prove the profit maximization of the consumer i.e. mu=p, accordintg to cardinalist using mathmatical derivation.
First we need to understand "cardinalist", someone in favour of the cardinal utility theory, namely utility (happiness) is quantifiable, [ and can be mathematically represented by a curve, and diffentiable, most of the time. ]
https://www.enotes.com/homework-help/discuss-cardinal-utility-theory-how-do-cardinalist-115449
We can consider the typical consumer to make choices in small units, and having limited budget. So each next unit of money will be spent on the most profitable (maximum utility/unit price, most happiness/unit price) item. Since utility varies (usually decreases) with quantity, the choice of the next unit of spending may be different. So the choice depends on maximum (additional) utility/unit price, or mu/p.
A more detailed explanation can be found in the following presentation (requires Microsoft Powerpoint or MPP reader).
gottheil.swcollege.com/gottheil3e/powerpoint/marg_util_powerpoint.ppt
We can consider the typical consumer to make choices in small units, and having limited budget. So each next unit of money will be spent on the most profitable (maximum utility/unit price, most happiness/unit price) item. Since utility varies (usually decreases) with quantity, the choice of the next unit of spending may be different. So the choice depends on maximum (additional) utility/unit price, or mu/p.
To prove profit maximization for a consumer according to the cardinalist approach, we need to assume that utility (u) is a measurable and quantifiable concept. In this approach, utility is represented by a cardinal-number scale, allowing us to assign numerical values to different levels of utility.
Let's suppose a consumer's utility function is given by U(x1, x2), where x1 and x2 are quantities of two goods consumed by the consumer. The consumer has a fixed income (I) and faces prices (p1, p2) for the goods. The goal is to maximize utility, subject to the constraint of the consumer's income.
Step 1: Set up the consumer's optimization problem:
Maximize U(x1, x2) subject to the constraint p1x1 + p2x2 = I.
Step 2: Set up the Lagrangian function:
L(x1, x2, λ) = U(x1, x2) - λ(p1x1 + p2x2 - I), where λ is the Lagrange multiplier.
Step 3: Take the first-order conditions by differentiating the Lagrangian function:
∂L/∂x1 = ∂U/∂x1 - λp1 = 0
∂L/∂x2 = ∂U/∂x2 - λp2 = 0
∂L/∂λ = p1x1 + p2x2 - I = 0 (Budget constraint)
Step 4: Solve the first-order conditions simultaneously:
∂U/∂x1 - λp1 = 0 -> Equation (1)
∂U/∂x2 - λp2 = 0 -> Equation (2)
p1x1 + p2x2 - I = 0 -> Equation (3)
Step 5: Rearrange Equation (1) and Equation (2):
∂U/∂x1 = λp1 -> Equation (4)
∂U/∂x2 = λp2 -> Equation (5)
Step 6: Divide Equation (4) by Equation (5):
∂U/∂x1 / ∂U/∂x2 = p1/p2
Step 7: Rearrange the equation to obtain:
∂U/∂x1 = p1/p2 * ∂U/∂x2
Step 8: Rearrange Equation (3):
p1x1 = I - p2x2
Step 9: Plug the rearranged Equation (3) into Equation (8):
p1(I - p2x2) = p1x1 = I - p2x2
Step 10: Simplify the equation:
Ip1 - p1p2x2 = I - p2x2
Step 11: Cancel out terms and rearrange:
Ip1 = I - p2x2 + p1p2x2
Ip1 = I(1 - p2) + p1p2x2
Step 12: Divide by I and rearrange:
p1/I = 1 - p2 + p1p2x2/I
Step 13: As p1/I = λ (from Equation 4), we can rewrite the equation as:
λ = 1 - p2 + λp2x2/I
Step 14: Rearrange the equation:
p2 - λp2x2/I = 1 - λ
Step 15: Divide by p2(I-λ):
1/x2 = 1/p2 - λ/I
Step 16: Rearrange the equation to get the marginal utility of x2:
∂U/∂x2 = λp2
From step 16, we can conclude that the marginal utility of good x2 is equal to the price of good x2 (p2) multiplied by the Lagrange multiplier (λ).
Step 17: Using the same logic, we can show that the marginal utility of good x1 is equal to the price of good x1 (p1) multiplied by the Lagrange multiplier (λ).
Step 18: Since the Lagrange multiplier (λ) times the price of good x1 (p1) gives the marginal utility of good x1, and the Lagrange multiplier (λ) times the price of good x2 (p2) gives the marginal utility of good x2, we can conclude that for a consumer to maximize utility, the marginal utility of each good should be proportional to its price.
Therefore, the proof shows that, according to the cardinalist approach, in order to maximize their utility, consumers should allocate their budgets such that the ratio of the marginal utilities of goods consumed is equal to the ratio of their prices. In other words, consumers should allocate their budgets in a way where the marginal utility per dollar spent is equal across all goods.
To prove the profit maximization of a consumer according to cardinal utility theory using mathematical derivation, you need to understand the underlying assumptions and concepts of this theory.
Assumptions:
1. Consumers are rational: They aim to maximize their utility or satisfaction from consuming goods and services.
2. Consumers have complete and transitive preferences: They can rank different combinations of goods and services.
3. Marginal utility diminishes: As a consumer consumes more of a particular good, the additional satisfaction derived from each additional unit consumed decreases.
The cardinal utility theory assumes that utility can be measured numerically, representing the consumer's level of satisfaction. The utility function is typically denoted as U(x, y, z, ...), where x, y, z represent the quantities consumed of different goods.
Let's consider a consumer's utility function as U(x, y), representing two goods, x and y. The consumer's budget constraint is given by the equation p1x + p2y = M, where p1 and p2 are the prices of goods x and y, respectively, and M is the consumer's income.
To derive the profit maximization condition, we use the concept of marginal utility (MU), which measures the change in utility resulting from a one-unit change in consumption of a good.
The consumer maximizes utility subject to the budget constraint by allocating their income in such a way that the marginal utility per dollar spent on each good is equal. This condition is represented as:
MU(x) / p1 = MU(y) / p2
In this equation, MU(x) represents the marginal utility of good x, MU(y) represents the marginal utility of good y, p1 represents the price of good x, and p2 represents the price of good y.
The equation above states that the consumer will allocate their spending so that the marginal utility per dollar spent on good x is equal to the marginal utility per dollar spent on good y. This ensures that the consumer obtains the maximum possible utility given their budget constraint.
Note that this derivation assumes that the consumer can assign cardinal values to their utility and measure the utility at the marginal level accurately.
By solving the above equation, you can find the optimal quantities of goods x and y that will maximize the consumer's utility while satisfying the budget constraint. This represents the profit maximization of the consumer according to cardinal utility theory.
It is important to note that cardinal utility theory has been widely debated, and alternative theories, such as ordinal utility theory, provide different perspectives on consumer behavior.