How much is an asset worth that produces a constant income?

Question

Say you have an asset that produces for you £1k per year profit (adjusted for inflation) at a constant rate? (The proverbial immortal goose that lays golden eggs).

Logic would say it's potential is infinite.

But for an individual of age 30 it is only worth up to £50k in their lifetime. And they may care less about what happens to it after they die.

But to a company who buys it, companies can live essentially forever.

Answer 1

An asset that gives you a constant sum of nominal income every year is called a perpetuity. (The UK government used to issue such a product and called it a consol.)

Everyone "knows" that £1,000 received next year is worth less than £1,000 received this year. But how much less?

To try to formalize this notion, economists use the concepts of discount rate and present value. The discount rate is the rate at which you discount future returns.

If for example your discount rate is $5\%$, then this means you treat £1,000 received next year as equivalent to £950 received this year. (And an economist would say that the present value of £1,000 received next year is £950.)

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We can now answer your specific numerical question:

• For simplicity, suppose your discount rate is fixed as $\delta$.

Then the present value of this asset is: $$1000+(1 -\delta)1000+(1 -\delta)^21000+\dots = \frac{1000}{\delta}.$$

• More generally, suppose your discount rate each year is $\delta_t$ ($t=0,1,2,\dots$).

Then the present value of this asset is: $$1000+(1 -\delta_1)1000+(1 -\delta_1)(1 -\delta_2)1000+\dots.$$

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The brief discussion above is usually covered in greater depth in a course on Intermediate Microeconomics. See e.g. Varian (2014, 9e) Ch. 10.