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As the title states, I want to know how to derive the hicksian demand of perfect complements $\text{min} \, \{x1,x2\}$. Thanks in advance.

Also, no price is given, or budget. my main question is how do i do this considering there really is no utility function?

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  • $\begingroup$ This is a fine question, but please, next time use MathJax formating :) $\endgroup$ – Pedro Cavalcante Sep 19 '18 at 20:39
  • $\begingroup$ @PedroCavalcanteOliveira Why is this a "fine question"? Do you agree with the claim that there "there really is no utility function"? $\endgroup$ – Giskard Sep 20 '18 at 6:30
  • $\begingroup$ @CR12 You are supposed to give a parametric solution, that is assume an income of $I$, and prices $p_1,p_2$. Do you know what Hicksian demand is? If yes, what have you tried so far? $\endgroup$ – Giskard Sep 20 '18 at 6:31
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Hicksian demand is the consumption bundle that minimizes the expenditure of the consumer subject to the constraint that he attains some target level of satisfaction in equilibrium. In the problem, the expenditure on any bundle $(x, y)$ is given by $p_Xx + p_Yy$ and the target level of satisfaction is $\mu$. Given that the utility function is $u(x, y) = \min(x, y)$, the expenditure minimization problem is formulated as:

\begin{eqnarray*} \min_{x, y} \ \ & p_Xx+p_Yy \\ \text{s.t.} \ \ & \min(x, y) \geq \mu\end{eqnarray*}

Solution to this problem is a function known as Hicksian demand function :

$x^h (p_X, p_Y, \mu) = \mu $

$y^h (p_X, p_Y, \mu) = \mu $

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  • $\begingroup$ Thank you. I ended up deriving this answer but was unsure if it was correct. your explanation helps me greatly. $\endgroup$ – user19452 Sep 19 '18 at 19:34
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Although one can derive the Hicksian demands by solving the expenditure minimisation problem, but the Leontief function is not differentiable at the 'kink', or at the 'point of optimality'. Thus one has to resort to simpler algebraic manipulations to find out the Hicksian demands.

Let $\textbf{p}\in\mathbb{R_{++}^2}$(that is both $p_1 \ \& \ p_2$ are $>0$). We know that for $U(x,y)=min\{x,y\}$, the optimal consumption bundle is at the point where $x=y$. Let the budget correspondence be $p_1x+p_2y\leq w$, where $w$ is the income level. Optimal consumption bundle occurs when all the income is used up. Thus, we have $p_1x+p_2x= w$ (since $x=y$ at point of optimality). This gives us the demand correspondences as $x(\textbf{p},w)=y(\textbf{p},w)=\frac{w}{p_1+p_2}$. Using the demands, we can find out the indirect utility as $v(\textbf{p},w)=U(x(\textbf{p},w),y(\textbf{p},w))=min\{\frac{w}{p_1+p_2},\frac{w}{p_1+p_2}\}=\frac{w}{p_1+p_2}$.

Next, we turn towards the duality principle,i.e; for a given level of utility, we have $v(\textbf{p},e(\textbf{p},u))=u$. Thus, we have $\frac{e(\textbf{p},u)}{p_1+p_2}=u$, or $e(\textbf{p},u)=u(p_1+p_2)$. Next, we appeal to the Shephard's Lemma,i.e; $\frac{\partial e(\textbf{p},u)}{\partial p_i}=h_i(\textbf{p},u)$. Notice that the expenditure function is differentiable in prices. Thus, we get $h_1(\textbf{p},u)=u$ and $h_2(\textbf{p},u)=u$.

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