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Show * 2aXj dDi = * 2aXi dDi+ tfaWi axi dDi or ( 7) £ L = A £ W_+ >> dC_ , ( i = l, ..., n). dDi dDi dDi Equation ( 7) reveals that the total change in income with respect to a small change in the i*-* 1 final demand item is equal to X ( the universal marginal value of water in terms of our welfare index-- household income) times the total change in total water intake with respect to a small change in the i" 1 final demand item plus y , ( the Universal marginal value of water facility resources in terms of income) times the total change in total water facility resources relative to a small change in D^ In other words, an equilibrium or optimum is achieved when for each i ™ final demand item its marginal value, in terms of income, is just compensated for by its opportunity marginal cost in terms of water plus its oppor • tunity marginal cost in terms of resources available for water facilities. The X and y are universal shadow prices ( in terms of income) of water and water facility resources, respectively. The term, HJ^ L. , in ( 7) gives the dDi total increase in water intake-- that required directly by the increase in Di and that required indirectly by the increase in all the gross outputs connected with the production of Di for final use. It, therefore, gives the increased demand for water intake, given an increase in the i^ 1 final demand item. Since water has alternative uses, it has alternative values in terms of income. The universal or common alternative or opportunity value is given by X . Therefore, X is the opportunity marginal water cost of Di in terms of income ( welfare) foregone, In a similar way, we can explain the terms 22- and }> 2£_ . There- dDi dDt fore, in equilibrium, the marginal value of the i*" final demand, 2- L. must dDi 12 |