Calculate the optimal consumption bundles for the following scenarios:

Question

(a) U(x,y) = x+47y−3y2. Income is $107. The price of x is $1 and the price of y is $23.

Answer 1

To find the optimal consumption bundles, we need to maximize utility subject to the budget constraint.

The budget constraint is given by:
px * x + py * y = I,
where px is the price of x, py is the price of y, and I is the income.

In this scenario, px = $1, py = $23, and I = $107.

The utility function is U(x, y) = x + 47y - 3y^2.

To solve for the optimal consumption bundle, we can use the Lagrange multiplier method.

The Lagrangian is defined as:
L(x, y, λ) = U(x, y) - λ(px * x + py * y - I).

Taking partial derivatives with respect to x, y, and λ, and setting them equal to zero, we get:

∂L/∂x = ∂U/∂x - λ * px = 1 - λ = 0 -> λ = 1,
∂L/∂y = ∂U/∂y - λ * py = 47 - 6y - 23 = 0 -> 6y = 24 -> y = 4,
px * x + py * y = I -> 1 * x + 23 * 4 = 107 -> 92 + x = 107 -> x = 15.

Therefore, the optimal consumption bundle is x = 15 and y = 4.