Option A (Series 04)
June 26, 2010

Time :  3 Hours Maximum Marks :  100

General Instructions.  Please read the following...
Part I

Instructions. 

0 Check‘that this booklet has pages 1 through 22. Also check that the bottom of
each page is marke...
(d) the impact on government seigniorage is ambiguous

QUESTION 4. When the nominal interest rate changes but the real rat...
Q‘. 

I 7“ EEE 2010 A 04 4



 
 
 
  
 
 
 
  
 
 
  
   
    
 
   
 
  
  
 
  
  
 
 
 
   
 
 
 
 
  
   
  
  

(b) ...
(a) The unique median of the distribution is 1

(b) The unique median of the distribution is 0

(c) The unique median of t...
Suppose :1: > y,  y ~ z and 9: ~ z.  Then

M57» is transitive 17% 1/ ’»‘ / ‘V ' ’
(b) N is transitive " " ‘7'

(c) both >-...
Part II

Instructions. 

o This part of the eicamination consists of 40 multiple-choice questions.  Each question
is follo...
The next four questions are based on the following information.  Consider an
economy with an aggregate production function...
(c)' shift to the left
(d) shift to the right

The next four questions are based on the following information.  Consider a...
(a) None
/ (5') At most one
(c) One

(d) At least-one

QUESTION 32. Suppose a function f 2 92 -> R is differentiable at :5...
Which of the following statements is correct? 
(a) g is increasing in z and decreasing in y
(b) g is increasing in both 1 ...
QUESTION 39. Consider f as described in the previous question.  Then, 
/ (/8) f(a; ) 5 0 for every 1 E [a. , b]
(b) f (1) ...
., _'. .| :  ‘M. ../ ea

QUESTION éIySuppose player 1 has fiye coii/19and player 2 has four coins.  Both players

‘ toss a...
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. . " V;  r -— » _, r ‘
(a) 0 '‘‘i ‘ it  "
(b) —‘2/25  ‘V ;  r (C,  .   .  _
...
(d) PI/ py > 0

QUESTION 53. Now suppose agent 1 lexicographically prefers y to I and agent 2 treats I
and y as perfect su...
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l Delhi School of Economics

Department of Economics

Entrance Examination for M.  A.  Economics
Option B
June 26, 2010

T...
Part I

Instructions. 

- Check that this examination booklet has pages 1 through 22. Also check that tlw
bottom of each p...
(d) At least one

QUESTION 3. Consider a twice differentiable function f :  ER —» 8? and a, b E 8? sin-I1
that a < b,  f(a...
(c) sin‘l(1)/7r + 1/2
(d) sin"1(: c)/1r + 1r/2

d random variable with mean (3 mid

QUESTION 8. Let X be a normally distri...
Part II

Instructions. 

ur out of Questions 11, 12, 13, 14 and 15 in the space following the
e provided for this purpose....
QUESTION 12. Consider an open set C C ER and f :  C -9 R.  C is said to be a convex
set if r, y E C and t 6 (0,1) implies ...
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QUESTION 13. (Notation:  D¢ denotes the derivative of a function ¢>. )

(A) A function f 2 ER“ ——» Ris...
QUESTION 14. Let U be a finite dimensional vector space with inner product (. ,.). 
Given a vector subspace S of U,  let 0(...
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Delhi School of Economics Entrance Exam (2010)

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This is the entrance exam paper for the Delhi School of Economics for the year 2010. 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 (2010)

