1. Utility is maximized when MRS = Px/Py
MRS = MUx/MUy =
y/x = Px/Py
So, y = xPx/Py
Budget constraint: Px*x + Py*y = I
So, xPx + Py(xPx/Py) = I
So, 2xPx = I
So, x = I/2Px
y = xPx/Py = (I/2Px)(Px/Py)
So, y = I/2Py
2. U = x1/2y1/2 = x1/2(xPx/Py)1/2 = x1/2x1/2(Px/Py)1/2
So, (Py/Px)1/2U = x(1/2)+(1/2)
So, x = (Py/Px)1/2U
y = xPx/Py = (Py/Px)1/2U(Px/Py) =
So, y = (Px/Py)1/2U
3. Px1 = 1; Py = 1; I = 20
x = I/2Px1 = 20/2*(1) = 10
y = I/2Py = 20/2(1) = 10
Px2 = 2; Py = 1; I = 20
x' = I/2Px2 = 20/2*(2) = 5
y' = I/2Py = 20/2(1) = 10
Total effect on x = x' - x = 5 - 10 = -5
Total effect on y = y' - y = 10 - 10 = 0
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