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Utility functions help us understand how individuals derive satisfaction or happiness from consuming goods or services. These functions are mathematical representations that describe the preference of individuals over a set of goods. In Mark and Judy's case, their utility functions express how much satisfaction each gets from their respective income levels. For Mark, the utility function is given by \( U_M = 100 \times I_M^{1/2} \), and for Judy \( U_J = 200 \times I_J^{1/2} \). These functions show diminishing returns, meaning as their incomes rise, the additional satisfaction gained from an additional unit of income decreases.

By using these functions, we measure how much "happiness" Mark and Judy derive from their portion of the total income, and how this distribution impacts the overall social welfare function.

Income distribution refers to how the total income in a society is divided among its members. In many economic models, we assume that income needs to be divided in a way that maximizes total social welfare. In our exercise, Mark and Judy must divide a total of \\(300. The challenge is to find the allocation that maximizes their combined satisfaction – as expressed by the social welfare function \( W = U_M + U_J \).

To determine an optimal distribution of income, it's essential to account for each individual's utility function. By examining the given utility functions, we can assess how different allocations between Mark and Judy affect total societal welfare. In this case, solving the utility maximization problem helps in deciding how the \\)300 should be split between them.

Calculus plays an important role in economic theory, especially in optimization problems. By using calculus, economists can determine how changes in one variable impact others and find the optimal solutions. In our problem, we're tasked with maximizing the social welfare function \( W = 100 \times I_M^{1/2} + 200 \times I_J^{1/2} \) subject to the constraint that \( I_M + I_J = 300 \).

Calculus helps us find this maximum by taking derivatives. In this exercise, we take the derivative of \( W \) with respect to \( I_M \), set it equal to zero, and solve for \( I_M \). This allows us to identify the income level at which the total welfare is maximized. Such applications demonstrate the utility of calculus in economic modeling and decision-making.

Marginal utility refers to the additional satisfaction someone receives from consuming an additional unit of a good or service. In our exercise, this concept is illustrated by how Mark and Judy's additional satisfaction changes with each extra dollar of income.

Mathematically, the marginal utility can be derived from the utility functions. For example, the marginal utility of Mark's income is represented by the derivative of his utility function: \( \frac{dU_M}{dI_M} = \frac{50}{I_M^{1/2}} \), indicating Mark's additional satisfaction from an extra unit of income. Similarly, Judy's marginal utility from income is \( \frac{dU_J}{dI_J} = \frac{100}{I_J^{1/2}} \).

Analyzing these helps us decide how the additional income should be allocated. The goal is to equalize the marginal utilities to achieve the optimal distribution that maximizes the social welfare function.