Value & Capital, CHAPTER XVIII, Section 2

In this section, the author starts from the assumption, made near the end of the previous section, that people "do plan, more or less consciously, and more or less definitely, those parts of future expenditure which are relevant to current expenditure."  He therefore argues that the assumption of a complete plan of future expenditures, if used only for determining "the details of current expenditure alone," is not unreasonable.

His analysis works by assuming "an individual who possesses, at the planning date, a certain stock of durable consumption goods;  who is receiving a sum of money R₀ in the current week ... and who expects to receive a series of sums R₁, R₂, R₃, ... in the same way in the following weeks."

The individual's expenditures in the coming weeks are assumed to be (in monetary terms) the sums E₀, E₁, E₂, E₃, ... .  The difference between receipts and expenditures in each week will cause an incremental change in the individual's holding of money or of securities (for simplicity, Professor Hicks assumes only the latter). 

Hicks asserts that the stream of differences R₀ - E₀, R₁ - E₁, R₂ - E₂, R₃ - E₃, ... may be regarded as a stream of lendings (which is reasonable terminology since investments in securities are made in hopes of receiving future payment).   

If the plan is to be carried on for a fixed but arbitrary number of weeks, say n, then at the end of that time the individual can expect to have accumulated, from carrying out his plan, a sum C_n that is available as part of his resources for future consumption or investment.  As the author then explains,

If we regard the provision of such a capital sum as one of the things to which expenditures can be devoted in the last week of the plan, we have an accounting device which enables us to reduce the whole problem to one of distributing expenditure between the n weeks.

Using this sum with the notation defined above, the author's "stream of lendings" becomes

R₀ - E₀, R₁ - E₁, R₂ - E₂, R₃ - E₃, ... , R_n - E_n - C_n

If the sum of E_n and C_n were indeed spent in the last week, then the stream of receipts and the stream of expenditures (adjusted to include C_n) would exactly cancel out, and the capital value of the adjusted stream of lendings must equal zero.  (The author spells out in a footnote his assumption that "the securities initially held are expected to retain the same value at the end as they possessed at the beginning.")

The author concludes the section by describing the equality of the streams of receipts and (adjusted) expenditures as "the clue which enables us to reduce the planning of expenditure (just as we reduced the planning of production) into terms of a problem we have already solved in static theory."