FInd demand function for Y?

Question

U=(2X+8) (Y+5)^1/2

8 UNITS OF X

10 UNITS OF Y

PX=6

PY=3

FIND MRS AT EQUILIBRIUM AS WELL AS EQUATION ENGEL CURVE.

THANKS

Answer 1

du/dx=2(y+5)^1/2

du/dy=1/2(2x+8)(y+5)^-1/2

therefore, MRS= - (du/dx)/(du/dy)

                          = - [2(y+5)^1/2][{2(y+5)^1/2}/(2x+8)]

                          = - 4 (y+5)/(2x+8)

                          = -2(y+5)/(x+4)

substituting the values of x n y in this eqn,

                  MRS= - 2(10+5)/8+4

                          = -15/6

now moving to the engel curve........

frm the eqm. condn.,

         (du/dx)/(du/dy)=px/py

   hence,

           4(y+5)/(2x+8)=px/py

       or,                    y=[1/4(px/py)(2x+8)]-5..............(A)

      

we know that the budget constraint to be M=px.x+py.y.......(B)

wer M is the income of the consumer.now substituting the value of y frm (A) in (B),

       M= px.x+py[1/4(px/py)(2x+8)-5]

          = px.x+1/4px(2x+8)-5py

          = px.x+1/2px.x+13

          = 3/2px.x+13

or,3/2px.x=M-13

or, x= 2/3(M-13)/px.....................(C)

(C) is the reqd. eqn of the engel curve.

i have jumped some of the steps while calculating the engel crv....hope u get that.if u have anymore prob keep in touch cos m going thrgh the consumer behaviour presently. good luck

(to cola..........u r welcum)

Answer 2

dU/dX = 2(Y+5)^1/2

dU/dY = 1/2(2X + 8)(Y + 5)^(-1/2)

            = (2X+8)/2(Y+5)^1/2



dU/dX.dX + dU/dY.dY = 0

Hence

dY/dX = -(dU/dX) / (dU/dY) = MRS

            = -2(Y+5)^1/2  x  [2(Y+5)^1/2]/(X+4)

            = -2(Y+5)/(X+4)

Substitute in X = 8, 10 = Y

dY/dX = -30/12 = -15/6

Haven't touch consumer theory for quite some time, but this is roughly what I remember, maybe you can check with textbook to double confirm... Engel curve wise, I'm sorry, cos I've forgotten.. Not really certain about the derivation, so better not mislead you. :) (To Durba: oops, made a mistake, thanks for verifying :) )