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Tutorial 2 Questions

1. Optimization with inequality constraints. Find

Approach this formally via Lagrangean. Write all the Karush-Kuhn-Tucker condi-tions. Argue that the non-negativity constraints will not bind and that the x+y2 ≤ 2 constraint will hold as equality. Solve the resulting system of equations.

2. Lagrange multiplier as shadow price of income. Consider a consumer with utility function u (x1; x2) = x1x2. She has income M and faces prices p1; p2: Using Lagrange method derive her optimal consumption bundle (x*1 ; x*2 ) and the level of utility at the optimum u(x*1 ; x*2 ): This is a function of (p1; p2; M): Show that du(x*1 ; x*2 )=dM = λ; the value of the Lagrange multiplier.

3. Exercise 2.2 from Lengwiler.

4. Exercise 2.3 from Lengwiler.

5. Consider consumers A and B with uilities

(a) What is the exchange economy equilibrium here.

(b) What is a Pareto e¢ cient allocation? The social welfare is maximized by allocating the goods to consumers A and B; and the allocation has to respect the resourse constraints. Formulate the search for such allocations as an optimization program.

(c) What can be said about the equilibrium prices at that allocation?

6. Consider the same consumers A and B that live for two periods and suppose now !1 = 6 and owned by A. This is the entire crop of this economy at date 1. Suppose also that ω2 = 12 and owned by B. This is the entire crop of this economy at date 2. The good is perishable and cannot store.

(a) Suppose A and B invented a way to trade over times 1 and 2. Formulate the consumer choice problems for A and B when such trade is possible.

(b) What is the equilibrium?

(c) Verify that trading over time allows to achieve Pareto e¢ ciency.