I found an interesting 1970 AER paper that adds land to the Solow model in continuous time and verifies the result, discussed last week, that as the rate of return on capital approaches the growth rate of the economy the price of land will approach infinity. The paper is a mess – the notation is disgusting, and doing continuous time instead of discrete time adds nothing but pain.
The bottom line is this. The rate of return on land and capital should be equal in equilibrium (factor price equalization). If the interest rate – i.e., the rate of return on capital – is low, then the rate of return on land must be lower.
First, the rate of return on land and capital should be equal on the balanced growth path (equation 6). Where K is capital, L is land, P is the price of land in terms of goods, and g is the rate of exogenous economic growth, we have the equilibrium condition:
F_K = F_L/P + (∂P/∂t)/P
Translated to English: the marginal product of capital must equal the marginal product of land plus the rate of increase in the price of land. Perhaps even better: the interest rate on capital equals the rent from land plus the capital gains from land. Or, one last rephrasing: the return on capital equals the return on land, where part of the return on land is price appreciation.
From this, an equilibrium relationship between effective capital (k = K/N) and effective land (l = PL/N) can be derived with a little algebra (equation 8):
l = F_L / [(F_K - g)*β]
Where beta is a constant (the ratio of initial effective labor to land).
This is the key result. If the rate of return on capital is equal to the growth rate, effective land goes to infinity: dividing by zero. And thus the price of land goes to infinity, as the supply of land is perfectly inelastic, i.e. fixed.
To repeat what I wrote last week: we don’t see the price of landing going to infinity, which would seem to be a challenge for secular stagnationists.