  1. 1. Option A (Series 04) June 26, 2010 Time : 3 Hours Maximum Marks : 100 General Instructions. Please read the following instructions carefully : 0 Check that you have a bubble-sheet accompanying this examination booklet. 1'10 not break the seal on this booklet until instructed to do so by the invigilatm, 0 Immediately on receipt of this booklet, fill in your Signature, Name, Roll number and Answer sheet number (see the top left-hand-side of the bubble-sham‘) in the space provided below. 0 This examination will be checked by a machine. Therefore, it is very imporumi that you follow the instructions on the bubble-sheet. 0 Fill in the required information in Boxes 1, 2, 4, 5 and 6 on the bubble-slum. The invigilator will sign in Box 3. 0 Do not disturb your neighbours at any time. the invigilator. Signature Name Roll number Answer sheet numbe JMEO EEE 2010 A 04 1 xiskysoft SI
  2. 2. Part I Instructions. 0 Check‘that this booklet has pages 1 through 22. Also check that the bottom of each page is marked with EEE 2010 A 04. 0 This part of the examination consists of 20 multiple-choice questions. E. ‘(‘l « 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 correzsl. answers, 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 slusding the appropriate bubble. 0 For each question, you will get 1 mark if you choose only the best answm‘. ‘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 —‘/3 mark for that question. 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 not be read or checked. You may begin now. Good luck! QUESTION 1. In models where expectations adjust slowly, a progressive income in: -: schedule is , _ (a) a means to maximize tax revenue ‘J’, . (b) a means to ensure tax compliance 2 (‘I (c) always an automatic stabilizer in the long run (d) often an automatic stabilimer in the short run _ QUESTION 2. Open market operations decrease the supply of base money by (a) selling government bonds (b) selling gold (c) reducing foreign currency holdings c/ W’) all of the above QUESTION 3. If an economy is experiencing hyperinflation, then (a) government seigniorage goes up (b) government seigniorage goes down (c) government seigniorage remains unchanged EEE 2010 A 04 2 iskysoft
  3. 3. (d) the impact on government seigniorage is ambiguous QUESTION 4. When the nominal interest rate changes but the real rate of interest remains unchanged (a) it affects both the investment function as well as the money demand function (b) it affects the investment function but does not affect the money demand function / (c) it affects the money demand function but does not affect the investment function (d) neither the investment function nor the money demand function gets affected QUESTION 5. Ceteris paribus, higher velocity of money circulation leads to (a) an increase in both the real and the nominal demand for money (b) an increase in the real demand for money and a decrease in the nominal demand for money (c) a decrease in the real demand for money and an increase in the nominal demand for money tel) a decrease in both the real and the nominal demand for money QUESTION 6. Consider a 3 X 3 nonsingular matrix A with real entries. If the matrix B is derived from‘A by interchanging the first and last columns of A, then the determinant of B, denoted det B, is egual to (a) detA /83) — (let A (C) 0 (d) 1/detA QUESTION 7. The sequence (—1). "(1 + 1/n), for positive integers n, (a) has limit point 1 (b) has limit point -1 (c) has limit points 1 and —1 (d) has no limit points QUESTION 8. Consider the following two games: Hawk Hawk Dove Game 1: Enter (—-L1) Game 2: Enter <~l,1 I_3,3 > Not enter 0,6 Not enter 0,6 0, 7 In every payoff pair “a: ,y”, z is the payoff of the row-player and y is the payoff of the column—player. Analyze these games. These games illustrate that, in a strategic situation (a) an expanded set of strategic options can be disadvantageous EEE 2010 A 04 3
  4. 4. Q‘. I 7“ EEE 2010 A 04 4 (b) a contracted set of strategic options can be advantageous _ (c) one should view the situation from one’s own, as well as from one’s opponent’s. perspective (d) all of . the above are true QUESTION 9. Ioe-cream vendors A and B know that they have to locate simultaneously on a beach. The beach is identified with the interval [0, 1] and at every point in [0, 1] there is a person who wants exactly one ice-cream oone. Each person will buy the ice-cream from the nearest vendor; if there are equidistant vendors, then the buyer randomizes among them with equal probabilities. Each vendor wants to maximize his own expected market share. The vendors will locate at (a)0and1 _ (b) 1/4 and3/4 ’ g , - 7’, ,~- _ . : v(p)’l/2 and-1/2 -. /,; N; - >/ jffp, W, -.7. ; (d) 1/3 and 2/3 /5 9’ ' _a7_ QUESTION 10. Suppose there is only one future period and’ he (presently unknown) state of the world in that period can be either 51 or s2. The future return on a share of a given company is 5 in state s1 and —1_in statesg. The future return on a government bond is 1 independent of the state. Suppose a third asset is offered on the market whose return is 3 in state s1 and 0 in state s2. The current prices of the stock and the bond are 3 and 1 respectively. If the price of the new asset "rules out the possibility of arbitrage profit (which arises when portfolios of assets that are identical in terms of returns have different prices), what is the price of the new asset? f C 7 7/5 : I S = 3‘ (a) It depend on the probabilities of the future statesE3 5 l 95f PG) -'0 -P)_ . gF_* Fr éf ) (b) Strictly between 2 and 3 . (c) Strictly between 1 and 2 E/ :4: Q? J‘) 2 1/2 . QUESTION 1}. /‘A number, say X1, is chosen at random from the set {L2}. Then a number, say X2, is chosen at random from the set {1, X1}. The probability that X; = 2 given that X; =1 is _, VET ,9 1, If ,9? 