<p>Here is a short argument <a href="https://doi.org/10.1007/s00199-007-0233-8" rel="noreferrer">due to</a> Maskin and Roberts. I'll give it in the simpler context of a pure exchange economy.</p>
<p>Suppose you have a Pareto efficient allocation, and an equilibrium <span class="math-container">$(x,p)$</span> exists starting with this allocation as the endowment specification <span class="math-container">$e$</span>. Then taking these equilibrium prices and everyone just consuming their endowment is an equilibrium; <span class="math-container">$(e,p)$</span> is an equilibrium. Indeed, since an agent's endowment is always in the budget set, and everyone chooses optimally, we must have <span class="math-container">$x_i\succeq_i e_i$</span> for every agent <span class="math-container">$i$</span>. But we cannot have <span class="math-container">$x_i\succ_i e_i$</span> for some <span class="math-container">$i$</span>, because then <span class="math-container">$x$</span> would be a Pareto improvement over <span class="math-container">$e$</span>, which is impossible by assumption. So <span class="math-container">$x_i\sim_i e_i$</span> for every <span class="math-container">$i$</span>, and just consuming the endowment is optimal given <span class="math-container">$p$</span>. Since <span class="math-container">$e$</span> is also clearly feasible, we are done.</p>
<p>The argument is so simple because I have just assumed that the equilibrium <span class="math-container">$(x,p)$</span> exists. The standard proofs of the second welfare theorem include an existence proof, and that is also why the usual additional assumptions are needed.</p>

CC BY-SA 4.0
2023-02-11T17:08:41.867Z
2023-02-11T17:08:41.867Z
Michael Greinecker
<p>Could you provide the intuition behind the second welfare theorem, and prove in a few rows the theorem itself?</p>

<microeconomics><general-equilibrium>
Second welfare theorem
John M.
# Second welfare theorem

(Asked by: username_0 on 2023-02-11)

Could you provide the intuition behind the second welfare theorem, and prove in a few rows the theorem itself?

**Tags:** microeconomics, general-equilibrium


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## 1 answers

### Answer by username_1 on 2023-02-11. Score: 5 | Accepted answer

Here is a short argument [due to](https://doi.org/10.1007/s00199-007-0233-8) Maskin and Roberts. I'll give it in the simpler context of a pure exchange economy.

Suppose you have a Pareto efficient allocation, and an equilibrium $(x,p)$ exists starting with this allocation as the endowment specification $e$. Then taking these equilibrium prices and everyone just consuming their endowment is an equilibrium; $(e,p)$ is an equilibrium. Indeed, since an agent's endowment is always in the budget set, and everyone chooses optimally, we must have $x\_i\succeq\_i e\_i$ for every agent $i$. But we cannot have $x\_i\succ\_i e\_i$ for some $i$, because then $x$ would be a Pareto improvement over $e$, which is impossible by assumption. So $x\_i\sim\_i e\_i$ for every $i$, and just consuming the endowment is optimal given $p$. Since $e$ is also clearly feasible, we are done.

The argument is so simple because I have just assumed that the equilibrium $(x,p)$ exists. The standard proofs of the second welfare theorem include an existence proof, and that is also why the usual additional assumptions are needed.


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