# Marginal Revenue Interpretation of Auctions

I'm reading Klemperer's survey on auction theory (pages 17-18) which discusses the relation between traditional microeconomic price theory and the revenue equivalence result. Firstly, I don't quite understand the analogy: marginal revenue is the revenue due to additional quantity sold, but an auction might just sell a single non-divisible object.

In particular, it is helpful to focus on bidders’ ‘‘marginal revenues’’. Imagine a firm whose demand curve is constructed from an arbitrarily large number of bidders whose values are independently drawn from a bidder’s value distribution. When bidders have independent private values, a bidder’s ‘‘marginal revenue’’ is defined as the marginal revenue of this firm at the price that equals the bidder’s actual value.

I'm not sure why "the marginal revenue of this firm at the price that equals the bidder’s actual value". Does this not assume that the bidder actually bids his actual value? This surely isn't the case for e.g. first-price auctions.

Bulow and Roberts follow Myerson to show that under the assumptions of the revenue equivalence theorem the expected revenue from an auction equals the expected marginal revenue of the winning bidder(s).

I'm confused with this (which probably stems from my confusion with the earlier parts): if you sell 1 item, then trivially the marginal revenue equals the expected revenue. Surely I am missing something here.

## 1 Answer

This analogy is due to Bulow and Roberts.

Think of a diagram from Econ 101 with price on the vertical axis and quantity on the horizontal axis and consider a monopolist with constant marginal cost of zero. Given a demand curve, you can draw the marginal revenue and find the monopoly quantity where this function intersects the axis.

Now take that same diagram and interpret "quantity" as "probability that a buyer accepts price $$p$$", i.e., $$q = 1- F(p)$$. Now, maximize revenue $$pq=q F^{-1}(1-q)$$ wrt $$q$$ to reach some form of the well known marginal revenue = marginal cost. The derivative of the revenue above is $$\frac{\partial pq}{\partial q} = F^{-1}(1-q) - \frac{q}{f(F^{-1}(1-q)} = v - \frac{1-F(v)}{f(v)},$$ where I replaced $$q=1- F(p)$$ and set $$p=v$$. This is Myerson's virtual value. The expected revenue of an optimal auction is the expectation of the highest virtual value. The part $$\frac{1-F(v)}{f(v)}$$ is the information rent, which takes care of the fact that you don't bid your true value in a first-price auction.