^\beta)$

$Mu_x= \alpha X^{\alpha-1}Y^\beta$

$\alpha-1$ Gives a negative figure since $\alpha$ is less than 1

$\delta Mu_x=\frac{\delta Mu_x}{\delta X}= \alpha(\alpha-1)X^{\alpha-2}Y^\beta$

$=$ $\frac{\alpha(\alpha-1)Y^\beta}{X^{\alpha-2}}$

Since $\alpha-1$ is negative, then $\delta Mu_x$ is negative indicating a diminishing marginal utility

$Mu_y= \beta X^\alpha Y^{\beta-1}$

$\beta-1$ Gives a negative figure since $\beta$ is less than 1

$\delta Mu_y=\frac{\delta Mu_y}{\delta y}$

$= \beta(\beta-1)X^\alpha Y^{\beta-2}$

$=$ $\frac {\beta (\beta-1) X^\alpha} {Y^{\beta-2}}$

Since $\beta-1$ is negative, then $\delta Mu_y$ is negative indicating a diminishing marginal utility