Cost-benefit analysis
   Cost-benefit analysis, CBA, is the social appraisal of marginal 
   investment projects, and policies, which have consequences over time
   It uses criteria derived from welfare economics, rather than commercial 
   criteria. 
   CBA seeks to correct project appraisal for market failure
   Environmental impacts of projects/policies are frequently externalities, 
   both negative and positive
   CBA seeks to attach monetary values to external effects so that they can be 
   taken account of along with the effects on ordinary inputs and outputs to 
   the project/policy 
   CBA is the same as BCA – Benefit-cost analysis. 
                           
                                   Intertemporal efficiency
     Given that CBA is concerned with consequences over time, and based in welfare 
     economics, a key idea is that of intertemporal efficiency. 
                              UA UA(CA,CA)
                                             0    1      (11.1)
                              UB UB(CB,CB)
                                             0    1
       An allocation is efficient if it is impossible to make one individual better 
       off without thereby making the other worse off.
       Intertemporal efficiency requires the satisfaction of 3 conditions
                 Equality of individuals’ consumption discount rates
                 Equality of rates of return to investment across firms
                 Equality of the common consumption discount rate with the common 
                 rate of return   
                                                     
                                                                        Discount rate equality
   MRUS  A                    MRUS B              otherwise one could be made better off without making the other 
             C0,c1     =              C0,c1
                                                  worse off
            A
       r                           MRUSA 1                        defines A’s consumption discount rate
           C0,C1            ≡                   C0,C1
         Then the first intertemporal efficiency condition is stated as
           A      B
         r =r  = r        (11.2)
         Note: consumption discount rates are not constants. 
                                                                                                       
                                         Shifting consumption over time
                                                         Foregoing Cb Ca  makes Ca Cb  available next 
                                                                            0   0              1   1
                                                         period. The rate of return to, on,investment is 
                                                         defined as            C  I
                                                                                   1          0
                                                                                     I0
                                                                       where ΔC1 is the second period 
                                                                       increase in consumption, Ca Cb , 
                                                                                                              1    1
                                                                       resulting from the first period 
                                                                       increase in investment ΔI , Cb Ca . 
                                                                                                           0     0   0
                                                                       For ΔI0 = ΔC0, this is
                                                                            ΔC  ( ΔC )           ΔC ΔC              ΔC
                                                                      δ 1                   0   1             0         1  1
                                                                                  ΔC                  ΔC             ΔC
                                                                                         0                   0              0
               which is the negative of the slope of the transformation frontier minus 1, which can 
               be written
                             1 s
                  where s is the slope of the frontier.
                                                                       
                           Rate of return equality
         If each firm were investing as indicated by C 1b and C 2b, then period 1 
                                            0     0
         consumption could be increased, without loss of period 0 consumption, by 
         having firm 1, where the rate of return is higher, increase investment by the 
         amount firm 2, where the rate of return is lower, reduced its investment.
         Only where rates of return are equal is this kind of period 1 gain 
         impossible. For N firms, the second intertemporal efficiency condition is 
            ,i1,...,N                       (11.3)
             i