Prove the profit maximization of the consumer i.e mu=p ,according to cardinalist using mathematical derivation
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To prove the profit maximization of the consumer using the concept of cardinal utility, we need to derive the condition known as the marginal utility equals price (MU = P).
Here's a step-by-step mathematical derivation to explain this concept:
1. Cardinal utility is a measure of utility that quantifies the satisfaction a consumer derives from consuming a good or service. It assumes that utility can be measured numerically.
2. The law of diminishing marginal utility states that as a consumer consumes more and more units of a good, the additional satisfaction derived from each additional unit decreases.
3. Suppose a consumer has a certain level of income (I) to spend on two goods, X and Y, and they want to maximize their utility from consuming these goods. Let's assume the prices of X and Y are Px and Py respectively.
4. Let's start by assuming that the consumer's utility function is given by U(X, Y), where X and Y are the quantities of goods consumed.
5. The consumer's budget constraint can be represented as: PxX + PyY = I, which states that the total expenditure on X and Y should equal the consumer's income.
6. To maximize utility subject to the budget constraint, we need to consider the consumer's marginal utilities of X and Y. The marginal utility of X (MUx) is the additional utility gained from consuming an extra unit of X, while the marginal utility of Y (MUy) is the additional utility gained from consuming an extra unit of Y.
7. Mathematically, the consumer's optimization problem can be stated as:
Maximize U(X, Y) subject to the budget constraint PxX + PyY = I.
8. To solve this problem, we introduce Lagrange multipliers. Let λ be the Lagrange multiplier associated with the budget constraint. The consumer's Lagrangian can be written as:
L(X, Y, λ) = U(X, Y) - λ(PxX + PyY - I).
9. To find the maximum utility, we differentiate the Lagrangian with respect to X, Y, and λ and set the derivatives equal to zero.
∂L/∂X = ∂U/∂X - λPx = 0 ...(1)
∂L/∂Y = ∂U/∂Y - λPy = 0 ...(2)
∂L/∂λ = PxX + PyY - I = 0 ...(3)
10. Equation (1) represents the marginal utility of X relative to its price, and equation (2) represents the marginal utility of Y relative to its price. Both equations state that the marginal utilities of X and Y are equal to the Lagrange multiplier times the respective prices.
11. From equations (1) and (2), we can write:
∂U/∂X = λPx ...(4)
∂U/∂Y = λPy ...(5)
12. Dividing equation (4) by equation (5), we get:
(∂U/∂X) / (∂U/∂Y) = (λPx) / (λPy)
Or, (∂U/∂X) / Px = (∂U/∂Y) / Py ...(6)
13. Equation (6) states that the marginal utilities of X and Y are "priced" in such a way that the ratio of their marginal utilities is equal to the ratio of their prices.
14. For profit maximization, the consumer will adjust their consumption until this equality holds. In other words, they will allocate their income in a way that the marginal utility per unit of expenditure is equal for all goods.
15. This condition can be expressed as:
MUx / Px = MUy / Py ...(7)
16. Equation (7) represents the condition of profit maximization in terms of the relationship between marginal utility and price. It states that the consumer will allocate their expenditure across goods X and Y in such a way that the marginal utility per dollar spent is equal for both goods.
Hence, by following the above mathematical derivation, we have proven that profit maximization for the consumer in cardinal utility theory is achieved when the condition MU = P (marginal utility equals price) is satisfied.