70 Minutes There are two questions. Answer both. Mar

ECONOMICS 001 Microeconomic Theory

Summer 2016-17 Mid-semester Exam 2

Time: 70 Minutes There are two questions. Answer both. Marks are given in parentheses.

1. Consider the following 2×2 economy. The utility functions are: u¹(.) = x¹₁x¹₂ and u²(.) = x²₁+ 2x²₂, where xⁱ_j denotes the quantity of jth good consumed by ith individual; i = 1,2 and j = 1,2. Let the initial endowments be e¹(.) = (3/4,3/8) and e²(.) = (1/4,5/8), respectively. Assume that individuals act as price-takers.

(a) Is (e¹(.), e²(.), as above, a Pareto optimum allocation?

(b) Does there exist a Walrasian equilibrium for the above economy?

(c) Is the Walrasian equilibrium unique?

Prove your claims. (2 + 1 + 2)

2. Consider two ‘small’ and ‘open’ economies; A and B. Each economy produces two goodsf and c. A good is produced using two factors of production labour;

l, and capital, t. There is free flow of outputs (goods) across countries. But, factors of production cannot move across countries. With the help of a suitable general equilibrium model answer the following:

(a) Demonstrate the effect of changes in the relative output prices on the rel- ative factor prices, assuming that in both economiesf is ‘labour intensive’

and cis ‘capital intensive’.

(b) Suppose in economy A, f is labour intensive but c is capital intensive.

However, in economy B, f is capital intensive and c is labour intensive.

Discuss the effect of changes in factor endowment on the output levels for the two economies.

Provide complete formulation for your answer. (4 + 6)

Microeconomic Theory Summer 2015-16 Mid-semester Exam 2

Time: 70 Minutes There are two questions. Answer both. Marks are given in parentheses.

1. Consider an economy consisting of two individuals; i = 1,2. There are two goods; x₁ and x₂. Utility functions are: uⁱ(xⁱ₁.xⁱ₂) =xⁱ₁.xⁱ₂, for i= 1,2. Endow- ments are: e¹ = (2,8), e² = (8,2), respectively. For this economy:

(a) Does allocationx= (x¹,x²), such that x¹ = (4,4) and x² = (6,6), belong to the Core?

(b) Does the Core have an envy-free allocation? Is it unique?

(c) Next, let us expand the above economy by including individuals 3 and 4.

So, now the total number of individuals is four. For individuals 3 and 4, utility functions are: uⁱ(xⁱ₁.xⁱ₂) = xⁱ₁.xⁱ₂, for i = 3,4. Endowments are:

e³ = (2,8), and e⁴ = (8,2). Does allocation x = (x¹,x²,x³,x⁴) belong to the Core, when x¹ = (4,4) = x³ and x² = (6,6) = x⁴?

Prove your claims.

(2+3+3) 2. Consider a pure exchange economy consisting of n individuals, and m goods, ({uⁱ(xⁱ)}ⁿ_i=1,{eⁱ}ⁿ_i=1), whereuⁱ(xⁱ) :R^m₊ 7→Ris continuous, strongly increasing and strictly quasi-concave for each i = 1, .., n. Further, endowment eⁱ ∈ R^m₊ are such that Pn

i=1eⁱ >> 0. Let z(p) :R^m₊₊ 7→R^m denote the excess demand function. Let P be the set of Pareto optimum allocations. Assume that for every profile of endowments (e¹,e², ...,eⁿ), such thatPn

i=1eⁱ >> 0, there exists a price vector p^∗ ∈R^m₊₊ such thatz(p^∗) =0. Prove that:

(a) The set of Pareto optimum allocations, i.e., P, is non-empty.

(b) Everyy∈P can be supported as a Walrasian equilibrium through a suit- able transfers of initial endowments.

(Note: You are not allowed to use any other information/results, apart from those mentioned here.)

(3+4)

ECONOMICS 001 Microeconomic Theory

Summer 2014-15 Mid-semester Exam 2

Time: 70 Minutes There are two questions. Answer both. Marks are given in parentheses.

1. Consider an economy consisting of four individuals; i = 1, ...,4. There are two goods; x₁ and x₂. Utility functions are: uⁱ(xⁱ₁.xⁱ₂) =xⁱ₁.xⁱ₂, for i = 1, ...,4. En- dowments are: e¹ = (1,9), e² = (9,1), e³ = (1,9), ande⁴ = (9,1), respectively.

(a) Does allocationx= (x¹,x²,x³,x⁴) belong to the Core, whenx¹ = (3,3) = x³ and x² = (7,7) = x⁴?

