1. Two consumers each have an income of $300, with which they buy good X (costing $5) and good Y (costing $4). If A's preferences are given by UA = X2Y and B's preferences are given by UB = X(Y+100), what is the optimal amounts of X and Y each will purchase?

2. A elderly consumer's only income is his monthly Social Security check. Last year, food cost $5 and clothing cost $10 per unit, and he bought 100 units of food and 25 units of clothing each month, exhausting his monthly Social Security check of $750. This year, the price of food is $8 and the price of clothing is $12. So that he can afford the combination of food and clothing he purchased last year, the Social Security Administration has raised his monthly check to $1100. How might the consumer adjust the amounts of food and clothing he buys this year? Will he be better off, equally well off, or worse off this year in comparison with last year?

3. A very poor consumer spends his daily income of $6 (I) on: bread (X) which costs 25¢ per unit (PX) and milk (Y) which costs $1 per unit (PY). His preferences are described by the utility function: U = X3/4Y 3/4.

a) What is his marginal rate of substitution (MRSYX), his budget constraint (BC), his optimal condition, and his optimal consumption of bread and milk (X*1, Y*1)?

On an indifference curve (IC) diagram, show:

the consumer's budget constraint (with intercepts and slope), opportunity set, indifference curve with marginal rate of substitution, and the optimal point.

b) Because this consumer is so poor, he must worry about getting the minimum daily amount of an important nutrient that he needs to stay alive. He gets 1 unit of the nutrient per unit of bread (NX) and 1 unit of the nutrient per unit of milk (NY).

He needs at least 10 units of this nutrient per day to stay alive (J).

What is his subsistence constraint?

Does the consumption bundle you found in part (a) satisfy both constraints?

Sketch the consumer's new opportunity set.

c) Now the price of bread rises to 50¢ per unit (PX2).

Sketch the consumer's new opportunity set (with all constraints).

What is the consumer's new optimal amounts of bread and milk (X*2, Y*2)?

d) In part (a) are bread and milk normal or inferior goods? Are they Giffen goods?

In part (c), does bread behave like a Giffen good? Explain your answers.

4. A consumer's preferences for X and Y are described by the utility function: U = X2/3Y1/3.

Price of X is $8 and price of Y is $3, and her income is $150.

a) What is the equation for her income expansion path?

What is the equation for her Engel curve for X?

b) How will her budget share spent on X change if her income changes?

What is her income elasticity of demand for X? And the same for Y?

How does the budget share spent on Y change as income changes?

c) What is the equation for her demand curve for X?

How does her demand curve for X shift when the price of Y rises?

5. Pat divides a $900 monthly income between the consumption of food (X) and all else (Y).

Pat's preferences can be described by the following utility function: U = XY.

a) If the price of food is $1, and of all else is $2, how much of each good will Pat choose?

b) If now the price of food rises to $2, how much of each good will Pat choose?
c) Explain this change in terms of income and substitution effects.
d) How much would Pat be willing to pay to avoid this price increase?

6. Chris's utility from present consumption (CP) and future consumption (CF) is: U = CPCF.

Chris has an income of $100 today, and will have no income (aside from current savings and accrued interest) in the future. The interest rate is 10% on which a 30% tax is levied.

a) What is the future value of $1 of present consumption?
b) What is Chris's constraint on future consumption, in terms of present consumption?
c) How much will be consumed today and in the future, under optimal conditions? Sketch this constraint and the optimal point.
d) To encourage savings, the President eliminates the tax on Chris's interest income. Will this policy be successful? What changes under this policy?

Explain the net effect of this policy in terms of income and substitution effects.

Graduate-level question (to be answered by all students taking the course for graduate credit):

7. Using the Lagrange Method, maximize U = X1/2Y
1/2 subject to X + Y = 4.
(This is an advanced method for solving 2-variable constrained optimization problems with an equality constraint, and is especially useful for multiple and complex constraints. See Ch. 4 Appendix in Besanko & Braeutigam or Pindyck & Rubinfeld for more details.)

a) Write the objective function, U(X,Y), and the constraint, L(X,Y) = 0.

Write the Lagrangian function, Φ (or Λ) = U(X,Y) - λ × L(X,Y).

b) Find the first-order conditions (F.O.C.) for the Lagrangian: ∂Φ/∂X, ∂Φ/∂Y, ∂Φ/∂λ = 0.

Solve for λ and find Y in terms of X

c) What is MUX, MUY in terms of λ? What is MRSYX? What is the optimal condition and optimal consumption?

d) State the dual of this consumer optimization problem.