<p>If your objective is to enforce that shares are comprised between zero and one, there are several nonlinear transformations of your linear model available, as for instance
<span class="math-container">$$s_i= \min\{ \max\{ \alpha_i - \sum_{j=1}^n{\beta_{ij} \log p_j}, 0 \}, 1\} ,$$</span>
<span class="math-container">$$s_i= \big( \sin( \alpha_i - \sum_{j=1}^n{\beta_{ij} \log p_j} ) + 1 \big) / 2,$$</span>
<span class="math-container">$$s_i= F( \alpha_i - \sum_{j=1}^n{\beta_{ij} \log p_j} ) ,$$</span>
where <span class="math-container">$F$</span> represents any cdf of your choice.
However, usually violations of the theoretical properties indicate that your model is misspecified, and you should consider either to remove your homotheticity assumption, or include fixed (individual and time) effects or generalize the specification of your error term.</p>

CC BY-SA 4.0
2024-01-17T20:02:51.690Z
2024-01-17T20:02:51.690Z
Bertrand
<p>Usually these systems approximate the &quot;true&quot; demand model at a given point (obs. Shares and prices), and the approximation becomes worse the further you are away from this point. I don't think there's a demand system which provides a global approximation for the &quot;true&quot; demand system.</p>

CC BY-SA 4.0
2023-12-18T16:22:09.833Z
Some have global regularity properties which mean they behave reasonably for all positive prices. Whether this is a good prediction is another question.
cfp
CC BY-SA 4.0
2023-12-18T15:02:15.197Z
2023-12-18T15:02:15.197Z
Claudio Ribeiro de Lucinda
<p>I am performing a demand system estimation, and a counter-factual exercise which involves estimating demands at prices quite far from the observed ones. I know a priori that preferences are homothetic, but I want full flexibility given this restriction.</p>
<p>(See Barnett and Serletis (2008; JoE) for a review of the demand system estimation literature.)</p>
<p>Given these constraints, one possibility would be to use the homothetic translog specification in which the shares are given by:</p>
<p><span class="math-container">$$s_i=\alpha_i - \sum_{j=1}^n{\beta_{ij} \log p_j},$$</span></p>
<p>where <span class="math-container">$\sum_{i=1}^n {\alpha_i}=1$</span> and <span class="math-container">$\sum_{i=1}^n {\beta_{ij}}=0$</span> for all <span class="math-container">$j$</span>, and where <span class="math-container">$\beta_{ij}=\beta_{ji}$</span> for all <span class="math-container">$i,j$</span>.</p>
<p>However, this can easily produce negative shares in counter-factual exercises, when demand is evaluated far from the observed prices.</p>
<p>Is there an approach for homothetic demand system estimation that ensures positivity of the shares?</p>
<p>The logit approach seems to have issues with symmetry (see e.g. Dumagan and Mound (1996; EM)). The homothetic normalized quadratic flexible functional forms from Diewert and Wales (1988; JBES) do not seem to guarantee positive shares. The miniflex Laurent (Barnett and Lee 1985; ECMA) does not seem to have a natural homothetic special case. (I imagine it is not valid just to set income to one in the share equations, replacing <span class="math-container">$v_i$</span> by <span class="math-container">$p_i$</span>?)</p>
<p>Any suggestions would be appreciated!</p>

<microeconomics><demand><demand-system-estimation>
Ensuring positive shares in homothetic demand system estimation
cfp
# Ensuring positive shares in homothetic demand system estimation

(Asked by: username_0 on 2023-12-18)

I am performing a demand system estimation, and a counter-factual exercise which involves estimating demands at prices quite far from the observed ones. I know a priori that preferences are homothetic, but I want full flexibility given this restriction.

(See Barnett and Serletis (2008; JoE) for a review of the demand system estimation literature.)

Given these constraints, one possibility would be to use the homothetic translog specification in which the shares are given by:

$$s\_i=\alpha\_i - \sum\_{j=1}^n{\beta\_{ij} \log p\_j},$$

where $\sum\_{i=1}^n {\alpha\_i}=1$ and $\sum\_{i=1}^n {\beta\_{ij}}=0$ for all $j$, and where $\beta\_{ij}=\beta\_{ji}$ for all $i,j$.

However, this can easily produce negative shares in counter-factual exercises, when demand is evaluated far from the observed prices.

Is there an approach for homothetic demand system estimation that ensures positivity of the shares?

The logit approach seems to have issues with symmetry (see e.g. Dumagan and Mound (1996; EM)). The homothetic normalized quadratic flexible functional forms from Diewert and Wales (1988; JBES) do not seem to guarantee positive shares. The miniflex Laurent (Barnett and Lee 1985; ECMA) does not seem to have a natural homothetic special case. (I imagine it is not valid just to set income to one in the share equations, replacing $v\_i$ by $p\_i$?)

Any suggestions would be appreciated!

**Tags:** microeconomics, demand, demand-system-estimation

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## 2 answers

### Answer by username_1 on 2024-01-17. Score: 4

If your objective is to enforce that shares are comprised between zero and one, there are several nonlinear transformations of your linear model available, as for instance
$$s\_i= \min\{ \max\{ \alpha\_i - \sum\_{j=1}^n{\beta\_{ij} \log p\_j}, 0 \}, 1\} ,$$
$$s\_i= \big( \sin( \alpha\_i - \sum\_{j=1}^n{\beta\_{ij} \log p\_j} ) + 1 \big) / 2,$$
$$s\_i= F( \alpha\_i - \sum\_{j=1}^n{\beta\_{ij} \log p\_j} ) ,$$
where $F$ represents any cdf of your choice.
However, usually violations of the theoretical properties indicate that your model is misspecified, and you should consider either to remove your homotheticity assumption, or include fixed (individual and time) effects or generalize the specification of your error term.


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### Answer by username_2 on 2023-12-18. Score: 0

Usually these systems approximate the "true" demand model at a given point (obs. Shares and prices), and the approximation becomes worse the further you are away from this point. I don't think there's a demand system which provides a global approximation for the "true" demand system.

> **Comments:**
> **username_0**: Some have global regularity properties which mean they behave reasonably for all positive prices. Whether this is a good prediction is another question.  

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