Question

In: Economics

For a demand function u (x, y) = xy, show the demand functions for good x...

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

Solutions

Expert Solution

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


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