Question

The utility function U(X,Y)=X^aY^1-a where 0≤a≤1 is called the Cobb-Douglas utility function. MUx=aX^a-1Y^1-a MUy=(1-a)X^aY^-a

(note for those who know calculus MUx=∂U∂x and MUy=∂U∂y)

• Derive the demand functions for X and Y
• Are X and Y normal goods? If the quantity of the good increases with income a good is a normal good. If the quantity decreases with income the good is an inferior good.
• Describe in words the preferences corresponding to a=0, a=1, a=.5

Answer 1

Utility function is U(X,Y) = X^a*Y^(1-a)

MUx = aX^(a-1)*Y^(1-a)

MUy = (1-a)X^a*Y^(-a)

MRSx,y = MUx/MUy

=aX^(a-1)*Y^(1-a)/((1-a)X^a*Y^(-a))

=(a/(1-a))(Y/X)

The Budget equation is: PxX + PyY = M

At equilibrium,MUx/MUy = Px/Py

This implies,

(a/(1-a))(Y/X) = Px/Py

aYPy = (1-a)XPx

YPy = ((1-a)/a)XPx

Thus, the demand equation is:

PxX + PyY = M

PxX +((1-a)/a)XPx = M

XPx(1+((1-a)/a)) = M

XPx(1/a)=M

X = aM/Px

Y= ((1-a)/a)XPx/Py

Y = ((1-a)/a)*aM/Py

Y = (1-a)M/Py

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Thus, the demand function of X = aM/Px

The demand function for Y = (1-a)M/Py

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Yes, good X and Y are normal goods as the quantity demanded of X and Y each is directly proportional to the income, implying the quantity of the good increases with income.

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When a = 0

X = aM/Px = 0

Y = (1-a)M/Py = M/Py

When a=1

X = aM/Px = M/Px

Y = (1-a)M/Py = 0

When a = 0.5

X = aM/Px = M/(2Px)

Y = (1-a)M/Py = M/(2Py)

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