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  1. I have to find the consumer's demand at prices $\textbf{p} = (1,1,1)$ and budget $m=4$ when the utility function is given by $$u(x_1,x_2,x_3) = 3 \sqrt{x_1+x_2+2x_3}$$ or when transformed $$\hat{u}(x_1,x_2,x_3) = x_1+x_2+2x_3 $$

    I know the answer is $\textbf{x} = (0,0,4)$

What I don't understand: For perfect substitutes the demand is given by $x_1^* = \frac{m}{p_1}$ so the demand should have been $\textbf{x} = (4,4,4)$. But why is $x_1^*,x_2^* = 0$?

  1. For the same utility function I want to find the demand at prices $p=(2,3,p_3) >>0$ and budget $m>0$.

    The answer is \begin{equation} \textbf{x} = \left(0,0, \frac{m}{p_3} \right) \quad \text{when} \quad p_3<4 \end{equation} \begin{equation} \textbf{x} = \left(\frac{m}{2},0,0 \right) \quad \text{when} \quad p_3>4 \end{equation} \begin{equation} \textbf{x} = \left(\frac{\alpha m}{2}, 0, \frac{(1-\alpha)m}{4} \right) \quad \text{when} \quad p_3=4 \end{equation} where $\alpha \in [0,1]$ because the consumer is indifferent between the bundles

What I don't understand: Why is the demand different depending on what $p_3$ is and how do I know when the demand is different? Personally, I would've said the demand is $\textbf{x} = \left( \frac{m}{2}, \ \frac{m}{3}, \ \frac{m}{p_3} \right)$

Stranqer95
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  • What have you tried to do? Have you written and solved the consumer's problem? Otherwise, where does the formula $x_1^{*}=\dfrac{m}{p_1}$ come from? Hint: do you think the bundle $x=(4,4,4)$ is feasible when $p=(1,1,1)$ and $m=4$? – Oliv Jun 18 '15 at 16:09
  • $x_1^* = \frac{m}{p_1}$ is from my book (Intermediate Microeconomics, Varian). Hm, it's true that it isn't feasible. But it might as well have been $x=(1,1,2)$. – Stranqer95 Jun 18 '15 at 16:22
  • You should examine Varian a bit more carefully, that is not the demand function for all values of $p_1$. – Giskard Jun 18 '15 at 16:24
  • Oh right, I found this the other day $x_1 = \frac{m}{p_1}$ when $p_1 <p_2$, $x_1 =$ any number between 0 and $m/p_1$ when $p_1 = p_2$ and $x_1 = 0$ when $p_1 > p_2$, but I wasn't sure how to apply it.. – Stranqer95 Jun 18 '15 at 16:27
  • @Stranqer95 now you can generalize the formula for three goods instead of two and you'll get the result. – Oliv Jun 18 '15 at 16:58
  • I can see why I get the demands stated above for $p_3 < 4$ and $p_3>4$. But I'm can't quite figure out the ones for $x=(0,0,4)$. Why is $x_1,x_2 = 0$. The book says "any number between 0 and $m/p_1$". Then couldn't it be $x=(1,1,2)$ as well. Also, for $p_3=4$ it's the same thing. Why does $x_1 = \frac{\alpha m}{2}$ and $x_2=0$ when it might as well be something else? – Stranqer95 Jun 18 '15 at 17:23
  • You are trying to apply a formula that holds in the two-goods case to a situation where there are three goods. Instead of using the formula you should try to understand the logic behind it: where does this result come from? I suggest that you try to write the consumer's problem and solve it, and report your progress here to receive feedback and advice. – Oliv Jun 18 '15 at 18:41

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Regarding the first question, @Oliv suggestion in the comments to formally write down the consumer's problem, budget constraint, Kunh-Tucker multipliers and all, and solve it (carefully -linear objective functions easily mislead when we are used to deal with non-linear ones), is the way to go in order to see how the mathematical set-up will lead you to the answer that stares you in the face:

Our purpose here is to maximize the utility index. Given unitary prices, isn't it obvious that the multivariable function

$$\hat{u}(x_1,x_2,x_3) = x_1+x_2+2x_3$$

will take its maximum value if we purchase only good $x_3$? For any tiny amount $\Delta x_3$ of $x_3$ that you don't purchase, you lose $2\Delta x_3$ in utility terms, while you gain (in utility terms) only $\Delta x_1 + \Delta x_2 = \Delta x_3$ for any combination of $\Delta x_1$ and $\Delta x_2$. Since $2\Delta x_3 > \Delta x_3$ it will be suboptimal to purchase anything else than $x_3$. That was the "plain logic" part, and I hope it answers the "what I don't understand..." aspect of your question. Put the model in math to reconcile this obvious arithmetic with the formal approach.

Alecos Papadopoulos
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