Delhi School of Economics Entrance Exam (2013)

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This is the entrance exam paper for the Delhi School of Economics for the year 2013. It contains both options A and B. Exam papers for other years are available as well here. Much more information on the DSE Entrance Exam and DSE Entrance preparation help available on www.crackdse.com

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Delhi School of Economics Entrance Exam (2013)

  1. 1. Entrance Examination for M. A. Economics, 2013 Option A (Series 04) 1 [)2 491 Time. 3 hours l«Iaximum marks. 100 General Instructions. Please read the following instructions carefully: » Check that you have a"bubble—sheet accompanying this booklet. Do not break the seal on this booklet until instructed to do so by the invigilator. K Immediately 'o"I"1’ re’céip't'of this’ booklet, fill i11 your Signature, Name, Roll number and Booklet number (see the top corners of this Booklet) in the space provided below. a This examination will be checked by a machine. Therefore, it is very important that you follow the instructions on the bubble—sheet. . :. Fill in the required information in Boxes on the bubble—sheet. Do not write anything in Box 3 — the invigilator will sign in it. Do not disturb your neighbours at any time. «. Make sure you do not have mobile, papers, books, etc. , on your person. The exam does not require use of a calculator. However, you can use non—programmable, non—alpha—numeric memory simple calcula- tor. Anyone engaging in illegal practices will be immediately evicted and that person’s candidaturewill be canceled. <: You are not allowed to leave the examination hall during the first 30 minutes and the last 15 minutes of the examination time. -‘ When you finish the examination, hand in this booklet and the bubble- sheet to the invigilator. Name , Signature Roll number Booklet number “A -Z W; _V. t7_*. '.’. «'. ‘F'lF‘. ':'t‘W">'| '/Frrrrvv--u~--4 . ., _.. . . . . ,.. . . . . .. . .. ... .. _,. .. , ’
  2. 2. EEE 2013 A 04 1 Before you start 0 Check that this booklet has pages 1 through ‘.27. Also check that the top of each page is marked with EEE 2013 A 04. Report any inconsistency to the invigilator. 0 You may use the blank pages at the end of this booklet, marked Rough work, to do your calculations and drawings. No other paper will be provided for this purpose. Your “Rough work” will be neither read nor checked. You may begin now. Best VVishes! Part I o This part of the examination consists of 20 multiple—choice questions. ‘ Each question is followed by four possible answers, at least one of which is correct. If more than one choice is correct, choose only the ‘best one’. The ‘best answer’ is the one that implies (or includes) the other correct answer(s). Indicate your chosen best answer on the bubble- sheet by shading the appropriate bubble. o For each question, you will get: 1 mark if you choose only the best answer; 0 mark if you choose none of the answers. However, if you choose something other than the best answer or multiple an- swers, you will get -1/3 mark for that question. QUESTION 1. Exchange rate overshooting occurs: (a) under fixed exchange rates when the central bank mistakenly buys or sells too much foreign exchange (b) under fixed exchange rates as a necessary part of the adjustment . pro— cess for any monetary shock (C) under flexible exchange rates when the exchange rate rises (depreciates) above and then falls down to equilibrium after a monetary expansion ‘¥iSkysoft
  3. 3. EEE 2013 A 04 E (d) under flexible exchan ‘ domestic economy have very ’ my with a system of flexible exchange rates , an expansionary monetary policy: (a) causes the domestic currency to appreciate (b) has a greater impact on income than in a closed economy (c) increases capital inflows into the country (d) induces a balance of payments deficit n aggregate supply curve is upward sloping because if (a) lower price level creates a wealth effect (b) lower taxes motivate people to work more (0) money wages do not immediately change when the price level changes (d) most business firms operate with long—term contracts for output but not labour UESTION 4 The term seignora e is associated with S (a) inflation generated by printing new money (b) real revenue created by printing new money (c) public indebtedness created by printing new money (d) none of the above QUESTION 5. If money demand 1S stable, an open market purchase of government securities by the central bank Wlll ~‘-. .' . -1"-, ‘.'*- "'-2-'3-E4:': '}'. ".-: '.': .-. '-: -'~: -~'-'—: ~
