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  =

(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