<p>Since <span class="math-container">$x({\mathbf p},u)=u$</span> and <span class="math-container">$y({\mathbf p},u)=u/2$</span> do not depend on prices <span class="math-container">${\mathbf p}$</span>, homogeneity of degree zero in prices is trivially satisfied for this special case: <span class="math-container">$x(t{\mathbf p},u)=x({\mathbf p},u)=u$</span> and <span class="math-container">$y(t{\mathbf p},u)=y({\mathbf p},u)=u/2$</span>.</p>
<p>For the general case, homogeneity of degree zero follows from the observation that the slope of the budget line is given by <span class="math-container">$-p_x/p_y=-(tp_x)/(tp_y)$</span> for <span class="math-container">$t&gt;0$</span>.</p>

CC BY-SA 4.0
2022-09-12T13:56:58.533Z
2022-09-12T13:56:58.533Z
VARulle
<p>Given a utility function <span class="math-container">$u:\mathbb{R}^L_+\rightarrow\mathbb{R}$</span>, price vector <span class="math-container">$p\in \mathbb{R}^L_{++}$</span> and target level of utility <span class="math-container">$\mu\in\mathbb{R}$</span>, expenditure minimisation problem is defined as follows:
<span class="math-container">\begin{eqnarray*}\min_{x\in\mathbb{R}^L_+} &amp; p\cdot x \\ \text{s.t. }&amp; u(x) \geq \mu\end{eqnarray*}</span>
Let <span class="math-container">$x^h(p,\mu)$</span> (Hicksian demand) denotes a solution to the above problem.
Observe that the solution to the above problem is also a solution to the problem below:
<span class="math-container">\begin{eqnarray*}\min_{x\in\mathbb{R}^L_+} &amp; \lambda p\cdot x \\ \text{s.t. }&amp; u(x) \geq \mu\end{eqnarray*}</span>
for any <span class="math-container">$\lambda &gt; 0$</span>.</p>
<p>In other words, <span class="math-container">$x^h(p,\mu) = x^h(\lambda p,\mu)$</span>. Therefore, Hicksian demand is homogeneous of degree <span class="math-container">$0$</span> in prices. Consequently, the expenditure function defined as the optimal level of expenditure <span class="math-container">$e(p, \mu) = p\cdot x^h(p, \mu)$</span> satisfy the following:</p>
<p><span class="math-container">$e(\lambda p, \mu) = \lambda p \cdot x^h(\lambda p, \mu) = \lambda p \cdot x^h( p, \mu) = \lambda e(p, \mu)$</span></p>
<p>and is therefore, homogeneous of degree <span class="math-container">$1$</span> in prices.</p>

CC BY-SA 4.0
2022-09-13T06:30:19.163Z
"observe" is a cheat word in math proofs. Also I think you would need the set of solutions to be identical, not just that those of the first problem also solve the second problem.
Giskard
CC BY-SA 4.0
2022-09-13T06:48:42.320Z
It follows from the above argument that the set of solutions is identical. You need to "observe" more carefully :)
Amit
CC BY-SA 4.0
2022-09-13T06:23:18.917Z
2022-09-13T06:23:18.917Z
Amit
<p>In my assignment I have a Leontief (perfect complements) function u(x,y)=min(x,2y). Keeping utility fixed, we minimize the expenditure. Since we have a Leontief function, at a fixed level of utility u(x,y)=u expenditure is minimized at the angle point, where x=2y. Since utility is fixed at u=min(x,2y), hence u=min(2y,2y)=2y and u=2y=x. Therefore Hicksian demands will be x=u and y=u/2.</p>
<p>The next part of my assignment is to prove that compensated demand function is homogeneous of degree 0 in prices both for this case and in general. However, in my case (as well as for the standard Leontief function) compensated demand function does not depend on prices. It is unlikely that I derived the wrong functions, since for the standard Leontief function compensated demand also does not depend on the prices.</p>

