Marketability

The prompt is based on this.

A marketable good is one that can be quickly sold at an “economic” price. One might think that an “unmarketable” good is one that cannot be sold, but if the asking price is low enough, virtually any good can be sold. Even so, there is certainly a sense in which a telescope (say) is much less marketable than wheat: the person selling the telescope (in barter) can get a much better price if he is willing to spend weeks looking for prospective buyers, whereas the seller of wheat will probably find his best offer within hours. Attempt to develop a measure of marketability in this sense.

A useful way to formalize marketability here is to treat it as a time-price tradeoff.

The idea is not “can it be sold at all?” but:

How much price must the seller sacrifice in order to sell within a short time?

That captures why wheat is highly marketable and a telescope is less so.

1. Define the sale environment

Let \(p(t)\) be the best expected price a seller can obtain if he is willing to search for buyers for up to time (t).

Then:

• \(p(0)\): immediate-sale price, or “spot liquidation” price
• \(p(t)\): expected best offer found within (t)
• \(p(\infty)\): the full-information or long-search price, the best price attainable if the seller can wait “long enough”

For a highly marketable good, \(p(t)\) rises quickly toward \(p(\infty)\). For a poorly marketable good, \(p(t)\) rises slowly.

2. A simple measure of marketability

One natural index is

\[ M(t)=\frac{p(t)}{p(\infty)} \]

This is the fraction of the full-search price obtainable within time \(t\).

Interpretation:

• If \(M(1\text{ day}) = 0.98\), the good is very marketable over a one-day horizon.
• If \(M(1\text{ day}) = 0.40\), the good is not very marketable over that horizon.

In this sense wheat would have a high \(M(t)\) even for small \(t\), while a telescope would have a much lower \(M(t)\).

3. “Instant marketability” or liquidity discount

An even more intuitive version is the discount for quick sale:

\[ D(t)=1-\frac{p(t)}{p(\infty)} \]

This is the percentage sacrifice required to sell within time \(t\).

• Small \(D(t)\): very marketable
• Large \(D(t)\): poorly marketable

Example:

• Wheat: \(p(\text{2 hours}) = 99\), \(p(\infty)=100\), so \(D=1%\)
• Telescope: \(p(\text{2 hours}) = 50\), \(p(\infty)=100\), so \(D=50%\)

That captures the intuition almost exactly.

4. A single-number summary

Since marketability depends on the time horizon, there is no single absolute number unless you specify a target.

Two convenient summaries are:

A. Time to sell near the best price

Define the time needed to get, say, 95% of the long-search price:

\[ \tau_{0.95}=\inf{t: p(t)\ge 0.95,p(\infty)} \]

Smaller \(\tau_{0.95}\) means greater marketability.

B. Price sacrifice for a standard horizon

Choose a standard time horizon \(T\) and define

\[ M_T=\frac{p(T)}{p(\infty)} \]

This lets you compare goods directly.

5. Search-theoretic interpretation

Why does this work? Because sale opportunities arrive over time.

Suppose buyers arrive randomly, and each buyer’s offer is drawn from some distribution. Then the more time the seller waits, the more buyers he sees, and the higher the expected best offer becomes.

A good is more marketable when:

• buyers arrive frequently,
• bid prices are tightly clustered,
• market participants know the good well,
• there are low costs of inspection, transport, and verification,
• there is a thick resale market.

A good is less marketable when:

• few buyers exist,
• finding them is costly,
• quality is hard to judge,
• each buyer values the good idiosyncratically,
• the spread between quick-sale and patient-sale price is large.

So marketability is closely related to:

• buyer-arrival rate
• thickness of market
• price dispersion
• ease of standardization

6. A refined measure

If you want a more structural definition, define marketability as the speed with which the expected sale price approaches its maximum:

\[ \text{Marketability} \propto \frac{d}{dt}p(t) \quad\text{for small } t \]

or, better, the proportional speed:

\[ \mu(t)=\frac{p'(t)}{p(\infty)-p(t)} \]

If \(\mu(t)\) is high, each extra unit of search time closes the gap to the best attainable price quickly. That means the good is highly marketable.

7. Best concise formulation

The cleanest measure is:

The marketability of a good over horizon \(t\) is the proportion of its full-search price that can be realized within \(t\).

\[ M(t)=\frac{p(t)}{p(\infty)} \]

Equivalent wording:

A good is more marketable the smaller the discount required for a quick sale.

\[ D(t)=1-\frac{p(t)}{p(\infty)} \]

This matches the telescope/wheat example:

• wheat: high \(M(t)\), low \(D(t)\), short \(\tau_{0.95}\)
• telescope: low \(M(t)\), high \(D(t)\), long \(\tau_{0.95}\)

So marketability, in this sense, is essentially time-conditioned realizability of value.