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The following is an attempt to explain the prices of clamps, using data gathered at Brimfield, 2003 Spring, on the Friday. Please note the special nature of the sample, and do not over-generalize the findings.
The raw data gives the details. I have noted the maker, the model, and the condition for each clamp I saw.
The weather was cool, but not too windy. So I went down for a quick check, which lasted 6 hours, and which involved about eighteen clamps at ten dealers.
See a tabulation of the data. Note that there was only one dealer who specialized in tools, and his pricing was basically a reverse auction. He said that he had very few clamps this year, and they were still there near the close of business.
My basic assumption is that the price of a clamp can be estimated from the product of two functions.
Price = Size_Function * Condition_Function
I constrain the Condition_Function so that "good" is worth 1, and the function is monotonic, that is, better condition must not imply lower prices. The conditions are defined to be poor, fair, good, and very good.
In order to establish a model, we must define what it means for an estimate to "be close" to the asking price. Errors can be defined in either of two ways: absolute, or relative. Absolute error implies we think an estimate that is a dollar off a $5 asking price is as good as an estimate that is a dollar off a $25 asking price. Relative error implies we think that being off by 10% is equally good, no matter the asking price.
The usual technique is to compute the square root of the average of the squares of the errors. (This is called the rms error, for root mean square error.) This implies that we believe a too low estimate is as bad as a too high estimate, and that all estimates are of equal importance.
The rms error is a measure of how close the estimates are. Approximately two thirds of the estimates will be within the rms error of the true value. Clearly, smaller rms errors are better.
It's an easy matter to set up a spread sheet of asking prices, initialize the model parameters, and to use the Excel Solver function to vary the model parameters in order to the minimize the rms error.
Gross? Dealers asked $15 to $25 for any clamp that worked.
If we assume that each inch of jaw adds an equal value to the clamp, then we attempt to fit a linear function, to connect size and value. Usually, the fit is poor, because there are only two free parameters. This year, the root mean square error was 32%.
Size is irrelevant; all clamps are about $21 (actually, $20.89).
Even if we consider $30 for a 6 incher to be an outlier (the single offering of a very specialized dealer), and remove it, nothing much shifts.
Poor condition is worth 48% of Fair condition, and Fair condition is equal to Good condition.
If we assume that each size of jaw has its own value, then we have a fitted function, to connect size and value. Usually, the fit is excellent, because there are a great many free parameters. This year, the root mean square error was 21%.
Six inch models are worth $22; eight and a half inch models are worth $25; ten inch models are worth $22; sixteen inch models are worth $15. The other models have too little data to use.
Poor condition is worth 40% of Fair condition, and Fair condition is equal to Good condition.
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