The learning curve
On a P per cent learning curve, every time the length of the production run is doubled, the unit cost is reduced by a factor p.
(p being the percentage P expressed as a fraction)
The cost, Cn of the nth unit is given by:
- Cn= C1.n-b
- b = (-logp)(log2)
The Cobb-Douglas production function
The Cobb-Douglas function has the form:
- Y = A. Lα . Cβ,
- Y = output, C = capital input, L = labour input,
- and A, α and β are constants determined by the technology employed.
If α = β = 1, the function represents constant returns to scale,
If α + β < 1, it represents diminishing returns to scale, and,
If α + β > 1, it represents increasing returns to scale.
It can be shown that, in a perfectly competitive economy, α is labour's share of the value of output, and β is capital's share.
Piero Saffra objected to the law of supply and demand on the grounds that, by bidding up the prices of inputs to suppliers of substitutes, the increased output of a product expansion could increase the demand for that product, thus violatimg the necessary condition that demand must be independent of supply . Jacob Viner had justified the long-run diminishing returns thesis by arguing that competitors for the required inputs would bid up their prices , but Lionel Robbins argued that Viner's justification was incomplete in cases where the market did not contain other users of an input and raised a number of other more complex objections .
Joan Robinson's objection to the production function equation was that she could not envisage a unit of measurement that could be applied both to output on the one, side and to labour and capital on the other side of the equation. Sir Henry Phelps Brown maintained that the apparent empirical support of the Cobb-Douglas function was in fact merely the consequence of an accounting identity  (a conclusion that was later supported by Felipe and McCombie ). Anwar Shaikh claimed that the belief that the Cobb-Douglas function has been empirically confirmed is mistaken because it is in fact consistent with a wide variety of possible data .
- "The Laws of Return Under Competitive Conditions", The Economic Journal December 1926
- Jacob Viner: "Cost Curves and Supply Curves", in Readings In Price Theory, edited by G. J. Stigler and K. E. Boulding. Irwin, 1952.
- Lionel Robbins: "Remarks Upon Certain Aspects of The Theory of Costs", Economic Journal March 1934.
- Henry Phelps Brown " The Meaning of the Fitted Cobb-Douglas function" 'Quarterly Journal of Economics vol 70 1957
- Jesus Felipe and J McCombie:"How Sound are the Foundations of the Aggregate Production Function?" September ISS 0111-1760, 2001
- Anwar Shaikh: "The Humbug Production Function II: The Laws of Production and the Laws of Algebra", Review of Economics and Statistics February 1974