Relationships between Demand, Indirect Utility, and Expenditure Functions

Assumption

1. $u(\cdot )$ is a continuous utility function representing the locally nonsatiated preferences $\succsim$ (defined on the consumption set $X={\mathbb{R}}_{+}^L$​).
2. $p\gg 0$.
3. $\succsim$ is strictly convex, so that the Walrasian and Hicksian demands, x(p, w) and h(p, u), are single-valued.

Hicksian Demand and the Expenditure Function

Proposition 3.G.1 (Relationships between HD and EF). Suppose that (1) $u(\cdot )$ is a continuous utility function representing the locally nonsatiated preferences $\succsim$ (defined on the consumption set $X={\mathbb{R}}_{+}^L$​), that (2) $p\gg 0$, and that (3) $\succsim$ is strictly convex, so that the Walrasian and Hicksian demands, x(p, w) and h(p, u), are single-valued. For all $p$ and $u$​, the Hicksian demand $h(p,u)$​​ is the derivative vector of the expenditure function with respect to prices:

$$\begin{array}{}\text{(3.G.1)}& h(p,u)={\mathrm{\nabla }}_{p}e(p,u).\end{array}$$

That is, $h_{\ell }(p,u)=\partial e(p,u)/\partial p_{\ell }$​ for all $\ell =1,...,L$.

Proposition 3.G.2 (properties of the price derivatives of the Hicksian demand function $D_{p}h(p,u)$). Suppose that

1. $u(\cdot )$ is a continuous utility function representing the locally nonsatiated preferences $\succsim$ (defined on the consumption set $X={\mathbb{R}}_{+}^L$​)
2. $p\gg 0$.
3. $\succsim$ is strictly convex, so that the Walrasian and Hicksian demands, x(p, w) and h(p, u), are single-valued.
4. $h(\cdot ,u)$ is continuously differentiable at $(p,u)$, and denote its $L\times L$ derivative matrix by $D_{p}h(p,u)$.

Then

1. $D_{p}h(p,u)=D_p^{2}e(p,u)$.
2. $D_{p}h(p,u)$ is a negative semidefinite matrix.
3. $D_{p}h(p,u)$​ is a symmetric matrix.
4. $D_{p}h(p,u)p=0$.

Definition (substitutes and complements). We define two goods $\ell$ and $k$ to be substitutes at $(p,u)$ if $\partial h_{\ell }(p,u)/\partial p_k\ge 0$ and complements if this derivative is nonpositive [when Walrasian demands have these relationships at $(p,w)$, the goods are referred to as gross substitutes and gross complements at $(p,w)$, respectively].

The Hicksian and Walrasian Demand Functions

Although the Hicksian demand function is not directly observable (it has the consumer’s utility level as an argument), we now show that $D_{p}h(p,u)$ can nevertheless be computed from the observable Walrasian demand function $x(p,w)$​ (its arguments are all observable in principle).

Proposition 3.G.3 (The Slutsky Equation).

Suppose that

1. $u(\cdot )$ is a continuous utility function representing the locally nonsatiated preferences $\succsim$ (defined on the consumption set $X={\mathbb{R}}_{+}^L$​)
2. $p\gg 0$.
3. $\succsim$ is strictly convex, so that the Walrasian and Hicksian demands, x(p, w) and h(p, u), are single-valued.
4. $h(\cdot ,u)$ is continuously differentiable at $(p,u)$, and denote its $L\times L$ derivative matrix by $D_{p}h(p,u)$.

Then for all $(p,w)$, and $u=v(p,w)$,we have

$$\begin{array}{}\text{(3.G.3)}& \frac{\partial h_{\ell }(p,u)}{\partial p_k}=\frac{\partial x_{\ell }(p,w)}{\partial p_k}+\frac{\partial x_{\ell }(p,w)}{\partial w}{x}_k(p,w)\text{for all }\ell ,k\end{array}$$$$\underset{\text{total effect }}{\underbrace{\frac{\partial x_l(p,w)}{\partial p_k}}}=\underset{\text{substitution effect }}{\underbrace{\frac{\partial h_l(p,w)}{\partial p_k}}}-\underset{\text{income effect }}{\underbrace{\frac{\partial x_l(p,w)}{\partial w}{x}_k(p,w)}}$$

• Intuition: If $p_k$ increases, two effects on demand for good $\ell$:

  • Substitution effect:$\frac{\partial h_{\ell }(p,w)}{\partial p_k}$
    • Movement along original indifference curve.
    • Response to change in prices, holding utility fixed.
  • Income effect: $\frac{\partial xl(p,w)}{\partial w}{x}_k(p,w)$
    • Movement from one indifference curve to another.
    • Response to change in income, holding prices fixed.

  Both the substitution effect and the income effect can have either sign.

  • Substitution effect is positive for substitutes and negative for complements.
  • Income effect is negative for normal goods and positive for inferior goods.

  By symmetry of Slutsky matrix, good $i$​​ is a substitute for $j$​, $⟺$ $j$ is a substitute for $i$​.

  Not true that $i$ is a gross substitute for $j$ , $j$​ is a gross substitute for $i$​.

  • Income effects are not symmetric.

or equivalently, in matrix notation,

$$\begin{array}{}\text{(3.G.4)}& D_{p}h(p,u)=D_{p}x(p,w)+D_{w}x(p,w)x(p,w{)}^T\end{array}$$

Walrasian Demand and the Indirect Utility Function

Proposition 3.G.4 (Roy's Identity).

Suppose that

1. $u(\cdot )$ is a continuous utility function representing a locally nonsatiated and strictly convex preference relation $\succsim$ defined on the consumption set $X={\mathbb{R}}_{+}^L$.
2. the indirect utility function is differentiable at $(p,w)\gg 0$.

Then

$$x(\bar{p},\bar{w})=-\frac{1}{{\mathrm{\nabla }}_{w}v(\bar{p},\bar{w})}{\mathrm{\nabla }}_p(\bar{p},\bar{w}).$$

That is, for every $\mathcal{l}=1,...,L$:

$$x_{\ell }(\bar{p},\bar{w})=-\frac{\partial v(\bar{p},\bar{w})/\partial p_{\ell }}{\partial v(\bar{p},\bar{w})/\partial w}.$$