Monday, 1 September 2025

Rates of return, net present values and cost-benefit ratios

 1. suppose the government spends £x on some investment, say a road or R&D.  What is the rate of eturn/ 

2. Following some excellent notes from the OBR, we have that the IRR of that spend is the discount rates that solves the equation

 Title: N P V equals sum from n equals 0 to infinity of B over open parentheses 1 plus r times close parentheses to the power of n minus x equals 0 - Description: {"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi><mi>P</mi><mi>V</mi><mo>=</mo><munderover><mo>&#x2211;</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mo>&#x221E;</mo></munderover><mfrac><mi>B</mi><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>r</mi><mo>&#xB7;</mo></mrow></mfenced><mi>n</mi></msup></mfrac><mo>-</mo><mi>x</mi><mo>=</mo><mn>0</mn><mspace linebreak=\"newline\"/><mspace linebreak=\"newline\"/></math>","origin":"MathType Legacy","version":"v3.19.0"}

 that, is the rate of return that sets this number to zero.

3. the Benefit from spending on some investment is more GDP.   That increase in GDP is dY/dK.=alphaY/K where alpha is the elasticity of output with respect to capital.  If Y/K is constant and we imagine spending £1 investment we have 


which says that the IRR is the marginal product of the project less the depreciation rate.  Or, the marginal product of the product is the gross rate of return. 

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