  4. 4. EEE 2013 A 04 3 (a) 4 (b) 20 (C) 6 ( d) None of the above necessarily holds. QUESTION 7. Suppose f : 3? —+ 3% (i. e. it is a real-valued function defined on the set of real numbers). If f is differentiable, f (3) = 2, and 3 3 f’ 5 4, for all av, then it must be that f (5) lies in the following interval. (a) [8, 10] (b) l0»8) (c) (10,oo) (Cl) [34] QUESTION 8. The function defined by f = :c(a: — 10)(: c — 20)(: c — 30) has critical points (i. e., points where f’ = 0)_ (a) at some a: < O and some tr > 30. (b) at an :3 < 0, and at some as between 0 and 30. (c) between 0 and 10, 10 and 20, 20 and 30. (d) None of the above captures all the critical points. QUESTION 9. The area of the region bounded above by f = ' 2:2 +1 and below by g(x) = ac — 6 on the interval [——1, 3] is (a) 50/3. (b) 22. (c) 31. (d) 100/3. QUESTION 10. liin, ,_,3 (’”2’5"‘2)1/3 equals (a) -2. — (b) -4/3. (C) (-4/3)”? (d) (2/3)”? iskysoft
  5. 5. EEE 2013 A 04 4 QUESTION 11. Consider an individual A with utility function u(a: ) = 10/ E, where : r denotes the amount of money available to her. Suppose. she has Rs 100. However, she has option of buying a lottery that will cost her Rs 51. If purchased, the lottery pays Rs 351 with probability p, and pays 0 (nothing) with remaining probability. Assume that A is expected utility maximizer. Which of the following statements is correct? A will (a) not prefer to buy the lottery at all as long as p < 1 (b) certainly prefer to buy the lottery as long as p > O (c) prefer to _b_}iy the lottery if and only if p > 51 / 351 (d) prefertio thleulottery if and only if p > 51/221 The next TWO questions are based on the following model: Suppose that there are two goods, which are imperfect substitutes of each other. Let p1,p2 denote the price of good 1 and good 2, respectively. Demand of good 1 and good 2 are as follows Di(Pi. P2) = G ‘ P1 + 19192; D2(10i, P2) = (1 ‘ P2 + 5191 where a > 0 and 1 > b > 0. Both of the goods can be produced at cost c per unit. QUESTION_ 12. Find the equilibrium prices, when good 1 and good 2 are ' produced by two different monopolists. QUESTION 13. Find the equilibrium prices, when both the goods are produced by single monopolist. (a) pi = 192 = “§°_", ,"° (b) pi = P2 = “+1°_‘, ,"° (c) P1 = p. = ; ;;: ,:; " (d) pl ___ p2 = a+C2—bc soft 3 r7- ‘, __? _,; .__, ~‘___. ':, ,__w, ,,1 , _ . ,_. _t, V 3 V i. ;.. .<-, ..; .1.«. -r. . r. .—. .m_. .7—_»—. ... _.. _.. . . . . . . . , . . fiisky
  6. 6. EEE 2013 A 04 5 The next TWO questions make use of the following notations: H stands for the Headcount Ratio of Poverty (total number of poor divided by to- tal population); C for Mean Consumption per—capita; E for Elasticity of H with respect to C; NSS for National Sample Survey; and NAS for National Accounts Statistics. QUESTION 14. Table 1: (Source: Datt and Ravallion; EPW 2010) Rate of Change of-~H (% per—a. nn1-rm) Rate of Population Growth (% per annum) -1.1 2.2 -2.4 1.7 Pre—1991 Post- 199 1 From the data in Table 1, we can conclude that the total number of poor people was (a) Rising before 1991 but falling after 1991 (b) Falling before 1991 but rising after 1991 (c) Falling both before and after 1991, but at different rates (d) Rising both before and after 1991, but at different rates ‘ QUESTION 15. Table 2: (Source: Datt and Ravallion; EPW 2010) E when C is based on NSS data E when C is based on NAS data ’ -1.6 ‘ ‘ -1.0 -2.1 -0.7 Pre-1991 Post- 1991 From the data in Table 2, we can conclude that both before and after 1991, mean consumption per—capita according to NAS was: (a) Lower than mean consumption per—capita according to NSS b) Growing faster than mean consumption per—capita according to NSS ( (c) Growing slower than mean consumption per—capita according to NSS (d) None of the above
  7. 7. EEE 2013 A 04 6 QUESTION 16. If two balanced dice are rolled, the sum of dots obtained is even with probability QUESTION 17 . A population is growing at the instantaneous growth rate of _1_.5_ per cent, The time taken (in years) for it to double is approximately QUESTION 18. A linear regression model y = a + fix + 5 is estimated using OLS. It turns out that the estimated B equals zero. This implies that: (a R2 is zero QUESTION 19. An analyst has data on wages for 100 individuals. The arithmetic mean of the log of wages is the same as: (a) Log of the geometric mean of wages (b) Log of the arithmetic mean of wages (c) Exponential of the arithmetic mean of wages (d) Exponential of the log of arithmetic mean of wages QUESTION 20. A certain club consists of 5 men and 5 women. A 5- member committee consisting of 2 men and_ 3 women has to be constituted. HOV’ many ways are there of constituting this committee?
