Top of Site > Clamps as Things > Prices for Clamps > 2001 Fall : All Models


Table of Contents

The following is an attempt to explain the prices of clamps, using data gathered at Brimfield, Fall 2001, on the Saturday. Please note the special nature of the sample, and do not over-generalize the findings.

Intro to a Model of Prices

Data

The raw data gives the details. I have noted the maker, the model, and the condition for each clamp I saw.

Assumptions

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.

The following present the results for various models.

Findings for a Linear Model, (2001 September)

A simple model is that each inch adds a certain value to a clamp. That is, the Size Function is linear. (Everybody knows that dealers charge more for big clamps than for little clamps, right?). An alternative is explored in the Fitted Model

Size Function

The answer depends on whether we use absolute errors, or relative errors. To minimize rms of the absolute errors, compute the value as $6.36 for existance, plus $0.78 for each inch.
To minimize rms of the relative errors, compute the value as $0.88 for existence, plus $0.85 for each inch.

The tabulated value for each size is

Value of Size, Linear Model
Using Absolute Error $10.25 $11.02 $11.80 $12.58 $14.13 $15.68 $17.24 $20.34
Using Relative Error $05.11 $05.96 $06.81 $07.65 $09.35 $11.04 $12.73 $16.12
Jaw Size (inches) 5 6 7 8 10 12 14 18

Condition Function

The condition function has constant value 1 for poor, fair, and good, that is, condition does not impact price. It did not matter whether we consider absolute errors or relative errors. (Well, actually, for relative error, the value for poor was 0.99; call it 1, and be off a dime on a $10 prediction.)


Findings for a Fitted Model, (2001 September)

A more complicated model is that each size of clamp has a distinctive value. That is, the Size Function is fitted (Everybody knows that dealers charge a lot for some popular sizes of clamps than for other sizes, right?)

An alternative is explored in the Linear Model.

Because a fitted model has more parameters (compared to a linear model) that it can adjust, we expect that the rms for a fitted model to be smaller than the rms for a linear model.

Size Function

The exact answer depends on whether we use absolute errors, or relative errors. But both agree that the mid-sizes (10, 12, and 14 inches) were most highly valued, with the 18 inch size next.

The tabulated estimate for each asking price is

Using Absolute Error $10.50 $06.00 $08.50 $05.00 $20.63 $15.75 $17.00 $14.00
Using Relative Error $07.24 $06.00 $06.07 $05.08 $17.34 $13.27 $13.43 $11.89
Jaw Size (inches) 5 6 7 8 10 12 14 18

Condition Function

The condition function has constant value 1 for poor, fair, and good, that is, condition does not impact price. It did not matter whether we considered absolute errors or relative errors.


Summary of Results

Asking Linear
abs
Linear
rel
Fitted
abs
Fitted
rel
$ 5.00 $11.80 $ 6.81 $ 8.50 $ 6.07
$ 5.00 $12.58 $ 7.65 $ 5.00 $ 5.08
$ 5.00 $12.58 $ 7.65 $ 5.00 $ 5.08
$ 5.00 $11.80 $ 6.81 $ 8.50 $ 6.07
$ 6.00 $10.25 $ 5.11 $10.50 $ 7.24
$ 6.00 $11.02 $ 5.96 $ 6.00 $ 6.00
$ 6.00 $11.02 $ 5.96 $ 6.00 $ 6.00
$ 8.00 $17.24 $12.73 $17.00 $13.43
$10.00 $20.34 $16.12 $14.00 $11.89
$10.00 $15.68 $11.04 $15.75 $13.27
$12.00 $11.80 $ 6.81 $ 8.50 $ 6.07
$12.00 $11.80 $ 6.81 $ 8.50 $ 6.07
$12.00 $14.13 $ 9.35 $20.63 $17.34
$14.00 $14.13 $ 9.35 $20.63 $17.34
$14.00 $15.68 $11.04 $15.75 $13.27
$15.00 $17.24 $12.73 $17.00 $13.43
$15.00 $17.24 $12.73 $17.00 $13.43
$15.00 $15.68 $11.04 $15.75 $13.27
$15.00 $14.13 $ 9.35 $20.63 $17.34
$15.00 $10.25 $ 5.11 $10.50 $ 7.24
$18.00 $17.24 $12.73 $17.00 $13.43
$18.00 $17.24 $12.73 $17.00 $13.43
$18.00 $20.34 $16.12 $14.00 $11.89
$20.00 $17.24 $12.73 $17.00 $13.43
$24.00 $14.13 $ 9.35 $20.63 $17.34
$24.00 $15.68 $11.04 $15.75 $13.27
$24.00 $14.13 $ 9.35 $20.63 $17.34
$24.00 $14.13 $ 9.35 $20.63 $17.34
$24.00 $14.13 $ 9.35 $20.63 $17.34
$25.00 $17.24 $12.73 $17.00 $13.43
$28.00 $14.13 $ 9.35 $20.63 $17.34
average average average average average
$14.58 $14.06 $ 9.59 $14.36 $11.69
rms rms rms rms
$ 6.39 % 43. $ 4.54 % 31.

last revised and validated

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