# Why do higher real interest rates increase land value?

I'm using this equation to calculate market value of land based on the levied land value tax:

y = market land value
x = land value tax
r = real interest rate
a = real time-independent land value
i = tax revenue

$$y + (y*x)/r = a$$ $$y = (r*a)/(r+x)$$

But something can't be right here because if I lower the real interest rate (to imply an expensive future), then the market value of land should go up since the future rent of the land becomes more important in the market's mind. Instead, the market value of land goes down! I guess this is because future taxes are taken into account as well and out-weigh any future rent. When rates go up, are the higher taxes, resultant from higher land values, also dismissed easier and thus making higher land values more tolerable so land owners don't care about the future tax cost as much?

Another question is calculating the optimal tax rate to maximize tax revenues:

$$yx = x(r*a)/(r+x)$$

It seems that charging a tax higher than 100% will actually result in higher tax revenue. Is that because the real interest rate (~1%) has an effect on the future tax revenue so it's better to collect taxes as quick as possible or the tax base will not tolerate higher taxes from higher land values? Also, it seems interest that a 1% land value tax only collects half the tax revenue that it potentially could.

A final question, is how would I calculate exclusivity value? A 1% land value tax should add value intrinsically due to its dampening effects on future risk; ie, it's easier to maintain ownership over land if only a small amount of future value appreciation had to be paid for, than if the entirety of future value appreciation had to be paid for. Ex: paying 10k a year knowing it will never go up, vs paying 9k a year knowing it will go up.

• [...] if I lower the real interest rate, then the market value of land should go down to imply the cheap future and thus a lack of interest in the future use of land, right? Nop. The higher the discount rate, the bigger the lack of interest in the future use of land. See, e.g. this – keepAlive Dec 13 '18 at 21:30
• Also, note that the correct way of conceiving $a$ is as a price, and $y$ as a rent. The rent is assumed to be time-constant with no finite horizon, hence the perpetuity calculation that is performed here. – keepAlive Dec 13 '18 at 21:35
• @Kanak, I edited the question. I didn't mean what I said originally; accident. "y" is the price you would find on the market. "a" is calculated by taking perpetual tax payments into account and adding that perpetuity onto the market value. "a" is what you would be willing to pay at 0% land tax. – EternalPropagation Dec 13 '18 at 22:42
• never mind; figured it out. – EternalPropagation Jan 2 '19 at 19:28