Prove that the profit maximation of the consume i.e MU=P,according to cardinalist using mathematical derivation.
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mu=p
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To prove that profit maximization occurs when marginal utility (MU) is equal to price (P) according to cardinalist theory, we need to use a mathematical derivation.
The basic idea behind cardinalist theory is that utility can be quantified and measured numerically. This means we can assign numerical values to utility levels.
Let's assume that the consumer is maximizing their profit by consuming a single good. The consumer's utility function is represented by U(Q), where Q is the quantity consumed.
The marginal utility (MU) can be defined as the additional utility gained from consuming an additional unit of the good. It can be expressed as:
MU = dU(Q)/dQ
The price (P) represents the cost of each unit of the good.
To maximize profit, the consumer should choose the quantity of the good that maximizes the difference between the total utility obtained and the total cost incurred. Mathematically, this can be represented as:
Profit (π) = U(Q) - P * Q
To find the quantity that maximizes profit, we need to take the derivative of the profit function with respect to Q and set it equal to zero. This is because profit is maximized when its derivative is zero.
dπ/dQ = dU(Q)/dQ - P = 0
Now, rearranging the equation, we have:
dU(Q)/dQ = P
This equation shows that profit is maximized when the marginal utility (dU(Q)/dQ) is equal to the price (P). In other words, the consumer should continue to consume more of the good until the additional utility gained (MU) equals the price they have to pay for it.
Therefore, we have mathematically derived that profit maximization occurs when MU = P according to cardinalist theory.