Rent As Joint Production

I here think about Sraffa's price equations for rent. I try to formulate them explicitly as the system of price equations in the chapter on joint production. I suppose this is a case of me setting out something that many take as obvious.

The data for Sraffa's price equations include the specification of a number of processes producing commodities. Suppose the l process is operated by a capitalist farmer. And they pay rent on the uth type of land. The price equation is:

(p₁ a_1,l + p₂ a_2,l + ... + p_n a_n,l)(1 + r) + rho_u c_u,l + w a_0,l = p_l

Above, rho_u is rent per acre, and c_u,l are the acres of land per unit output for this process. I hope the remaining symbols are obvious.

Let f_u be the purchase price of an acre of land of the uth type. The above equation can be rewritten as:

(p₁ a_1,l + p₂ a_2,l + ... + p_n a_n,l + f_u c_u,l)(1 + r) + w a_0,l = p_l + f_u c_u,l

where:

rho_u = f_u r

The above equality is equivalent to an infinite sum:

f_u = rho_u/(1 + r) + rho_u/(1 + r)² + rho_u/(1 + r)³ + ...

In an explicit formulation of a model of rent as one of joint production, no elements of the input and output matrices, as seen in the price system, are negative. The price of an acre of land is the present value of the stream of rent payments expected to be received on that land in the future.

The above works for intensive rent. One of the lands has a price of zero in the case of extensive rent. How is that expressed in the price equations? I suppose only commodities, that is, goods with positive prices, enter into the equations. Air, being free, does not appear in the price equation for some chemical process that requires nitrogen in some way, for example. But then the solution to a problem of the choice of technique is taken as given. I have a difficulty with many of the chapters in the part on joint production in Sraffa's book.