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### Problem Set 3: Posted on November 26

Advanced Microeconomics I (Fall, 1st, 2013) 1. Question 1 (6 points)

Consumers have the following Cobb-Douglas utility functions: U (x1, x2) = α ln x1 + (1 − α) ln x2,

where xt is consumption in period t. Consumers vary in their preference parameter α. We will refer to a consumer with parameter α as a type α consumer. There is a single commodity. Types are distributed continuously over the interval [0, 1]. Type α has density f (α) = 1 over this interval so that the total mass of types is 1. Each type has the same income in each of two periods.

(a) If the interest rate is r, which types will be savers and which will be borrowers? (b) Calculate the amount of net saving for type α consumer, s(α).

(c) Solve for the total saving S by all types who save and the total borrowing B by all types who borrow. Depict S(r) and B(r) in a neat figure.

2. Question 2 (5 points)

Consider the following (social welfare) maximization problem where ui is a strictly increasing and continuous function for all i ∈ I and (λ1, · · · , λI) ∈ RI+\ {0}.

max

x

XI i=1

λiui(xi)

s.t. XI

h=1

xkh ≤ ek for k = 1, ..., n.

(a) Show that any solution of the above maximization problem (you may denote x) must be Pareto efficient.

(b) Find an example of Pareto efficient allocation that cannot be the solution of the maximization problem whichever (λ1, · · · , λI) ∈ RI+\ {0} will be chosen. 3. Question 3 (6 points)

Consider an exchange economy with two goods, x and y. Suppose that individuals have the following symmetric utility functions and symmetric initial endowments:

u(x, y) = x2+ y2x, ωy) = (1, 1)

(a) Assume there are only two individuals in this economy. Then, draw the Edgworth-box and show the contract curve. Find a competitive equilibrium if it exists. If there is no equilibrium, explain the reason.

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(b) Now suppose there are n(> 2) individuals. Then, can we find a competitive equilibrium? (How) Does your answer depend on n?

4. Question 4 (8 points)

Consider a production economy with two individuals, Ann (A) and Bob (B), and two goods, leisure x1 and a consumption good x2. Ann and Bob have equal en- dowments of time (= ω1) to be allocated between leisure and work, so the total endowment of time is 2ω1 units. There is no (initial) endowment of consumption good. Each individual has a common utility function U (x) = ln x1+ 2a ln x2. Sup- pose that only Ann owns the firm and its production function is y2 = √z1, where y2 is the output of consumption good and z1 is the input of (total) labor. Let the price of x2 be normalized by 1 and the price of labor, i.e., wage, be denoted by w.

(a) Solve the profit maximization problem of this firm.

(b) Solve the utility maximization problem of each individual.

(c) Derive the competitive equilibrium (both price w and allocation x).

(d) Now consider an exchange economy with n consumers and k goods. We de- note the bundle of total endowments by ω = (ω1, . . . , ωk). Suppose that all consumers have identical (strictly) convex preferences. Then, show that equal division of total endowments, i.e., xi = ω/n for all consumer i, is always a Pareto efficient allocation.

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