Why does the Substitution effect always have to be negative?

Question

I understand why the income effect can be positive or negative depending if the good is normal or inferior. but why does the substitution effect always have to be non-positive?

Answer 1

Consider the following utility maximisation problem of the consumer with utility function $u:\mathbb{R}^n_+\rightarrow\mathbb{R}$: \begin{eqnarray*} \max_{x\in\mathbb{R}^n_+} & u(x) \\ \text{s.t.} & \ p\cdot x\leq M\end{eqnarray*} where $p\in\mathbb{R}^n_{++}$ and $M\in\mathbb{R}_{+}$. Let $x^d(p,M)$ denotes the solution to the above problem. It is referred to as demand.

Consider the following expenditure minimisation problem of the consumer with utility function $u:\mathbb{R}^n_+\rightarrow\mathbb{R}$: \begin{eqnarray*} \min_{x\in\mathbb{R}^n_+} & p\cdot x \\ \text{s.t.} & \ u(x)\geq \mu\end{eqnarray*} where $p\in\mathbb{R}^n_{++}$ and $\mu\in\mathbb{R}$. Let $x^h(p,\mu)$ denotes the solution to the above problem. It is referred to as Hicksian demand.

Theorem 1. Suppose consumer's preferences represented by $u$ satisfy local non-satiation (LNS), then $x^h(p, u(x^d(p,M)))=x^d(p,M)$

Proof. Simple. Try this on your own.

Now Consider a change of price to $p'\in\mathbb{R}^n_{++}$.

Lemma 2: For all $p\in\mathbb{R}^n_{++}, p'\in\mathbb{R}^n_{++}, \mu\in\mathbb{R}$, the following inequality is true: $(x^h(p,\mu)-x^h(p',\mu))\cdot(p-p')\leq 0$

Proof: Observe that $p\cdot x^h(p,\mu)\leq p\cdot x^h(p',\mu)$ and $p'\cdot x^h(p',\mu)\leq p'\cdot x^h(p,\mu)$. Adding them, we obtain $p\cdot x^h(p,\mu)+p'\cdot x^h(p',\mu)\leq p\cdot x^h(p',\mu)+p'\cdot x^h(p,\mu)$ or equivalently, $(x^h(p,\mu)-x^h(p',\mu))\cdot(p-p')\leq 0$.

Definition. Hicksian demand curve for commodity $i$ is defined as the relationship between $x_i^h(p,\mu)$ and $p_i$ holding other prices, and $\mu$ fixed.

Definition. Hicksian demand curve for commodity $i$ is said to have a non-positive slope if for all $p=(p_1,p_2,\ldots, p_i, \ldots, p_n)\in\mathbb{R}^n_{++}$, $p'=(p_1,p_2,\ldots, p_i', \ldots, p_n)\in\mathbb{R}_{++}^n$ and $\mu\in\mathbb{R}$, $(x^h_i(p,\mu)-x^h_i(p',\mu))(p_i-p'_i)\leq 0$.

Theorem 3. Hicksian demand curve for commodity $i$ has a non-positive slope.

Proof. Consider any $p=(p_1,p_2,\ldots, p_i, \ldots, p_n)\in\mathbb{R}^n_{++}$, $p'=(p_1,p_2,\ldots, p_i', \ldots, p_n)\in\mathbb{R}_{++}^n$ and $\mu\in\mathbb{R}$, $(x^h_i(p,\mu)-x^h_i(p',\mu))(p_i-p'_i)=(x^h(p,\mu)-x^h(p',\mu))\cdot(p-p')\leq 0$ (by Lemma)

Suppose only price of $i$ changes from $p_i$ to $p_i'$.

Definition. Hicksian Substitution effect for commodity $i$ is $x_i^d(p,M)-x_i^h(p',u(x^d(p,M)))$.

Definition. Hicksian Substitution Effect for commodity $i$ is said to be non-positive if for all $p=(p_1,p_2,\ldots, p_i, \ldots, p_n)\in\mathbb{R}^n_{++}$, $p'=(p_1,p_2,\ldots, p_i', \ldots, p_n)\in\mathbb{R}_{++}^n$ and $M\in\mathbb{R}_+$, $(x_i^d(p,M)-x_i^h(p',u(x^d(p,M))))(p_i-p'_i)\leq 0$.

Theorem 4. Suppose consumer's preferences represented by $u$ satisfy local non-satiation, then Hicksian Substitution Effect for commodity $i$ is non-positive.

Proof. By Theorem 3, Hicksian demand curve for commodity $i$ has a non-positive slope i.e. $(x^h_i(p,\mu)-x^h_i(p',\mu))(p_i-p'_i)\leq 0$ for all $\mu, p, p'$. By Theorem 1, $x^h(p, u(x^d(p,M))=x^d(p,M)$. Therefore, $(x^d_i(p,M)-x^h_i(p',u(x^d(p,M))))(p_i-p'_i)=(x^h_i(p,u(x^d(p,M)))-x^h_i(p',u(x^d(p,M))))(p_i-p'_i)\leq 0$.