Derive the solution to the utility maximization problem by using different ways.?

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Utility Maximization Problem

Utility maximization is a fundamental concept in economics that involves determining the optimal allocation of resources to maximize an individual's satisfaction or utility. The problem can be mathematically represented as:

Maximize U(x₁, x₂, ..., x_n)

subject to

p₁x₁ + p₂x₂ + ... + p_nx_n = I

Method 1: Lagrange Multiplier Approach

The Lagrange multiplier approach is a common method to solve constrained optimization problems. To solve the utility maximization problem using this method, we introduce a Lagrange multiplier (λ) and set up the following Lagrangian:

L(x₁, x₂, ..., x_n, λ) = U(x₁, x₂, ..., x_n) - λ(p₁x₁ + p₂x₂ + ... + p_nx_n - I)

We then take partial derivatives of the Lagrangian with respect to each variable and the Lagrange multiplier, setting them equal to zero:

∂L/∂x_i = ∂U/∂x_i - λp_i = 0 (for i = 1 to n)

∂L/∂λ = p₁x₁ + p₂x₂ + ... + p_nx_n - I = 0

Solving these equations simultaneously will yield the optimal values for the consumption of each good (x₁, x₂, ..., x_n) that maximize utility.

Method 2: Marginal Utility Approach

Another approach to solving the utility maximization problem is by using marginal utilities. The idea behind this method is to allocate resources in a way that the marginal utility per dollar spent on each good is equal.

Mathematically, this can be expressed as:

MU(x₁) / p₁ = MU(x₂) / p₂ = ... = MU(x_n) / p_n

where MU(x_i) represents the marginal utility of good i and p_i represents its price.

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