Question

A consumer lives on an island where she produces two goods, x and y according to production possibility frontier x² +y² ≤200, and she consumes all the goods herself. Her utility function is U=xy³ . She also faces an environmental constraint on her production as x+y≤20.

(a) Find her optimal x and y.

Answer 1

The question involves optimizing a utility function with constraints, which requires the use of Lagrangian multipliers to find values of x and y that maximize the consumer's utility while satisfying both the production possibility frontier and the environmental constraint.

Step-by-step explanation:

The student's inquiry relates to finding the optimal points of production (x and y) on an island with a given utility function U = xy³, subject to two constraints: the production possibility frontier (x² + y² ≤ 200) and an environmental constraint (x + y ≤ 20). The problem is a classic example of how theoretical models can simplify complex real-world issues to facilitate understanding, much like the simplified two-goods economy models used in economics to illustrate trade-offs and resource allocation.

To solve for the optimal x and y, we'd typically set up a Lagrangian with the utility function and the constraints, and find the maximum utility by taking the partial derivatives with respect to x, y, and the Lagrange multipliers associated with the constraints. The optimal x and y would satisfy both constraints while yielding the maximum possible utility for the consumer. This is an application of constrained optimization in microeconomic theory, specifically dealing with consumer choice theory.