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# Delhi School of Economics Entrance Exam (2008)

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This is the entrance exam paper for the Delhi School of Economics for the year 2008. 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 (2008)

2. 2. PART I: Onc~mark questions Instructions: 0 First check that this booklet has pages numbered 1 through 24. Also check that the bottom ofevery page is marked AS02. Bring any missing pages to th attention ofthe invigilator. - This part ofthe examination consists of20 multiple-choice questions. Each question is followed by four possible answers, one of which is correct. Indicate the correct answer on the bubble sheet, NOT on this booklet. - Each correct choice will cam you 1 mark. However, you will lose 1/3 mark jb each incorrect choice. If you shade none ofthe bubbles, or more than one huhhlc, you will get 0 for that question. 0 You may begin now. Good luck. ' l/ /lln the context ofideal price index numbers, consider the following statements: ' S1: The index number should be invariant to the choice of base, i. e. P, ,* P5, = 1. S2: An index that satisﬁes the circularity condition Pm “ P5. = P", r? ‘ t, need not satisfy S 1. S3: [fall prices change in the same proportion 1, then the index should equal 1. S4: If we change the units of measurement ofthe prices, but not those of the quantity weights, that should not affect the index. Given these statements, which ofthe following is true? (:1) SI and S2 (b) S2 and S4 (c) Sl, S2, and S3 (d) SI and S3 2. Suppose that the probability that any particle emitted by a radioactive material will penetrate a given shield is .0 1. If ten particles are emitted, what is the probability that exactly one of the particles will penetrate the Shield? 0 . o ’ or 97) 7 (a) (. o1)(o.99)’ ’ ‘I ( U ( Vets) (o. l )(o.99)’ (C) 0.1 (d) l/9 3. Consider a symmetric 90% conﬁdence interval for the population 9;; -._an of a normal distribution with unknown variance, constructed using a random sample of 400 observations. Which of tla9fBllowing‘changesz, ceteris paribus, would shorten the length of the conﬁdence interval b9Qltugr'¢a: t~_ie§; 't_§ii; ')_tlnt’? .£l'«. '1>_-1,, -__ _ (a) The conﬁdence level is chat! ' {yo ' : 43 3 M090 V05) The sample mean is halfits origina valt-ii*f8Ta4‘3=" 339 (c) The sample size is four times itsoriginal value. (d) The sample standard deviation is one third its original value. iaskyson
3. 3. ilnsidcr the lollowing statements‘ I ll‘. - and ll are independent, the)‘ ll ll} and B are mutuallyexclusix lll. lliAand Bare ll-lllCllOlill1CZll1l) e statements are true‘? _: i I and l’ if‘) H and lll. at ill and IV. i. li : :-Jtlc‘ ofthe statements are true. ' Consider a random variable ‘I0. and whose probability distribution l he an appropriate definition lbr go‘), the taigg») : A/ ‘(JD Vy= 1,4, . .. .4oo and g(y)= rb')g(1')= I/55)]: Vy=1,4.. . (C)g()')" 2f(U> Vy= I.4.. .., d)gor>= t/ flﬁl Vy=1.4,. ... '* . consumer has a utility function U(. , y) = lolltm ing statements must be true‘? la) The goods are imperfect substitutes (b) The goods are perfect substitutes lei The goods are perfect complements id) None olithc above‘ 7.Suppose a consumer has a utility function U = subject to his budget constraint and consumes (x", litlloivirig stzitements must be true? tn) price olgootl x is necessarily equal to b (c) price ofgood x is less than or equal to td none olithe above. Suppose zi monopolist sells his product in tw one murket, there is no possibility ol‘it bei the following statements must be true‘? (21) Prices in both markets must be equal w (b) Price must be higher in the market w” / (1.‘) l’ri<; e must be higher in the market with (d) None ofthe above. - lL‘l A and B be any two events. each ot'hieh has X which can take on only nonzerointcger values from “£0 . .illed the / /m/ m/)1"/ I/)' mu. s.s'fu/ Iclimi ol‘ X, gives the . l. l_. . , 20. Now considcrthe rzmdom variable Y = XI. Which ol't ) price olgood x is double the price olgood y ) a positive probability ofoeeurring. niust be n1utua| l_ e. clusie e. they must be independent. independent, they cannot be mutually exclusive. l’. ll‘A and B are mutually e. clusi'e, they cannot be independent. to is symmetric around 0. Suppose the function / /V/ . probability that X = .r. V x ~ It) he following : ilLi probability mass function for Y’? 0 otherwise. (r ‘J’ 'l / V ( ’ V , /l, I . . 400 and go‘) = 0 otherwise. / ~ ' 400 and g(}') = 0 otherwise. / ‘ 400 and gﬁv) = 0 otherwise. l0(x2 + 4xy+-’1y3) + 20. Which one olithe 2 . i( V min {x+y, 2y}. He maximizes his uIilit_* y*) = (3, 3). Which one ofthe price olgood _’ price of good y 0 separate markets. Alter the product is sold in ng resold in the other market. Which one or 1th higher price elasticity ofdemand. lower price elasticity ofdcmand. 7onsider a firm producing a single good with the cost function ts, f I: 0 Cl’) : i I A Ram lt 10+lO. r,i'/ x>0 eshwari Photocopy Service Delhi School of Economics 9310867818. 9871975493 I AF )2 1 1 / , ; ./ >' Y ‘gr ~ —.
