a)
U(X,Y) = (X + 2)(Y +1)
U(2,8) = (2 + 2)(8 + 1)
= 36
Thus, 36 = (X + 2)(Y +1)
36/(X+ 2) = Y + 1
Y = 36/(X + 2) - 1
Hence equation of Indifference curve Y = 36/(X + 2) - 1
b) Px = 1 Py = 1 M = 11
Budget constraint
XPx + YPy = M
X(1) + Y(1) = 11
X + Y = 11
c) U/X = MUx = (Y + 1)
U/Y = MUy = (X + 2)
MRS = MUx/MUy
= (Y + 1)/(X + 2)
d) At optimal choice MRS = Px/Py
(Y + 1)/(X + 2) = 1/1
(Y + 1)/(X + 2) = 1
Y + 1 = X + 2
Y = X + 2 - 1
Y = X + 1
Put Y = X + 1 in budget constraint
X + Y = 11
X + X + 1 = 11
2X = 11 - 1
2X = 10
X = 5
Y = 5 + 1
= 6
Thus optimal bundle is (5,6)
e) Again using optimal choice MRS = Px/Py
(Y + 1)/(X + 2) = Px/Py
Y + 1 = (X + 2)(Px/Py)
Y = (X + 2)(Px/Py) - 1
Put Y = (X + 2)(Px/Py) - 1 in budget constraint
XPx + YPy = M
XPx + Py[ (X + 2)(Px/Py) - 1] = M
XPx + (X+ 2)Px - Py = M
XPx + XPx + 2Px - Py = M
2XPx = M - 2Px + Py
X = M/2Px - 1 + Py/2Px
Thus, demand function of x
X(Px, Py,M) = M/2Px - 1 + Py/2Px
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