7-3 ( « . rm » 4 " 4 / (331 re). .. "V11. / (b) 1/2 .1, __L _—'-e _ / (Z)/ f/3 °_. (L_ "‘, ‘“i +/ ,‘‘ 1 , a (d) 1/4. KM ) elf? » .2 '4 Ir QUESTION 12. Suppose a random variable X takes values -2, 0, 1 and 4 with proba- bilities 0.4, 0.1, 0.3 and 0.2 respectively. ’ ’ D O s F I —' " ( ' . X’ /
  5. 5. (a) The unique median of the distribution is 1 (b) The unique median of the distribution is 0 (c) The unique median of the distribution lies between 0 and 1 (d) The distribution has multiple medians QUESTION 13. Two persons, A and B, shoot at a target. Suppose the probability that A will hit the target on any shot is 1/3 and the probability that B will hit the target on any shot is 1/4. Suppose A shoots first and they take turns shooting. What is the probability that the target is hit for the first time by A’fll_1irgl shot? (a) 1/24 /1 .2 1 . 1/12 (0) 1/ 5 (d) 1/3 QUESTION 14. A fair coin is tossed repeatedly until a head is obtained for the first time. Let X denbte the number of tosses that are required. The value of the distribution function of X at 3 is i (a) 3/4 (b) 1/2 (9) 7/8 (d) 7/16 QUESTION 15, Using a random sample, an ordinary least squares regression of Y on X yields the 95% confidence interval 0.43 < fl < 0.59 for the slope parameter /3. Which of the following statements is false? 7 7(5) This interval contains the parameter 5 with probability 0.95 (b) The point estimate of B obtained from our regression always lies within this interval (c) The 90% confidence interval for fl is a subset of the interval obtained above (d) If such intervals are constructed from repeated samples drawn from the, population in question, then on average Qfnout of 100 of these intervals are likely to contain the true parameter value. QUESTION 16. Consider a binary relation 5 defined on the set A = {: z:, y,z}. Define relations >— and ~ on A by: for a, b E A, a>-bifandonlyifafibandnotbta. and a~bifandonlyifa: bandbta EEE 2010 A 04 5
  6. 6. Suppose :1: > y, y ~ z and 9: ~ z. Then M57» is transitive 17% 1/ ’»‘ / ‘V ' ’ (b) N is transitive " " ‘7' (c) both >- and ~ are transitive (d) we cannot conclude anything about the transitivity of > and ~ QUESTION 17. The utility function 'u(:1:, 'y) = -(: r + y)‘/2, for (:5, y) 2 (0,0), exhibits (a) diminishing marginal rate of substitution and diminishing marginal utilities (b) increasing marginal rate of substitution and diminishing marginal utilities , (<. ')’constant marginal rate of substitution and diminishing marginal utilities (d) increasing marginal rate of substitution and constant marginal utilities QUESTION 18. Suppose there are just two goods, say st, ‘ and 12. Consider a consumer who chooses : c2 = 0 for all income levels 19 > 0 and all prices 1lL> O and P2 > 0. These J choices are consistent with the consumer 0' g 11 , _ 1 4,, 72 r, (a) having utility function 'l. I.(! I:1,! l32) = 2:1 + 22:2 5 9; VT’ ‘ i (b) having utility function u(: z:1,: r2) = 22:; + 132 72' "(em 70. f ; (c) lexicographically preferring $2 to :51 (d) lexicographically preferring 2'; to 1:2 QUESTION 19. Consider a Coumot duopoly with firms 1 and 2 that produce a homo- geneous good. The inverse demand curve for this good is given by P(a: ) = 5 — z, where 2: is the total output of the two firms. Firm 1 has a constant average cost 5/2 and firm 2 has a constant average cost 3/2. In equilibrium, (a) only firm 1 produces a positive output (b) only firm 2 produces a positive output I (c) both firms produce positive outputs with firm 1 producing more than firm 2 (9/) both firms produce positive outputs with firm 2 producing more than firm 1 QUESTION 20. If all input prices double, then what happens to the minimum cost of producing a given output? (a) It doubles V (b) It more than doubles (c) It less then doubles (d) It depends on the production function End of Part 1. Proceed to Part II of the examination on the next page. EEE 2010 A 04 I 6 iSkysC>ft