4. 4. 'l‘his lirm's sunk cost and fixed cost are respectively (a) l0 and l0 (b) l0 and 0 (c) 0 and 10 a , = K p (d) 5 and t0 f " '7 /7 -5.. r a ’r; 'f; “- ‘ ‘ l0. The elasticity of substitution of the production functionj(x. y) = cx"y" is (a) crab : «r/ »>-:1 7 ' ‘:2’ (b) ab/ c ° (c) a + b Via) 1 V1 1: U '9. ‘< VI : o : r . - l l. In the standard lS-LM framework if you introduce endogenous money money su I de ends ositivel on the nominal rate ofinterest, the corrcs ondin A . PP)’ P P Y 5 . P 8 . curve '7 ~ A / ¢V»c‘4~ Q (a) becomes steeper / (,b')’ becomes ﬂatter (c) becomes horizontal (d) remains unchanged l2. Consider a simple Keynesian model where equilibrium output is determined by aggre demand. investment is autonomous and a constant proportion of the income is saved. I‘ this framework an increase iii the savings propensity has the following effect: (a) it leads to higher level ofoutput in the new equilibrium c(/ U) it leads to lower level of output in the equilibrium (c) the level of output in the new equilibrium remains unchanged (d) the level of output in the new equilibrium may increase or decrease depending on th degree of increase in the savings propensity. avings propensity has the following 13. In the Solow model of growth, an increase in the s impact: (a) it leads to a (b) it leads to a y) the steady state rate of gr (d) the steady state rate of growth may increase increase in the savings propensity. higher steady state rate of growth lower steady state rate of growth owth remains unchanged or decrease depending on the degree of cred interest parity prevails between two countries, (a) nominal interest rates of the two countries must be equal. , (b) expected currency depreciation must equal the interest rate differential. (c) expected currency depreciation must equal the interest rate differential plus the risk premium. (d) interest rate di 14. lfcov fferential must equal the risk premium. l5. Which of the following constitute the “impossible trinity" whigh cannot be simultaneously ensured in an open economy? l. Fixed exchange rate 2. Balance in the balance of payments 3. Free intcmational capital mobility 4. lndependent monetary policy (a) l, 2 and 3 (b) l, 2 and 4 , €c)l,3and4 (d)2,3and4 3 H‘ I , _ ‘ff-' iskvsoft
5. 5. ~'- (Juisitlet the liillmving two ftmctions mapping points on the plane back to points on the plane. ti) / '(. ‘, ,.r2) : (x, l, x_-3 +1) and (ii) g(. ‘[ . .‘2) : (. ‘2,. '| ). Which of the above functions is a linear function? M Both functions are linear. (h) Neitlter function is linear. (c)_/ is a linear function, g is not linear. (ti; ;: in a linear funetion, f is not linear. '7. Consider the equation X)’. +yz + Z" = /( , delined for all positive values of. ' and). and I where k is a given positive constant. The partial derivative 52/ax then equals (a) —(xz"' +y‘ lny)/ (yx"" + 2‘ In 2) (b) —-(yx' '+ 2‘ lnz)/ (xz"' +y‘ lny) (c) —(zy’ ' +. t" lnx)/ (xz"“ +y" lny) ld) —(zy"' +2‘ lnz)/ (.t'z"' +2‘ lnz) l8. The derivative ofthe function defined by f(x) = sin? ‘ .1‘ 4- C082 x is (a) an increasing function of): (b) an oscillating function ofx Rameshwari Photocopy Service tl/5) ‘=1 t~‘0"—*'l11'“ Delhi School of Economics (d) a decreasing function ofx 9310357313, 9871975493 ’9. Consider an I! x n real matrixA with n > 4. lntcrchanging the positions oftwo columns ’(, a) will change the sign ofdet A (b) will not change the sign ofdet A (c) may or may not change the sign ofdet A, depending on the value ofn (d) may or may not change the sign ofdet A, depending on the positions ofthe two columns A - - . . . . Sly Suppose we have a chair with /1 legs and It stands with all l[S legs touching the floor, regardless ofthe floor quality, i. e., evenness, smoothness. etc. Then, 11 is (3)? - (b 3 , (0)4 gfymzgumy —”¢; Cf)7.1/{AI : { r :3 ((1)5 AS02 us
