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I am supposed to determine if shocking either the income or preferences would certainly change the optimum solution in case of a corner solution?

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I'm not sure what you mean by "shocking the preferences". But it's very easy to find an example in which the optimal solution is still at the corner after a (small) positive income shock.

Suppose $u(x,y)=\sqrt x+y$, $p_x=p_y=1$, and income is $m=0.1$. Performing utility maximization subject to budget constraint, we get $$ x^*=m,\qquad y^*=0. $$ Now give income a positive shock: $m'=m+\delta$. As long as $\delta\le0.15$, the optimal solution is still going to be a corner solution: spend all the income on $x$ and zero on $y$.

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  • $\begingroup$ Hi. I really appreciate the answer. By preference shock, I mean what if the consumer's preferences change slightly, would the optimum bundle still remain the same? $\endgroup$ – Kartik Feb 23 '17 at 22:51
  • $\begingroup$ @Kartik: Depends on what you mean exactly by "preferences change slightly". One way of interpreting this would be, for example, that $u(x,y)=\alpha\sqrt x+\beta y$, and you give shocks to $\alpha$ and $\beta$ (currenlty $\alpha=\beta=1$). If you choose the shocks carefully, the conclusion in my answer will likely remain the same. $\endgroup$ – Herr K. Feb 23 '17 at 23:12
  • $\begingroup$ Thank you. Could you clarify me one more thing, where is this number 0.15 coming from? $\endgroup$ – Kartik Feb 26 '17 at 13:27
  • $\begingroup$ @Kartik: If you do the maximization, you'll see that if $m$ is large enough, $x^*=0.25$ and $y^*=m-x^*$. Since we assumed $m=0.1$ and we want $y^*=0$, we must have $\delta\le0.15$. $\endgroup$ – Herr K. Feb 26 '17 at 18:19

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