<p>There are few things that need to be clarified before we want to understand <em>the nature of goods.</em></p>

<p>First concept is <strong>elasticity</strong>: which is just change of one variable in response to change in another variable, here increase/decrease in <em>demand for good</em> as a result of change in its <em>price</em> or consumer <em>income</em>. </p>

<p>Mathematically:</p>

<ul>
<li><strong>Price (point) elasticity</strong>: $\varepsilon_{price} = \frac{P}{Q_{d}} \frac{\partial Q_{d}}{\partial P}$ </li>
</ul>

<p>If certain good has $|\varepsilon_{price}| &lt; 1 $, it is inelastic meaning that any change in price will have low impact on your quantity demanded (think about drugs, which are very inelastic). Opposite works for $|\varepsilon_{price}| &gt; 1 $.</p>

<ul>
<li><strong>Income elasticity</strong>: $\varepsilon_{income} = \frac{I}{Q_{d}} \frac{\partial Q_{d}}{\partial I}$</li>
</ul>

<p>The idea here is the same as in case of price elasticity but now income will have and impact on your demand.</p>

<p>In example you provided the prices are constant, therefore we cannot measure price elasticity of demand. However, it is relatively simple to calculate income elasticity of demand. We can either calculate it by hand or use simple OLS model. For sake of convenience I will use the latter (as it also was what you asked for). </p>

<p>I create simple linear model where my dependent variable is the <em>demand for good x</em> and independent variable is <em>income</em>. The result will be just measuring unit impact. In case you are interested in measuring % change you have to take logarithms on both sides. </p>

<p>$$demand_{i} = \alpha + \beta_1 income_i + \varepsilon_i$$</p>

<p>Good 1 (normal):</p>

<pre><code>            Estimate Std. Error t value Pr(&gt;|t|)
(Intercept)  5.31579   20.82775   0.255    0.841
income       0.03421    0.13674   0.250    0.844
</code></pre>

<p>Good 2 (inferior):</p>

<pre><code>            Estimate Std. Error t value Pr(&gt;|t|)
(Intercept) 22.89474    6.24832   3.664     0.17
income      -0.09474    0.04102  -2.309     0.26
</code></pre>

<p>As you can see there are only 3 observations so the results are not very robust but should give you main idea. Furthermore, as there is not enough data we can only hypothesize about characteristic of a good, there is more information needed to distinguish between normal and luxury goods. </p>

<p>Nevertheless, current data shows that demand for first good increases as income increases. The case is opposite for the second good.</p>

<p><strong>EDIT:</strong> As noted by <code>@luchonacho</code> in order to generalize into income and price elasticity we would have to include price into our regression. The resulting model would look as follows:</p>

<p>$$demand_{i} = \alpha + \beta_1 income_i + \beta_2 price_i + \varepsilon_i$$</p>

<p>Now one has to be careful with interpretation of the coefficients. $\beta_{i}$ shows and effect of 1 unit increase in income\price on the demand for good (elasticity), <strong>keeping all other variables constant</strong> (<em>ceteris paribus</em>).</p>

CC BY-SA 3.0
2017-07-20T10:10:41.627Z
Welcome to Econ.SE! Great answer. Since the data the OP provided is just a sample (which happened to have constant prices), the regression the OP would need to use must include prices in order to get the _ceteris pairubs_ elasticity. I would add this to the answer.
luchonacho
CC BY-SA 3.0
2017-07-20T09:12:21.700Z
2017-07-20T10:38:55.127Z
2017-07-20T10:38:55.127Z
An economist
<p>I have some data on household income and goods purchased. The first three columns and rows are:</p>

<pre><code>Income    Good1    Good2  Price1  Price2
  100       4        12      1.5    2
  140       18       12      1.5    2
  200       9        3       1.5    2
</code></pre>

<p>I'm interested in understanding how I would use regression analysis/econometrics to understand the nature of the goods in question (i.e are they normal, ordinary, inferior, etc).</p>