6. 6. - acli correct choice will im-urrucl c/ mice. l f you s hade get 0 for that question. 2!. The frequency 100 households distribution of varia is as follows: X No. ofliouseholds The median monthly family expenditure (:1) Rs 2000 to 3000 (b) Rs 3000 to 4000 (c) Rs 4000 to 5000 (d) Rs 5000 to 6000 To two decimal places, the/ mean and varia (a) 0.95 and 0.50 9 (b) 0.50 and 0.95 / (r/ )”0.95 and 0.55 / ea) 0.95 and 0.45 earn you 2 mar ' none of ti ble X (monthly family expenditure in Rs. ‘000) Fox is in the range: nee ofY (in that order) are 04”, . '1; _ Q. .’ 2:xv=35o, 2x=5o, ZY=60, 27:5 » A 5 6 unit change in Y, and a one Ullll ange in X , _" ; ; )7 - ; o — ZT0 ’ 5"’ / he origin 9% 3 1 I ‘N AS02
7. 7. . In a random sample ot'40O mangoes selected from a large consignment, 30 were found rotten. The null hypothesis is that the proportion Ir of rotten mangoes in the consignment is |0°/ o. It is true that: la)Given HA: it : = 0.1, we can‘t reject the null at the l% level ofsigniticanee, and the probability of error type l ofthis test is 0.005 lb)Given HA: 72' < 0.l . we reject the null at the 5% level ofsignilieanee. and the probability oferror type l ofthis test is 0.05 (C) Given llA: 71' : t 0.1, we reject the null at the 10% level otlsignilicanee. and the probability olierror type I ofthis test is (). l (d) IFHA: 71 > (). l , the power of the associated test is higher than ifHA: /I < 0.1 ‘I In a sample of l000 mangoes, the mean weight ofa mango is 210 g, and the standard deviation 9.5 g. hi‘ another sample of I200 mangoes, the mean is l80 g and the standard deviation l l .5 g. Assume that the respective populations from which these samples are drawn have the same variances. Given the null hypothesis H0: ;1, — , u2 = 0 _ where , u[ and , u2 are the population means, it is true that: /5 ‘ /7’ (a) lfHA: p, ~ , u3 at O , we reject the null hypothesis at the l5"/ o, l3°/ o, 12% and 8% ‘ 4: levels of signiﬁcance. (b) lfHA: /1‘ ~ /12 at 0 , we do not reject the null hypothesis at the l% level of signiﬁcance. (C) The appropriate estimator for testing whether the samples are from essentially the same population is , u, — , u2. (_d) If HA: ,ul — pl ¢ 0 , and we conduct a test at the 7% level olisigniﬁcance, the probability of error type l ofthis test is 0.035. ".7. Data on lndia’s exports oflute and Tea for the years 2000-2003 are as follows: 2000 2001 2002 2003 Jute Q‘. '““‘i‘y (000 0 37‘ 706 724 627 Rameshwari Photocopy Service l; rlCe (RS ‘C0000/l000 Q) 202 320 31 l 302 Deuﬁ schoO1 of Economics ea ' Quantity (‘O00 i)_ 199 179 214 288 9810867818’ 9871975493 Price (Rs ‘Q0000/l000 t) 577 767 884 799 Wliich of the following are true? (a) The chain-base price indices with 2000 as base year are = l46.3, Tm = 152.2. and Pm = 144.9 (b) The chain-base price indices with 2000 as base year are Fm = l48.3, T-’02= 152.2. and E, = 146.9 (c) The chain-base price indices with 2000 as base year are PM = l48.3, [702 = 154.2, and F0, = 144.9 (d) The ehain—base price indices with 2000 as base year are E“ = 148.3, F01 = lS6.2, and P, ,,= 146.9 28. Suppose that 80% of all statisticians are shy, whereas only 15% of all economists are shy. Suppose also that 90% of the people at a large gathering are economists and the other Wit 7 ASWZ
8. 8. l0% are statisticians. lfyou meet a shy person at random at the gathering, what is the probability that the person is a statistician? (a) 3/9 f(s)€) ; 40:; 33> 2 Sin: (5) 0,3 ’ ””__/0? ? w r/ K0 (“~03 _ . LW§v+ 1°‘; - —e= s/ I/ fd) 80/2i) /97) /5,9 /01'-7 /09 E; 9 : . D, 29. Each cell ofthe following table provides the probability ofthejoint occurrence ofth corresponding pair of values ofthe random variables X and Y. Consider thcofollowing statements about X and Y : >‘l. l’r(Y=2)>Pr(X= l) 0" - _ r/ ll. l’r(Y= l|X=2)= Pr(Y= ||X= l) —_‘, ~~q '_; _ “ill. The events