ECON 4402 - Microeconomic Theory Homework 3
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ECON 4402 - Microeconomic Theory
Homework 3 - answer key
2022
1. Show that the Marshallian demand function satisfies homogeneity of degree zero in (p,y), that is, x(p,y) = x(αp,αy) for any (p,y) and a scalar α > 0. Do the same for the indirect utility function.
Just verify that first order conditions are not affected when (p,y) is multiplied by a scalar α > 0. In this case, the demand function does not change. As a consequence, the indirect utility function also remains the same.
2. Consider the following utility function: U = min{x1 ,x2 }
(a) Find the Marshallian demand function of each good x1 and x2 .
Because this utility function is not differentiable in all points, we state the condition that x1 = x2 . Substituting in the budget constraint p1 x1 + p2 x2 = ω , where ω is the consumer’s income, we find that p1 x1 + p2 x1 = ω implies the following Marshallian demand function:
x1 = x2 =
ω |
p1 + p2 |
(b) Use the Slutsky equation to calculate the substitution and income effects on the
demand of good 1 caused by a change in p1 .
Writing the Slutsky equation in its ’short’version, we have:
∂h1 ∂x1 ∂x1
∂p1 ∂p1 ∂ω
Because goods 1 and 2 are perfect complements, the substitution effect must be zero, and the Slutsky equation becomes:
∂x1 ∂x1
∂p1 ∂ω
The derivative of the Marshallian demand function with respect to ω is −1/(p1 + p2 ), which is substituted along with x1 in the previous equation:
∂x1 1 ω ω
∂p1 p1 + p2 p1 + p2 (p1 + p2 )2
All the impact of a price change is generated by the income effect. No substitu- tion occurs, as the two goods are perfect substitutes. Intuitively, the consumer is ’tied’by the complementarity of the goods, and does not make any substitution after a change in prices.
3. Let xj (p,ω) and ν(p,ω) represent the Marshallian demand function of good j and the indirect utility function, respectively, where p stands for price and ω for income. The Roy’s identity is described by the following relation:
∂ν(p,ω)/∂pj
xj (p,ω) = −
Verify the Roy’s identity for the utility function u(x1 ,x2 ) = (x1(ρ) + x2(ρ))1/ρ . The indirect utility function is:
ω
p1(ρ)/(ρ − 1) + p2(ρ)/(ρ − 1) (ρ − 1)/ρ
We calculate the derivatives of the indirect utility function with respect to income:
∂ν(p,ω) 1
∂ω p1(ρ)/(ρ − 1) + p2(ρ)/(ρ − 1) (ρ − 1)/ρ
and with respect to price:
∂ν(p,ω) ωp 1(1)/(ρ − 1)
∂p1 p1(ρ)/(ρ − 1) + p2(ρ)/(ρ − 1) − 1/ρ
Now we calculate the ratio of these two derivatives as equal to:
∂ν(p,ω)/∂ω ωp 1(1)/(ρ − 1)
∂ν(p,ω)/∂p1 p1(ρ)/(ρ − 1) + p2(ρ)/(ρ − 1)
which coincides with the Marshallian demand function.
2022-04-06