In: Economics
For a demand function u (x, y) = xy, show the demand functions for good x and good y. (Remember that MRS = (du/dx) / (du/dy) = px / py in the point of interest, the tangency point of budget line and indifference curve. The budget condition is given by pxx + pyy = m)
for u (x, y) = x1/3 y2/3
U(x,y) = xy
MUx = y
Muy = x
MRS = MUx/Muy
= y/x
At optimal choice MRS = Px/Py
y/x = Px/Py
y = (xPx/Py)
put y = (xPx/Py) in budget equation
Pxx + Pyy = m
Pxx + Py(xPx/Py) = m
Pxx + xPx = m
2xPx = m
x = m/2Px
y = (xPx/Py)
= x(Px/Py)
= ( m/2Px )(Px/Py)
= m/2Py
Thus demand functions of xand y
x = m/2Px
y = m/2Py
Now U(x.y) = x1/3y2/3
MUx = (1/3)x- 2/3y2/3
MUy = (2/3)x1/3 y- 1/3
MRS = MUx/Muy
= [(1/3)x- 2/3y2/3 ] /[(2/3)x1/3 y- 1/3 ]
= y/2x
At optimal choice MRS = Px/Py
y/2x = Px/Py
y = 2xPx/Py
put y = 2xPx/Py in budget equation
Pxx + Pyy = m
Pxx + Py(2xPx/Py ) = m
Pxx + 2xPx = m
3xPx = m
x = m/3Px
y = 2xPx/Py
= (m/3Px )( 2Px/Py)
= 2m/3Py