<p>You need to add <span class="math-container">$x_0 v_0$</span> to your social welfare, where <span class="math-container">$v_0$</span> is the value the seller assigns to keeping the good and <span class="math-container">$x_0=1$</span> in that case. Then, the efficient (social-welfare-maximizing) allocation is that the highest-value buyer gets the good if her value is above <span class="math-container">$v_0$</span> - otherwise it is efficient that the seller keeps the good.</p>
<p>Similarly, in a <a href="https://economics.stackexchange.com/questions/40412/auction-with-one-buyer-and-multiple-sellers/40435#40435">procurement auction</a>, we have one buyer who values the good to be procured at <span class="math-container">$v_0$</span> and <span class="math-container">$n$</span> sellers who can produce the good at private cost <span class="math-container">$c_i$</span>. Social welfare is then
<span class="math-container">$$\sum_{i=1}^n x_i (v_0 - c_i),$$</span>
and, hence, it is efficient to let the lowest-cost firm produce the good given its cost is below <span class="math-container">$v_0$</span> - otherwise it is efficient that the good is not produced.
For sufficiently high values <span class="math-container">$v_0$</span>, you can also define it as <span class="math-container">$\sum_{i=1}^n x_i (- c_i).$</span></p>

CC BY-SA 4.0
2020-11-06T18:11:29.177Z
2020-11-06T18:11:29.177Z
Bayesian
<p>In single item forward-auctions, the <em>social welfare</em> is defined as:
<span class="math-container">$$\sum_{i=1}^{n} v_i x_i$$</span>
where <span class="math-container">$v_i$</span> is buyer <span class="math-container">$i$</span>'s valuation for the item, and <span class="math-container">$x_i$</span> is a binary variable indicating whether or not <span class="math-container">$i$</span> gets the item.</p>
<p>What is definition of social welfare in a reverse auction(procurement auction)?</p>

<auctions>
What's the definition of social welfare in a procurement auction?
Mengfan Ma
# What's the definition of social welfare in a procurement auction?

(Asked by: username_0 on 2020-11-06)

In single item forward-auctions, the *social welfare* is defined as:
$$\sum\_{i=1}^{n} v\_i x\_i$$
where $v\_i$ is buyer $i$'s valuation for the item, and $x\_i$ is a binary variable indicating whether or not $i$ gets the item.

What is definition of social welfare in a reverse auction(procurement auction)?

**Tags:** auctions

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## 1 answers

### Answer by username_1 on 2020-11-06. Score: 4 | Accepted answer

You need to add $x\_0 v\_0$ to your social welfare, where $v\_0$ is the value the seller assigns to keeping the good and $x\_0=1$ in that case. Then, the efficient (social-welfare-maximizing) allocation is that the highest-value buyer gets the good if her value is above $v\_0$ - otherwise it is efficient that the seller keeps the good.

Similarly, in a [procurement auction](https://economics.stackexchange.com/questions/40412/auction-with-one-buyer-and-multiple-sellers/40435#40435), we have one buyer who values the good to be procured at $v\_0$ and $n$ sellers who can produce the good at private cost $c\_i$. Social welfare is then
$$\sum\_{i=1}^n x\_i (v\_0 - c\_i),$$
and, hence, it is efficient to let the lowest-cost firm produce the good given its cost is below $v\_0$ - otherwise it is efficient that the good is not produced.
For sufficiently high values $v\_0$, you can also define it as $\sum\_{i=1}^n x\_i (- c\_i).$


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