4.17 Problems

  1. For the following utility functions,

    • Find the marginal utility of each good.

    • Determine whether the marginal utility decreases as consumption of each good increases (i.e., does the utility function exhibit diminishing marginal utility in each good?).

    • Find the marginal rate of substitution.

    • Discuss how MRSXY changes as the consumer substitutes X for Y along an indifference curve.

    • Derive the equation for the indifference curve where utility is equal to a value of 100.

    • Graph the indifference curve where utility is equal to a value of 100.

    1. U(X,Y) = 5X + 2Y

    2. U(X,Y) = X0.33Y 0.67

    3. U(X,Y) = 10X 0.5 + 5Y

  2. Suppose that Maggie cares only about chai and bagels. Her utility function is U = CB, where C is the number of cups of chai she drinks in a day, and B is the number of bagels she eats in a day. The price of chai is $3, and the price of bagels is $1.50. Maggie has $6 to spend per day on chai and bagels.

    1. What is Maggie’s objective function?

    2. What is Maggie’s constraint?

    3. Write a statement of Maggie’s constrained optimization problem.

    4. Solve Maggie’s constrained optimization problem using a Lagrangian.

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

    UA(x,y) = x 0.5y 0.5

    UB (x,y) = x 0.8y 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’s call her C ). You don’t 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. Katie likes to paint and sit in the sun. Her utility function is U(P, S ) = 3PS + 6P, where P is the number of paint brushes and S is the number of straw hats. The price of a paint brush is $1 and the price of a straw hat is $5. Katie has $50 to spend on paint brushes and straw hats.

    1. Solve Katie’s utility-maximization problem using a Lagrangian.

    2. How much does Katie’s utility increase if she receives an extra dollar to spend on paint brushes and straw hats?

  5. Suppose that a consumer’s utility function for two goods (X and Y) is

    U(X,Y) = 10X0.5 + 2Y

    The price of good X is $5 per unit and the price of good Y is $10 per unit. Suppose that the consumer must have 80 units of utility and wants to achieve this level of utility with the lowest possible expenditure.

    1. Write a statement of the constrained optimization problem.

    2. Use a Lagrangian to solve the expenditure-minimization problem.