206 AGRICULTURAL ECONOMICS 



the net annual income derived from a piece of land is six dollars 

 per acre, and the rate of discount is five per cent, the present 

 capital value of the land would be one hundred and twenty 

 dollars per acre. One hundred and twenty dollars is, then, 

 the amount of money which, if lent at five per cent, would 

 yield an annual net income of six dollars. This is usually spoken 

 of as the capital value of the land. 



That this simple method of dividing the six-dollar net rent 

 by the prevailing rate of discount to find the capital value of a 

 piece of land is equivalent to finding the sum of an infinite 

 series of prospective net annual three-dollar rents discounted 

 at the same rate may be demonstrated as follows : 



The present value of a dollars due in / years if the interest 



be compounded annually at the rate of r would be , . t since 



X dollars compounded at rate r would give X(i-\-r) 1 , and if 



X(i-\-r) l = a then X = -, — ; — r,. If then the net income of a 



(i+r)' 



farm be a dollars a year its value would be expressed by the 



equation : V = — — \- , , ., + . , .. + , . N4 + ad inf. 

 ^ i+r (i+r) 2 (i+r) 3 (i+r) 4 



This is an infinite " geometrical " progression with first term 



and ratio — 7— . The limit of the sum of such a series is 



i+r i+r 



a 



I ~*~ r which reduces to - . We have then the formula for 

 1 r 



1 — 



i+r 



the value : V = - which is the ordinary method of capitalizing 

 r 



rent. 



As a matter of fact, however, the present capital value of the 



land as determined in this way does not often correspond with 



the price which is paid for land. There are several important 



reasons for this difference. First it is not certain that the annual 



income that can be drawn from one hundred and twenty dollars 



will always be six dollars. The rate of interest may fall to four 



