# performance as utility function / price?

I am trying to define a 'model' for the perceived performance of a product based on this definition:

Presumably, customers take both price and quality into account as they form an overall evaluation about a product's performance. To avoid a confounding of the two, each was measured in light of the other— perceived performance is thus measured by price (given quality) and quality (given price). (Fornell, 1992)

I thought that a utility function might be a good way to describe the quality of a product, so ended up with:

$$P(x) = \frac{u(x)}{c}$$

Where the perceived performance P of a product x is equal to its utility function u(x) divided by its total price c. Is this more or less ok? Or should I include the price variable in a different way, such as e.g.:

$$P(x) = u(x,c)$$

...or even simpler

$$P(x) = u(x)$$

As cost might be assumed to be part of the trade-offs of the function? Any ideas or help with the notation or function would be really appreciated.

• Is $x$ a number of the identifier of the product? If you want to model product quality $x$ could measure that but in your text you call it product $x$ which implies the latter. Both $\frac{u(x)}{c}$ and $u(x,c)$ are fine, the second is more general. Commented Dec 22, 2015 at 14:27
• Thanks, that really helpful. My idea was to use x as identifier for the product and then u(x) could be interpreted as the quality of x. Is it ok to use P<sub>i(x)</sub>=u(q<sub>x</sub>)/p<sub>x</sub> to say that the perceived performance of a user i for a product x is equal to its quality divided by price?
– vabm
Commented Dec 23, 2015 at 0:55

There is a long tradition in economics (some 50 years worth of research) called "Hedonic Price Analysis".

It consists of "allocating" (through regression estimation) the price of a product to its various qualitative features ("characteristics") quantified, that presumably are valued by the consumers, i.e. those that are utility-enhancing. These characteristics are either quantified (e.g. Computer RAM measured in GB), or "1/0" dummy variables (the characteristic exists/it doesn't). Note that in the related empirical studies actual transacted prices are used, i.e. prices that the consumers have accepted to pay.

The estimated coefficients from the regression provide the monetized "willingness to pay" per unit of characteristic (or per existence of it). Once you have them, you can extrapolate outside the sample to find the total "willingness to pay" for a product $i$, say $W_i$, by multiplying its characteristics by the marginal willingness to pay for each characteristic.

In essence, in this way you transformed something that is considered ordinal and un-measurable (utility) into a monetized equivalent (using the "revealed preference" approach, since you used prices that the consumers actually paid).

If you then compare this magnitude to some advertised price of this product, $P_{ai}$, then a ratio like

$$\text{Prf}_i = \frac {W_i}{P_{ai}}$$

could be your "Performance" index. If for example it is higher than unity, it means that the product "is a good bargain" for the consumers, because the monetary value of the utility they will get out of it, is higher than the money they will have to pay to acquire it.

NOTE: Hedonic Price Analysis is not just something that academic economists do. In many nations, official Price Indexes are for several years now adjusted for quality, using the methods developed by economists throughout the years.

• Thanks, this is really useful. Does hedonic price addresses user satisfaction or expectations in any way? I am trying to model satisfaction as a function of perceived performance and expectations. Will have a look.
– vabm
Commented Dec 23, 2015 at 1:01
• @vabm Expectations, no. What is "user satisfaction"? How does it differ from utility? Commented Dec 23, 2015 at 2:15
• according to my survey data, satisfaction can be explained as a relationship between expected and actual performance, so I am using expectation disconfirmation theory as general framework. I am trying to use utility/price to explain perceived performance, but I also need a way to add expectations to the model (rational expectations?). Any ideas on potential approaches?
– vabm
Commented Dec 23, 2015 at 3:04