# What is a good way to generate realistic utility curves?

I am aiming to program a basic simulation of a simplified economy to look at the impact of various interventions.

The economy will have N groups of homogeneous consumers and M producer / employer sectors. Each producer sector makes one type of good that the consumers can purchase, so there are M goods. To try to keep it (relatively) simple, the goods will all be independent with no close substitutes or compliments. E.g. goods will be steak and diapers, not steak and chicken, nor steak and potatoes.

Each group of consumers will need to have a demand curve for each of the M goods, as well as a "demand" for leisure.

Given the marginal utilities for each good, you can calculate a demand curve for that good. So, the approach I'm thinking is to generate utility curves for each good, then use those utility curves to derive demand curves which then drive consumer purchasing behavior and other downstream effects.

This brings me to the question, what's a good way to randomly generate a lot of plausible utility curves?

For example, if I wanted a linear utility curve I could randomly generate a slope and an intercept. That is simple and easy to do, but it misses diminishing marginal utility which is one of the most important features.

What's a form that's easy to work with, has relatively few parameters to specify, and still captures most of the key features of utility curves?

## 3 Answers

It's somewhat simple, actually. You can just use a Cobb-Douglas with $m+1$ goods available, the last one being leisure. So, you should have something like:

$$U(x_1,x_2,...,x_m,x_{m+1}) = \prod_{i=1}^{m+1} x_i^{\alpha_i}$$

Subject to $\sum_{i=1}^{m+1} \alpha_i = k$, where $k$ is the degree of homogeneity of the utility function. Usually, $k=1$ is considered reasonable.

• "$k = 1$ is considered reasonable"? If the ratio of the $\alpha_i$ parameters is constant, any positive $k$ will result in the same preferences. – Giskard Aug 12 '18 at 16:26
• Yeah, but isn't it obvious that the ratio of each $\alpha_i$ should be good $i$'s participation in the observed consumer bundle? Should have I explicitly said that? – Pedro Cavalcante Aug 12 '18 at 17:17
• I don't think you should have said that, and neither have I. I am trying to point out that the functions $U(x_1,x_2) = x_1x_2$ and $\hat{U}(x_1,x_2) = x_1^2x_2^2$ represent the same preferences despite having different degrees of homogeneity. In case of utility functions $k = 1$ is not a "reasonable" choice but a matter of convenience. – Giskard Aug 12 '18 at 18:29
• Does a Cobb Douglas work in this case? At minimum it has the limitation that you can’t have a zero quantity of any good. Among a diverse group of consumers and a diverse basket of goods, it is often the case that a given consumer will purchase zero of some good. – Scathelock Aug 14 '18 at 16:41
• Is consuming zero quantity of some goods an essential part of consumer behavior in your model? Does it critically change your results? If it doesn't, then it's just an useful simplification. – Pedro Cavalcante Aug 14 '18 at 17:14

You could also consider any additively separable utility function (the Cobb Douglas form can be transformed to be additively separable). Then marginal utility of a good depends only on the consumption of that good, which makes it very easy to find the demands. If for whatever reason, you aren't happy with Cobb-Douglas, this is a nice generalization. You can also pick functions that go to $$-\infty$$ when the input is 0 to ensure interior solutions.

If a functional form satisfies the Inada conditions, it will be "well behaved" in the way you want. The Wikipedia article speaks to production functions, but absent specific real-world data to the contrary, the only differences between a production function and a utility function are who is maximizing it and whether cardinality matters.