  7. 7. Part II Instructions. o This part of the eicamination 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. Among the correct answers, the best answer is the one that implies (or includes) the other correct a. nswer(s). Indicate your chosen best answer on the bubble-sheet by shading the appropriate bubble. c 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. I-Iowever, if you choose something other than the best answer orlmultiple answers, then you will get -2/3 mark for that question. The following notational conventions apply wherever the following symbols are used. ER denotes the set of real numbers. Given a function . f , D f (:5) and D2 f (:1:) denote the first and second derivatives of f (if they exist), respectively, evaluated at 3:. QUESTION 21. Consider a closed economy. If the nominal wage is flexible and nominal money supply is increased, then which of the following will be true in equilibrium? (a) Real wage decreases and real money supply decreases (b) Real wage decreases and real money supply increases , (c’) Real wage is unchanged and real money supply is unchanged (d) Real wage decreases and real money supply is unchanged QUESTION 2 . ' Suppose an economy is at than full employment and it consists of an aggrega e “worker” and~an aggregate “capitalist”, with the former having a higher marginal propensity to consume from his disposable income. Suppose both agents pay income tax according to the same linear schedule. If the government’s budget is in balance and a lump-sum income transfer is made from’ thecapitalists to the workers, then. the government’s (a) budget will go into deficit -, J,b)' budget will go into surplus (c) income and expenditure will be unchanged (d) income and expenditure will change but the budget will stay in balance EEE 2010 A 04 7 *iSkysoft
  8. 8. The next four questions are based on the following information. Consider an economy with an aggregate production function Y = aK + , BL, where or and fl are positive constants, K is capital, L is labour and Y is output. K is fixed in the short run. Perfectly competitive producers take the nominal wage rate W and the price level P as given, and employ labour so as to maximize profit. This generates the labour demand schedule. The labour supply schedule is L5 = ——’y + 6W/ P, where 7 and 6 are positive constants. Producers and workers have perfect information about P and W. QUESTION 23. The labour market will clear if and only if Van / i > 7/6 (b) B < 7/6 (c) fl > 6/7 (d) B -< 6/7 QUESTION@/ Assume that the required parametric condition of the previous question holds and that the nominal wage rate is fixed. The short -run aggregate supply schedule for this economy, with P along the vertical axis and Y along the horizontal axis, will look as follows: (a) for high values of P it will be horizontal; for some mid—range values of P it will be downward sloping; for low values of P it will be horizontal again (b) for high values of P it will be horizontal; for some mid~range values of P it will be upward sloping; for low values of P it will be horizontal again (c) for high values of P it will-be vertical; {or some mid-range values of P it will be downward sloping; for low values of P it will be vertical again (d). for high values of P it will be vertical; for some mid-range values of P it will be upward sloping; for low _values of P it will be vertical again QUESTION 25. If there is a one shot increase in the fixed stock of the capital stock, then the short run aggregate supply schedule will (a) shift up A (b) shift down (c) shift to the left ofd) shift to the right QUESTION 26. If there is a one shot increase in the fixed nominal wage rate, then the short run aggregate supply schedule will VCA) shift up (b) shift down EEE 2010 A 04 8 iskysoft
  9. 9. (c)' shift to the left (d) shift to the right The next four questions are based on the following information. Consider a C. 'o. ~‘~: *ri economy simple Keynesian model of the goods market, where prices are fixed and output in equilibrium is determined by aggregate demand. Investmentis fixed at I '. There is no government sector. Suppose there are two groups of households, called A and B, and the total income Y is distributed among these two groups in such a way that group .4 gets Y‘ = AY, and group B gets Y3 = (1 — A)Y, where A 6 (0, 1) is a constant. Tin" consumption function of group A is C‘ = c + CAYA and the consumption function of group B is C3 = c+ c3YB, where0 < c4 < cg < 1, i. e., the two groups have differeivz‘ consumption propensities. QUESTION 27. The value of the investment multiplier in. this economy is given by (8') l—Acfi—Ac3 (bl WX>T. 'T—7E; (d) 1.. ..» —1.‘i)(. .,_. _~, .) QUESTION 28. If there is a one shot increase in the parameter A, (a) equilibrium output unambiguously increases equilibrium output unambiguously decreases (c) equilibrium output remains unchanged (d) equilibrium increases or decreases depending on whether A 2 1/2 QUESTION 29. If there is a one shot increase in the parameter cA, vter) equilibriumioutput unambiguously increases (b) equilibrium output unambiguously decreases (c) equilibrium output remains unchanged (d) equilibrium increases or decrmes depending on whether A 2 1/2 QUESTION 30. If there is a one shot increase in the parameter C3, / (5.) equilibrium output‘ unambiguously increases (b) equilibrium output unambiguously decreases (c) equilibrium output remains unchanged ' (d) equilibrium increases or decreases depending on whether A 2 1/2 QUESTION 31. Suppose the function f : R —v R is given by f(1:) = 1:3 — 3:1: + b. Find the number of points in the closed interval [-1, 1] at which f (1) = 0. EEE 2010 A 04 ‘J fiskysoit
