Question

Problem Show that each of the follwing utility functions has a diminishing MRS but that they exhibit constant, increasing, and decreasing marginal utility, respectively. a) UE,y) = ry b) U(I,y) = rºy c) U(x,y) = ln I + Iny W Pyoblem A consumer is willing to trade 3 units of for 1 unit of y when she has 6 units of rand 5 units y. She is also willing to trade in 6 units of for 2 units of y when she has 12 units of rand 3 units of y. She is indifferent between bundle (6,5) and bundle (12, 3). What is the utility function for goods and y? Hint: What is the shape of the indifference curte? Problem A consumer is willing to trade 4 units of 1 for 1 unit of y when she is consuming bundle (8, - She is also willing to trade in 1 unit of 1 for 2 units of y when she is consuming bundle (4. 4). She is indifferent between these two bundles. Assuming that the utility function is Cobb-Douglas funtion of the form U(z,y) = y. where a and 8 are positive constants, what is the utility function for this consumer?

Answer 1

Answering only first question

Moy ЭХ MWу & uxx,y) = xy MRS= MW x = 9/x a) EMRS = AMRS - 40 _Y/X2 <o, thus diminishing MRS Proved. 2x S Mux=ý, a Mwx =0, 1 Moy=x, amuy =o. Constant mo. b.) U=X2Y2 mux= 2x12 May = 2x2y. MRS= MUX = QX Y 2 hr = Yx , = 4 OMRS --Yx2 <0. Diminishing Amox= 24? , amo y = 2x² 0 MRS Pored ax >0 9 so exhibit Increasing mo : (.) u = ln xteny, mux= ay = 1/x Moy= dy=1/4 D MRS = MUX = Y/X ƏMRS = -4/X2 Lo, Diminshing Mrs. Proved ƏMUX = -\x2 <0, amoy = -V4240, exhibit diminishing mo 2X²Y ЭХ ду Mix au 2x a X ax