<p>Among the simplest demand functions allowing for arbitrary price and income elasticities there is:
<span class="math-container">$$ x_1^M(p_1,p_2,y) = \alpha + \beta\ln(\frac{p_1}{p_2}) +  \gamma\ln(\frac{y}{p_2}),
$$</span>
or its log version:
<span class="math-container">$$ \ln( x_1^M(p_1,p_2,y) ) = \alpha + \beta\ln(\frac{p_1}{p_2}) +  \gamma\ln(\frac{y}{p_2}).
$$</span></p>
<p>The elasticity parameters wrt own-price and income are denoted by <span class="math-container">$\beta$</span> and <span class="math-container">$\gamma$</span> and can take (almost) arbitrary values. Note that the elasticity wrt <span class="math-container">$p_2$</span> cannot be arbitrary as a consequence of homogeneity of degree zero in <span class="math-container">$(p_1,p_2,y).$</span> The expression for <span class="math-container">$x^M_2$</span> is obtained from the budget constraint.</p>

CC BY-SA 4.0
2022-02-07T14:21:19.947Z
This looks like what I'm looking for. A few questions. Is it still possible to normalise p2 as the price of "all other goods"? Is it possible to interpret alpha as something meaningful here? In cobb-douglas it's possible to say something about budget share for each good, I assume that is replaced by being determined with the elasticities?
Victor Nielsen
CC BY-SA 4.0
2022-02-07T15:37:52.677Z
@Victor Nielsen: yes, normalization is possible. The budget share is easily obtained using its definition $p_1x^M_1/y$ and it is not constant but depends upon $(p,y)$. Regarding the utility function generating such demands, I am afraid that it is not easy (not possible ?) to give its expression.
Bertrand
CC BY-SA 4.0
2022-02-15T11:34:43.143Z
Quick question. What values can alpha take and what does it tell us?
Victor Nielsen
CC BY-SA 4.0
2022-02-15T13:05:49.897Z
@Victor Nielsen: it depends which commodity you consider. Both $\beta$ and $\gamma$ can be positive or negative depending on whether the good is normal/inferior, Giffen, luxury, etc.
Bertrand
CC BY-SA 4.0
2022-02-15T17:45:07.997Z
I understand beta and gamma, but alpha seems to be a constant by itself.
Victor Nielsen
CC BY-SA 4.0
2022-02-15T18:08:02.633Z
The constant part of demand $\alpha$ may be zero for several goods, but this is not necessarily the case for commodities with a subsistence level, or for which the State guarantees a minimum consumption level (medicaid, housing).
Bertrand
CC BY-SA 4.0
2022-02-05T20:54:01.327Z
2022-02-07T15:32:05.693Z
2022-02-07T15:32:05.693Z
Bertrand
<p>Cobb-Douglas utility functions assume price elasticity of <span class="math-container">$-1$</span> and income elasticity of <span class="math-container">$1$</span>.</p>
<p>Are there any utility functions with two goods that lead to a demand function, where you have the choice of changing the price and income elasticities?</p>
<p>Say you wanna test a model that has a demand function, where the price elasticity is <span class="math-container">$-0.5$</span> and income elasticity is <span class="math-container">$2$</span>. Or <span class="math-container">$-0.25$</span> and <span class="math-container">$0.75$</span>.</p>

CC BY-SA 4.0
2022-02-05T05:17:51.203Z
Are you talking about point or arc elasticities?
Giskard
CC BY-SA 4.0
2022-02-05T05:21:13.763Z
Do you want the utility function to result in an isoelastic demand function, or should it just take the chosen elasticity value locally? It seems like you want the former, but you can't have a demand function with constant income elasticity $\mu > 1$.
Giskard
CC BY-SA 4.0
2022-02-05T15:22:05.987Z
To be completely honest, I'm not sure how to answer those questions. Maybe it would help if I could explain how I plan to use it?
Victor Nielsen
CC BY-SA 4.0
2022-02-05T18:00:52.603Z
Maybe! Please [edit the additional information into your question](https://economics.stackexchange.com/posts/50316/edit). Keep it clear, focused and do not solicit opinions.
Giskard
<microeconomics><mathematical-economics><utility><elasticity><demand>
What is a simple demand function that allows for different price and income elasticities than 1 and -1?
Victor Nielsen
# What is a simple demand function that allows for different price and income elasticities than 1 and -1?

(Asked by: username_0 on 2022-02-04)

Cobb-Douglas utility functions assume price elasticity of $-1$ and income elasticity of $1$.

Are there any utility functions with two goods that lead to a demand function, where you have the choice of changing the price and income elasticities?

Say you wanna test a model that has a demand function, where the price elasticity is $-0.5$ and income elasticity is $2$. Or $-0.25$ and $0.75$.

**Tags:** microeconomics, mathematical-economics, utility, elasticity, demand
> **Comments:**
> **username_1**: Are you talking about point or arc elasticities?  
> **username_1**: Do you want the utility function to result in an isoelastic demand function, or should it just take the chosen elasticity value locally? It seems like you want the former, but you can't have a demand function with constant income elasticity $\mu > 1$.  
> **username_0**: To be completely honest, I'm not sure how to answer those questions. Maybe it would help if I could explain how I plan to use it?  
> **username_1**: Maybe! Please [edit the additional information into your question](https://economics.stackexchange.com/posts/50316/edit). Keep it clear, focused and do not solicit opinions.  

---

## 1 answers

### Answer by username_2 on 2022-02-05. Score: 4 | Accepted answer

Among the simplest demand functions allowing for arbitrary price and income elasticities there is:
$$ x\_1^M(p\_1,p\_2,y) = \alpha + \beta\ln(\frac{p\_1}{p\_2}) + \gamma\ln(\frac{y}{p\_2}),
$$
or its log version:
$$ \ln( x\_1^M(p\_1,p\_2,y) ) = \alpha + \beta\ln(\frac{p\_1}{p\_2}) + \gamma\ln(\frac{y}{p\_2}).
$$

The elasticity parameters wrt own-price and income are denoted by $\beta$ and $\gamma$ and can take (almost) arbitrary values. Note that the elasticity wrt $p\_2$ cannot be arbitrary as a consequence of homogeneity of degree zero in $(p\_1,p\_2,y).$ The expression for $x^M\_2$ is obtained from the budget constraint.

> **Comments:**
> **username_0**: This looks like what I'm looking for. A few questions. Is it still possible to normalise p2 as the price of "all other goods"? Is it possible to interpret alpha as something meaningful here? In cobb-douglas it's possible to say something about budget share for each good, I assume that is replaced by being determined with the elasticities?  
> **username_2**: @Victor Nielsen: yes, normalization is possible. The budget share is easily obtained using its definition $p_1x^M_1/y$ and it is not constant but depends upon $(p,y)$. Regarding the utility function generating such demands, I am afraid that it is not easy (not possible ?) to give its expression.  
> **username_0**: Quick question. What values can alpha take and what does it tell us?  
> **username_2**: @Victor Nielsen: it depends which commodity you consider. Both $\beta$ and $\gamma$ can be positive or negative depending on whether the good is normal/inferior, Giffen, luxury, etc.  
> **username_0**: I understand beta and gamma, but alpha seems to be a constant by itself.  
> **username_2**: The constant part of demand $\alpha$ may be zero for several goods, but this is not necessarily the case for commodities with a subsistence level, or for which the State guarantees a minimum consumption level (medicaid, housing).  

---

