Microeconomic Theory Summer 206-7 Mid-semester Exam 2 There are two questions. Answer both. Marks are given in parentheses.. Consider the following 2 2 economy. The utility functions are: u (.) = x x 2 and u 2 (.) = x 2 + 2x 2 2, where x i j denotes the quantity of jth good consumed by ith individual; i =, 2 and j =, 2. Let the initial endowments be e (.) = (3/4, 3/8) and e 2 (.) = (/4, 5/8), respectively. Assume that individuals act as price-takers. (a) Is (e (.), e 2 (.), 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 + + 2) 2. Consider two small and open economies; A and B. Each economy produces two goods f 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 relative factor prices, assuming that in both economies f is labour intensive and c is 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 205-6 Mid-semester Exam 2 There are two questions. Answer both. Marks are given in parentheses.. Consider an economy consisting of two individuals; i =, 2. There are two goods; x and x 2. Utility functions are: u i (x i.x i 2) = x i.x i 2, for i =, 2. Endowments are: e = (2, 8), e 2 = (8, 2), respectively. For this economy: (a) Does allocation x = (x, x 2 ), such that x = (4, 4) and x 2 = (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: u i (x i.x i 2) = x i.x i 2, for i = 3, 4. Endowments are: e 3 = (2, 8), and e 4 = (8, 2). Does allocation x = (x, x 2, x 3, x 4 ) belong to the Core, when x = (4, 4) = x 3 and x 2 = (6, 6) = x 4? Prove your claims. (2+3+3) 2. Consider a pure exchange economy consisting of n individuals, and m goods, ({u i (x i )} n i=, {e i } n i=), where u i (x i ) : R m + R is continuous, strongly increasing and strictly quasi-concave for each i =,.., n. Further, endowment e i R m + are such that n i= ei >> 0. Let z(p) : R m ++ R m denote the excess demand function. Let P be the set of Pareto optimum allocations. Assume that for every profile of endowments (e, e 2,..., e n ), such that n i= ei >> 0, there exists a price vector p R m ++ such that z(p ) = 0. Prove that: (a) The set of Pareto optimum allocations, i.e., P, is non-empty. (b) Every y P can be supported as a Walrasian equilibrium through a suitable transfers of initial endowments. (Note: You are not allowed to use any other information/results, apart from those mentioned here.) (3+4)
Microeconomic Theory Summer 204-5 Mid-semester Exam 2 There are two questions. Answer both. Marks are given in parentheses.. Consider an economy consisting of four individuals; i =,..., 4. There are two goods; x and x 2. Utility functions are: u i (x i.x i 2) = x i.x i 2, for i =,..., 4. Endowments are: e = (, 9), e 2 = (9, ), e 3 = (, 9), and e 4 = (9, ), respectively. (a) Does allocation x = (x, x 2, x 3, x 4 ) belong to the Core, when x = (3, 3) = x 3 and x 2 = (7, 7) = x 4? (b) Does allocation x = (x, x 2, x 3, x 4 ) belong to the Core, when x = (3, 3), x 2 = (7, 7), x 3 = (4, 4), and x 4 = (6, 6)? (c) Suppose ( p, p 2 ) is an equilibrium price vector when the economy consists of only the first two individuals. Is ( p, p 2 ) is also an equilibrium price vector for the above economy consisting of all four individuals? Prove your claims. OR (2+3+3) Consider a pure exchange economy consisting of two individuals, and 2. There are two goods; x and y. Individual 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) = αx 2 + βy 2, α, β > 0. The endowments are e = (e x, e y) >> (0, 0) and e 2 = (e 2 x, e 2 y) >> (0, 0). Moreover, e x + e 2 x > e y + e 2 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)
2. Consider a pure exchange economy consisting of N individuals and M goods, ({u i (x i )} N i=, {e i } N i=), where u i (x i ) : R M + R is continuous, strongly increasing and strictly quasi-concave for each i =,.., N. Further, endowment e i R M + are such that n i= ei >> 0. Let C(e) be the set of Core allocations. For this economy, prove the following. There exists an allocation (x, x 2,..., x N ) C(e) with following properties: For all i, j {,..., N} (a) [u i (.) = u j (.) and e i = e j ] [x i = x j ] (b) [u i (.) = u j (.) and e i e j ] [u i (x i ) > u j (x j )] (c) e i = e j [u i (x i ) u i (x j )], and e i e j [u i (x i ) > u i (x j )] Note: x y holds if every component of vector x is at least as large as the corresponding component of vector y, but one or more components are strictly greater. (3+2+2) 2
Micro Economics Summer 203-4 Mid Semester Exam 2 Answer question 3, and EITHER question OR question 2. Consider a two-person two-goods pure exchange economy. The initial endowment vectors are e = (, 0) and e 2 = (0, ). The two individuals have identical preferences represented by the utility functions: {, when x + y < u (x, y) = u 2 (x, y) = x + y, when x + y, 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, assuming p = p 2 =. (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 efficiency of Walrasian/competitive equilibria and equal division of initial endowments. (2+3+2) 2. Consider a pure exchange economy; (u i (.), e i ) i I. Let x = (x, x 2,..., x I ) be a feasible allocation. Suppose x is Pareto superior to e. However, there exists a blocking coalition, S {,..., I}, for x = (x, x 2,..., x I ). Which of the following is necessarily true? (a) There exists at least one allocation z = (z, z 2,..., z I ) such that z x, and z is Pareto superior to e. (b) The allocation z = (z, z 2,..., z I ), as in part (a) above belongs to the Core. Explain your answer. (4+3)
3. Consider a two person two goods production economy. The goods are; x and y. The utility functions are: u (x, y ) = ln x + ln y u 2 (x 2, y 2 ) = x α 2.y2 α, where x i and y i is the quantity consumed by person i of good x and y, respectively, and α =. The initial endowments are 4 e (.) = e 2 (.) = ( 2, 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 functions. (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) 2