<p>Suppose there is an item that deteriorates with use, its condition expressed as a number between <span class="math-container">$0$</span> (completely broken condition) and <span class="math-container">$1$</span> (perfect condition), for which I want to assign a monetary value.  To make the units nicer, let's think of the condition also as a monetary value; i.e., someone has guaranteed to pay me <span class="math-container">$T$</span> dollars if I want to sell the item at any time, where <span class="math-container">$T$</span> is its condition.  <span class="math-container">$M(T)$</span> is an implicit multiplicative markup on <span class="math-container">$T$</span> (i.e., <span class="math-container">$M(T) = 2$</span> would mean I value the item at twice the guaranteed resale value), and my overall value is <span class="math-container">$V(T) = M(T) * T$</span>.</p>
<p>The first intuition (which I'm treating as an axiom) is that <span class="math-container">$V(T)$</span> should be monotonically increasing in <span class="math-container">$T$</span>. The second is that <span class="math-container">$M(T)$</span> should be monotonically decreasing in <span class="math-container">$T$</span>.</p>
<p>I can also estimate my <span class="math-container">$V$</span> at a handful of points in <span class="math-container">$T$</span>.  So far, I have tried Lagrange Polynomial Interpolation on such a handful of points to output a candidate <span class="math-container">$V(T)$</span>, but when I divide this by <span class="math-container">$T$</span> to produce the corresponding <span class="math-container">$M(T)$</span>, the result violates the second axiom; namely, polynomials produce non-decreasing regions in <span class="math-container">$M(T)$</span> around the LPI input points.</p>
<p>Are there ways to construct a <span class="math-container">$V(T)$</span>, in elementary but perhaps non-polynomial functions, so that the two axioms are satisfied, and it also passes through a handful of input points (the points can be trusted to themselves satisfy the axioms)?  Outside the regions corresponding to <span class="math-container">$0 \leq T \leq 1$</span>, of course, the behavior of <span class="math-container">$V(T)$</span> and <span class="math-container">$M(T)$</span> do not matter.  Is there a standard way to express valuations for these sorts of items?</p>

CC BY-SA 4.0
2023-03-09T14:56:44.927Z
How many is "a handful" of points in $T$? Does your $V(T)$ have to exactly fit the estimated points or is an approximate fit enough?
VARulle
CC BY-SA 4.0
2023-03-09T15:22:34.773Z
@VARulle An approximate fit is enough; the points are just subjectively determined by asking "about how much would I be willing to pay for the item at this particular condition?"  I was using 5 points with my Lagrange Polynomial Interpolation, but I guess up to around 20 points would be manageable.
user10478
CC BY-SA 4.0
2023-03-10T09:29:48.773Z
O.k., but then instead of exact interpolation why not just use a simple regression model? Even linear regression could do, depending on how your data points look like.
VARulle
CC BY-SA 4.0
2023-03-10T19:24:08.570Z
@VARulle Linear regression would be incompatible with the two axioms, wouldn't it?  To be clear, I'm not looking to express merely a trend in my data points.  I want a function I could hand to someone and say, "this is what I'm willing to pay for any $T$."  It's just that I wouldn't be very sensitive to, i.e., a $2-3\%$ error.
user10478
CC BY-SA 4.0
2023-03-12T13:12:56.870Z
Linear regression for V. If you get a positive intercept, then dividing by T gives you a decreasing M.
VARulle
<microeconomics><finance><financial-economics><value><financial>
Constructing Valuation As a Function of Condition
user10478
# Constructing Valuation As a Function of Condition

(Asked by: username_0 on 2023-03-09)

Suppose there is an item that deteriorates with use, its condition expressed as a number between $0$ (completely broken condition) and $1$ (perfect condition), for which I want to assign a monetary value. To make the units nicer, let's think of the condition also as a monetary value; i.e., someone has guaranteed to pay me $T$ dollars if I want to sell the item at any time, where $T$ is its condition. $M(T)$ is an implicit multiplicative markup on $T$ (i.e., $M(T) = 2$ would mean I value the item at twice the guaranteed resale value), and my overall value is $V(T) = M(T) \* T$.

The first intuition (which I'm treating as an axiom) is that $V(T)$ should be monotonically increasing in $T$. The second is that $M(T)$ should be monotonically decreasing in $T$.

I can also estimate my $V$ at a handful of points in $T$. So far, I have tried Lagrange Polynomial Interpolation on such a handful of points to output a candidate $V(T)$, but when I divide this by $T$ to produce the corresponding $M(T)$, the result violates the second axiom; namely, polynomials produce non-decreasing regions in $M(T)$ around the LPI input points.

Are there ways to construct a $V(T)$, in elementary but perhaps non-polynomial functions, so that the two axioms are satisfied, and it also passes through a handful of input points (the points can be trusted to themselves satisfy the axioms)? Outside the regions corresponding to $0 \leq T \leq 1$, of course, the behavior of $V(T)$ and $M(T)$ do not matter. Is there a standard way to express valuations for these sorts of items?

**Tags:** microeconomics, finance, financial-economics, value, financial

> **Comments:**
> **username_1**: How many is "a handful" of points in $T$? Does your $V(T)$ have to exactly fit the estimated points or is an approximate fit enough?  
> **username_0**: @username_1 An approximate fit is enough; the points are just subjectively determined by asking "about how much would I be willing to pay for the item at this particular condition?"  I was using 5 points with my Lagrange Polynomial Interpolation, but I guess up to around 20 points would be manageable.  
> **username_1**: O.k., but then instead of exact interpolation why not just use a simple regression model? Even linear regression could do, depending on how your data points look like.  
> **username_0**: @username_1 Linear regression would be incompatible with the two axioms, wouldn't it?  To be clear, I'm not looking to express merely a trend in my data points.  I want a function I could hand to someone and say, "this is what I'm willing to pay for any $T$."  It's just that I wouldn't be very sensitive to, i.e., a $2-3\%$ error.  
> **username_1**: Linear regression for V. If you get a positive intercept, then dividing by T gives you a decreasing M.  

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## 0 answers

