Suppose I have a utility function of Cobb-Douglas form
$$U(x, y) =x^{0.2}*y^{0.8}$$
I want to describe it in words. I would say like:
The utility of consumer is captured by number of good x and number of good y. The indifference curve is downward sloping and convex, which means the marginal rate of substitution is diminishing as x increases. The consumer prefer y to x since y gives a higher utility level.
Is this correct and conprehensive? Any help will be appreciated.
The best way to go about it is to examine the marginal rate of substitution.
Take $$ x^\alpha y^\beta = C $$ and use the implicit function theorem to derive the slope of the indifference curve: $$ x^\alpha y(x)^\beta = C $$ $$ \alpha x^{\alpha-1} y(x)^{\beta} + x^\alpha \beta y(x)^{\beta-1} y'(x) = 0 $$ $$ MRS(x) = y'(x) = -\dfrac{\alpha y(x)}{\beta x} = -\dfrac{y}{4x}. $$ This gives the rate at which the consumer is willing to give up $x$ for $y$ and stay on the same indifference curve. In words, "If the consumer currently has the bundle $(3,4)$, he is willing to give up $4/12$ of a unit of the $x$ good for a bit of the $y$ good."
You might worry that the MRS does not full capture the Cobb-Douglas utility function, but you can recover Cobb-Douglas preferences by "solving" the differential equation $y'(x) = -\alpha y(x)/\beta x$, since it can be rewritten $$ \dfrac{y'(x)}{y(x)} = - \dfrac{\alpha}{\beta x} $$ and the left-hand side is the derivative of $\log(y(x))$ and the right-hand side is the derivative of $\log(x)$, implying that $$ \log(y(x)) = - \dfrac{\alpha}{\beta}\log(x) + v $$ so that $$ \log(y(x)^\beta x^\alpha) = v $$ and $$ y^\beta x^\alpha = e^v = u(x,y) $$ so that the MRS fully characterizes the Cobb-Douglas utility function.
