Final answer:
The Lagrangian for the given utility maximization problem, with the utility function U = x1^1/2 + x2, prices p1 and p2, and income I, is written as L = x1^1/2 + x2 + λ(I - p1x1 - p2x2), where λ is the Lagrange multiplier.
Step-by-step explanation:
The student is asking about how to formulate the Lagrangian for a utility maximization problem given a utility function, prices, and income constraints. In this context, the utility function is given as U = x11/2 + x2, where x1 and x2 represent the quantities consumed of two goods.
Prices for these goods are p1 and p2, and the consumer has a fixed income I.
In economics, the goal of the utility maximization problem is to find the combination of x1 and x2 that delivers the highest possible utility within the constraints of the consumer's budget.
The Lagrangian helps in solving this optimisation problem by combining the utility function and the budget constraint I = p1x1 + p2x2 into one function to work with.
The general rule, as described, tells us that at the utility-maximizing point, the ratio of the prices of the goods should equal the ratio of their marginal utilities.
To write down the Lagrangian, it's necessary to include a Lagrange multiplier, often denoted by the symbol λ (lambda), which represents the shadow price of the income constraint.
The Lagrangian (L) for this problem is formulated as:
L = x11/2 + x2 + λ(I - p1x1 - p2x2)
This equation incorporates both the utility to be maximized and the budget constraint that must be satisfied. By finding the first-order derivatives of L with respect to x1, x2, and λ, and setting them to zero, the optimal consumption bundle that maximizes utility can be determined.