Advanced Economic Theory I

ECON 629

Homework 7

Fall 2024 Due: Monday, December 2, 2024

1.  Exercise 11.3 Consider a pure exchange economy with two consumers and two commodi- ties. Suppose that free disposal is allowed in this economy, and the consumption set for each consumer is R . Consumer 1 has a utility function given by:

u1 (x1 , y1 ) = 200 - (10 - x1 )2  - (10 - y1 )2 .

Consumer 2 has lexicographical preferences, with x2 primary andy2 secondary. The two consumers have the same endowments given by:

wi  = (10, 10),    i = 1, 2.

(a)  Draw an Edgeworth Box diagram for this economy.

(b)  Find all Pareto optimal allocations. Give the reason if there are none.

(c)  Find competitive equilibria in this economy, if any. If there are none, explain why.

2.  Exercise 11.5 (Harmful Goods) Consider a pure exchange economy with two consumers.

The utility functions for consumers 1 and 2 are given by: ui(xi(1), xi(2)) = xi(1)(4 - xi(2)),

where x2 is a “harmful product," and its price should be negative. The consumption set is [0, 5] × [0, 3] ≤ R . The endowments are w1  = (1, 3) and w2  = (3, 1), respectively.

(a)  Prove that an allocation x is Pareto efficient if and only if x 1(1) + x1(2)  = 4. (b)  Solve for the competitive equilibrium.

(c)  Draw the contract curve and the offer curves in the Edgeworth Box.

3.  Exercise 11.9 Consider a pure exchange economy with n consumers and two commodi- ties: the consumption set for each consumer is R , and each consumer i’s utility function is

ui(xi) = max{xi(1), xi(2)}.

(a)  If there are only two consumers 1 and 2, with endowments w1   =  (1, 1) and w2  = (1, 1), respectively, what are the Walrasian equilibria?

(b)  In the previous question, what is the set of Pareto efficient allocations? What is the set of weakly Pareto efficient allocations?

(c)  Now, if there are three consumers, with endowments w1  =  (1, 1), w2  =  (1, 1), and w3  = (1, 1), respectively, what is the set of Pareto efficient allocations?

4.  Exercise 11.13 (The First Welfare Theorem with Convex Preferences) Consider a pure exchange economy. Suppose that ≥i is convex, and (x, p) ∈ RL  × R Wal- rasian equilibrium. Show that the Walrasian equilibrium allocation x is Pareto optimal.

5.  Exercise 11.21 (Indivisible Goods) Consider an economy with two commodities x and y and an arbitrary number of agents.  Assume that preferences are strictly increasing in each good. Consider two cases:

(a)  x is perfectly divisible, but y comes only in integer units.

(b)  Both x andy come only in integer units. Answer the following questions:

(a)  Could we have a competitive equilibrium allocation that is not Pareto optimal? (An- swer separately for case (1) and case (2)).

(b)  If your answer is “yes," provide an example with a competitive equilibrium allo- cation that is not Pareto optimal.   (Justify both of the following facts:  (i) that the allocation in your example is a competitive equilibrium allocation; and (ii) that it is not Pareto optimal.)

(c)  If your answer is “no," provide a proof to justify it.

6.  Exercise 11.28 Consider a production economy that can produce food (product 1) and electricity (product 2) with labor (L) and capital (K). The production functions of the two products are:

y1  = √L1K1,  y2  = min{L2, K2 }.

The economy has a representative economic agent whose endowments are 1 unit of labor and 1 unit of capital. The utility function is u(x1 , x2 ) = √x1 x2 .

(a)  Find competitive equilibrium.

(b)  Find the Pareto efficient allocation of the economy.