## The Utility Surface (and corresponding Indifference Curves)

© 1999-2020, Douglas A.Ruby (06-04-2020)

Given a utility function: U = f(x,y), we can diagram this as a utility surface with the two goods
measured along the base and utility measured as the height as shown in figure 1:

**figure 1 -- the Utility Surface**

This utility surface is a concave function based on our assumptions of diminishing marginal utility.
Changes in elevation represent increasing levels of utility. Traversing the surface such that we hold elevation
constant help to identify one particular indifference curve as a convex set in the x-y plane.

Using this utility function, we calculate separate expressions for the
marginal utility of good-x and good-y:

dU/dx = MU_{x} -- *holding 'y' constant*

or dU_{x} = MU_{x}dx -- the equivalent of climbing the hill by traveling due east
(along the 'x' axis).

dU/dy = MU_{y} -- *holding 'x' constant*

or dU_{y} = MU_{y}dy -- the equivalent of climbing the hill by traveling due
north (along the 'y' axis).

These marginal utilities represent a measure of the slope or steepness of the hill as we
traverse in the due-east or due-north directions. With respect to the notion of utility, we can think of these
marginal utilities as the rate by which consumption of either good is transmitted (or converted) into additional
into utility or satisfaction.

Combining these separate transmission mechanisms, we can identify how utility changes with
consumption of one or both goods:

dU_{total} = dU_{x} + dU_{y}
= MU_{x}dx + MU_{y}dy -- climbing the hill by moving
in the north-east direction.

Holding utility constant (dU_{total} = 0) or traversing the surface allows us to write:

0 = MU_{x}dx + MU_{y}dy

or

dy/dx = -MU_{x}/MU_{y}

Thus, the slope of an indifference curve at a given point in the x-y plane represents a measure of
the ratio of marginal utilities which is also known as the marginal rate of
substitution (MRS). This represents the rate at which one good must be substituted for another in
order to keep the level of utility constant.

Concepts for Review:
- Indifference Curve
- Marginal Rate of Substitution
- Marginal Utility
- Utility

© 1999-2020, Douglas A.Ruby (05-20-2020)