Over the relevant period the railway system in the West was being rapidly expanded. As a result, the net of transportation cost revenue received by farmers was rising. Suppose that, at a particular location, the net of transportation cost revenue that a farmer would have received from his crops at the end of year t was R(t) = 10 (1 + g)^{(t - 1862)}, where g is the growth rate of net revenue, and that the opportunity cost of the farmer's labor was constant over time at 13. Then the surplus received by the farmer at the end of year t (the crop revenue received by the farmer at the end of year t, net of transportation costs and net of the opportunity cost of the farmer's labor time) would have been S(t) = 10(1 + g)^{(t - 1862)} - 13. From the formulae derived in the lecture on discounting, the value of land at time t would have been V(t) = 10(1 + g)^{(t - 1862)}/(r - g) - 13/r . Assume that the discount rate, r, was 5% and that the growth rate of net crop revenue received by farmers, g, was 1%.

a) At the time the Homesteading Act was introduced, in 1862, what would have been the market value of land?

b) Under the Homesteading Act, the land would have been settled as soon as its market value became positive (According to my calculations, t would have been 1866). Why?

c) It would have been efficient for the land first to be settled as soon as S(t) rises to zero. (According to my calculations, t would have been 1889). Why?

Thus, according to Andersen and Hill's argument, at this particular location, land would have been settled 23 years earlier than was efficient.