  8. 8. EEE 2013 A 04 (a) 20 (b) 100 (c) 150 (d) None of the above g End of Part I. Proceed to Part II of the examination on the next page. fiskysoft
  9. 9. EEE 2013 A 04 8 _m Part II 0 This part of the examination consists of 40 multiple—choice questions. Each question is followed by four possible answers, at least one of which is correct. If more than one choice is correct. choose only the ‘best one’. The ‘best answer’ is the one that implies (or includes) the other correct answer(s). Indicate your chosen best answer on the bubble- sheet by shading the appropriate bubble. ‘. “"For”éach question, you will get: 2 marks if you choose only the best answer; 0 mark if you choose none of the answers. However, if you choose something other than the best answer or multiple an- swers, then you will get —2/3 mark for that question. 0 The following notational conventions apply wherever the following sym- bols are used. ER denotes the set of real numbers. 3?” denotes the n- dimensional vector space. The next THREE questions are based on the following information: Con- sider an open economy simple Keynesian model with autonomous investment (1 People save a constant proportion (s) of their disposable income and con- sume the rest. Government taxes the total income at a constant rate (T) and spends an exogenous amount (G) on various administrative activities. The level of export (X) is autonomous at the fixed real exchange rate (normalized to unity). Import (M) on the other hand is a linear function of total income with a constant import propensity m. Let I=3200; G=4000; X=800; szl; 7'=2; and m= —1—. 2 5 10 QUESTION 21. The equilibrium level of income is given by: (a) 8,000 (b) 16,000 (c) 10,000 (d) None of the above # i Skysoft
  10. 10. EEE 2013 A 04 9 QUESTION 22. At this equilibrium level of income (a) there is a trade surplus (b) there is a trade deficit (c) trade is balanced (d) one cannot comment on the trade account without further information QUESTION 23. Now suppose the government decides to maintain a bal- anced trade by appropriately adjusting the tax rate 7' (thereby affecting do- mestic absorp. tio. n.) --. ... with1o. ut changing the exchange rate or the amount of government expenditure. Values of other parameters remain the same. The government (a) can attain this by decreasing the tax rate to 1/ 5 (b) can attain this by increasing the tax rate to 4/5 (c) can attain this by simply keeping that tax rate unchanged at 2 / 5 (d) can never attain this objective by adjusting only the tax rate QUESTION 24. Suppose in an economy banks maintain a cash reserve ratio of 20%. People hold 25% of their money in currency form and the rest in the form of demand deposits. If government increases the high-powered money by 2000 units, the corresponding increase in the money supply would be (a) 5000 units (b) 2000 units (c) 7200 units (d) None of the above QUESTION 25. Consider the standard IS—LM framework with exogenous money supply. Now suppose the government introduces an endogenous money supply rule such that the money supply becomes an increasing function of the interest rate. As compared to the standard IS—LM case, now (a) the IS curve will be flatter and fiscal policy" would be more effective (b) the IS curve will be steeper and fiscal policy would be less effective (c) the LM curve will be flatter and fiscal policy would be more effective (d) the LM curve will be steeper and fiscal policy would be less effective
  11. 11. EEE 2013 A 04 10 The next FIVE questions are based on the following information: Con- sider an economy where aggregate output is produced by using two factors: capital (K) and labour Aggregate production technology is given by the following production function: Y, = 01K; + BL“ where a, i3 > 0. At every point of time both factors are fully employed; each worker is paid a wage rate 5 and each unit of capital is paid a rental price or. A constant proportion s of total output is saved and invested in every period - which augments the capital stock in the next period (no depreciation of capital). Labour force grows at a constant rate 71. QUESTION 26. This production function violates (a) the neoclassical property of constant returns to scale (b) the neoclassical property of diminishing returns to each factor (c) the neoclassical property of factor returns being equal to the respective marginal products . . (d) the neoclassical property of substitutability between capital and labour QUESTION 27. Let kt = K, /Lt. The corresponding per capita output, yt, is given by ‘which of the following equations? (3) 3/1 = CY+5kt QUESTION 28. The dynamic equation for capital accumulation per worker is given by ’ i (a) % = sfikf’ — nkt . iskysoft -. 3 £3
  12. 12. EEE 2013 A 04 11 (b) 51,5 = sak, +36 — nkt (c) %‘ = sakt — nk; (d) ‘% = saktfl - nk, 4' 2 steady state value of capital per worker is given by . . . - . . (a) 8 (b) 36 (C) (4)5 ( d) There does not exist any well-defined steady state value QUESTION 29. Let or = %; ,8 = 12; 3 = 1' n = 5. The corresponding QUESTION 30. Consider the same set of parameter values as above. An increase in the savings ratio (a) unambiguously increases the steady statevalue of capital per worker (b) unambiguously decreases the steady state value of capital per worker (c) leaves the steady state value of capital per worker unchanged (d) has an ambiguous effect; a steady state may not exist if the savings ratio increases sufficiently QUESTION 31. The function defined by f = 3:5 + 7x3 -1- 132: — 18 (a) may have 5 real roots. (b) has no real root. (c) has 3 real roots. (d) has exactly 1 real root. QUESTION 32. The function f(: i:) = fix‘ — -%: z:3 — 3:132 + 6 is (a) concave on (-oo, 2) and convex on (2,oo). (b) concave on (—1,2), convex on (-00, -1) and (2, oo). (c) convex on (-1, 2), concave on (-oo, -1) and (2, (d) convex on (-00, 2) and concave on (2,oo).
