Question

# 1. Emily has preferences for two goods x, y while her marginal rate of substitution (MRS)...

1. Emily has preferences for two goods x, y while her marginal rate of substitution (MRS) between x and y is given by 3y/x. Her budget constraint is m ≥ pxx + pyy, where m = income and px, py are prices of x, y respectively. (a) Emily's expenditure on y (i.e., pyy) is 1/3 of her expenditure on x (i.e., pxx). Is this true Are x and y normal goods? (b) Andrew has different preferences: his marginal rate of substitution (MRS) between x and y is equal to 3 for all x, y > 0. His budget constraint is the same as Emily's. What can you say about his demands for x, y?

(a) MRS = 3y / x

At optimality MRS = Px/Py => 3y / x = Px / Py

=> 3Y * Py = XPx

=> YpY = XPx / 3 ....... (1)

Budget constraint: XPx + YPy = M

From eq (1)

YPy = 1/3 * (XPx)

where YPy is expenditure on Y and XPx is the expenditure on X.

So it is True.

(b) MRS = 3

Budget constraint: XPx + YPy =M

MRS = MUx / MUy = 3 => MUx = 3MUy

If Px> 3Py then X= 0 and Y = M/Py

If Px < 3Py then X= M/Px and Y=0

If Px=3Py then XPx + YPy = M

3Py X + YPy = M

3X + Y =M/Py