How do I calculate price? October 5, 2012 9:53 AMSubscribe

Introduction to Markets/Auctions: I need a short book or paper that covers bids / asks

I have some limited data on historical markets: selling offers, and whether/when they expired/sold. I would like to use this to construct a theory for my optimal selling price of a given volume.

This seems like a simple task, but it'd be nice to see the logic worked out clearly. Is there a go-to text for this?

I think you would probably get more useful responses if you could be a little more specific. For example, there is a huge literature on auction theory in economics (where it's considered a subfield of game theory), and one (of the most) important insight(s) of that literature is that the specifics (in how the market is set up, the type and number of bidders, etc.) turn out to matter a lot. If you're looking at a financial market, you may want to consult literature specific for that specific type of market.

That said, I'm not even sure this is actually an auction problem, so here I will give you my basic economic/data crunching perspective. First, I will assume that your data looks something like (price you offer, quantity you offer, sold or not dummy). On the whole, I think that your problem simplifies significantly if you could just choose a single per unit price, and then charge a total of unit price*quantity. To test whether this is possible, I would run the following regression (as a kind of "diagnostic test"):

(sold dummy) = b0 + b1*(price/quantity) + b2*quantity + u

You can dress up this regression buy using, say, a probit/logit model but I doubt it'd make a huge difference.

In this model, I'd expect b1 to be negative -- the lower the unit price the higher the probability that the buyer accepts the trade. If b2=0, then the probability of buying depends only on the unit price (price/quantity) and you can just choose a single price (yay!). If b2 is significantly different from b2 from zero, then you're in one of these cases:

(1) If b2 is negative, then buyers wanting larger quantities are less likely to accept a given unit price, i.e., they want a quantity discount. This makes sense.

(2) If b2 is positive, then buyers wanting larger quantities are more likely to accept a given unit price, i.e., they're ready to take a "price penalty". Unless your market is super illiquid, this would be a very strange result as they could simply split up their purchase.

If you're in case (1), you have a simple model of the probability of acceptance of different offers. You can include some (quantity)^2, (quantity)^3 or maybe go non-parametric (probably not so good if you have little data) to extend your model.

The second part of your problem involves setting up a maximization problem to find the optimal price(s). How to do this depends on your circumstances and objectives. Do you have a given quantity of stuff to sell in order to give you the highest revenue? Or can you provide more stuff at some (variable?) cost and you want to maximize profit? What is the cost of the good not being sold (is it perishable, or does it depreciate?)? Once you know stuff like this, you can set up and solve a problem to pick either a per-unit price (or a price for each quantity level, i.e., a function) that maximizes the stuff you care about (profit, revenue, etc.). posted by yonglin at 1:05 PM on October 5, 2012

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a lot. If you're looking at a financial market, you may want to consult literature specific for that specific type of market.That said, I'm not even sure this is actually an auction problem, so here I will give you my basic economic/data crunching perspective. First, I will assume that your data looks something like (price you offer, quantity you offer, sold or not dummy). On the whole, I think that your problem simplifies significantly if you could just choose a single per unit price, and then charge a total of unit price*quantity. To test whether this is possible, I would run the following regression (as a kind of "diagnostic test"):

(sold dummy) = b0 + b1*(price/quantity) + b2*quantity + u

You can dress up this regression buy using, say, a probit/logit model but I doubt it'd make a huge difference.

In this model, I'd expect b1 to be negative -- the lower the unit price the higher the probability that the buyer accepts the trade. If b2=0, then the probability of buying depends only on the unit price (price/quantity) and you can just choose a single price (yay!). If b2 is significantly different from b2 from zero, then you're in one of these cases:

(1) If b2 is negative, then buyers wanting larger quantities are less likely to accept a given unit price, i.e., they want a quantity discount. This makes sense.

(2) If b2 is positive, then buyers wanting larger quantities are more likely to accept a given unit price, i.e., they're ready to take a "price penalty". Unless your market is super illiquid, this would be a very strange result as they could simply split up their purchase.

If you're in case (1), you have a simple model of the probability of acceptance of different offers. You can include some (quantity)^2, (quantity)^3 or maybe go non-parametric (probably not so good if you have little data) to extend your model.

The second part of your problem involves setting up a maximization problem to find the optimal price(s). How to do this depends on your circumstances and objectives. Do you have a given quantity of stuff to sell in order to give you the highest revenue? Or can you provide more stuff at some (variable?) cost and you want to maximize profit? What is the cost of the good not being sold (is it perishable, or does it depreciate?)? Once you know stuff like this, you can set up and solve a problem to pick either a per-unit price (or a price for each quantity level, i.e., a function) that maximizes the stuff you care about (profit, revenue, etc.).

posted by yonglin at 1:05 PM on October 5, 2012