  13. 13. EEE 2013 A 04 12 QUESTION 33. Consider the function f(. "lI) = gr‘? /3) - §:1:(5/3) for all :5 in the closed interval [-1, 5]. f(5) is approximately equal to 4.386. f(I) attains a maximum on the interval [-1, 5] at (a) :1: = -1. (b) :1: = 2. (c) : c = 3. (d) at = 4. . QUESTION 34. Consider the function f defined by = 2:6 + 5x4 + 2, for all 3: Z 0. The derivative of its inverse function evaluated at f = 8, that is, (f"1)'(8) equals (a) 1/7. (b) 1/15. (c) 1/26. (d) 1/20. QUESTION 35. Consider the following functions. A_/ ' : ?R2 —> 323 defined by f(x, y) 2 (20 + 2y, av - y, -215 + 3y). And g : $12 —> 3?? defined by g(a: ,y) : (2: +1,y + 2). Then (a) Both f and g are linear transformations. (b) f is a linear transformation, but g is not a linear transformation. (c) f is not a linear transformation, but g is a linear transformation (d) Neither f nor g is a linear transformation. . QUESTION 36. A six meter long string is cut in two pieces. The first piece, with length equal to some x, is used to make a circle, the second, with length (6 - :13), to make a square. What value of x will minimize the sum of the areas of the circle and the square? (:3 is allowed to be 0 or 6 as well). )3: = 247r/ (1-l 47r). 93 QUESTION 37. The repeated nonterminating decimal 0.272727. . . , ;.3,. .e. -fi-; ;,; .g. mee¢, ,q, ,., ,.. ,., ... ,.. ,_, ._. _.. .._. _.. .., .._-. ... ... ... .. . . , . £5, / ,
  14. 14. EEE 2013 A 04 13 (a) cannot be represented as a fraction. (b) equals 27/99. (c) lies strictly between 27/99 and 27/100. (d) is an irrational number. The next THREE questions are based on the following situation: Suppose three players, 1, 2 and 3, use the following procedure to allocate 9 indivisible coins. Player 1 proposes an allocation (CII1,IL'2,1L'3) where at, is the number of coins, giye_n_ _t_o_player i. Players 2 and 3 vote on the proposal, saying either Y (Yes) or N (No). If there are two Y votes, then the proposed allocation is implemented. If there are two N votes, the proposal is rejected. If there is one Y vote and one N vote, then player 1 gets to vote Y or N. Now, the proposal is accepted if there are two Y votes and rejected if there are two N votes. ‘ If 1’s proposal is rejected, then 2 makes a proposal. Now, only 3 votes Y or N. If 3 votes Y, then 2’s proposal is accepted. If 3 votes N, then the proposal is rejected and the allocation (3, 3, 3) is implemented. Assume that, if the expected allocation to be received by a particular player by voting Y or N is identical, then the player votes N. QUESTION 38. If 1’s proposal is rejected and 2 gets to make a proposal, her proposal will be QUESTION 39. 1’s proposal will be (a) (5.0.4) (b) (4.05) (c) (3,6,0) (<1) (13.3.0) QUESTION 40. Consider the following change of the above situation. If 2 . makes a proposal and 3 votes Y, then 2’s proposal is implemented. However, iskysoft
  15. 15. EEE 2013 A 04 14 if 3 votes N, then 1 gets to choose between 2’s proposal and the allocation (3, 3, 3). If 1’s proposal is rejected and ‘2 gets to make a proposal, her proposal will be (a) (4,5,0) (b) (0,5,4) (c) either (a) or (b) (d) neither (a) nor (b) QUESTION 41. Utility of a consumer is given by u($1, $2) — min{$1, $2}. His income is M, and price of good 2 is 1. There are two available price schemes for good 1: per unit price 2 and a reduced per unit price 2 - 0 along with a fixed fee T. A consumer would be indifferent between the above schemes if (a) 9 = 2T/ M (b) 9 = 3T/ M (c) 0 = T/ M (d) 9 = (T+1)/ M The next TWO questions are based on the following model: Suppose, an individual lives for two periods. In each period she consumes only one good, which is rice. In period 2, she can costlessly produce 1 unit of rice, but in period 1 she produces nothing. However, in period 1 she can borrow rice at an interest rate r > 0. That is, if she borrows .2 units of rice in period 1, then in Period 2, she must return z(1 + 7“) units of rice. Let $1 and $2 denote her consumption of rice in period 1 and period 2, respectively; $1,$2 2 0. Her utility function is given by U ($1, $2) = $1 + B$2, where B is the discount factor, 0 < B < 1. Note that there are only two sources through which rice can be available; own production and borrowing. QUESTION 42. Find the interest rate r, at which the individual would borrow % unit of rice in period 1. (a) 2 (b) 2
