Jill has a utility function defined on two goods x1 and x2 J

Jill has a utility function defined on two goods: x_1 and x_2. Jill\'s utility function is of the form u(x_1,y_1) = 2x_1^1/2 + x_2, His budget constraint is of the form I = p_1 x_1 + p_2 x_2 where I is income, p_1 is the price x_1 of and p_2 is the price x_2.Find Jill\'s demand for x_1 and x_2

Solution

U = X₁^1/2 X₂^1/2

    = X₁^0.5 X₂^0.5

The budget constraint is: I = P₁X₁ + P₂X₂

Then, MU_X1 = 0.5X₁^-0.5 X₂^0.5

            MU_X2 = 0.5 X₁^0.5 X₂^-0.5

For utility maximization

MU_X1/P₁ = MU_X2/P₂

0.5X₁^-0.5 X₂^0.5 /P₁ = 0.5 X₁^0.5 X₂^-0.5 /P₂

Solving for X₂

X₂^0.5/X₁^0.5P₁ = X₁^0.5/X₂^0.5P₂

X₂P₂ = X₁P₁

X₂ = X₁(P₁/P₂)

Substituting the value of X₂ into budget constraint

I = P₁X₁ + P₂X₂

   = P₁X₁ + P₂ X₁(P₁/P₂)

   = 2X₁P₁

Then, X₁ = I / 2P₁   (This is the demand curve for good X₁)

Similarly,

For utility maximization

MU_X1/P₁ = MU_X2/P₂

0.5X₁^-0.5 X₂^0.5 /P₁ = 0.5 X₁^0.5 X₂^-0.5 /P₂

Solving for X₁

X₂^0.5/X₁^0.5P₁ = X₁^0.5/X₂^0.5P₂

X₂P₂ = X₁P₁

X₁ = X₂(P₂/P₁)

Substituting the value of X₁ into budget constraint

I = P₁X₁ + P₂X₂

   = P₁ X₂(P₂/P₁) + P₂ X₂

   = 2X₂P₂

Then, X₂ = I / 2P₂   (This is the demand curve for good X₂)