X = 3 and Y = 3 are mutually éxclusivc. [V. X and Y are independent. Which ofthe above statements are true? (a) only I and ll. Xb/ ) only ll and ll_l. (c) only Ill and IV. (d) only ll, lll and IV. 30. A survey of asset ownership in poor households in rural UP and Bihar found that 40% of P (R) 1 Ll/ '0 W the households own a radio, 15% own a television and 60% own a bicycle. it also found Ty) = lg/ ' that 5% ofthe households own both a radio and a television, 26% own both a radio and a t’( . - l ( (7,) . — C, /I0 bicycle, 5°/ o own both a television and a bicycle, and l% own all three. lfa randomly ; P . . v) - 5/Mm selected poor householdtn these areas IS found to own exactly one ofthcsc three assets, F (Rn-r. r O/ We what is the probability that it is a bicycle? A e(€N3"2 (a)2()/23 — . §9’.5.‘i¢+J- _ g '5 ll, 3., sﬁo'°‘ ‘ lU‘0 in rcro | °"° “ , W-. ».. ; r (5 / We (b) I7/23 . . - ~_, _w» \$5 23 . ' “ (_ P_<. ){C>7'' JQIS/23 _| p_+ \$2 4 ﬁg . lilo (d) 12/23 In in) ‘W3 3|. Consider an exchange economy with two persons and two goods. Person l's utility function is u. (x. y) = x +y and person 2's utility function is u; (x. y) = e’z"2"2”. Person l's endowment is (1, 0) and person 2's endowment is (0, l). Denote person l's allocation by (x. , y. ) and person 2's allocation by (x2,y2). The set ofefﬁcient allocations ((x. , y. ), (X), y; )) is such that .9) (xz. yz)= (l -xi. 1 -yr) forallxuxz 5 [Or 1] r I (b) (x; ,y2)= (l —x. , l—y, ) for all x. , x; e (0, l) (C) (Xi. ,V: ) = (1/2. 1/2) = (X2. Y2) (d) (. '. ,y. ) = (e‘', e‘) and (x2,y; ) = (1 - e", I — e'’) for all a, b e (-00,0) 32. Consider an exchange economy with persons l and 2 and goods xand y. Person l's utility} function is u. (x, y) = xzyz and person 2's utility function is u; (x, y) = em’. The total endowment ofthe economy is (2, l). The allocation (x. , y. ) = (1, 1) and (x2, y; ) = (l, 0) (a) Pareto inefficient ‘ (b) Pareto eﬁicient or inefficient depending on the endowments of the two persons (c) Pareto efficient or inefficient depending on the state of the world t, (d)_ Paretoxefﬁcient iskysoft
9. 9. t ‘V Consider the situation of the preceding question. lfperson l's endowment is (l, l) and 3's endowment is (1. 0), then the following allocation is a competitive equilibrium: (a) (. '1. _i'. ) = (3/2, 1/2) and (x2, )’2) = (l/2, l/2) ‘ / (l>)(xi. _n)= (l. |)and(xz. .)'2) = (l.0) (C) (M. y. ) = (3. 0) and (X2. , ‘z) = (0. l) . (d) <. x-. . ya : (2. 1) and (-‘2- ya = (0. 0) 1» 54. Consider the situation ofthe preceding question. Which ofthe following is an equilibrium price vector‘? (11) ([)l) P1) 7‘ (1. 0) lb) (P). /72) : (0» 1) U3’) (f)i. P2) : (1) l) (d) none of the above I / {Consider an exchange economy with the same two agents and utility functions as the last thiee questions, but now the endowments ofthe two persons are (0, l) and (2, 0). Which ofthe following pre-trade lump-sum transfers ofwealth will lead to allocations (x. , y, ) T (l, 1) and (x2, yz) = (1, 0) and prices (pl, P2) = (1, l) being a competitive equilibrium’? (a) subsidies of l to both persons (Y. .. ; M: — it (b) taxes of l on both persons . (ﬁ’a subsidy of l to person I and a tax of l on person 2 (d) a subsidy of l to person 2 and a tax of l on person l 36. There are 3 items of choice, x, y z, and Ms. A has 4 possible choice situations: in 3 ofthem. she is asked to choose one or more items from the 3 possible pairs of items {x, y}, {y, z} and {x, 2}. She chooses the items . ', y, and 2 respectively in these 3 situations. ln the 4th situation she must choose one or more items from the set {x, y, 2}; we are not told directly what her choice is. Which ofthe following is correct’? (21) Ms A’s choices violate the weak axiom of revealed preference. lb) Ms A’s choices are consistent with the weak axiom of revealed preference. (c) We can’t say (a) or (b) because we don‘t know her choice from the set {. t‘. y. 2}. (d) We can't say (a) or (b), even though we can deduce her choice front the set {. '. J’. 