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Microeconomics Theory: Assignment 1

Please read the following instructions:

.  Please try to answer all questions.  Your assignment will be graded based on effort rather than accuracy.

.  If you discuss with your classmates to finish the assignment, please list their names on your assignment. You should still hand in your own version of assignment.

.  The due date is February 6th before the class starts. Email submission is not accepted.

1. Prove the following proposition.

Proposition: If 之 is rational, then:

(a)  ≻ is both irreflexive (x ≻ x never holds) and transitive (if x ≻ y and y ≻ z, then x ≻ z).

(b)  ~ is reflexive (x ~ x for all x), transitive (if x ~ y and y ~ z, then x ~ z), and symmetric (if x ~ y, then y ~ x).

(c)  if x ≻ y ≿ z, then x ≻ z.

2.  Let X = {x, y, z}, and consider the choice structure (B, C(·)) with

B = {{x, y}, {y, z}, {x, z}, {x, y, z}}

and C({x, y}) = {x}, C({y, z}) = {y}, and C({x, z}) = {z}. Show that (B, C(·)) must violate WARP.

3.  Consider the following utility functions defined over a consumption set X ⊆ R2+.  For each utility function,  assess its represented preference’s desirability (monotonic,  strongly monotonic,  LNS, or non-monotonic), convexity (convex, strictly convex, or non-convex), and draw the indifference curves.

(a)  U(x1, x2) = min{4x1 , 6x2 }

(b)  U(x1, x2) = 100 − (x1 − 10)2 − (x2 − 10)2

(c)  U(x1, x2) = x 1 + ln(x2 + 1)

(d)  U(x1, x2) = max{x1, x2}

4. Consider the following preference relation defined on X ⊆ R2+ . A bundle (x1, x2) ≿ (y1, y2) if and only

if min{6x1 + 4x2 , 4x1 + 6x2 } ≥ min{6y 1 + 4y2 , 4y 1 + 6y2}.

(a)  For any given bundle (x1, x2), draw the upper contour set, the lower contour set, and the indifference set of this preference relation.

(b)  Check if this preference relation is rational.

5.  A consumer has preferences summarized by the utility function U(x1, x2) = x 1 + 4√x2.

(a) Find the expression of the consumer’s Marginal Rate of Substitution.

(b)  In a diagram, draw the consumer’s indifference curves for utility level equal to 1 and equal to 5.

(c)  Would this consumer ever choose to consume only Good 2 and no Good 1? Would she ever choose to consume only Good 1 and no Good 2? Explain.

(d) Write the consumer’s Lagrangian function and Kuhn-Tucker conditions.  Let the prices of Good 1 and 2 be p 1 and p2. The wealth available is w.

(e)  Solve the utility maximization problem and find the consumer’s optimal consumption bundle (Consider only interior solutions).