[0,1]∞5 , with g 5 G0 and g 5 G1, and P: [0,1]∞5 , with p 5 P0 and

C . Blackorby, D. Donaldson Mathematical Social Sciences 37 1999 139 –163 147 1 H H h y 5 y , . . . , y is the economy’s income vector and y 5 o y is aggregate income, h 51 with H h y 5 O y 5 P p 1 p ? v 1 p ? g. 26 h 51 Equilibrium requires, for each j 5 1, . . . ,m, H h h D p, y:5 O D p, y 5 P p 1 v 1 g . 27 j j j j j h 51 b We consider a public project that moves the public input–output vector g from g a before to g after. A path of integration is described by the continuous functions m b a m b G: [0,1]∞5 , with g 5 G0 and g 5 G1, and P: [0,1]∞5 , with p 5 P0 and a b a b a p 5 P1. Equilibrium prices corresponding to g and g are p and p . The income 1 H vector along the paths G and P is Yt 5 Y t, . . . ,Y t, t [ [0,1], and is given by b a 25 evaluated at g 5 Gt and p 5 Pt with y 5 Y0 and y 5 Y1. Aggregate income H h b a along the paths is Yt 5 o Y t with Y0 5 y and Y1 5 y , and is given by 26 h 51 evaluated at g 5 Gt and p 5 Pt. The Hicksian partial-equilibrium claim discussed in Section 2 above can be demon- strated in this general equilibrium model. The aggregate Dupuit–Marshall consumers’ surplus, allowing for income change, is a a p , y m s 5 E 2 O D p, ydp 1 dy F G ba j j j 51 b b p , y 1 m 5 E 2 O D Pt,YtdP t 1 dYt . 28 F G j j j 51 1 H It is equal to the sum of the individual consumer’s surpluses s , . . . ,s , and ba ba a a p , y H H m h h h h s 5 O s 5 O E 2 O D p, y dp 1 dy F G ba ba j j h 51 h 51 j 51 b b p , y 1 H m h h h 5 OE 2 O D Pt,Y tdP t 1 dY t . 29 F G j j h 51 j 51 Theorem 1 shows that this consumers’ surplus can be computed in another way in an undistorted competitive economy: one in which all agents are price takers, producer and consumer prices are equal, and all taxes and or transfers are lump-sum. The general equilibrium analogue to the market-demand-curve surplus of 7 is computed along a particular path where all markets are in equilibrium. Along a market-equilibrium path, M M 1M H M Pt 5 P t and Yt 5 Y t 5 Y t, . . . ,Y t, where 148 C . Blackorby, D. Donaldson Mathematical Social Sciences 37 1999 139 –163 H M M h M hM D P t,Y t 5 O D P t,Y t j j h 51 M 5 P P t 1 v 1 G t, 30 j j j j 5 1, . . . ,m, for all Gt, t [ [0,1]. It is possible for these paths to be different even if b a the end points g and g are the same because the paths hGtut [ [0,1]j can be different, t [ 0,1. Theorem 1. For any market-equilibrium price path , the aggregate Dupuit–Marshall consumers ’ surplus 28 is equal to 1 H m h M a a b b s 5 O s 5 2 EO G tdP t 1 [ p ? g 2 p ? g ] 31 ba ba j j h 51 j 51 or , equivalently, 1 H m h M s 5 O s 5 EO P tdG t. 32 ba ba j j h 51 j 51 Proof. Along the market-equilibrium path, using 29 and 30 and noting that Yt 5 H h b a o Y t, Y0 5 y , and Y1 5 y , h 51 1 H m h M M M M s 5 O s 5 E 2 O D P t,Y tdP t 1 dY t F G ba ba j j h 51 j 51 1 1 1 m m m M M M M a b 5 2 EO P P tdP t 2 EO v dP t 2 EO G tdP t 1 [ y 2 y ] j j j j j j j 51 j 51 j 51 1 m a b a b M a b 5 2 [ P p 2 P p ] 2 [ p ? v 2 p ? v] 2 EO G tdP t 1 [ y 2 y ]. 33 j j j 51 Using 26, the change in income is a b a b a b a a b b y 2 y 5 [ P p 2 P p ] 1 [ p ? v 2 p ? v] 1 [ p ? g 2 p ? g ], 34 and 33 becomes 1 m M a a b b s 5 2 EO G tdP t 1 [ p ? g 2 p ? g ], 35 ba j j j 51 which is 31. Integrating 35 by parts yields C . Blackorby, D. Donaldson Mathematical Social Sciences 37 1999 139 –163 149 1 m a a b b M a a b b s 5 [ p ? g 2 p ? g ] 1 EO P tdG t 2 [ p ? g 2 p ? g ], 36 ba j j j 51 which yields 32. j It is straightforward to interpret the results of Theorem 1. Eq. 31 is the general equilibrium analogue of 7, the partial-equilibrium result. Because p ? g is government- a a b b sector profit, [ p ? g 2 p ? g ] is the change in profit. It results in a change in consumers’ incomes. The first term is easiest to interpret if the government sector produces a single good j 51 with a change in its price but no change in the prices of government inputs other prices may change. In that case, the surplus is simply 1 M a a b b 2 E G tdP t 1 [ p ? g 2 p ? g ]. 37 1 1 The first term corresponds to the area to the left of the market demand curve—ACED in Fig. 1—and it should be added to the change in government-sector profit. The summation sign in 31 indicates that, in each market in which the government participates, consumers’ surpluses should be computed for each output or input whose price changes. It is true, of course, that there may be private-sector involvement in some or all of the goods including inputs that correspond to the non-zero elements of g. Eq. 31 indicates that this can be accounted for completely through the market equilibrium M path hP tut [ [0,1]j. Eq. 32 is an area under market demand curves. Suppose that some prices b a ¯ j [ h1, . . . ,mj are unaffected by the change in g, then p 5 p 5 p for those j j j a b ¯ values of j, and the corresponding terms in 32 are o p [ g 2 g ], the change in j [ j j j government-sector profit. If 5 h1, . . . ,mj, this is the usual cost-benefit test in an undistorted competitive economy. In the case where all prices can change, it is possible to approximate the path hPt, t [ [0,1]j linearly with P t 5 a G t 1 b , 38 j j j j b a j 5 1, . . . ,m, P0 5 p , P1 5 p . In this case, 32 becomes b a p 1 p a b F G ]] s 5 ? [g 2 g ]. 39 ba 2 b a This is the change in government-sector profit computed at the average of p and p , before and after prices. Such averaging, based on partial-equilibrium arguments, is common in cost-benefit analysis and 39 provides a general-equilibrium justification.

4. Interpretation