EOC501 Microeconomic Theory
Problem Set 1
1. A consumer has a utility function u : R2
+ ! R defined as for each (x1, x2) 2 R2
+,u(x1, x2) = px1 + px2. The prices for commodities 1 and 2 are p1 > 0 and p2 > 0,respectively, and the consumer has a wealth of w > 0. Let x ⌘ (x1, x2) and p ⌘(p1, p2).1
(a) Write down the consumer’s utility maximization problem (be sure to include the non-negativity constraints).
(b) Let the budget set be B(p, w) ⌘ {x 2 R2
+ : p · x w}. Show that B(p, w) is compact. Using the latter observation, show that the maximization problem has a solution. Hint: Recall the Weierstrass theorem.
(c) Set up the Lagrangian function and derive the first-order condition (FOC) for maximization.
(d) Is the budget constraint p · x w binding at a solution? Why or why not?
(e) Are the non-negativity constraints (x1, x2 ≥ 0) binding at a solution? Why or why not?
(f) Using the observations in (d) and (e), find a solution satisfying the FOC in (c).
(g) Does the solution in (f) satisfy the second-order condition for maximization?
2. Consider a lexicographic preference relation defined over R2
+. Sketch typical upper and lower contour sets and indi↵erence sets.
3. It is known that for rational preference relations, continuity is a sufficient condition for utility representation (Proposition 3.C.1). Prove or disprove that continuity is also necessary.
4. Solve the following exercises from the textbook: 3.B.2, 3.C.2
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