2}- Questions 37-39 use the following information: A chemicalfaclory produces (I (. ‘llL‘IIlfL'£Il K ‘and an cf/ Iucnl If which i! dump. ) in :1 river. /1 down. r!re(iniﬁ. rlicrjv proiliictnsﬁxli F and in’ uu. '! .s‘ arc uffeclcd h y the level of cfﬂuenl in the river. The lwuﬁrnis are competitive amlfurc unit prices PK = 10 and l’, .- = 20 for the chemical and rhcﬁsh respectively. The cosi_fu))cIimi of the chemical factory is C(K, E) = K 2 [(5 — E)2 + l]. The coslfzmction of the ﬁsliery is E(1«“,5) = F2153 - 37. lfthe chemical factory chooses levels of chemical and efﬂuent to maximize proﬁts, and the fishery chooses the level of fish to maximize proﬁts, the chosen levels ofthe chemical, effluent and fish (i. e. K, E,F) are respectively (a)5,0, 5/8 7,, . ,0,» g (b)3,0,2/5 I, V . _ , , ; ;a»\$,5,2/5 , ’ P t . « ’ " d 5,5,5/3 ' . I , . ( ) pf . . 6, , /, A502 5] 9
10. 10. 38‘ Suppose the socially optimal levels ofthe chemical, effluent and ﬁsh, denoted by K, E, F respectively, are the levels that maximize the joint proﬁt of an integrated ﬁrm consisting ofthe chemical factory and the fishery. Then I-.7 is equal to (3) 5/<2 / (K2 — F’) l ‘ (b) sf! /(1?’+F1) 3 (c)5F1/(1?’—F3) I was/ ?-’/ (1?’+rT1) 39. Suppose a government knows the firms‘ production functions and t 3; can costlessly monitor the amount ofeffluent released by the chemical factory. Ifit sets a ‘ tax I per unit ofefﬂuent produced by the chemical factory, the level ofthis tax that will I result in the socially optimal efﬂuent level .7: is equal to . ; (a)z=1o/ ?r72/(/ ?2+F’) ‘ (b): =1o1?’F’/ (/? ’—rT') ' ‘N , mi=1oI? *F3/(/ ?’+rT’) (d) r =5I? F’ / (E1 +F’) heir output prices, and 40‘ Consider a firm with one output and two possible choices ofcapital stock, say I and 2. ; The associated cost functions are C(x, l) = 2 + 2x and C(x. 2) I capital stock, the firm's cost function is 4 + x. Before choosing its 1 _ 2+2x, ifxe[0,2] v, __:7A, /_‘+. (_ ‘I V/ @)C*(x)—{4+x. Ifx>2 ~l (b)C"(x)=2+2.x ' ‘. 3‘ (c) C“(x) = 4 + x _ 4+x, i/xe[O,2] (d) C*(x)—{2+2x, if. :>2 = AL‘: (K’_)I_a, where L is the total employment of labour; V ‘A K7 is the total capital stock (which is fixed in the short run); parameters of the system. aggregate price level and the money wage rate respectively, Assume economy maximize proﬁt in a perfectly competitive set up. and/ (>0 and 0<a<l are Let P and W denote the that the producers in the
11. 11. 1 I i n it” ‘r 944." t’i. iu)’ '1? i" I ' l to) [fl = i Kijl agﬂjlﬂl , .. ia(t<)"" / ’ i_<l, iNui1col‘tlie abmc ll there is it niic—slii>t ll1Cl'CélSL‘ in the stock oi‘ capital (l: )_ the tlClll2|ll(l for l. '=i7’llll' schedule. as derived above. /(a) shilits up ; lb) shifts down '(_C) does not shift Rameshwari Photocopy Service Delhi School of Economics 9810867818, 9871975493 (ii) the information available so far is not adequate to answer this question Vzippose the above economy is characterized by a single hoiiseho/ d which takes the aggregate "rice level and the money wage rate as given and decides on its consumption ants’ siipply by nia.1'i/ nizing its utility subject /0 its budget constraint. The household has milownicnt o units 0 labour time, 0 which it su )lies L5 units to the market : Pl viwzey wage rate W, and enjoys the rest as leisure. Its utility depends on its consumption and ; ‘eisiii'e in the following way. " U = (C)/ Y +([j—Ll-)3,‘ O < /3 < l. The only . Y()1t. "i"t’ of 1l1L'()m(. ’ of the household is the wage income iiviszinzption goods at the price I’. and it_s/ )ends its entire wage earning in L. ‘ -1.‘. The corresponding supply of labour schedule as a function of the real wage rate lS bx": vQ. ''<Vi1 / [' /3 1: C‘/ €—, ' I {—”-/ —j]—/3 , _, . _ ﬂap _A ri’ Jan/ . — J; ’ ll’ 1,5 I ~ / ' iiei ) Z L‘= — ‘b’ . 5,, 1+ L/ _Jl~/3 l’ (d) None ofthe above J-l. lfthere is an exogenous increase in the total endowment of labour time (Z), the " V i l“ . . , i._ of labour schedule, as derived above, ASS ll
