# Do assets without rental income streams appreciate relative to assets with rental income streams?

In his book "The Armchair Economist", economist Steve Landsburg critiques an op-ed concerning the relative value of stocks and real estate:

James K. Glassman wrote a piece in The New Republic to prove that stocks are better investments than real estate. He calculates that "if you bought a 200,000 dollar home in Foggy Bottom [a neighborhood in Washington, D.C.] in 1979, it would have been worth 316,000 dollars [ten years later]. But if you bought 200,000 dollars worth of stock in 1979, it would be worth 556,000 dollars [ten years later]—and you'd have another 68,000 dollars in dividend income." Well, yes, but if you'd bought the house you would have had a place to live for those ten years, whereas if you'd bought the stock you'd have been making rental payments to a landlord. This renders Glassman's comparison meaningless. All he shows is that if you compare all of the benefits of owning stock to some of the benefits of owning real estate, then the stock comes out ahead. Big deal.

Glassman's piece has a place of honor in my Sound and Fury File because his conclusion is so exactly the opposite of the truth. He explains that "stocks appreciate faster than real estate; they always have and they always will. The reason is that a share of stock is a piece of a company in which minds are producing value. Real estate just sits there." The truth is that stocks appreciate faster than houses precisely because a house does not just sit there; it provides shelter, warmth, and closet space every single day that you own it. Stocks need to appreciate faster to compensate for the fact that they don't provide any comparable stream of services. If stocks and real estate appreciated at the same rate, nobody would own stocks.

My question is, what are the mathematical details of Landsburg's claim in bold? If A is an asset that produces a stream of rental income, and B is an asset that does not produce a stream of rental income, then must the relative price of B with respect to A be constantly increasing? Is there a model that shows this? And what exactly causes this relative price change?

Let $P_t$ denote the price of an asset in year $t$, $d_t$ the dividend (or rental income or cashflow) you get from said asset in year $t$ and let $r$ denote the interest rate (which I assume to be constant for simplicity's sake). Then the price of the asset today is $$P_1 = \frac{d_1}{1+r} + \frac{P_2}{1+r}.$$ Using iteration $$P_1 = \sum_{t=1}^{n - 1} \frac{d_t}{(1+r)^t} + \frac{P_n}{(1+r)^{n-1}}.$$ If you compare two assets with $$P_1^A = P_1^B$$ where $$\forall t: \ d_t^A > 0, \ d_t^B = 0,$$ then or all $t>1$ you will get: $$P_t^B > P_t^A.$$ This is somewhat paradoxical, as a there is no clear reason why $P_t^B$ should be positive if it never pays dividends. You could assume that it has not yet payed any dividends but will start doing so soon.
However I think Landsburg's main argument is that $d_t>0$ holds for the house as well, not just the stock.
• If you make the assumptions that there is no arbitrage and there are no pricing bubbles then yes. If one sort of investment would have a higher yield then in a perfectly rational world people would buy that investment. This would drive up today's price and thus drive the yield of that investment down until the yields of the two investments are equal. In my notations yearly yield is equal to $r$. Given $P_1, d_1$ and $P_2$ you can determine the yield from $$P_1 = \frac{d_1}{1+r} + \frac{P_2}{1+r}.$$ – Giskard May 27 '15 at 4:43