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As far as I know, both of the first welfare theorem and the Walras law are closely tied to the invisible hand. what is the difference between them?

thank you very much for your help

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Walras' law describes market equilibria conditions, which states roughly speaking that if there exists within an exchange economy market equilibria for $n-1$ good markets, then the last good market $n$ is also in equilibrium. Thus, we have a market system with $n$ equations and $n-1$ independent variables, where the last market is linear dependent from the other $n-1$ markets.

Now, the first Welfare Theorem of Arrow states that for every exchange economy with convex preferences, every Walrasian equilibrium is Pareto efficient.

The implication of the first Welfare Theorem is that we need no central planner to achieve a Pareto efficient outcome in an exchange economy. The Pareto efficient outcome is based on decentralized decisions within the exchange economy. If we now introduce transaction costs and incomplete information into the system, then it should be obvious that every decentralized economy is doing better than every centralized one due to the information problem.

Update:

I forget to mention some interesting implications of Walras' law. The law reveals to us that we have only one degree of freedom. This implies that a central instance can only fix one price level, for instance, that of the numeraire good, the others prices of the markets are determined by the market forces. Hence, a central instance has no control on them.

To make this argument more precise, let us apply this law to a central bank policy. We have only one degree of freedom, the central bank can now fix either the interest rate of the money market or a price index, but not both. If it fixes the interest rate, it has to accept the price levels and therefore the inflation rate, and vice versa for the price index. This also implies that a central bank cannot simultaneously targeting the price index and the unemployment rate of an economy.

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They are hugely different!! The Walras law is a mathematical result that simplifies the task of characterizing prices and maybe it’s deeper meaning is that price levels are irrelevant in a perfectly competitive economy, rather relative prices is what matters.

In contrast, the first welfare theorem is way more than a mathematical result. It states that any equilibrium of a perfectly competitive market is Pareto efficient. This one is indeed related to the invisible hand argument in favor of capitalism, It’s implication is basically the markets are awesome, since they always lead to socially desirable outcomes.

Of course both theorems depend on assumptions (as they should), and these are debatable, but their implications are hugely distinct.

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    $\begingroup$ What you claim to be Walras's law is just a consequence of the law. $\endgroup$ – Giskard May 17 at 7:23
  • $\begingroup$ en.wikipedia.org/wiki/Walras%27s_law ... In general equilibrium, if budget constraints are binding, the value of excess demand sums to zero. I don't really see how am I mischaracterizing it, do you mind explaining, or otherwise reversing your vote? $\endgroup$ – Regio May 18 at 14:08
  • $\begingroup$ I did not vote on this answer. (I can prove it by adding a downvote if you wish.) $\endgroup$ – Giskard May 18 at 14:51
  • $\begingroup$ 1. In your comment you write about budget constraints, in your answer you do not. 2. The Wikipedia article you linked gives the actual statement of Walras's law: "[in equilibrium] the values of excess demand (or, conversely, excess market supplies) must sum to zero." $\endgroup$ – Giskard May 18 at 14:53
  • $\begingroup$ So it is more of a mathematical result (or technical result if you want), rather than a result with deep economic intuition. I understand all theorems are mathematical results, but I'm sure you understand that some have larger economic implications than others, and the first welfare theorem is clearly way more than a technical result, and that was the point of my answer. As of your voting, I concede I confused correlation with causality, but feel free to vote as your judgement dictates :) . $\endgroup$ – Regio May 18 at 15:00

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