128 MULTIPLE PURPOSE RIVER DEVELOPMENT 



and A I — perpetual future consumption stream that / can enjoy. 

 Two identities are implicit in these definitions: 



(1) Yi = Ci + Si, ^nd 



(2) iiASi = AAi. 



The identity (2) defines the rate at which saving produces a future^ 

 consumption stream. We assume a utility function 



(3) Ui = Ui{Ci,Ai), 



which reflects I's present valuation of current consumption and con- 

 sumption in the future. He maximizes his utility function subject 

 to his income and interest constraints. This is equivalent to maxi- 

 mizing the Lagrangean expression 



(4) cl> = Ui (Ci,A,) - X (Yi-Ci-Si) - ^ {i,ASi-AA,), 

 which has the first-order maximum conditions 



(5) -^TT^ + A =0, A - ^^i = O, and ^ + ^ = O. 



Therefore, 



(6) — — = —p.1.1 and -— - = — ^c, and so 



dUi 



(7) ^T77 = 'I' or -— - - ii 



d^i dCi a^i 



a^-^i 



A change in taxation is a change in disposable income, part of 

 which will change consumption, part saving. Thus, 



(8) AYi = AC, -f AS,, 

 and the change in I's utility is 



(9) Af/j = -— AC, -f — - AAi, 



neglecting higher order terms on the grounds that they will be of 

 the second order of smalls. (9) is equivalent to 



