Question

3) A consumer's utility function is u(x,y)22 (a) Find the consumer's optimal choice for x, y as functions of income I and (b) Sketch the demand curves for x, y as functions of income I when prices prices pa,Py. (Be careful!) are p 16,Py 2. (Be careful!)

Answer 1

1. Consumer's utility function:

Maximise the utility function $Q+S5J+3VRluWwAAAABJRU5ErkJggg==$

Subject to $FsFeE5vvPZEAAAAASUVORK5CYII=$

Maximize with respect to x and y

Max L = + λ (pxx+ pyy-I)

= 1+ λpx =0          (1)

   = + λpy =0    (2)

  (3)

From equation 1 ,

$Cmd6j4XYY1YAAAAASUVORK5CYII=$        (4)

From equation 2,

    (5)

Equating (4) and (5),

- $GvzzsEwgA5euFqYAAAAABJRU5ErkJggg==$        = $+gAAAABJRU5ErkJggg==$

Putting this value in (3), we get

Taking px common,

$YRxACVHZvEfBHrDAWNtkRkAAAAASUVORK5CYII=$

The utility function so defined in question is quasi-linear type of utility function. The demand of y is independent of income I. If thpriceses remaiconstant asas given, then he will consume 2 units of output.The consumerer spends all its income, after spending on y, on good x.

2. Given : px=16 and py=2

$o9ldLyNnJEUAAAAASUVORK5CYII=$

y=2 units

$zs4AAAAASUVORK5CYII=$