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The Marshallian demand function, also known as the ordinary demand function, represents the relationship between the quantity of a good a consumer demands and the prices of goods, as well as the consumer’s income, while holding preferences constant. This concept is fundamental in microeconomic analysis as it illustrates how consumers allocate their income across different goods to maximize utility under a budget constraint. To derive the Marshallian demand functions for two goods, X and Y, we start by considering a consumer who aims to maximize their utility subject to a budget constraint. Let us assume the consumer has a utility function $U(X, Y)$, where $X$ and $Y$ are quantities of goods X and Y respectively. The utility function captures the satisfaction or happiness the consumer derives from consuming different combinations of X and Y. The consumer faces prices $P_X$ for good X and $P_Y$ for good Y, and has a total income $M$. The consumer’s budget constraint can be expressed as $P_X X + P_Y Y = M$, which means the total expenditure on goods X and Y cannot exceed the consumer’s income. To find the Marshallian demand functions, the consumer’s problem is formulated as a constrained optimization problem: maximize $U(X, Y)$ subject to $P_X X + P_Y Y = M$. A common method for solving this is using the Lagrangian technique. The Lagrangian function is given by: $$ \mathcal{L} = U(X, Y) + \lambda (M - P_X X - P_Y Y) $$ where $\lambda$ is the Lagrange multiplier that measures the marginal utility of income. To find the optimal quantities of X and Y, we take the partial derivatives of the Lagrangian with respect to $X$, $Y$, and $\lambda$ and set them ____ _______ _____ __________ __________ ____ ______ ______ ___.
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