Derive mathematically the consumer equilibrium conditions based on ordinary utility. Assume the goods x1 and x2 are consumed at price p2 and p2 respectively ?

To derive the consumer equilibrium conditions based on ordinary utility, we need to consider the consumer's budget constraint and their utility maximization problem.

Let's assume the consumer has a fixed income (I) and consumes two goods, x1 and x2, at prices p1 and p2 respectively.

The consumer's budget constraint can be expressed as:
p1x1 + p2x2 = I

The consumer's utility function can be represented as U(x1, x2).

To maximize utility subject to the budget constraint, we can set up the Lagrangian function:

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

Where λ is the Lagrange multiplier.

To find the consumer equilibrium conditions, we need to solve the following equations:

1. First-order condition with respect to x1:
∂L/∂x1 = ∂U/∂x1 - λp1 = 0

2. First-order condition with respect to x2:
∂L/∂x2 = ∂U/∂x2 - λp2 = 0

3. Budget constraint:
p1x1 + p2x2 = I

Solving these equations simultaneously will give us the consumer equilibrium conditions.

Note: The specific form of the utility function U(x1, x2) is not provided, so we cannot derive the exact equilibrium conditions without knowing the functional form of the utility function.