Utility

Utility is just a label for the ranking

A utility function assigns a number to every bundle so that more-preferred bundles get higher numbers. That’s all it does. The size of the number carries no meaning — only the ordering does. This is called ordinal utility.

The clean consequence: if you take any utility function and apply a positive monotonic transformation (anything that preserves order — adding a constant, multiplying by a positive number, taking logs, squaring positive numbers), you get a different utility function that represents the exact same preferences. Same indifference curves, same choices — only the labels on the curves change.

A Cobb–Douglas example: \[ U(x_1,x_2) = x_1^{a}\,x_2^{1-a}. \] Take logs and you get \(a\ln x_1 + (1-a)\ln x_2\) — a monotonic transform, so it draws the identical indifference map. The MRS is the real object, and it’s invariant: \[ \text{MRS} = -\frac{MU_1}{MU_2} = -\frac{a}{1-a}\,\frac{x_2}{x_1}, \] which comes out the same whichever representation you differentiate.

Try it. Switch the representation (U, ln U, U²) — the curves don’t move, only the numbers labelling them change. Then slide a to re-weight the two goods and watch the whole map tilt.

What to notice

  • Flip between U, ln U, and U² — the indifference curves stay put. The numbers on them change, but the ordering (which curve is higher) never does.
  • That’s why we never ask “how many utils?” — only “which bundle is preferred?” Utility is a ranking device, not a measurement.
  • The MRS is the economically meaningful slope, and it survives any monotonic relabelling — so every prediction about choice is unaffected by which representation you pick.