Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

ECON 3101 - Intermediate Microeconomics

Problem Set 2

1. (POINTS: 15)

(a) Explain what is means when preferences are:

i. complete (Points: 3.5)

ii. transitive (Points: 3.5)

(b) Prove the following statement: if preferences are represented by an utility function u, then they must be complete and transitive. (Note, this is to say completeness and transitivity are necessary if preference relations admit utility function representation.) (Points: 8)

Hint: Remember the definition of representation by utility function.

2. (POINTS: 10)

For each utility function below: draw a xy-diagram; plot at least 3 IC; draw arrows pointing to the direction of utility growth; and justify, or provide an example why not, if they are: i) weakly monotone/strongly monotone; ii) convex/strictly convex. Assume all utility functions are real valued functions defined on .

(a) u(x, y) = 3x+2y (Points: 5)

(b) u(x, y) = 3x (Points: 5)

3. (POINTS: 40) A consumer has preferences over sandwiches (x) and juice (y), represented by the utility function:

Assume I > 0, and prices, and , are positive.

(a) Consider the general budget constraint, and set up this consumer’s utility maximization problem. (Points: 4)

(b) Will the BC be binding or not? Explain. (Points: 4)

(c) Can you say whether the optimal demand is interior or not? Explain. (Points: 4)

(d) Set up this consumer’s Lagrangian. (Points: 4)

(e) Obtain the system of FOCs for the Lagrangian, then derive the tangency condition. (Points: 4)

(f) Derive the consumer’s Marshallian demands for sandwich and juice. (Points: 4)

(g) Now, assume the following values

and draw a xy-diagram, plot the budget constraint, and the IC of the optimal bundle (Points: 4)

(h) Imagine now the consumer is offered a new promotion: for the first 3 sandwiches the price is $4, but for every sandwich after that the price is $1. Draw in your diagram above the new budget constraint, and below write its formula. (Points: 10)

Hint: The drawing will have a kink, and the budget constraint will have two equations, dependent on the value of x.

(i) Given the new pricing, will the consumer choose a different bundle or not? (Points: 2)

Hint: 

4. (POINTS: 15)

(a) Write the UMP for a consumer with perfectly complements preferences, with a = b = 1, assuming the general budget constraint. (Points: 3)

(b) Obtaining the Marshallian demand for this consumer. (Points: 4)

(c) Calculate the price/cross-price/income elasticities of demand for good x and classify the good accordingly. (Points: 8)

5. (Points: 20) Consider a consumer with preferences represented by the following utility function:

(a) Write this consumer’s EMP, given a level of utility . (Points: 3)

(b) Explain why you can set up a Lagrangian for this problem, that is, why all constraints will be satisfied as equalities or strictly inequalities at the optimum. (Points: 3)

(c) Write the consumer’s Lagrangian. (Points: 4)

(d) Derive the system of FOCs, and obtain the tangency condition. (Points: 6)

(e) Obtain the compensated demand for goods x and y. (Points: 4)