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I've been going over the concept of Walras's Law and often references to Say's Law. I've heard and have seen online that they are really the same thing however by looking at the definitions I notice a common difference.

Walras’s law:
“For any price vector $p$, we have $𝑝𝑧(𝑝)≡0$; the value of the excess demand is identically zero.”

Where excess demand $z(p)$ is defined as: $$z(p)=\sum_{i=1}^n(x_i(p,m)-\omega_i)$$ where $x_i(p,m)$ is our marshallian demand for good $i$ and $\omega_i$ is the initial endowment of good $i$.

Conversely,

Say's Law:
"aggregate production necessarily creates an equal quantity of aggregate demand" (from Wikipedia).

or $$Q_s(p)=Q_d(p)$$

Would it be safe to say that Walras's Law makes reference to only consumer demand (which is transferred through a market and is not "consumed" by the firm themselves via inventory investment), While Say's Law is a statement regarding both demand by consumers and producers regardless of whether or not it goes from producers to consumers or producers back to themselves in form of inventory investment?

The rationale for this argument comes from the fact that we do conciser inventory investment to be included in demand even though it does not change hands in a market.

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The way you have defined excess demand, it is only consumer excess demand. But Walras's law holds in any private ownership economy at all prices (at which demand and supply are well defined). Walras's law is basically equivalent to consumers spending their budget fully.

Let there be $l$ commodities, so every commodity bundle is an element of $\mathbb{R}^l$. There are $m$ consumers and $n$ firms. The set of production plans firm $j$ can produce in terms of net-output is $Y_j$. With this convention, if $p\in\mathbb{R}^l$ is the price system and $y\in Y_j$ a production plan, $p\cdot y=\sum_{k=1}^l p_k y_k$ is the resulting profit. Every consumer has an endowment $\omega_i\in\mathbb{R}^l$ and a (potentially zero) share $\theta_{ij}$ in firm $j$ and the resulting profit. Every firm is privately owned, so each firm's shares sums to one, $\sum_{i=1}^m\theta_{ij}=1$.

At the price system $p$, the budget of each consumer is $p\cdot \omega_i +\sum_{j=1}^n \theta_{ij} p\cdot y_j$. If the consumer $i$ demands $x_i$ and spends their whole budget, we have $p\cdot x_i=p\cdot \omega_i +\sum_{j=1}^n \theta_{ij} p\cdot y_j$. Summing up and using that prices are linear, $$p\cdot\sum_{i=1}^m x_i=p\cdot\sum_{i=1}^m\omega_i+p\cdot \sum_{i=1}^m\sum_{j=1}^n \theta_{ij}y_j$$ $$=p\cdot\sum_{i=1}^m x_i=p\cdot\sum_{i=1}^m\omega_i+p\cdot \sum_{j=1}^n\sum_{i=1}^m \theta_{ij}y_j$$ $$=p\cdot\sum_{i=1}^m x_i=p\cdot\sum_{i=1}^m\omega_i+p\cdot \sum_{j=1}^ny_j$$ $$=p\cdot\bigg(\sum_{i=1}^m x_i-\sum_{i=1}^m\omega_i-\sum_{j=1}^ny_j\bigg)=0,$$ so the value of aggregate excess demand is zero and Walras's law holds. But nothing in the argument requires prices to clear markets so that supply equals demand, $$\sum_{i=1}^m x_i-\sum_{i=1}^m\omega_i-\sum_{j=1}^ny_j=0.$$ The latter is clearly a sufficient condition for excess demand to have the value zero, but very far from necessary.

Walras's law is a provable result under some standard assumptions (that guarantee that consumers spend their whole budget), but Say's law is not a formal result in any way.

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  • $\begingroup$ This may be a good discussion of Walras' law, but does it engage with the question (on whether Walras' law and Say's law are equivalent)? You write that Say's law is 'not a formal result in any way', but is it similar in content to Walras' law? $\endgroup$ – afreelunch Oct 23 '18 at 16:30
  • $\begingroup$ @afreelunch Whatever Say's law is meant to say exactly, I'm sure nobody would see it as amounting to people spending their budget. $\endgroup$ – Michael Greinecker Oct 23 '18 at 17:44

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