  16. 16. EEE 2013 A 04 15 (c)fi—1 (<1); - QUESTION 43. Now suppose that there are N agents in the above two- period economy. The agents are identical (in terms of production and utility function) except that they have different discount factors. Suppose that B follows uniform d2'stm'butz"on in the interval [%,1]. Assuming 7“ 3 1, the demand function for rice in period 1 will be (1—r) (3) N17?) 0» Niki (l+r) (C)N ‘ 1+1‘ (<1) N3 The next TWO questions are based on the following situation: Consider a two-person two—good exchange economy: persons/ agents are A and B, and goods are 1 and 2. The agents have the following utility functions: It/ i(H? ,-T2) = CW1 + -772. UB(y1,1/2) :1/11/2 where $1 and $2 denote the allocation to A of good 1 and good 2, respec- tively. Similarly, y1 and y2 denote the allocation to B of good 1 and good 2, respectively. There are 5 units of each good; i. e., $1+y1 = 5 and $2+y2 = 5. Now, consider the following allocation: Agent A gets 4 units of good 1 only, but agent B gets 1 unit of good 1 and 5 units of good 2. QUESTION 44. Suppose an agent 1 is said to envy agent j, if 7'. strictly prefers j ’s allocation over her own allocation. And, an allocation is called ‘No-envy allocation’ if none of the agents envies the other. In that case, (a) the above allocation is always ‘No-envy allocation’ (b) the above allocation is never ‘No-envy allocation’ _ (c) the above allocation is ‘No-envy allocation’ if a 2 2 (d) the above allocation is ‘No-envy allocation’ if or 3 QUESTION 45. The above allocation is
  17. 17. EEE 2013 A 04 16 (a) always Pareto optimal (b) never Pareto optimal (c) Pareto optimal if oz 3 5 (d) Pareto optimal if a 2 5 The next TWO questions are based on the following situation: There are two goods: a basic good, say a car, and a complementary good, say car audio. Suppose that the basic good is produced by a monopolist at no cost and the complementary good is produced by a competitive industry at cost c per unit. Let p be the price of the basic good. Each consumer has three choices: (2) consume nothing, which gives 0 utility consume one unit of the basic good, which gives 0 — p utility consume one unit of the basic good and one unit of the complementary good (called bundle), which gives in - p - c utility. Assume w > v > 0. Next, suppose there are two types of consumers of cars: Middle Class: They have valuations '01 and 2121 for the basic good and the bundle, respectively. Rich: They have valuations 122 and 1112 for the basic good and the bundle, respectively. Suppose, '02 > 01 and 102 - 122 > c > 101 — ’U1. QUESTION 46. Find the socially efficient consumption. (a) Rich choose and Middle class choose (b) Rich choose and Middle class choose (c) Rich choose and Middle class choose (d) Both Rich and Middle class choose QUESTION 47. Suppose that the monopolist can distinguish between two types of consumers. What prices would she charge? _ (g 122 from Rich, and 121 from Middle Class Wiskysoft
  18. 18. EEE 2013 A 04 17 (b) 1122 — c from Rich, and 1.111 — c from Middle Class (c) 1111 - c from Rich as well as Middle Class (d) 102 — c from Rich, and v1 from Middle Class QUESTION 48. Suppose that a city can be described by an interval [0,1]. Only two citizens, A and B, live in this city at different locations; A at 0.2 and B at 0.7. Government has decided to set up a nuclear power plant in this city but is yet to choose its location. Each citizen wants the plant as far as possible from her home and hence both of them have the same utility function, u(d) = cl, where (1 denotes the distance between the plant and home. Find the set of Pareto optimal locations for the plant. (a) All locations in the interval [0,1] are Pareto optimal (b) All locations in the interval [0.2, 0.7] are Pareto optimal (c) 0.5 is the only Pareto optimal location (d) 0 and 1 are the only Pareto optimal locations The next TWO questions are based on the following situation: Consider an exchange economy with agents 1 and 2 and goods :1: and 3/. Agent 1 lexicographically prefers the good $: when offered two non-identical bundles of $ and y, she strictly prefers the bundle with more of good $, but if the bundles have the same amount of good :5, then she strictly prefers the bundle with more of good y. However, Agent 2 lexicographically prefers good y. QUESTION 49. Suppose 1’s endowment is (10,0) and 2’s endowment is (0,10). The vector (p, ~,, py) is a competitive equilibrium vector of prices if and only if (a)pe=1=py (b)pI>0andpy>0 QUESTION 50. Suppose we make only one change in the above situation: Person 1 lexicographically prefers y and 2 lexicographically prefers $. The vector (pmpy) is a competitive equilibrium vector of prices if and only if
