Question

2. Show that each of the following utility functions has a diminishing MRS. Do they exhibit constant, increasing, or decreasing marginal utility? Is the shape of the marginal utility function an indicator of the convexity of indifference curve? a. (2) U(X,Y) = XY b. (2) U(X,Y) = x2y2 c. (2) U(X,Y) = In X + In Y

Answer 1

mux= // question (2) (a) OCX,Y)= XY mux=du - Y dx muy do = X dy MRS - Mux muy X as x increases Since it is in denoménator of mos So as x increases MRS decreases. (b) (x,y) = X Y MOx= Jo= 2xy² muy = 20 = 2x²y UX dy MRS - mux 2x Y muy 2x²y Х inco MRS decreases. U(X,Y)= lnx tiny muy = ! MRS = mux 1. Y=Y Y muy х X 2 2 2 2 y decreasing in x (C): .

MRS = Y X decreasing in x , as we consume The rate of substitution decreasel. more of Х Me testade of the man Convexity of incufference couve implices That The marginal rate of substitution of x for y falls as more of x is substituted for y. Coneider The de creasing MRS Y Х Suppose for X Х for convex Ic for of one more ciacition unit That is ax by I we have decreasin casing-ny. -oyl -oyd AX I-DY Дх ΔΧ Ic
for all of the 3 we have the MRS =Y/X implies as X increase the MRS will decrease !