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In the previous slide, we saw that the individual values goods using a measure that is distinct from market prices – the fact that two bundles cost the same does not mean they are worth the same to you. We use the concepts of utility function and indifference curves to represent the individual’s valuation system.
At the start, two properties of a simple utility function are worth noting. The first is non-satiation - more is always better. If we think of bundles that contain two goods – wraps and gasoline – more of either good, holding the other good constant, will increase the bundle’s utility.
The second property is diminishing marginal utility. In our two-good bundle, more of either good, holding the other good constant, will increase the utility of the bundle but at a decreasing rate. For example, in a bundle of six gallons of gasoline and four wraps, a fifth wrap (holding gasoline constant) will increase the bundle’s utility, a sixth wrap will further increase the bundle’s utility but not by as much as the fifth wrap did, a seventh wrap will increase the bundle’s utility but not by as much as the sixth wrap did, and so on.
Figures 1 is a three dimensional picture of gasoline, wraps and a utility function that displays these properties. In the figure, each contour line (marked by changing colors) represents the combinations of gasoline and wraps that give a particular constant level of utility.
Figure 1. Utility Surface
Figure 2 is a side view of this same utility function. Here, each of the black lines represents a particular “slice” of the utility function in which the amount of gasoline is held constant while the number of wraps increases. Each of these lines illustrates diminishing marginal utility – i.e. additional wraps increase utility but at a diminishing rate.
Indifference Curves are a way to work with our three-dimensional utility function in two dimensions. We do this by setting utility to an arbitrary level – say 2 units in the current example – and then using the utility function to map out all the combinations of wraps and gasoline that give two units of utility – one of the countours in Figure 1. We then repeat the process for a second level of utility – e.g. 4 units, and so on.
Figure 3 shows a set of indifference curves that come from this utility function for 2,4, 6,..... units of utility.
Note that the particular indifference curves in Figure 3 were chosen arbitrarily. Instead of drawing indifference curves for 2,4, 6, …. units of utility, we could have drawn indifference curves for 2.1, 3.9 and 37 units of utility. Whatever set we draw will be a partial representation of the whole utility function. In fact, every point in Figure 3 has a particular level of utility as well as one and only indifference curve running through it.
Figure 4. Utility
In Figure 4, move the red dot around to see how every point on the graph generates a particular level of utility. If you move the red dot along any of the three indifference curves, you can also see the property of diminishing marginal utility. As you move the red dot along the curve from right to left – i.e. toward bundles with more gasoline and fewer wraps – you can see that the marginal utility of gasoline – the utility of one more gallon of gasoline - declines while the marginal utility of wraps increases. Both marginal utilities reflect the fact that you already have a lot of gasoline (and so don’t value more highly) and relatively few wraps (so you value them a lot).