CC BY-SA 3.0
2017-07-19T17:32:30.563Z
What about regressing each good on income and every price? The coefficient of income would given you the result. Or am I totally wrong?
luchonacho
CC BY-SA 3.0
2017-07-20T08:19:50.860Z
@luchonacho no you are totally correct! For example, we define inferior good as one where demand drops as income increases. In given example I think we could disregard the price (since it is constant) and just measure effect of income increase on demand. We could make an adjustment by creating price/income ratio but it should not matter in given example as the prices are constant.
An economist
<consumer-theory><applied-econometrics>
Consumer Econometrics: Identifying nature of goods
EconJohn
# Consumer Econometrics: Identifying nature of goods

(Asked by: username_0 on 2017-07-19)

I have some data on household income and goods purchased. The first three columns and rows are:
Income    Good1    Good2  Price1  Price2
  100       4        12      1.5    2
  140       18       12      1.5    2
  200       9        3       1.5    2

I'm interested in understanding how I would use regression analysis/econometrics to understand the nature of the goods in question (i.e are they normal, ordinary, inferior, etc).

**Tags:** consumer-theory, applied-econometrics
> **Comments:**
> **username_1**: What about regressing each good on income and every price? The coefficient of income would given you the result. Or am I totally wrong?  
> **username_2**: @luchonacho no you are totally correct! For example, we define inferior good as one where demand drops as income increases. In given example I think we could disregard the price (since it is constant) and just measure effect of income increase on demand. We could make an adjustment by creating price/income ratio but it should not matter in given example as the prices are constant.  

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## 1 answers

### Answer by username_2 on 2017-07-20. Score: 3 | Accepted answer

There are few things that need to be clarified before we want to understand the nature of goods.
First concept is elasticity: which is just change of one variable in response to change in another variable, here increase/decrease in demand for good as a result of change in its price or consumer income. 
Mathematically:

Price (point) elasticity: $\varepsilon_{price} = \frac{P}{Q_{d}} \frac{\partial Q_{d}}{\partial P}$ 

If certain good has $|\varepsilon_{price}| < 1 $, it is inelastic meaning that any change in price will have low impact on your quantity demanded (think about drugs, which are very inelastic). Opposite works for $|\varepsilon_{price}| > 1 $.

Income elasticity: $\varepsilon_{income} = \frac{I}{Q_{d}} \frac{\partial Q_{d}}{\partial I}$

The idea here is the same as in case of price elasticity but now income will have and impact on your demand.
In example you provided the prices are constant, therefore we cannot measure price elasticity of demand. However, it is relatively simple to calculate income elasticity of demand. We can either calculate it by hand or use simple OLS model. For sake of convenience I will use the latter (as it also was what you asked for). 
I create simple linear model where my dependent variable is the demand for good x and independent variable is income. The result will be just measuring unit impact. In case you are interested in measuring % change you have to take logarithms on both sides. 
$$demand_{i} = \alpha + \beta_1 income_i + \varepsilon_i$$
Good 1 (normal):
            Estimate Std. Error t value Pr(>|t|)
(Intercept)  5.31579   20.82775   0.255    0.841
income       0.03421    0.13674   0.250    0.844

Good 2 (inferior):
            Estimate Std. Error t value Pr(>|t|)
(Intercept) 22.89474    6.24832   3.664     0.17
income      -0.09474    0.04102  -2.309     0.26

As you can see there are only 3 observations so the results are not very robust but should give you main idea. Furthermore, as there is not enough data we can only hypothesize about characteristic of a good, there is more information needed to distinguish between normal and luxury goods. 
Nevertheless, current data shows that demand for first good increases as income increases. The case is opposite for the second good.
EDIT: As noted by @luchonacho in order to generalize into income and price elasticity we would have to include price into our regression. The resulting model would look as follows:
$$demand_{i} = \alpha + \beta_1 income_i + \beta_2 price_i + \varepsilon_i$$
Now one has to be careful with interpretation of the coefficients. $\beta_{i}$ shows and effect of 1 unit increase in income\price on the demand for good (elasticity), keeping all other variables constant (ceteris paribus).

> **Comments:**
> **username_1**: Welcome to Econ.SE! Great answer. Since the data the OP provided is just a sample (which happened to have constant prices), the regression the OP would need to use must include prices in order to get the _ceteris pairubs_ elasticity. I would add this to the answer.  

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