prove the profit maximaization of the consume i.e mu=p, according tocardinalist using mathmaticale derivation
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prove the profit maximization of the consumer i.e mu p. according list using mathematical derivation ?
To prove the profit maximization of the consumer, we need to determine the consumer's optimal level of consumption. The concept of marginal utility (MU) and the principle that consumers maximize utility when MU is equal to price (P) can be used to achieve this.
Let's follow the steps below to derive mathematically the condition for profit maximization using cardinal utility theory:
1. Start with the utility function:
U = f(X, Y), where X represents the quantity of one good consumed and Y represents the quantity of another good consumed.
2. Determine the marginal utility of X (MUx) and the marginal utility of Y (MUy):
MUx = ∂U / ∂X and MUy = ∂U / ∂Y.
3. Set up the constraint equation:
Let's assume the consumer has a fixed budget (B), and the prices of X and Y are denoted by Px and Py, respectively. The constraint equation can be written as:
Px * X + Py * Y = B.
4. Apply the principle of utility maximization:
According to cardinal utility theory, a consumer maximizes utility when the marginal utility per dollar spent on each good is equal. Therefore, we have:
MUx / Px = MUy / Py.
5. Rearranging the equation:
We can rewrite the equation as:
MUx / MUy = Px / Py.
6. Simplifying further:
By cross-multiplying, we get:
MUx * Py = MUy * Px.
7. Dividing both sides of the equation by Py:
We obtain:
MUx = (MUy * Px) / Py.
8. Conclusion:
From the derived equation, we can see that the condition for profit maximization in cardinal utility theory states that the marginal utility of X (MUx) should be equal to (MUy * Px) divided by Py.
By satisfying this condition, consumers will have maximized their utility given their budget constraint. This equation shows that the consumer maximizes utility when MUx is equal to the ratio of (MUy * Px) to Py, indicating that the consumer values the goods based on their marginal utility relative to their prices.