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


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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