  19. 19. EEE 2013 A 04 18 (a)Pz=1=Py (b)p, >0andpy>0 (c)pI>py>0 ldlpy‘-'p1>0 QUESTION 51. Consider two disjoint events A and B in a sample space S. Which of the following is correct? (a) A and B are always independent (b) A and B cannot be independent (c) A and B are independent if exactly one of them has positive probability (d) A and B are independent if both of them have positive probability QUESTION 52. A bowl contains 5 chips, 3 marked $1 and 2 marked $4. A player draws 2 chips at random and is paid the sum of the values of the chips. The player’s expected gain (in $) is (a) less than 2 (b) 3 (c) above 3 and less than 4 (d) above 4 and less than 5 QUESTION 53. Consider the following two income distributions in a 10 person society. A : (1000,1000,1000,1000,1000,1000, 1000,2000,2000, 2000) and B : (1000,1000, 1000, 1000, 1000, 2000, 2000, 2000, 2000,2000). Which of the following statements most accurately describes the relationship between the two distributions? _ (a) The Lorenz curve for distribution A lies to the right of that for distri- bution B ' (b) The Lorenz curve for distribution B lies to the right of that for distri- bution A (c) The Lorenz curves for the two distributions cross each other (d) The Lorenz curves for the two distributions are identical for the bottom half of the population *¥iSkysoft ’
  20. 20. EEE 2013 A 04 19 QUESTION 54. A certain club consists of 5 men and 5 women. A 5- member committee consisting of 2 men and 3 women has to be constituted. Also, suppose that Mrs. F refuses to work with Mr. M. How many ways are there of constituting a 5-member committee that ensures that both of them do not work together? (a) 50 (b) 76 (c) 108 ‘ “None of ' the above QUESTION 55. Suppose, you are an editor of a magazine. Everyday you get two letters from your correspondents. Each letter is as likely to be from a male as from a female correspondent. The letters are delivered by a post- man, who brings one letter at a time. Moreover, he has a ‘ladies first’ policy; he delivers letter from a female first, if there is such a letter. Suppose you have already received the first letter for today and it is from a female corre- spondent. What is the probability that the second letter will also be from a female? (a) 1/2 lb) 1./4 (0) 1/3 (d) 2/3 QUESTION 56. On an average, a waiter gets no tip from two of his cus- tomers on Saturdays. What is the probability that on next Saturday, he will get no tip from three of his customers? QUESTION 57. A linear regression model y = 0' -1- B$ + S is estimated using OLS. It turns out that the estimated R2 equals zero. This implies that: (a) All $’s are necessarily zero
  21. 21. EEE 2013 A 04 20 (b)B=1andy= d+$ (c) B = 0 or all $’s are constant A (d) There are no implications for B QUESTION 58. Using ordinary least squares, a market analyst estimates the following demand function logX = a + BlogP + 5 where X is the output, and P is the price. In another formulation, she estimates the above function after dividing all prices by 1000. Comparing the two sets of estimates she would find that (a) 6» and B will be the same in both formulations (b) (3: and B will differ across both formulations (c) a will change but B will not (d) B will change but a will not QUESTION 59. A linear regression model is estimated using ordinary least squares y = a + B$ + 8. But the variance of the error term is not constant, and in fact varies directly with another variable 2, which is not included in the model. Which of the following statements is true? ’ (a) The OLS estimated coefficients will be biased because of the correlation between $ and the error term ‘ (b) The OLS estimated coeflicients will be unbiased but their estimated standard errors will be biased (c) The OLS estimated coefficients will be unbiased and so will their esti- - mated standard errors because the error variance is not related to $, but to z which is not included in the model (d) Both the OLS estimated coefficients and their estimated standard er- rors will be biased QUESTION 60. Suppose the distribution function F($) of a random vari- able X is rising in the interval [a, b) and horizontal in the interval [b, c]. Which of the following statements CAN be correct? *iSkysoft
  22. 22. EEE 2013 A 04 21 (a) F (:13) is the distribution function of a continuous random variable X . (b) c is not the largest value that X can take. (c) The probability that X = b is strictly positive. (d) Any of the above. End of Part II fiskysoft
  23. 23. Delhi School of Economics 8 Department of Economics 2 0 0 0 2 Entrance Examination for M. A. Economics Option B June 29, 2013 Time 3 hours Maximum marks 100 Instructions Please read the following instructions carefully. A Do not break the seal on this booklet until instructed to do so by the invigilator. Anyone breaking the seal prematurely will be evicted from the examination hall and his/ her candidature will be cancelled. —' Fill in your Name and Roll Number on the detachable slip below. c When you finish, hand in this examination booklet to the invigilator. - Use of any electronic device (e. g., telephone, calculator) is strictly prohibited during this examination. Please leave these devices in your bag and away from your person. ' . -’ Do not disturb your neighbours for any reason at any time. - Anyone engaging in illegal examination practices will be immediately evicted and that person’s candidature will be cancelled. Do not write below this line. This space is for official use only. Marks tally mks 3 1 1 1 1 1 1 EEE 2013 B 1
