Prove the profit maximization of the consumer i.e mu=p according to cardinalist using mathematical derivation.
no
answered
In cardinal utility theory, the consumer's objective is to maximize their total utility or satisfaction from consuming goods or services. The condition for profit maximization of the consumer is given by the equation: marginal utility (MU) equals price (P).
Step 1: Start with the utility function. Assume the consumer's utility function is given by U(x) where "x" represents the quantity consumed.
Step 2: Determine the marginal utility (MU) function. This represents the additional utility gained from consuming an additional unit of the good.
MU = dU/dx, where "dU" is the change in utility and "dx" is the change in quantity consumed.
Step 3: Assume the consumer's budget constraint. Let's say the consumer has a limited income (I) and the price of the good is "p". Therefore, the consumer's budget constraint can be expressed as: I = p * x, where "x" is the quantity consumed.
Step 4: Apply the Lagrange optimization technique. We need to maximize the consumer's utility function subject to the budget constraint. Let λ (lambda) be the Lagrange multiplier.
The Lagrangian function is defined as: L(x, λ) = U(x) - λ(I - p * x)
Step 5: Take the partial derivative of L(x, λ) with respect to "x" and set it equal to zero to find the critical point:
∂L/∂x = ∂U/∂x - λp = 0
Step 6: Rearrange the equation to get MU = λp.
This equation implies that the consumer maximizes their utility by equating the marginal utility (MU) with the price (p) of the good.
Hence, we have proven mathematically that in cardinal utility theory, the condition for profit maximization of the consumer is MU = p.