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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.

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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.

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  • $\begingroup$ Thanks for your answer. Is it also necessary to note that the indifference curve is convex? And do I need to explain the reason of the difference between power of x and y(I. E due to consumer's own preference, he prefers y to x)? $\endgroup$ – Alex Apr 5 at 15:25

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