<p>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. </p>

<p>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 $y$. </p>

CC BY-SA 3.0
2017-02-23T22:51:22.590Z
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?
Kartik
CC BY-SA 3.0
2017-02-23T23:12:02.493Z
@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.
Herr K.
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2017-02-26T13:27:25.070Z
Thank you. Could you clarify me one more thing, where is this number 0.15 coming from?
Kartik
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2017-02-26T18:19:03.483Z
@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 $\delta\le0.15$.
Herr K.
CC BY-SA 3.0
2017-02-23T22:39:50.967Z
2017-02-23T22:39:50.967Z
Herr K.
<p><br> I am supposed to determine if shocking either the income or preferences would <strong>certainly</strong> change the optimum solution in case of a corner solution?</p>

<microeconomics><consumer-theory>
Effects of shocking budget or indifference curves on optimum point?
Kartik
# Effects of shocking budget or indifference curves on optimum point?

(Asked by: username_0 on 2017-02-23)

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

**Tags:** microeconomics, consumer-theory

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## 1 answers

### Answer by username_1 on 2017-02-23. Score: 2

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 $y$.

> **Comments:**
> **username_0**: 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?  
> **username_1**: @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.  
> **username_0**: Thank you. Could you clarify me one more thing, where is this number 0.15 coming from?  
> **username_1**: @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 $\delta\le0.15$.  

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