Problem
A consumer has m dollars to spend on xylophones (x) and yardsticks (y), where m > 0 is an exogenously fixed number. Each xylophone costs $1 and each yardstick also costs $1. The consumer derives material welfare from these two goods according to the following utility function: u(x, y) = x 2 y. Non-integer units of consumption are allowed (even though these two items are naturally supposed to be purchased in discrete units). In this problem, assume that the consumer is required to spend all of her income, but disregard the non-negativity constraints on consumption.
1. Use Lagrange's method to solve for the optimal consumption bundle and the optimal value of the Lagrange multiplier (x ∗ (m), y ∗ (m), λ ∗ (m)) as functions of the consumer's income m.
2. Compute the value function, which is defined as the utility value of the best consumption bundle for every given level of income.