<p>If a functional form satisfies the Inada conditions, it will be "well behaved" in the way you want.  The <a href="https://en.wikipedia.org/wiki/Inada_conditions" rel="nofollow noreferrer">Wikipedia article</a> 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.  </p>

CC BY-SA 4.0
2019-12-06T18:35:32.633Z
2019-12-06T18:35:32.633Z
heh
<p>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 <span class="math-container">$-\infty$</span> when the input is 0 to ensure interior solutions.</p>

CC BY-SA 4.0
2018-10-12T00:01:44.013Z
2018-10-12T00:01:44.013Z
Mathew Knudson
<p>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:</p>

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

<p>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.  </p>

CC BY-SA 4.0
2018-08-12T16:26:03.123Z
"$k = 1$ is considered reasonable"? If the ratio of the $\alpha_i$ parameters is constant, any positive $k$ will result in the same preferences.
Giskard
CC BY-SA 4.0
2018-08-12T17:17:23.563Z
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
CC BY-SA 4.0
2018-08-12T18:29:27.030Z
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
CC BY-SA 4.0
2018-08-14T16:41:47.327Z
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
CC BY-SA 4.0
2018-08-14T17:14:43.207Z
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
CC BY-SA 4.0
2018-08-14T17:55:27.280Z
Hmm fair point. I don’t see having zero of any good as being essential to the main point. And I can make sure that there are no goods at such a high pice point that buying a single unit would be a distortion to people’s total budgets.
Scathelock
CC BY-SA 4.0
2018-08-12T16:13:57.327Z
2018-08-12T16:24:31.743Z
2018-08-12T16:24:31.743Z
Pedro Cavalcante
<p>I am aiming to program a basic simulation of a simplified economy to look at the impact of various interventions. </p>

<p>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.</p>

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

<p>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.</p>

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

<p>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.</p>

<p>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?</p>

<utility><supply-and-demand><simulations>
What is a good way to generate realistic utility curves?
Scathelock
# What is a good way to generate realistic utility curves?

(Asked by: username_0 on 2018-08-12)

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?

**Tags:** utility, supply-and-demand, simulations

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## 3 answers

### Answer by username_1 on 2019-12-06. Score: 1

If a functional form satisfies the Inada conditions, it will be "well behaved" in the way you want. The [Wikipedia article](https://en.wikipedia.org/wiki/Inada_conditions) 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.


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### Answer by username_2 on 2018-10-12. Score: 1

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.


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### Answer by username_3 on 2018-08-12. Score: 1

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.

> **Comments:**
> **username_4**: "$k = 1$ is considered reasonable"? If the ratio of the $\alpha_i$ parameters is constant, any positive $k$ will result in the same preferences.  
> **username_3**: 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?  
> **username_4**: 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.  
> **username_0**: 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.  
> **username_3**: 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.  
> **username_0**: Hmm fair point. I don’t see having zero of any good as being essential to the main point. And I can make sure that there are no goods at such a high pice point that buying a single unit would be a distortion to people’s total budgets.  

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