  24. 24. Part I Instructions. ~A Check that this examination has pages 1 through 22. I This part of the examination consists of 10 multiple-choice questions. Each question is followed by four possible answers, at least one of which is correct. If more than one choice is correct, choose only the best one. Among the correct answers, the best answer is theione that implies (or includes) the other correct answer(s). Indicate your chosen answer by circling (a), (b), (c) or rt For each question, you will get 2 marks if you choose only the best answer. If you choose none of the answers, then you will get 0 for that question. However, if you choose something other than the best answer or multiple answers, then you will get -2/3 mark for that question. QUESTION 1. Which of the following two numbers is larger: e" or 7r°'.7 (a) e-tr . (b) 1r“ (c) they are equal (d) it depends on the value of e QUESTION 2. Suppose there are n coins. Each coin has equal probability of falling Heads or‘ Tails. The coins are simultaneously tossed in rounds 1,2,3, . . . , until in some round, one coin toss has a different outcome from that of the other 71. — 1 coins. The probability of the coin-tossing ending in the k-th round is (a) 2“"(1 ~ 21'")'°“ (b) n2“"(1 — n21’")"‘1 (c) n2‘"(1 — n2‘")"‘1 (<1) 2‘"(1 ~ 2‘")'°“ QUESTION 3. There are three identical boxes, each with two drawers. Box A contains a gold coin in each drawer. Box B contains a silver coin in each drawer. Box C’ contains a gold coin in one drawer and a silver coin in the other drawer. A box is chosen, a drawer opened and a gold coin is found. What is the probability that the chosen box is A? (a) 2/3 (b) 1/3 (c) 1/2 EEE 2013 B 2
  25. 25. (d) 3/4 QUESTION 4. Consider the functions f : 5R2 —+ 3?’ and g : R2 —> 322, defined by f(: c, y) = (:1: + 2y, a: — y, -23 + 3y) and g(a: , y) = (I + Ly + 2). (a) Both f and g are linear transformations (b) f is a linear transformation, but g is not a linear transformation (c) f is not a linear transformation, but g is a linear transformation (d) Neither f nor g is a linear transformation QUESTION 5. A six meter long string is cut in two pieces. The first ‘piece, with length equal to some 2:, is used to make a circle, the second, with length (6 — 1:), to make a square. _ , _ What value of 1: will minimize the sum of the areas of the circle and the square? (a) : r=O (b) : z:=61r/ (4+7r) (c) :1:=6 (d) : r= 1/21r The next three questions are based on the following situation. Suppose players 1, 2 and 3 use the following procedure to allocate 9 indivisible coins. Player 1 proposes an allocation (: c1,: r2,a:3) where :5, is the number of coins given “to player 1'. Players 2 and 3 vote on the proposal, saying either Y (Yes) or N (No), If there are two Y votw, then the proposed allocation is implemented. If there are two votes, the proposal rejected. If there is one Y vote and one N vote, then player 1 gets to vote Y or N. Now, the proposal is accepted if there are two Y votes and rejected if there are two N votes. If 1’s proposal is rejected, then 2 makes a proposal. Now, only 3 votes Y or N. If 3 votes Y, then 2’s proposal is accepted. If 3 votes N, then the proposal is rejected and the allocation (3, 3, 3) is implemented. Assume that, if the expected allocation to be received by a particular player by voting Y or N is identical, then the player votes N. 1 QUESTION 6. If 1’s proposal is rejected and 2 gets to make a proposal, her proposal will be ' (a) (0, 5,4) (b) (0, 4, 5) (c) (0, 6, 3) (d) (0, 3, 6) Jest 3 y 39? T»Co? >“~f>‘~f: "=‘~f’=7‘. ‘?-FIT-3*}-T2’-T-’F'~"r"-’7‘7‘-"“"5"’ '
  26. 26. QUESTION 7. 1’s proposal will be (a) (5,0,4) (bl (4:05) (C) (3, 6, 0) (d) (6,3,0) QUESTION 8. Consider the following change of the above situation. If 2 makes a proposal and 3 votes Y, then 2’s proposal is implemented. However, if 3 votes N, then 1 gets to choose between 2’s proposal and the allocation (3, 3, 3). If 1’s proposal is rejected and 2 gets to make a. proposal, her proposal will be (a) (4. 5, 0) ’ (b) (0, 5,4) (c) either (a) or (b) (d) neither (a) nor (b) QUESTION 9. [0, 1] is the same as (3) Ui'. °=1l1/W: 1 - 1/71] (b) U? .°=1(1/Tl: 1 r 1/") (c)r'1;', “’= ,(——1/'rz,1 + 1/n) (d) none of the above QUESTION 10. (0, 1) is the same as (3') Ugilll/ nv 1 _ 1/nl . (b) r"I; ",°=1(—1/71., 1 + 1/11) (c) fl; °,°=1[—1/n, 1 + 1/11] (d) all of the above _______________________________________. _________________ Part II Instructions. 0 Answer any four of the following five questions in the space following the relevant question. No other paper will be provided for this purpose. You may use the blank pages at the end of this booklet, marked Rough work, to do calculations, drawings, etc. Your “Rough work” will not be read or checked. ' .0 Each question is worth 20 marks. . ' iskyscftsnzoias 4