12. 12. (21) shifts up jb’) shiﬁs down i (C) does not shift W 1+J§ "2 (b)——= ————— P 2 (c) %= l2—2«/ if/2 (d) %= l2+2~/ El/2 supply curve (outpu (a) upward sloping (b) downward sloping (c) vertical (d) horizontal the general pricre level is exogenous‘/ y ﬁxed at F . '/ ‘/at is given by: Y= C+I+G, income mand (I) is a function of t tal tax revenue in the economy = T.Y,0<‘r<l; ana'thegovern I ' autonomous such that G = 5. where the consutnp/ /, (Y—T) such ll/ Us he real rate of interw 3 (7') is a ﬁmction of the investment de d>0.' to aggregate real income such that T (r) such that I = I —— a’. r, where supply of money (A0 is exo balances (L) is (I f I. =a. Y—b. i interest and the expected rate Assume that the economy sta rt. ‘ lmth the gootls I71aI‘l-’(’I and the money market clear. 47. ln ceteris paribus, a unit increase in govemment expenditure has the following impact on the equilibrium income level: I (a) it increases by ———~————-—- units b[1— c(l — 1)]+ ad AS02 l’).
13. 13. l (b) it decreases by —? :?—: — units b[1— c(l — r)]+ acl . b . / (2') it increases by —-—~——— units b[1— c(l — r)]+ ad b ————-~— units 1 ' d b (()ll ecreases y b[l_c(1_r)]+ ad unit increase in the expected rate of inﬂation has the follmving um value ofthe real interest rate: [1 — c(l -. r)] (a) it increases by —~——————— units bil —- c(l - r)]+ ad a lll. In cetcris / )at'il)u. s‘, a impact on the equilibri units (b'd“ ' . 'b ————~ )tt ecreases y a[1_C(1_T)]+ad (c) it increases b I —b[1————2—_ c(1— T“ b[l — c(l - r)]+ ad . _ b[1— c(l — r)] }5L) it decreases by ———: ———b[1 _ C(1_ T)]+ ad the general price level has the following impact on units units 49. In ceteris paribus, a unit increase in the equilibrium value of the real interest rate: [17 / (Fﬂ units (a) it increases by bil — c(l — r)]+ ad Rameshwari ph t " —‘ S O °°°Py (sci. '. . MI I’ Ch ‘ ‘ -: [ ( units 98108678°1°8’°9f8ECOl1OlltlCS - 71975493 b ' d - . b ()it ecreases y b[1_C(1_T)]+ad [l ~c(l _ ail/ Vi/ (i'>)1] J/ (e) it increases by b[1_ c(1_ T)]+ ad units (d) it decreases by ~¢~ﬁ——— units 50. Suppose we now allow the gencr characteristics of the macro-economy, demanded as a function of the general price level) is (a) upward sloping Lcbr) downward sloping (c) vertical _ (d) horizontal Fix an m x n matrix A and an mtvector b. A condition that ensures the existence of a 51. ' rsolution (an n—vector x) of the equation A): = b is (€) det A i 0 (b) m = n ‘ (c) the columns of A are linearly independent ‘ i , (d) the rows of/1 are linearly independent AS02 al price level to be ﬂexible. Given the almx: the corresponding aggregate demand curve (outpu;
14. 14. 52. The rcal—valued functionj(x) = x‘ is / (’a) strictly convex . (b) strictly concave I, n 3 T 7 9 , (c) neither strictly concave nor strictly convex / ‘ L / ’ I I ‘ / / n] (d) convex but not strictly convex ~ 0 _, 1/ / Q U I 53. Consider the equation Ax = 0 where A is an n x n matrix such that a, , ,1‘ 0 for every ie {l, ... .,r1} and a. , = 0 whenever i >j. This equation has (a) n distinct solutions (b) an n e l dimensional vector space of solutions . (/ (E) exactly one solution (d) an :1 dimensional vector space of solutions. /, 4/ 9, / Q. 54. There is a pile of l7 matcltsticks on a table. Players 1 and 2 take turns in removing mzitchsticks from the pile, starting with Player l. On each turn. a player has to remove a number of sticks that equals the square ofa positive integer, such that the number of inatchsticks that remain on the table equals some non~negative integer. The player who cannot do so, when it is his/ her