  10. 10. (a) None / (5') At most one (c) One (d) At least-one QUESTION 32. Suppose a function f 2 92 -> R is differentiable at :5. Consider thr _ rc — 2: — _ statements: f( ) f( h) Df (13) = ,1g§g)- h (1) . I —- I __ “d f( + 2h) M an Df(I) = £5 —~—-2—h——— (11) In general, (M (1') is true and (ii) is false (b) (i) is false and (ii) is true (c) Both are true (d) Both are false QUESTION 33. Consider the statements: for : c,y E R, . P _ / - ‘ III — I11!" 5 Ir — 111 ('11) '9 and I l| r1_—1y| | = l3 - yl (ii) In general, / (at) (i) is true and (ii) is false (b) (i) is false and (ii) is true (c) Both (1') and (ii) are true (d) Both (2') and (ii) are false QUESTION 34. Suppose the function f : §B’_—;31js_in91:. easingJln. lb9t. l1_a. x'gugnts, i. e., f(; cly) is increasing inx and increasing y. For : c,y 6 32, let _ x fizgy zAy—{¢ fi: >y 3 Defineg: §R2—+§Rby V 4 . _ . f(3»I/ )—'1'f(3A3/Ia: /y)1 if-T29 ”($’y)_{%f(z/ y. -?c/ y), ifz<y EEE2010A04 : ”*': /‘- f%"~d>: % +77 * 71' 7/ o It yykl) (0 1/’ I
  11. 11. Which of the following statements is correct? (a) g is increasing in z and decreasing in y (b) g is increasing in both 1 and y K is increasing in 2: but may or may not be increasing in y (d) g may or may not be increasing in I The next three questions are based on the following definitions. Consider the set A = {(: z:, y) E R2 I 2:2 +y2 S 1}. Given (a, b) 6 ER2 such that (a, b) ;5 (0,0) and c > 1, let X = {(1.11) + (a, b) I (33:31) E A} Y = {(2~*/ ’<x + y>,2-We - y» I (Ly) e A} Z = {c(ziy) I (xvy) E ' QUESTION 35. Which set is not a disc (i. e., the region inside a circle)? (8) X (b) Y (0) Z , ‘/(d) None QUESTION 36. Which set has a larger area than A? (a) X (b) Y W (d) None QUESTION Which set does not contain all the points that belong to A? (a) X (b) Y / ; ’ (c) Z -'1 %(d) None QUESTION 38. Consider at twice differentiable function f : ER —» 9? and a, b 6 ? R such that a < b, f-(_a) = 0 = f(b) and D2f(: c) + Df(1:) — 1 = O for every 1: 6 [a, b]. Then, (a) f has a maximum but not a minimum over the open interval (a, b) (/ (5) f has a minimum but not a maximum over (a, b) (c) f has neither a maximum nor a minimum over (a, b) (d) f has a maximum and a minimum over (a, b) EEE 2010 A 04 " 11 iskysoft.
  12. 12. QUESTION 39. Consider f as described in the previous question. Then, / (/8) f(a; ) 5 0 for every 1 E [a. , b] (b) f (1) 2 0 for every : c E [a, b] (c) f(a: ) : — 0 for every x 6 [a, b] (d) f must take positive and negative values on the interval [a. , b] 19¢, QUESTION 40. Suppose f : [0,1] -+ [0, 1] is a continuous nondgcreasing function with A0.) = 0 and m) = 1. Define 9:10.11 ~ (0.1) by yo) = min{z e (0,1) I re) 2 y). Then, (a) g is non—decreasing (b) If g is continuous, then f is strictly increasing (c) Neither (a) nor (b) is true (; l) Both (a) and (b) are true ix‘ . QUESTION Suppose X; and X; are real-valued random variables with f as their common probability density function. Suppose ($1,222) is a sample generated by tl1c-. s.: - random variables. The expectation of the number of observations in the sample/ , that fall within a specified interval [a, b] is 2 (a) (1: ma) dz) (b) If 3%) are ‘/ (c) 2faf(1:)d: c (d) ff If(1:) dz: QUESTION Suppose X1,. . . ,X, . are observed completion times of an experiment with values in [0, 1]. Each of these random variables is uniformly distributed on [0,1]. If Y is the maximum observed completion time, then the mean of Y is ‘ . 1 (a){n/ (n+1)1’ , ’°(X>‘_¢“" - (b)n/ ?(n+1) 4 *1 9 (c) n/ (n+1) _ fL(](/ nllf i (d) 2n/ (n+ 1) o S {V QUESTION 43. Suppose the random vmiabbglfi takes values in the set {—1,0, 1} and the probability of each value is equal. Let Y = 2. Which) of the following statements is true? ' ' ‘ — (a) X and Y are correlated but independent 02) X and Y are uncorrelated but dependent (c) X and Y are dependent and have the same mean V i I3 (d) X and Y are correlated and have different means i ’) EEE2010 A 04
  13. 13. ., _'. .| : ‘M. ../ ea QUESTION éIySuppose player 1 has fiye coii/19and player 2 has four coins. Both players ‘ toss all their coins and observe the number that come up heads. Assuming all the coins are fair, what is the pr@bility that player lpbtains more heads than player 2? , _ J” 1/2 ’ nu»: .: co, u. (b) 4/9 c‘ 1. “.2. t, 1 (C) 'r/ av}; r. I-‘ A . .f. .,. fi9 .1, 2 3 (d) 4/5 _, ____. fi ‘D, QUESTION 45. Suppose 10 athletes are running ina race and exactly 2 of them are taking banned drugs. An investigator randomly selects‘_(2g, athletes for drug-testing. What is the probability that neither of the cheaters will be caught? ‘ f7 ’ , /4”’ (a) 16/25 "' r, _ 9, / lo . . ,4" *""‘3 (b) 4/5 .5 - _ I (c) 3/5 gy : . ' , Q, —-, 61 _, ‘ / QUESTION 46. Suppose 0 is a random variable with uniform distribution on the interval [—1r/2,1r/ 2]. The value of the distribution function of the random variable X = sin9 at I E [-1, is A (a) sin'1(: c) (b) sin-1(z) +7:/2 V , , « _, ”~ ’ h (c) si-n:1l(z)/1r+1/2 F; 0., ~ 3 V " (d) sin (z)/1r + 1r/2 : “ QUESTION 47,) Let‘ X be a normally distributed random variable with mean 0 and variance 0’. Then, the mean of X2 is Xrv ( 0/ 9' 2.3 -.18) 0 (b) 0 (c) 20 (d) <72 . The next three questions are based on the following data. The number of loaves of bread sold by. a bakery in a day is a random variable X. The distribution of X has a probability density function f given by / v , _. . 9 36. km, ifx E [0, 5) / ‘ f(z) = Ic(10 — x), if: E [5, 10) 0, ifa: E [10, oo) QUESTION 48. As f is a probability density function, the value of I: must be EEE 2010 A 04 5 1, 13 f/4 7 ~' J r '0 7 ) ) 0 ° 9 . - kx= V" ~ mp r» ° Iskysoft J» ~ "')~»_'p_ « -‘i C‘ -a-,1 (1‘rk— *9; .4 1 , , (9%: s I 3. _ T . / ‘3 - 74* 5 / . . « A ’, .Z)
