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In Syldavia the economists find that (annual) household consumption $c$ is related to (annual) income $y$ by the formula $c=\alpha +\beta y$, where $\alpha>0$ and $0<\beta<1$. Because of this, they argue, inequality of consumption must be less than inequality of income.

How do I provide an intuitive argument for this?

My work: enter image description here

$\frac{\partial c}{\partial y}=\beta $. So if $\beta$ increases then the $c$ curve shifts up. The change in $c$ is dependent on change in $\beta$ which is between $0$ and $1$. But I am not sure if I can intuitively explain why inequality of consumption must be less than inequality of income.

OGC
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    I am not sure how one measures these things. Here is a probabilistic viewpoint, probably not intended. Pick a random resident. Let random variable $Y$ be her income, and $C$ her consumption. Then $\text{Var}(C)=\beta^2\text{Var}(Y)\lt \text{Var}(Y)$. – André Nicolas Mar 06 '14 at 01:52
  • @AndréNicolas Neat stuff. You just used one of the properties of the variance of a random variable and showed me the light. Thanks. – OGC Mar 06 '14 at 01:58
  • You are welcome. I did not give this as an answer because it used non-economics language. One could use other measures of variability in this case, since the relationship is linear. – André Nicolas Mar 06 '14 at 02:00
  • @AndréNicolas What about the error term? In other words, you've calculated the variance of the conditional expected value of C, which is a kind of mean of C, rather than the variance of the random variable C, which should be much larger? – dimitriy Mar 06 '14 at 21:56
  • We have calculated the variance of the random variable $C$, in terms of the variance of $Y$. The variance of $Y$ cannot be determined from the information we have in this problem. – André Nicolas Mar 06 '14 at 22:13
  • @DimitriyV.Masterov The equation does not have an error term in it. I understand that you are referring to econometrics/statistics here, but I think the OLS assumption of heterskedasticity should take care of the error term. – OGC Mar 07 '14 at 00:52
  • @user36829 If by "find" you mean estimated, then I believe you need the adjective "expected" in front of consumption for the statement to be true. However, I may be being pedantic, especially since Syldavia is a fictional Balkan country, so one could expect the relationship to between income and consumption to be exact there. – dimitriy Mar 07 '14 at 01:38
  • @DimitriyV.Masterov Now look here this is just an exercise problem and does not reflect reality, so there is no need to be pedantic on this occasion. – OGC Mar 07 '14 at 01:59

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see your question has some missing points but nevertheless from pure intuition even at zero income consumption is happening(autonomous).that means even if you have zero income you are consuming something.that intuitively means your income inequality is relatively more than consumption inequality.to survive they will have to eat something(remember this is a short term keynesian consumption function that's why intercept is positive,that will vanish in long run)

paul
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