# Why does the Substitution effect always have to be negative?

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?

• There is more than one way to define substitution effects. And do you want intuition? A formal proof? At which level? Commented Dec 5, 2023 at 19:37
• Not necessarily. You can calculate the following two cases: 1) the good is actually a bad; 2) the preference is not convex. Check what do you find!
– dodo
Commented Dec 5, 2023 at 21:59
• @MichaelGreinecker we were looking at the slutsky equation. i see that it helps to isolate the change in demand due to the change of relative prices only and the change in purchase power of income. the income effect can be negative or positive depending on normal or inferior good, because we adapt income so that purchase power stays constant bla bla. now in our study slides it says that the substitution effect always has to be negative though. and that i don't get. formally nor intuitionally Commented Dec 5, 2023 at 23:07
• i assume we are looking at it a bit more superficially, because there's not much explanation to that part Commented Dec 5, 2023 at 23:08

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, $$(x^d_i(p,M)-x^h_i(p',u(x^d(p,M))))(p_i-p'_i)=(x^h_i(p,u(x^d(p,M)))-x^h_i(p',u(x^d(p,M))))(p_i-p'_i)\leq 0$$.