Question

Suppose u(x1, x2 ) = x1^ax2^1-a

1. (a) Find the optimal bundle x(p, w) and the indirect utility function v(p, w).

2. (b) Find the Hicksian demand function h(p, u) and the expenditure function e(p, u).

3. (c) For the remainder of the problem, suppose α = 4 and w = 5. If p = (2,1), what is5

   the optimal bundle? What is the utility of that bundle? [Leave your answer in terms of fractions and exponents]

4. (d) Suppose the price of good 2 increases, so that the new price vector is p′ = (2, 2).

   (i) Find the new bundle. What is the utility of that bundle? (ii) Compute the compensating variation of the price change.

   (iii) Compute the equivalent variation of the price change.

Answer 1

Here, utility function is given by : u(x1, x2 ) = x1^ax2^1-a

a) The optimum solution can be derived by solving the exercise:
Max u(x1, x2) subject to px ≤ w

Here, optimal bundle: and

the indirect utility function

b)  By deriving the first order conditions for the Expenditure Minimizing Procedure (EMP) and substituting from the constraints u (h₁ (p, u), h₂ (p, u) = u, we obtain the Hicksian demand functions.

where,

The expenditure function can be obtained as: e(p,u) = p.h(p,u)

Here,

c) NOTE: I am assuming you have mistakenly written a = 0.4 as 4

For the given information, X = (1,3) and U(1,3) = 1.933 = (1933/1000) = 1.933e + 3

d)

i) New bundle, X = (1,1.5)

and New utility, U(1,1.5) = 1.275 = 51/40

ii) Using the expenditure function, we get the CV as = (5.742 - 4.998) = 0.744

EV can be solved by solving the exercise :

Solving which, we get EV = 2.578

Answer 2

Here, utility function is given by : u(x1, x2 ) = x1^ax2^1-a

a) The optimum solution can be derived by solving the exercise:
Max u(x1, x2) subject to px ≤ w

Here, optimal bundle: and

the indirect utility function

b)  By deriving the first order conditions for the Expenditure Minimizing Procedure (EMP) and substituting from the constraints u (h₁ (p, u), h₂ (p, u) = u, we obtain the Hicksian demand functions.

where,

The expenditure function can be obtained as: e(p,u) = p.h(p,u)

Here,

c) NOTE: I am assuming you have mistakenly written a = 0.4 as 4

For the given information, X = (1,3) and U(1,3) = 1.933 = (1933/1000) = 1.933e + 3

d)

i) New bundle, X = (1,1.5)

and New utility, U(1,1.5) = 1.275 = 51/40

ii) Using the expenditure function, we get the CV as = (5.742 - 4.998) = 0.744

EV can be solved by solving the exercise :

Solving which, we get EV = 2.578