(b) Does allocationx = (x¹,x²,x³,x⁴) belong to the Core, when x¹ = (3,3), x² = (7,7), x³ = (4,4), andx⁴ = (6,6)?

(c) Suppose (¯p₁,p¯₂) is an equilibrium price vector when the economy consists of only the first two individuals. Is (¯p1,p¯2) is also an equilibrium price vector for the above economy consisting of all four individuals?

Prove your claims.

(2+3+3) OR

Consider a pure exchange economy consisting of two individuals, 1 and 2. There are two goods;xandy. Individual 1 strictly prefers bundle (a, b) to bundle (c, d) if, either a > c, or a =c and b > d. Individual 2’s preferences are represented by the utility functions u(x, y) = αx² +βy², α, β > 0. The endowments are e¹ = (e¹_x, e¹_y)>>(0,0) and e² = (e²_x, e²_y)>>(0,0). Moreover,e¹_x+e²_x> e¹_y+e²_y. For this economy:

(a) Find out the set of Pareto optimum allocations.

(b) Does a competitive equilibrium exist? [Hint: Think in terms of the location of the endowment vector in the Edgeworth box.]

Fully explain your answers.

(3+5)

({uⁱ(xⁱ)}^N_i=1,{eⁱ}^N_i=1), whereuⁱ(xⁱ) :R^M₊ 7→Ris continuous, strongly increasing and strictly quasi-concave for each i = 1, .., N. Further, endowment eⁱ ∈ R^M₊ are such that Pn

i=1eⁱ >> 0. Let C(e) be the set of Core allocations. For this economy, prove the following.

There exists an allocation (x¹,x², ...,x^N)∈C(e) with following properties: For alli, j ∈ {1, ..., N}

(a) [uⁱ(.) = u^j(.) and eⁱ =e^j]⇒[xⁱ =x^j]

(b) [uⁱ(.) = u^j(.) and eⁱ ≥e^j]⇒[uⁱ(xⁱ)> u^j(x^j)]

(c) eⁱ =e^j ⇒[uⁱ(xⁱ)≥uⁱ(x^j)], and eⁱ ≥e^j ⇒[uⁱ(xⁱ)> uⁱ(x^j)]

Note: x ≥ y holds if every component of vector x is at least as large as the corresponding component of vectory, but one or more components are strictly greater.

(3+2+2)

ECONOMICS 001 Micro Economics

Summer 2013-14 Mid Semester Exam 2

Time: 70 Minutes Answer question 3, and EITHER question 1 OR question 2

1. Consider a two-person two-goods pure exchange economy. The initial endow- ment vectors aree¹ = (1,0) ande² = (0,1). The two individuals have identical preferences represented by the utility functions:

u¹(x, y) =u²(x, y) =

1, when x+y <1 x+y, when x+y≥1,

where x is the quantity of the first good and y is the quantity of the second good. For this economy:

(a) Find out the set of Walrasian/competitive equilibria, assumingp₁ =p₂ = 1.

(b) Find out the set of Pareto optimum allocations.

(c) Will the equal division of the initial endowments be a Pareto efficient allocation?

Explain your findings, in view of the results/thoerems regarding Pareto effi- ciency of Walrasian/competitive equilibria and equal division of initial endow- ments.

(2+3+2) 2. Consider a pure exchange economy; (uⁱ(.),eⁱ)i∈I. Let x = (x¹,x², ...,x^I) be a feasible allocation. Suppose x is Pareto superior to e. However, there exists a blocking coalition,S ⊆ {1, ..., I}, forx= (x¹,x², ...,x^I). Which of the following is necessarily true?

(a) There exists at least one allocationz= (z¹,z², ...,z^I) such thatz6=x, and z is Pareto superior to e.

(b) The allocationz= (z¹,z², ...,z^I), as in part (a) above belongs to the Core.

Explain your answer.

(4+3)

The utility functions are:

u¹(x₁, y₁) = lnx₁+ lny₁ u²(x₂, y₂) = x^α₂.y₂^1−α,

where x_i and y_i is the quantity consumed by person i of good x and y, respec- tively, and α= ¹₄. The initial endowments are

e¹(.) = e²(.) = (1 2,1

2).

The production sector uses x to produce y, subject to a constant returns to scale technology. So, the profits are zero for each firm as well as for the entire production sector. For this economy,

(a) Derive the individual demand functions and then the excess demand func- tions.

(b) Find out competitive equilibrium price and allocation vectors when the production function is y = x, assuming that the production sector will meet all the demand as long as profits are non-negative.

(c) Find out competitive equilibrium price and allocation vectors when the production function is y = 3x, assuming that the production sector will meet all the demand as long as profits are non-negative.

(3+3+2)