tum, loses. Which of the following statements is true? (ra’) ll' Player 2 plays appropriately, he can win regardless ofhow 1 actually plays. l (b) lfPlaycr l plays appropriately, she can win regardless of how 2 actually plays. ,. / (c) Both players have a chance to win, ifthey play correctly. "y’, ,}, ,,y, a (d) The outcome of the game cannot be predicted on the basis of the data given. f , (/ //-M. / _ . _ ' S. lfA IS a set of real numbers, let g, be the function such that g,4(x) = I if x e A , and g, ,(x) 71: = 0 if x as A. With this notation, consider the infinite sequence of functionsﬁ, where I‘ f, .(x) = n g[0_, ,,, )(x) (i. e., n multiplied by gm, “ (x) ) for all real numbers x and for each n: = 1,2, 3 . ... Then (a) For every x, the‘ sequence of numbers (f, , (x)): , has a limit in the space of real numbers. (b) lim, __m f" (x) does not exist, for any x. _ (c) When lim, ,_m f, ,(x) exists, the actual limit depends on the x in question. (d) limmm f" (x) exists for all but a ﬁnite set of real numbers x. / S . Continuing with the sequence of functions above, we consider the sequence of real numbers (lfn): =, (that is, the sequence oftheir integrals). This sequence ofintegrals is (a) an increasing sequence (b) a decreasing sequence (c) a constant sequence 4 (d) an oscillating sequence ’ s 57. For each positive integer n = 1,2, 3, . .., let S, , be the set of points lying on the curve y = (1/x"), for all positive real numbers x. Then the intersection of these sets over all a, (that is, ﬁlls, ) is (a) a set with infinitely many points /66) a set having a single point (c) a set having exactly 2 points _(d) a set having more than 2, but a finite number of points iskysoft
15. 15. >78. Which ofthe following two numbers is larger: 8” or 71””? cm (b) 7r" (C) they are equal (d) it depends on the value olic Railing: :0 Delhi Schoo. c. ._, v-in-V-'»~’ _ __ , , sin (I / x), if . x' > 0 9810867818, 9871975493 :9 lhe function f(x)= f 0 a, 1 x: (a) is continuous or discontinuous at x : 0, depending on the value (I ) (b is continuous at x -= 0 it) is tliseontinuous at . t' 7- 0 (d) is upper hemicontinuous at x = 0 60 Suppose v, . V3 and v; are three vectors in 3-dimensional space, and Then the vectors v, + vg, I’; ‘t v3, and r. '-‘l' (a) are linearly independent ~ (b) may be linearly dependent J! ) are linearly dependent (d) are linearly ilepeinlciil ‘ are linearl_v independent, except when one ofthe vectors is zero. The remaining pages of th is booklet are for your rough work, which will not be checked. After you ﬁnish, hand in this booklet along witlz your bubble sheet to the in vig ilators before you leave the examination hall. AS02
16. 16. Delhi School of Economics Department of Economics Entrance Examination for M. A. Economics Option B . Iune 28, 2008 Time 3 hours! Maximum marks ‘1U0 General instructions. Please read the following instructions Carefully. 0 Check that your examination has pages 1 to 5 and you have been given :1 blank Answer booklet. Do not start writing until instructed to do so by the i11-'igilato1'. - Fill in your Name and Roll Number o11 the small slip attached to the Aiiswer booklet Do not write this information anywhere else in this booklet. 0 When you finish, hand i11 this Examination along with the Answer booklet to the iiivigilator. 0 Do not disturb your neighbors at any time. Anyone engaging i11 illegal exam- ination practices will be immediately evicted and that person‘s cmididaturo will be cancelled. o The e. 'an1ination has two sections. Follow the instructions given at the beginning ul‘e;1(: l1 section. Do not write below this line. This space is for oflicial use only Fictitious Roll Number Section/ Question Marks I II/11 - phogocopy Service , Rameshwari . 11/ 12 Deihl School of Economics H /13 9810867818. 9871975493 11/14 11/15 Total EEE 2008 B 1 Fl?