  14. 14. T Q A I ' r Cg" ~ V r I / I ” / . 1 ‘ '2 ) . . " V; r -— » _, r ‘ (a) 0 '‘‘i ‘ it " (b) —‘2/25 ‘V ; r (C, . . _ J2)”1/25 "35 (d) 2/75 f’‘’r~VP*’v’>. - 7. 1 7.’ QUESTION 49. Let A be the event that X 2 5 and let B be the event that X E [3, T, The probability of A conditional on B is ‘~’ I‘ 7 _: ’ if ll -‘E A (a) 16/37 ‘ '17 , /(l)}’l1/37 ' (c) 25/37 (d) 1 , ‘ , ;~V, ;. QUESTION 50. Events A and B are 1 I? ~ 3 g V , V - / ;-~ f’/ 'f; ) not independent ” 7 9: 4 (b) independent V V C I (c) conditionally independent ; (cl) unconditionally independent The next four questions are based on the following data. Consider an excliangw economy with agents 1 and 2 and goods : r and y. Agent 1’s endowment is (0, 1) (i. e.. . good 1 and 1 unit ofgood y) and agent 2’s endowment is (2, O) (i. e., 2 units ofgood 3: and no good y}. The agents can consume only nonnegative amounts ofa: and y. QUESTION 51. Suppose agent 1 lexicographically prefers 1.‘ to y, i. e., between any two ‘bundles of goods, she strictly prefers the bundle containing more of : c, and if the bundles contain equal amounts of :5, then she strictly prefers the bundle with more of y. Suppose agent 2 treats :1: and y as perfect substitutes, i. e., between any two bundles (r, y) and (1’, y’), she strictly prefers (I, y) if and only if : c + y > I’ + y’. The competitive equilibrium allocation for this economy is (a) 1 gets (0,1) and 2 gets (2,0) (b) 1 gets (2,0) and 2 gets (0,1) (c) 1 gets (3/2,0) and 2 gets (1/2,1) (d) 1 gets (1,0) and 2 gets (1,1) QUESTION 52. Suppose agents 1 and 2 have the preferences described above. The .5: of all possible competitive equilibrium prices consists of all p, > 0 and p, , > 0 such that ((41) 172/1711 = 1 t ‘ (bl P: /Py Z 1 1 (C) 111/12,, 5 1 EEE 2010 A of? )4
  15. 15. (d) PI/ py > 0 QUESTION 53. Now suppose agent 1 lexicographically prefers y to I and agent 2 treats I and y as perfect substitutes. The set of all possible competitive equilibrium prices consist ~ of all p, > 0 and p, , > 0 such that 1 FT‘ 7 I (a) P: /pg = 1 (bl 172/172: 2 1 L3) P: /Py S 1 (d) P: /Py > 0 QUESTION 54. Now suppose agent 1 lexicographically prefers y to I and agent 2 treats I and y as perfect complements. The set of competitive equilibrium allocations (a) includes the allocation (1,0) for agent 1 and (1, 1) for agent 2 (b)! includes the allocation (0,1) for agent 1 and (2,0) for agent 2 is empty (d) includes all allocations (I, 1) for agent 1 and (2 — I, 0) for agent 2, where I E [0, 2] QUESTION 55. Consider a person who chooses among lotteries. Each_lottery is of the form (p1,p2,p3), where pl is the probability of getting Rs. 5, p2 is the probability of getting Rs. 1 and p3 is the probability of getting Rs. 0. This person prefers lottery (0, 1, O) to lottery (0.1,0.89,0.01). If this person rnaximizes expected utility and is faced with the lotteries (0,0.1l,0.89) and (0.1, 0, 0.9), which lottery should he prefer? (/ a) The lottery (0, 0.11, 0.89) ' The lottery (0.1,0,0.9) ’ (c) He should be indifferent between these lotteries (d) There is insufficient data to decide QUESTION 56. Consider an economy with two agents, A and B, and two goods, 131 and 2122. Both agents treat 1171 and 122 as perfect complements. Suppose the total endowment of (I1 is 4 and the total endowment of 122 is 2. Which of the following allocations is not Pareto optimal? (Note that a bundle (a, b) represents a units of £121 and 17 units of 1132.) ‘V (a) A gets (1,1) and B gets (1, 1) (b) A gets (2,1) and B gets (3/2, 1) / (,2) A gets (1/2, 3/2) and B gets (3,1V/2) (d) A gets (3, 2) and B gets (0,0) QUESTION 57. A consumer has the utility function u(I, y) = Iy. Suppose the consumer demands bundle (: c‘, y‘). Now suppose the seller of good I offers a “buy one, get one free” scheme: for each unit of good I purchased, the consumer gets another unit of I for free. EEE 2010 A 04 15
