Question

Suppose you have the following preferences u(x,y) = x^ay^b. Calculate the optimal demand functions. Is good y a normal or inferior good?

Please show work.

Answer 1

Answer 2

To find the optimal demand functions, we need to maximize the utility function subject to the budget constraint. Let's assume the consumer has an income I, and the prices of goods x and y are Px and Py, respectively.

The consumer's optimization problem is as follows:

Maximize u(x, y) = x^a * y^b subject to the budget constraint: Px * x + Py * y = I

To find the optimal demand functions, we'll use the method of Lagrange multipliers. The Lagrangian function is defined as:

L(x, y, λ) = x^a * y^b + λ(I - Px * x - Py * y)

Now, we'll find the first-order conditions by taking partial derivatives with respect to x, y, and λ and setting them to zero:

∂L/∂x = a * x^(a-1) * y^b - λ * Px = 0 ... (1) ∂L/∂y = b * x^a * y^(b-1) - λ * Py = 0 ... (2) ∂L/∂λ = I - Px * x - Py * y = 0 ... (3)

From equations (1) and (2), we can solve for λ:

λ = a * x^(a-1) * y^b / Px ... (4) λ = b * x^a * y^(b-1) / Py ... (5)

Now, we'll equate the two expressions for λ:

a * x^(a-1) * y^b / Px = b * x^a * y^(b-1) / Py

Solving for x/y:

x / y = (b * Py) / (a * Px) ... (6)

Now, we'll use equation (3) to solve for x:

I = Px * x + Py * y I = Px * x + Py * (x / ((b * Py) / (a * Px))) [using equation (6)] I = Px * x + a * x x * (1 + Px/a) = I x = I / (1 + Px/a) ... (7)

Next, we'll find the optimal demand for y using equation (6):

x / y = (b * Py) / (a * Px) y = (a * Px * x) / (b * Py) [using equation (7)] y = (a * Px * (I / (1 + Px/a))) / (b * Py) y = (a * I) / (b * Py + a * Px) ... (8)

So, the optimal demand functions are:

Optimal demand for x: x = I / (1 + Px/a) Optimal demand for y: y = (a * I) / (b * Py + a * Px)

Now, let's analyze whether good y is a normal or inferior good. For that, we need to look at the income elasticity of demand for y. The income elasticity of demand (Ey) is given by:

Ey = (% change in quantity demanded of y) / (% change in income)

From the optimal demand function for y (equation 8), we can see that the income elasticity of demand for y is positive. This indicates that y is a normal good, as an increase in income will lead to a proportionate increase in the quantity demanded of y.