Question

Advanced Microeconomics

Instruction:

1.      The utility maximization problem of a consumer is given by

 

  max X1, X2

 

 U(X₁,X₂)=X₁^αX₂^1-α                               

 s.t. 𝑝₁x₁ + 𝑝₂x₂ = 𝑚

 

+

Where 𝛼 ∈ ,[0,1] &   𝑥₁, 𝑥₂  ∈ real number . Assume that price vectors is 𝑝 = (𝑝₁, 𝑝₂) > 0,

and income 𝑚 > 0.

 

a)            Find the Marshallian demand functions

b)            Find the budget share and price of x₁ and income elasticity

c)            Show that the Walrasian demand function is homogeneity of degree zero in

(p, m)

d)           Show that 𝑒(p, u) is homogeneous of degree one in 𝑝

e)           Show that 𝑣(p, m) is strictly increasing in m and non-increasing 𝑝

f)         Show that Hicksian demand functions is homogeneous of degree zero in p.

g)            Evaluate the Walrasian demands x(p, m) at m = e(p; u), and show that Walrasian and Hicksian demands coincide, that is,

 

x(p, e(p, u)) = h(p; u):

 

h)       Evaluate the Hicksian demands h(p, u) at u = v(p, m), and show that Hicksian and Walrasian demands coincide, that is,

 

h(p, v(p, m)) = x(p, u):

 

i)         Evaluate the indirect utility function v(p, m) at m = e(p, u), and show that  v(p, e(p, u)) = u:

j)        Evaluate the expenditure function e(p, u) at u = v(p, m), and show that e(p, v(p, m)) = m: