Value of Currency Itself
The value of something, in this case money, is based on it's demand; and since you want the value based on it's efficiency, not what it can purchase, you are looking for the value of money based on how efficient money is.
I believe I have just the answer for you. 

Your missing variable is the Velocity of Money. 
The demand for money and the Velocity of Money are inversely related. (your question emphasizes this point, "in contrast to the value it represents?".

The velocity of money (or the velocity of circulation of money") is a measure of how fast money passes from one holder to the next. It is most commonly measured as the income velocity of money, which is the frequency at which the average unit of currency is used to purchase newly domestically-produced goods and services within a given time period. In other words, it is the number of times one unit of money is spent to buy goods and services per unit of time.

Vt = PT/M
where 


Vt is the velocity of money for all transactions in a given time frame;
P is the price level;
T is the aggregate real value of transactions in a given time frame; and
M is the total nominal amount of money in circulation on average in the economy.
Thus PT is the total nominal amount of transactions per period.


Hope this helps.