Bads utility function

Question

The utility function for goods can be modeled using something like the exponential utility:

$$B(x)=\frac{1-\exp(-ax)}{a}$$ with $a>0$.

So that marginal benefits are modeled as being within $[0,1]$:

$$B'(x) = \exp(-ax)$$

This would look something like this:

So far, so good. Now, in the production of this good, there are costs, because there are bads which are produced as byproducts (think pollution) that we would like to model.

I am not from economics. I just think a project I am working on would benefit by us studying and using whatever is used in micro.

What are some good utility function for bads (no pun intended) or for costs?

Or if you could recommend some good textbook that covers this with enough mathematical rigor that would be fine as well. From Google Books, I have seen this could be a good bet, but it doesn't let me look at it without buying it:

Microeconomic Theory And Applications - Page 84
Agarwala S K - 2008
If we want more industrialisation we have no option but to
accept more pollution as well. Industrialisation leads to rise
in utility level while pollution leads to fall in utility level
It makes the ICs upward sloping to the right. GoodY (Bad) GoodY
...

It seems like it has a nice discussion on how to model bads/costs. Would you recommend it?

ps: I just realized maybe I could use $C(x)$ as being $B(x)$ but with $a<0$ for what I want...

Answer 1

The branch of microeconomics that you seem to be looking for is the study of externalities—activities that have side-effects that (positively or negatively) effect the welfare of others. The Wikipedia article on the topic is here.

Any good introductory/undergraduate level textbook will have a treatment of this topic. For example, Hal Varian's "Intermediate Microeconomics: A Modern Approach" (which would be my recommended first port of call) has a chapter on externalities. A slightly more specialised treatment can be found in a textbook on so-called public economics such as "Intermediate Public Economics" by Myles and Hindriks.

These books also contain examples of how goods with externalities can be formally modeled.

Alternatively, because externalities are a standard element of the economics curriculum, a Google search for "Externalities lecture notes" will turn up lots of freely-accessable PDFs that give a good overview.