Value & Capital, CHAPTER XIV, Section 4

This section continues -- seemingly in mid-paragraph -- the discussion of the previous section, by beginning, "For consider what happens, first, if interest rates are expected to change."  In this case, the author argues, "a definition based upon constancy of money capital becomes unsatisfactory."  He then constructs a numerical example in which the interest rate becomes fixed in the second time period at a rate that is twice the rate in the first period (and it remains at the second-period rate thereafter).  In his example, the individual can make the stream of expenditures

 £10, £20, £20, £20, ...  

and maintain his same quantity of capital, whereas that quantity of capital being available at the beginning of the second time period allows the individual to spend the stream

 £20, £20, £20, £20, ...  

The author notes that "It will ordinarily be reasonable to say that a person with the latter prospect is better off than one with the former."

Accordingly, the author arrives at the definition he calls "Income No. 2."  Namely, it is "the maximum amount the individual can spend this week, and still expect to be able to spend the same amount in each ensuing week."  For constant interest rates, this definition is the same as Income No. 1, but not when interest rates are expected to change.  The author concludes, "Income No. 2 is then a closer approximation to the central concept than Income No. 1 is."

Value & Capital, CHAPTER XIV, Section 3

This section considers the first of several approximations to the "central meaning" of the term income.  This first approximation "would make everything depend on the capitalized money value of the individual's prospective receipts."  The author explains by use of an example (in terms of pounds sterling, denoted by £):

Suppose that the stream of receipts expected by an individual at the beginning of the week is the same as that which would be yielded by investing in securities a sum of £M.  Then if he spends nothing in the current week, reinvesting any receipts which he gets ... he can expect that the stream which will be in prospect at the end of the week will be £M plus a week's interest on £M.  But if he spends something, the expected value of his prospect at the end of the week will be less than this.  There will be a certain particular amount of expenditure which will reduce the expected value of his prospect to exactly £M.  On this interpretation, that amount is his income.

It might be easier to understand this explanation by seeing it expressed mathematically.  Let r denote the rate of interest earned by investments; let M (as above) be the initial sum available for investment, and let y denote income.  Based on the author's description of income as an expenditure that will reduce the expected prospect to M, we have the following equation:

(M − y)⋅(1 + r) = M

Performing one step of multiplication, we get

M + r⋅M − y⋅(1 + r) = M

Subtracting M from both sides and rearranging, we get

r⋅M = y⋅(1 + r)

Therefore,

y = M⋅r / (1 + r).

The author notes that this definition makes sense in the case of income derived from property.  He goes on to note that it is "less obviously sensible" in the case of labor income but "is still quite consistent with ordinary practice."  The author refers to this definition as "Income No. 1" and notes that it is "the maximum amount which can be spent during a period if there is to be an expectation of maintaining intact the capital value of prospective receipts (in money terms)."  Looking at the equation above, we can see that y is less than a single period of interest on the original investment M because of the term of 1 + r in the denominator (assuming r is positive).  This makes sense because the expenditure of y would cause interest to be earned on less than the full investment of M.

The section concludes by noting that this definition is likely the one that "most people do implicitly use in their private affairs;  but it is far from being in all circumstances a good approximation to the central concept."