Homogeneity of compensated demand for Leontief (perfect complements) function

Question

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.

Answer 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 2

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.