  27. 27. QUESTION 11. Suppose $2 is given the Euclidean metric. We say that f : 9? —> 8? is upper semicontinuous at J: 6 9? if. for every 6 > 0, there exists 6 > 0 such that y 6 ER and Ix — yI < 6 implies f — f(: z:) < 5. VVe say that f is upper semicontinuous if it is upper semicontinuous at every 2 6 ER. (A) Show that, f is upper semicontinuous if and only if {as 6 $2 I f(: c) 2 r} is a closed subset of $2 for every 7' 6 52. (B) Consider a family of functions {f, - I i E I} such that f, - semicontinuous for every 2' E I and inf{f, -(2) f : §R——> §Rby f(: z:) = inf{f, -(:3) I i E I}. Show that {:1: 6 $1‘. I f(x) 2 7'} = r‘I¢e1{a: E 3% I f. -(at) Z 7'} for every__1_: _e. §R. .. . ,. .. (C) In the light of (A) and (B), state and prove a theorem relating the upper semi- continuity of f and the upper semicontinuity of all the functions in the family {f, - I 2’ 6 I ANSWER. : 9? —+ 9? is upper I 1' E I} E 9? for every :1: E W. Define EEE 2013 B 5
  28. 28. QUESTION 12. Let H be the Euclidean metric on R. Consider the function f 2 ER —> R. Suppose there exists /3 E (0.1) such that If(2:) — f(y)I 3 BI: — yI for all ; r.y 6 ER. Let 20 E ? R. Define the sequence (: r:, ,) inductively by the formula ;2:, , = f (: z:, ,_1) for n E N. Show the following facts. (A) (: r:, .) is a Cauchy sequence. (B) (: c,, ) is convergent. (C) The limit point of 1:, say 2:’, is a fixed point of f, i. e., 2:‘ = f(:1;‘). (D) There is no other fixed point of f. ANSWER. l9‘E'E 2013 B -8
  29. 29. QUESTION 13. Let V be a vector space and P : V ——> V a linear mapping with range space R( P) and null space A/ (P). P is called a projector if (a) V = 'R(P) $. IV(P), and (b) for every u E R(P) and w E A/ (P), we have P(u + w) = u. In this case, we say that P projects V on ’R(P) along N Show the following facts. (A) P is a projector if and only if it is idempotent. (B) If U is a vector space and X : U —) V is a linear mapping with ’R(P) = 'R. (X), then P is a projector if and only if PX = X . (C) P is a projector if and only if I — P is a projector. " I in I " ' Let W be a vector space and A : V —-> W a linear mapping. Let B : W —) V be a linear mapping such that ABA = A. Show the following facts. (D) p(A) = p(AB), where denotes the rank of the relevant linear mapping. (E) AB projects W on ANSWER. #. iSkysoftEEE2013B 11 II
  30. 30. QUESTION 14. Given : z:. y E 3?". define (33.31) = {tr+ (1 — t)y I t E (0, We say that C C ER" is a convex set if x. y E Cimplics (Ly) C C. We say that f : ER" —+ ER is a concave function if a: ,y E 3?“ and t E (0.1) implies f(t1: + (1 — t)y) 2 tf(: z) + (1 — t)f(y). (A) Show that f 2 ER" —> 32 is a concave function if and only if H(f) = {(: z,'r) E ER" xERIf(x) Zr} isaoonwzx set in? " X3}. (B) Consider a family of functions {f. - I -i E I } where f, - : ER" —> ER is a concave function for every 1' E 1. Suppose inf{f, (.1:) I116 I) 6 ER for every x 6 ER". Show that f : ER" ——> ER, defined by f (as) = inf{f. -(23) I 2' 6 I} is a concave function. (C) Consider concave functions fl : ER" —> ER and f2 : ER" —) R. Define f : ER" —> ER by f(: z:) = max{f1(: r), f2(: z:)}. Is f necessarily a concave function? Provide a proof or counter-example. (D) Show that, if f : ER" —> 3? is a concave function, then {x 6 ER" I f(:1:) Z r} is a convex set for every 1' 6 ER. (E) Is the converse of (D) true? Provide a proof or counter-example. ANSWER. EEE 2013 B 14
  31. 31. QUESTION 15. (A) An urn contains N balls, of which Np are white. Let S, , be the number of white balls in a sample of n balls drawn from the urn without replacement. " Calculate the mean and variance of S, .. (B) Let X and Y be jointly continuous random variables with the probability density function f(z. y) = §17;exp I-élzz +1/’)I (a) Are X and Y independent? (b) Are X and Y identically distributed? (c) Are X and Y normally distributed? (d) Calculate Prob [X2 + Y2 3 4]. ""‘ ' " " ' " ‘ (e) Are X 2 and Y2 independent random variables? (f) Calculate Prob [X2 3 2]. (g) Find the individual density function of X 2. ANSWER. I . 'Sk sof. EEE2/J13B 17 V »; .__. _4_. _.. _., .V. ,., ._. _.3.g, --. .~, . . —.. ,. ..— -- . .

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