# What's the definition of social welfare in a procurement auction?

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)?

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, 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 $$\sum_{i=1}^n x_i (- c_i).$$