This exercise adopts the common meaning of "rent", to denote the payment made to be able to use something (here, land) for a given amount of time (here, one year). The exercise illustrates: (i) how rent is determined in a competitive market, (ii) the relation between land, rent, and land price, and (iii) the distinction between "pure" rent and quasi-rent.

There is a fixed stock of unimproved land, L= 10. The annual value of marginal product of land is 20q, where q is the amount of land that is rented.

Denote the annual rent for a unit of land as f. What is relation between the value of marginal product of land and the inverse demand for land rental? Using the answer to this question, and the fact that the equilibrium rent equates supply and demand, find the equilibrium value of f.

(b) What price would someone with an annual discount rate of r be willing to pay to buy this land? Your answer gives the price as a function of the discount rate; recall equation 2.13. (The price of land is the amount that someone pays to buy the land; the land rent is the amount they pay to use it for a period of one year.)

(c) If the land is improved, its value of marginal product increases by 2. If q units are improved, the value of marginal product increases from 20 q to 22 q. What is the equilibrium annual rental rate, and what is the equilibrium price, if the entire stock of land is improved?

(d) Suppose that the cost of improving an acre is 15. What is the critical value of r at which the landowner is indifferent between leaving (all of) the land in its unimproved state, and improving all of it?

(e) If the land is improved, what fraction of the rent (or the price) is pure rent, and what fraction is quasi-rent.