  16. 16. ‘f as 1 5 3sE‘ li71:‘, (v1=l 3”)-‘H70 1~“", i.‘50, _- gb .9,‘ 3' . 1 WV) All 1 Dléliven this scheme, suppose the consumer buys bundle (Id_, y,l) and gets an 2_1_dClitlu1;: -.l . ,» to 1 for free. Which one of the following statements must be true? _ A‘ (a)I¢>: z:‘ and? /.x>y‘ ‘ -1-: Ifgpz 2. W)! I ‘ )Qa)I, l>'I‘andyd= y' X 1 PT! Olgfllél (c)Id>I‘ andy, l<y' Zl"fm— ‘>67-—-9:/ "‘“m(, dr)I, l=I‘andyd= y’ °Fi ; If A "QUESTION 58. Consider a Bertrand duopoly with firms 1 and 2 that produce a homo l LS0} geneous good and set prices pl and pg respectively. Suppose pl and pg have to be pm} .1 . integers. If pl < pg (resp. pl > 772), then firm 1 (resp. firm 2) sells 5 — pl (resp. 5 —r ; ,/_/ , (, /< '60, and the other firm sells nothing. If pl = pg, then each firm sells Firm 1 l'v: :.~: V W ~, " ” constant average cost 5/ and firm 2 has a coistant a: /efirage cost 3 . In equilibrium » ‘i : " 3) "/5. (3) P1": 2 = 172 , l H j / : — 1 (blP1=3=P2, g/ “,3, : J,o -: ‘ I ‘i ‘M (c) P1 = 3 and 112 = 2 I g: £1 , V", _»: ’r “l = 3 and P2 is 2 or 3 ‘ . (0,; M. QUESTION 59. Consider a Stackelberg duopoly wit and firm :2 as the follower. If (ql, qg) is the Stackelberg equilibrium, then (3l"Z" 3‘l,3° . _ q_ 2. . (;() firm 1’s optimal isoprofit curve and firm 2;s reaction curve intersect at (ql, ‘q-g) an-: 4.; are tangential at (ql, qg) (b) firm 2’s optimal isoprofit curve and firm 1’s reaction curve’ intersect at (ql, qg) 2:! ‘-I‘ are tangential at (ql, qg) (C) isoprofit curves of the (d) reaction curves of the two firms intersect at (ql, qg) two firms intersect at (ql, qg) and are tangential at (ql, qg) l/ QUESTION 60. Firm 1 is the potential entrant into a market in which firm 2 is the incumbent monopolist. Firm 1 moves first and chooses to “enter” or “not enter” If it then firm 1 gets profit 0 and firm 2 gets the monopoly profit 10. If fin. _ does “not enter” , ofit “enters”, then firm 2 chooses to “fight” or “not fight”. If firm 2 fights, then firm 1‘s pr is —2 and firm 2’s profit is 6. If firm 2 does “not fight’. ’, then firm 1’s profit is 2 and Firm ‘2’s profit is 8. Firm 2’s strategy of “fight” is best interpreted as (a) a. commitment , , Hi” I L, = : .4. ‘ '> I‘ — / ()3) a non—credible threat 1 "V V V (c) a punitive action ~"”“‘”'l“°'[ F3“! 1 (W ‘D -, )3. 5 5; u‘ (d) acquiescence ‘. "F -. Q 'a- ‘!9' 0 EEE 2010 A 04 , iii 9* ‘ . <, / iskysoft
  17. 17. l Delhi School of Economics Department of Economics Entrance Examination for M. A. Economics Option B June 26, 2010 Time. 3 hours Maximum marks. 100 Instructions. Please read the following instructions carefully. - 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. 0 Fill in your Name and Roll Number on the detachable slip below. 0 When you finish, hand in this examination booklet to the, ivigilator: EEE 2010 B 1 *iSkysoft
  18. 18. Part I Instructions. - Check that this examination booklet has pages 1 through 22. Also check that tlw bottom of each page is marked with EEE 2010 B. o 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 the one that implies (or includes) the other correct answer(s). Indicate your chosen best answer by circling the appropriate choice. 0 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. You may begin now. Good luck! _j? __j_ QUESTION 1. Suppose the function f : R2 —. v ER is increasing in both arguments, i. e., f(I, y) is increasing in I and increasing y. For I, y 6 33,. let _ I, ifrrsy I/ y_{y, ifI>y Define g : ER2 —v 3? by _. 1 ' > gay) = {{(I. y) 2 (I/ y. a:/ y). {fr _ y §f(I/ y, I/y), ifI<y Which of the following statements is correct? (a) g is increasing in I and decreasing in y (b) g is increasing in both I and y (c) g is increasing in 1‘ but may or may not be increasing in y (d) g may or may not be increasing in I QUESTION 2. Suppose the function f 2 R ——v 3? is given by f(I) = I3 — 31+ b. Find the number of points in the closed interval [-1, 1] at which f(: r) = 0. (a) None (b) At most one (c) One *iSkysoft EEE 2010 B 2
  19. 19. (d) At least one QUESTION 3. Consider a twice differentiable function f : ER —» 8? and a, b E 8? sin-I1 that a < b, f(a) = 0 = f(b) and D2f(: z:) + Df(a: ) — 1 = 0 for every :2: 6 [a, b]. Then, (a) f(z) g 0 for every :5 6 [a, b] (b) f(: r:) 2 0 for every x E [a, b] (c) f(a: ) = 0 for every 1' 6 [a, b] (d) f must take positive and negative values on the interval [a, b] QUESTION 4. Suppose X; and X2 are real-valued random variables with f as tlioir common probability density function. Suppose (11,222) is a sample generated by tlw. «:c-. random variables. The expectation of the number of observations in the sample that fall within a specified interval [a, b] is (a) (If f(z) dz)2 (b) If Mr) dz (c) 2 ff f(= c)'dI (d) f: :1:»f(: r)d:1: QUESTION 5. Suppose X1,. .., X,, are observed completion times of an experiment with values in [0,1]. Each of these random variables is uniformly distributed on [0, 1]. H‘ Y is the maximum observed completion time, then the mean of Y is (3) ln/ (71 + 1)l2 (b) n/2(n + 1) . (C) n/ (n + 1) (d) 2n/ (n + 1) QUESTION 6. Suppose player 1 has five coinsland player 2 has four coins. Both players toss all their coins and observe the number that come up heads. Assuming all the coins are fair, what is the probability that player 1 obtains more heads than player 2'? (a) 1/2 (b) 4/9 (C) 5/9 (d) 4/5 QUESTION 7. Suppose 6’ is a random variable with uniform distribution on the interval [—7r/2,1r/2]. The value of the distribution function of the random variable X = sin 0 at I E [-1, 1] is (a) sin‘l(: r) (b) sin‘l(x) + 1r/2 EEE 2010 B I5 fiskysort
