# Clarification of equations regarding capital switching and dated labor

Then, using this example (and further discussion), Samuelson demonstrates that it is impossible to define the relative "roundaboutness" of the two techniques as in this example, contrary to Austrian assertions. He shows that at a profit rate above 100 percent technique A will be used by a profit-maximizing business; between 50 and 100 percent, technique B will be used; while at an interest rate below 50 percent, technique A will be used again. The interest-rate numbers are extreme, but this phenomenon of reswitching can be shown to occur in other examples using more moderate interest rates.

The second table shows three possible interest rates and the resulting accumulated total labor costs for the two techniques. Since the benefits of each of the two processes is the same, we can simply compare costs. The costs in time 0 are calculated in the standard economic way, assuming that each unit of labor costs $w$ to hire: $cost = (1 + i)w×L_{–1} + (1 + i)^2w×L_{–2} + (1 + i)^3w×L–3$ (from http://en.wikipedia.org/wiki/Cambridge_capital_controversy#Reswitching)

I am not getting the final equation. Can anyone explain this? I mean, why is do we need to multiple 1+profit rate to wageg times labor input? I think this question can be answered in a simple manner, as I read what dated labor is in terms of Sraffa.

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Might be one for Quantitative Finance, but don't quote me on that – AakashM Dec 14 '12 at 9:33

The issue is one of compound interest. The present value of labour costs a year ago involve multiplying the labour costs then by $(1+i)^1$, the present value of labour costs two years ago involve multiplying the labour costs then by $(1+i)^2$, and the present value of labour costs three years ago involve multiplying the labour costs then by $(1+i)^3$.