Final Answer:
The consumer's optimal consumption bundle, considering the utility function U(x,y) = x + ln(y), is determined by the following conditions: x = 1 and y is as high as possible.
Step-by-step explanation:
The optimal consumption bundle can be found by maximizing the consumer's utility function subject to the budget constraint. In this case, the consumer aims to maximize U(x, y) = x + ln(y) given their budget constraint, which can be expressed as Pₓx + Pᵧy = M, where Pₓ and Pᵧ are the prices of goods x and y, and M is the consumer's income.
To solve for the optimal values of x and y, we set up the Lagrangian:
L(x, y, λ) = x + ln(y) + λ(M - Pₓx - Pᵧy)
Taking the partial derivatives with respect to x, y, and λ, and setting them equal to zero, we get the following system of equations:
1. ∂L/∂x = 1 - λPₓ = 0
2. ∂L/∂y = 1/y - λPᵧ = 0
3. ∂L/∂λ = M - Pₓx - Pᵧy = 0
From the first equation, we find that λ = 1/Pₓ. Substituting this into the second equation gives 1/y - (1/Pₓ)Pᵧ = 0, which simplifies to y = Pᵧ/Pₓ.
Now, substituting these values into the budget constraint, we find that x = M/Pₓ. Therefore, the optimal consumption bundle is x = M/Pₓ and y = Pᵧ/Pₓ, and the consumer's utility is maximized when x = 1 and y is as high as possible.