Problem 4: Constraint optimization - math

Suppose that there are two goods (and). The price ofis $2 per unit, and the price ofis $1 per unit. There are two consumers (A and B). The utility functions for the consumers are

UA(x,y)=(x^0,5)*(y^0,5)

UB(x,y)=(x^0,8)*(y^0,2)

Consumer A has an income of $100, and Consumer B has an income of $300.

1. Use Lagrangians to solve the constrained utility-maximization problems for Consumer A and Consumer B.
2. Calculate the marginal rate of substitution for each consumer at his or her optimal consumption bundles.
3. Suppose that there is another consumer (let us call her C). You do not know anything about her utility function or her income. All you know is that she consumes both goods. What do you know about C's marginal rate of substitution at her optimal consumption bundle?Why?
4. What is Consumer B's marginal utility ofandat the optimal consumption bundle? What is his benefit-cost ratio ofand? By how much does his utility increase if he receives an extra dollar to spend onand?