  20. 20. (c) sin‘l(1)/7r + 1/2 (d) sin"1(: c)/1r + 1r/2 d random variable with mean (3 mid QUESTION 8. Let X be a normally distribute variance 0?. Then, the mean of X 2 is (a) O (b) 0 (c) 20 (d) 02 The next two questions are based on the following data. The number of Imves of bread sold by a bakery-in a day is a ran e distribution of X has a probability density function f given by dom variable X. Th ‘ km, ifzE[0,5) f(: c)= k(l0-1), if: cE[5,10) 0, if : c E [10, oo) QUESTION 9. As f is a probability density function, the value of k must be 2’ (‘ll 0 A 2' 5 (b) ——2/25 (c) 1/25 ‘ (d) 2/75 ' A I" X / QUESTION 10. Let A be the event that X _ The probability of A conditionalon B is ff)/ ,5) I / (find) (a) 16/37 /9(5) , (b) 21/37 (C) 25/37 (d) 1 End of Part I. H he examination on the next page. Proceed to Part II of t >2 and let B be the event that X C B]. . EEE 2010 B iskysoft
  21. 21. Part II Instructions. ur out of Questions 11, 12, 13, 14 and 15 in the space following the e provided for this purpose. You may use arked Rough work, to do rough calculations, will not be read or checked. 0 Answer any fo relevant question. No other paper will b the blank pages at the end of this booklet, m drawings, etc. However, your “Rough work” - Each question is worth 20 marks. _______%_____________. __________~ [0,1] be a non-decreasing, right continuous function with QUESTION 11. Let G : $2 _. (: z:) = 1 for every :4: Z 1. Define b : [0,1] —-v ? R by C(25) = O for every :1: < 0 and G _ 0, if c = 0 bl‘) ’ {in£G—‘([c, y), if c e (o,1] bound, of A. ) (Notation: inf A denotes the infimum, or greatest lower Also show that b is nonnegative, (A) Show that G(b(c)) Z c for every c 6 [0,1]. increasing, bounded and left continuous. (B) Show that b is continuous if and only if (C) Show that G is continuous if and only i G is strictly increasing on [0,1]. f b is strictly increasing on [0, 1]. Answer. EEE 2010 B *iskysoi—t
  22. 22. QUESTION 12. Consider an open set C C ER and f : C -9 R. C is said to be a convex set if r, y E C and t 6 (0,1) implies tsc + (1 — t; )y 6 C. f is said to be a convex function if my 6 C and t 6 (0,1) implies f(t: r: + (1 — t)y) 3 tf(: c)+(1—-t)f(y). f is said to locally bounded if, for every :1: E C, there exists r > 0 and M 6 321,, such that | f(y)[ 3 M for every y E B, (a: ). (A) Suppose C is convex and f is convex. Show that f is locally hounded. (Hint: Find r > 0 such that B, ($) C C. Consider y E B, (a: ). Use the convexity off to get an upper bound on f(y) that is independent of y, say M. Noting that there exists 2 E B, .(. 'c) such that : c = y/2 + 2/2, use the convexity of f to get a lower bound on f(y) that is independent. of y. ) (B) By (A), there exists r > 0 and M 6 91+ such that B2,(r) C C and [f(y)] 3' M for every y E Bg, (m). Consider distinct y, z E B, (:r) and set 'u. ' : 2 + (T/ O)(Z ~ y), where 0- = [y ~ 2|. Show that w E B2,(x) and that z is a convex combination (i. e.. weighted average) of y and in. (C) Use the convexity of f to show that nz) ~ f(y) s ‘fl/ («oi — mm 3 $12 — yl (D) lnterchanging the roles of y and 2 in (B)_and (C), we have [f(z) ~ f(y)[ _<_ (2M/ r)[z — y| . Show that f is continuous. Answer. EEE 2010 B
  23. 23. iskysoft QUESTION 13. (Notation: D¢ denotes the derivative of a function ¢>. ) (A) A function f 2 ER“ ——» Ris said to be homogeneous of degree k > 0 if f(t$) = t'°f(a: ) for every :2: 6 ER" and t > 0. Given f : ER" -—> R and k > 0, show that f is homogeneous of degree k if and only if D f (: n). :c = k f (as) for every :1: E 31". (B) Consider a pair of time—dependent variables lc : 31+ —+ R. .. and q : 51+ —+ $24, whose values at time t satisfy the pair of ODES: Dk(t) = f (k(t)) - nk(t) and Dq(t) = q(t)l(n + P) - Df(k(t))l Suppose n > 0, p > 0 and f : R ——v ER is twice continuously differentiable such that f(0) = 0, Df(0) G (n + p, 00) and limk1°oDf(k) = 0. Moreover, suppose Df(lc) > 0 and D2f(lc) < 0 for every k E 32, and there exists k > 0 such that f(k) > nk. Show the existence of k‘ such that D f (k‘) = n + p. Show the existence of k" such that f(k“) = nlc". Show that lc“ > k‘. Using thesefacts, .draw the phase diagram for the above pair of ODES. Analyze the dynamics of (k, q) in this diagram if lc(0) 2 0 and q(0) > 1. Answer. EEE 2010 B
  24. 24. QUESTION 14. Let U be a finite dimensional vector space with inner product (. ,.). Given a vector subspace S of U, let 0(3) = Fl, eg{y G U | (at, y) = 0}. (A) Show that U = S 63 0(5). Let V and W be finite dimensional vector spaces with inner products (. , . )V and (. ,.)W respectively. Let £(V, W) be the space of linear transformations from V to W. Given A E . C(V, W), let 7?. (A) be the range space of A and let . // (A) be the null space of A. If A e z: (v, w), then A’ e c(w, V) is called the adjoint of A if (:5, A*y)V = (A: r,y)W forall: rEVandy€W. Prove the following propositions for A E £(V, W (B) A/ (A) = O(7Z(A")) and 'R. (A) = O(N(A*)). (C) A and A‘ have the same rank. (D) A is injective. if and only if A‘ is surjective. (E) ’R(A’A) = 7Z(A‘) and / /(A‘A) = A/ (A). Answer. —%°iskysori-. ..m . .

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