(3)Leontief/Fixed Proportions u(x1, x2,.xL)= min(ax1, a2x2,.a3xL) n this case, people always consume these proportions are independent of snd goods in exactly the same proportions prices (4)Cobb-Douglas( fixed budget shares u( x1,x℃2,XL ∏x"or∑a,log(x)(3) Leontief/Fixed Proportions ux1, x2,... xL  min1x1,2x2,...3xL In this case, people always consume goods in exactly the same proportions and these proportions are independent of prices. (4) Cobb-Douglas (fixed budget shares) ux1, x2,... xL   i1 L xi i or  i1 L i logxi