CC BY-SA 4.0
2022-09-12T11:04:57.703Z
Your Hicksian demand functions are not consistent with your utility function.
VARulle
CC BY-SA 4.0
2022-09-12T11:21:41.887Z
How to make them consistent?
Ksenia
CC BY-SA 4.0
2022-09-12T11:25:49.857Z
@Ksenia By doing the derivation correctly. Can you edit the question so that it includes your calculations?
Giskard
CC BY-SA 4.0
2022-09-12T11:50:46.803Z
I accidentally wrote the wrong utility function.
Ksenia
<microeconomics><demand><leontief><perfect-complements>
Homogeneity of compensated demand for Leontief (perfect complements) function
Ksenia
# Homogeneity of compensated demand for Leontief (perfect complements) function

(Asked by: username_0 on 2022-09-12)

In my assignment I have a Leontief (perfect complements) function u(x,y)=min(x,2y). Keeping utility fixed, we minimize the expenditure. Since we have a Leontief function, at a fixed level of utility u(x,y)=u expenditure is minimized at the angle point, where x=2y. Since utility is fixed at u=min(x,2y), hence u=min(2y,2y)=2y and u=2y=x. Therefore Hicksian demands will be x=u and y=u/2.

The next part of my assignment is to prove that compensated demand function is homogeneous of degree 0 in prices both for this case and in general. However, in my case (as well as for the standard Leontief function) compensated demand function does not depend on prices. It is unlikely that I derived the wrong functions, since for the standard Leontief function compensated demand also does not depend on the prices.

**Tags:** microeconomics, demand, leontief, perfect-complements

> **Comments:**
> **username_1**: Your Hicksian demand functions are not consistent with your utility function.  
> **username_0**: How to make them consistent?  
> **username_2**: @username_0 By doing the derivation correctly. Can you edit the question so that it includes your calculations?  
> **username_0**: I accidentally wrote the wrong utility function.  

---

## 2 answers

### Answer by username_1 on 2022-09-12. Score: 1

Since $x({\mathbf p},u)=u$ and $y({\mathbf p},u)=u/2$ do not depend on prices ${\mathbf p}$, homogeneity of degree zero in prices is trivially satisfied for this special case: $x(t{\mathbf p},u)=x({\mathbf p},u)=u$ and $y(t{\mathbf p},u)=y({\mathbf p},u)=u/2$.

For the general case, homogeneity of degree zero follows from the observation that the slope of the budget line is given by $-p\_x/p\_y=-(tp\_x)/(tp\_y)$ for $t>0$.


---

### Answer by username_3 on 2022-09-13. Score: 1

Given a utility function $u:\mathbb{R}^L\_+\rightarrow\mathbb{R}$, price vector $p\in \mathbb{R}^L\_{++}$ and target level of utility $\mu\in\mathbb{R}$, expenditure minimisation problem is defined as follows:
\begin{eqnarray\*}\min\_{x\in\mathbb{R}^L\_+} & p\cdot x \\ \text{s.t. }& u(x) \geq \mu\end{eqnarray\*}
Let $x^h(p,\mu)$ (Hicksian demand) denotes a solution to the above problem.
Observe that the solution to the above problem is also a solution to the problem below:
\begin{eqnarray\*}\min\_{x\in\mathbb{R}^L\_+} & \lambda p\cdot x \\ \text{s.t. }& u(x) \geq \mu\end{eqnarray\*}
for any $\lambda > 0$.

In other words, $x^h(p,\mu) = x^h(\lambda p,\mu)$. Therefore, Hicksian demand is homogeneous of degree $0$ in prices. Consequently, the expenditure function defined as the optimal level of expenditure $e(p, \mu) = p\cdot x^h(p, \mu)$ satisfy the following:

$e(\lambda p, \mu) = \lambda p \cdot x^h(\lambda p, \mu) = \lambda p \cdot x^h( p, \mu) = \lambda e(p, \mu)$

and is therefore, homogeneous of degree $1$ in prices.

> **Comments:**
> **username_2**: "observe" is a cheat word in math proofs. Also I think you would need the set of solutions to be identical, not just that those of the first problem also solve the second problem.  
> **username_3**: It follows from the above argument that the set of solutions is identical. You need to "observe" more carefully :)  

---

