# How to describe a utility function in words?

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.

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.