P = R/(T+C)

This post is to explain the equation P = R/(T+C), which explains what up-front price a land title will sell for if a tax is promised to be levied on that same price in perpetuity given the conditions below.

I came up with this formula a while ago. I recently learned I'm not the only one, so I'm certainly not saying that makes me special -- just that it's intuitive to me and I "get it."

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The conditions are as follows. These may be obvious, but I'm stating them in case they're not:

1. R is independent. It's what the bidder would bid as rent to leasehold the title at the time bidding on the up-front price takes place. It's the true value of the title -- again, at the time of the title auction. (Or "sale" as it might be called colloquially, since we don't have proper auctions in the modern world).

2. T is independent. It's the number the government has communicated it will apply to the selling price of the title in order to determine the periodic amount the title holder will pay to the government.

3. I'm used to writing the cap rate as C. In any case, it's also independent.

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There are a number of angles from which to approach an explanation. I'm unlikely to choose the one that works best for you, but I'll give it a shot.

Let's start with the usual price equation in the absence of taxes: P = R/C.

Well, the government has come in and said, "Potential landowners will no longer have the full 'R' to capitalize into a bid price. After bidding P_B, they will pay a monthly fee of 'TP_B' to hold the title."

In other words, any theoretical bid price, P_B, will tell the bidder what his periodic payment will be into the public purse: TP_B. The government has given the bidder all the information he needs in order to determine whether he, bidding P_B, can retroactively justify bidding that price. This is not a causal problem because each bidder can determine their range of justifiable bid prices before actually bidding.

I don't know how to write the chain of dependency mathematically, but in English, it is like this:

"A bid price of P_B will cause the landowner to have a periodic liability to the public of TP_B. This will leave the landowner with only (R-TP_B) periodically. In other words: (R-TP_B) will be all that's available to capitalize into a price right now. Thus any bid price P_B must be less than or equal to the capitalization of this still-available rent ... in other words, P_B must be less than or equal to (R-TP_B)/C."

• P_B immediately yields (R-TP_B)/C, which is essentially just a "check number".
• If that result, (R-TP_B)/C, is greater to or equal to the P_B that fed into it, that P_B can be afforded as a bid price.

The bidder can do a plug-and-chug spreadsheet, or he can just cut to the mathematical chase by skipping to the maximum acceptable bid price, the one where P_B yields (R-TP_B)/C such that P_B = (R-TP_B)/C.

At this point, I'm just going to chop the P_B back down to P to make this writing easier on myself:

• P = (R-TP)/C
• CP  = R-TP
• CP + TP = R
• P(T+C) = R
• P = R/(T+C)

The maximum price the bidder would pay is R/(T+C), if we're defining R as the rent that bidder would pay. If we're defining R as the market rent, then the formula can equally well be applied to determine what the up-front market price of the title will be.

(No up-front price on a land title, of course is a true "market price," since it represents force being used to prevent the landowner from paying the full market rent.)