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On wikipedia, the definition of Tobin's q is given as:

$$ q=\frac{\text{Market value of installed capital}}{\text{Replacement cost of capital}} $$

Imagine the market value of installed capital is 100\$, and that of replacement capital is \$10. Does $q$ mean that for each dollar of replacement, there are \$10 dollars of market value of that capital? Or simply that the market value of installed capital is 10 times that of replacement capital?

I frequently encounter this interpretation issue. Is there a rule of thumb about when to use which interpretation of division?

Kwame Brown
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    What's the difference? – aschepler Apr 15 '23 at 15:31
  • each dollar of replacement cost of capital yields $$10$ dollars of market value of that capital? -- This is definitely NOT necessarily true! All the quotient tells us is that the average yield is $10$ dollars per dollar of replacement cost, which could be the case if each of $9$ individual one-dollar replacement costs resulted in $$9$ of market value and a $10$th one-dollar replacement cost resulted in $$91$ of market value. Or other possibilities, which also don't have to be considered in separate one-dollar replacement cost amounts. – Dave L. Renfro Apr 15 '23 at 15:44
  • In the former case, I am asking how many dollars of market capital per unit of replacement capital. In the latter, I am asking how many 10 dollars of replacement capital fit into 100 dollars of market valued capital. Are they not conceptually different? – Kwame Brown Apr 15 '23 at 21:36

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From just the formula alone, I don't see how you can come up the word "yields".

It might be an idea of business that it does, or maybe should, yield that much, but that's outside of the mathematical meaning of the formula.

So the math meaning is very much the second one, and you should ask a business professor to what degree the first is intended.

ADDED after comments:

I mean, I can kind of see how you could view them as different concepts. "How many 10 dollars of replacement capital fit into 100 dollars of market valued capital?" is asking

$$ 100 = ? * 10$$

while "market capital per unit of replacement capital", because it uses the magic word "per", asks

$$ ? = 100/10$$

The math fact is that they have the identical solution. Thinking of them as two distinct concepts doesn't strike me as any more useful than distinguishing between "A is 3 inches taller than B" and "B is three inches shorter than A".

I'll also say that thinking of $A/B$ as asking "how many $B$ fit in $A$" is less common. Yes, if you ask "how many dozen is 48 eggs?", then I can see how that corresponds to counting the number of cartons-of-12 you've got. But google tells me that Tobin's Q is some times called the "Q ratio", and ratios almost always say how many A's per B, i.e. your first interpretation, so I'm going to say go with that one.

JonathanZ
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  • Thank you for that. Perhaps the word yield was wrong of me to use- I meant 10 dollars of market value per 1 dollar of replacement. – Kwame Brown Apr 15 '23 at 21:34
  • In the former case, I am asking how many dollars of market capital per unit of replacement capital. In the latter, I am asking how many 10 dollars of replacement capital fit into 100 dollars of market valued capital. Are they not conceptually different? – – Kwame Brown Apr 15 '23 at 21:36
  • Thanks a lot. Is there a rule of thumb about which interpretation to use depending on the context? – Kwame Brown Apr 16 '23 at 13:44
  • Hmm, I've never thought about that before. Off the top of my head, I'll say that if you see "per" or "ratio", it's going to be the first version. And that the second version is rarer, and I can't think of places other than the "how many dozens" type where I'd think of it that way. I'll also add that for me I've never felt the need to strongly distinguish between the two cases, but lots of people think about math in lots of different ways, so if it helps you, more power to you. – JonathanZ Apr 16 '23 at 17:08