Question

2. Consider the following four consumers (C1,C2,C3,C4) with the following utility functions:

Consumer   Utility Function

C1                 u(x,y) = 2x+2y

C2                 u(x,y) = x^3/4y^1/4

C3                u(x,y) = min(x,y)

C4                u(x,y) = min(4x,3y)

On the appropriate graph, draw each consumer’s indifference curves through the following points: (2,2), (4,4), (6,6) and (8,8), AND label the utility level of each curve. Hint: Each grid should have 4 curves on it representing the same preferences but with different utility levels.

3. In the following parts, you will graph budget constraints under different prices and then analyze the possible utility maximizing bundles. Assume that preferences are rational, continuous, strictly convex, and strongly monotone. Be sure to show your calculations for graphing the budget constraints.

(a) Suppose David wants to maximize his utility over bundles of apples (x) and oranges (y). If he has $12 to spend, and a bag of apples costs $3 and a bag of oranges costs $2, what would his budget constraint look like? Graph his budget constraint. Make sure to label and number the axes.

(b) Would we be guaranteed a unique utility maximizing bundle? Meaning, will there only be 1 combination of quantities of apples and oranges that will give David the highest utility level? Explain why.

(c) Now suppose that a hurricane damaged a lot of the apple trees in the US, so the price of apples increases to $6 per bag. On the same graph (from part a), graph the new budget constraint. Interpret what the change in the budget constraint (axis intercepts and slope) means for David. The price of oranges is the same as part a.

(d) What if instead of switching completely to the higher price, the local grocery store decided to have a deal for their loyal customers (like David) where the price of apples depended on the quantity the customer purchases. The deal is: the price of a bag of apples is $1 for each of the first 3 bags, and then for any additional bag the price would be $ 3. The price of oranges is the same as part a.

(e) Would we be guaranteed a unique utility maximizing bundle in this case? Meaning, will there only be 1 combination of quantities of apples and oranges that will give David the highest utility level? Explain why and if/how this case is different from part a.

4. Suppose there is a consumer that is trying to solve the following optimization problem where px = 2, py = 1, m = 6:

Max x+2y− 1 2 (2x2+y2)

s.t. 2x+1y≤6

(a) Use the Lagrangian method to solve the above utility maximization problem.

(b) Are the demands you solved for in part a the utility maximizing values for x and y? If yes, explain. If no, what are the utility maximizing values for x and y and why did the Lagrangian not give you the correct answer? (Hint: Check the marginal utilities for each good, are they positive for all values of x and y?)

5. Suppose a consumer the following utility function: u(x,y) = 3x2 3y2 3.

(a) Setup the consumer’s utility maximization problem.

(b) Can we use the Lagrangian method to solve this problem? Explain why. Be sure to include any traits of the utility function and problem setup that indicate you should use a Lagrangian.

(c) Does the consumer’s utility function satisfy the sufficient condition for an interior solution? Either if yes or no, prove it.

(d) What is the demand for good x and good y? That is, solve the utility maximization problem for x∗(px,py,m) and y∗(px,py,m).

(e) Compare the utility levels of the interior and corner solutions.

(f) Write the formulas and calculate the following elasticities of demand for good y.

•Price elasticity of demand,

•Cross price elasticity of demand,

•Income elasticity of demand,

6. Houdini sees bow ties with stripes(x) and bowties with polka-dots(y)as perfect substitutes according to the utility function u(x,y) = 2x+4y.

(a) Solve for Houdini’s ordinary demand for bow ties with stripes x∗ and with polka-dots y∗ as functions of Houdini’s income m and the prices of bow ties px and py. (Hint: No Lagrangian. Consider cases.)

(b) Suppose Houdini has m = 18 to spend on bow ties, and py = 6. Graph the utility maximizing demand function for bow ties with stripes (x) on the graph below.

(c) At which price for striped bowties, will Houdini demand 18 striped bowties? Explain.

7. Amy is studying at a local coffee shop and would like to buy coffee and biscuits. For every 3 cups of coffee (c), she wants 1 biscuit (b). These can be summarized with the following utility function

u(c,b) = min(c,3b)

(a) Setup Amy’s utility maximization problem where she has an income of m to spend on biscuits and coffee, the price of coffee is pc, and the price of a biscuit is pb. Then, solve for the optimal demand of coffee and biscuits. Make sure to show all your work.

(b) Does Amy see coffee and biscuits as complements or substitutes? Explain your answer using comparative statics to analyze the demand functions you found above.

(c) Suppose that m=4 and pb =3. Graph Amy’s Marshallian demand function for coffee as the price of coffee changes. Be sure to label and number your graph.

(d) Using the graph above and the demand equations your found in part a,explain if Amy’s demand for coffee follows the Law of the Demand. Define the Law of demand in your answer.

Answer 1

Answering only first question posted with 4 subparts.

1) U = 2X + 2Y

Now at( 2,2) , U = 8

(4,4) , U = 16

(6,6) ,U = 24

(8,8) , U = 32

Graphs

Q2) U = X^{​​​​​​3/4}Y^1/4

At (2,2), U = 2

(4,4),U = 4

(6,6), U = 6

(8,8) , U = 8

Graphs

3) at (2,2), U = min (2,2) =2

4) U = Min ( 4x, 3y)

At (2,2), U = min ( 8,6) = 6

(4,4) , U = Min ( 16,12) = 12

(6,6) U = min ( 24, 18) = 18

(8,8) , U = min ( 32,24) = 24