17. 17. Section I Instructions. Answer Questions 1 to 10. Each question is followed by {our asserts‘, (a) to (d), one of which is correct. Indicate the correct choice by circling 5’ Examination. Each correct choice will earn you 2 marks. However, you will lose 2/3 : :i. ;: for each incorrect. choice. Please ensure that you hand in this Examination nlm with the Answer booklet. Question 1. There is a pile of 17 matchsticks on a table. Players 1 and 2 take "v ‘ removing matchsticks from the pile, starting with Player 1. On cacli turn, a playr~r‘ to remove an mnnber of sticks that equals the square of a positive integer, such it number of matchsticks that remain on the table equals some non-negative integi» player who cannot do so, when it is his turn, loses. (a) lfPlz1yer 2 plays appropriately, he can win regardless of how 1 actually pl, -2-, ~ (b) If Player 1 plays appropriately, he can win regardless of how 2 actually pi iv (C) Both players have a chance to win, if they play correctly. (d) The outcome of the game cannot be predicterl on the basis of the data giv w Question 2. Given .4 C ER, let IA be the lunction deﬁned on .3}? by: 1,4(: :) : 3 ii and l_4(: r) = 0 if 17 Q’ /1. Consider the sequence of functions (f, ,) for positive iiitegrcw where f, ,(I) = nl[0_, /,, ](z) for I E 9?. (a) For every 2:, the sequence of numbers (f, ,(ar)) has a limit in FR. (1)) For every llHl, ,_. .°o f, ,(: L') does not exist. (C) (d) If lim, ,_. o° f, .(: c) exists, then the limit depends on an. lim, ,_. ,x, f, .(: :) exists for all but a ﬁnite set of I 6 FR. Question 3. Tlleije, argftwﬁgaidentical boxes, each with two drawers. Box A C(')Ill. l.1ll: gold co'ir'1‘_in'_ea. chfdi‘awer. .ﬁoﬁn Jcontains a silver coin in each drawer. Box C (‘. u‘lat: ~1.. ‘!. - a gold coin Siié di'i1a3v'€aE'Ihn<\$g§ silver coin in another drawer. A box is chosen, a dmu i opened and a gold coin is found. What is the probability that the chosen box is C‘? (8) 2/3 (b) 1/3 (C) 1/? ((l) 3/-1 Question 4. A random variable has outcomes Success and Failure with probabilities I; ,. and 1/4 respectively. A gambler observes the sequence of outcomes of this variable l»l. l.l EEE 2008 B
18. 18. receives a prize of '2” if n is the first time that Success occurs. What is the expected value of the ganibler‘s prize? (E1) 1 (L) (2 ((1) 3 (<1) ‘,1 I Question 5. A number of Inatliematieians in the middle of the 20th (: cntur_' contributed to a series of books published in the name of a fictitious mathematician called Bourbaki. Suppose a sociological critic of science asserts “Every book by Bourbaki contains a chapter such that the validity of every theorem in that chapter depends on the readers‘ gender. “ If every assertion by this critic is false, which of the following assertions must be true? (a) There exists a book by Bourbaki containing a chapter such that the validity of every theorem in that chapter is independent of the reader‘s gender. (b) Every chapter in every book by Bourbaki contains a theorem whose validity is independent of the reader’s gender. (c) There exists a book by Bourbaki such that every chapter in it contains a theorem whose validity is indepenrlcnt of the rea(ler‘s gentler. ((1) Every book by Bourbalii contains a cliapter such that the validit_' ol all the l. ll('. ()~ reins in it is independent of the reader’s gentler. Question 6. A family has 3 children. Suppose the probability that a child will be a girl is l/ '2 and that all births are iiitlopunrlunt. ll‘ the family has at least one girl, what is the probability that the family has at least one boy? (a) 5/7 Service - it \$331; R, y“*: :‘: “s': ::°: i:: °.: :§nomacs c 9 ((1) 4/6 9810867B18,98‘I1975493 Question 7. Suppose f : ER? —~ 9? is continuously differentiable and f(a, b) = 0. ‘A condition that ensures the existence of a unique continuously differentiable function d> such that b = ¢(a) and f(: c,d>(: c)) = 0 for all z sufficiently close to a. is (a) D1/'(a, b) > 1 (b) D1f(a, b) <1 (C) D2f(a. ,b) >1 (d) D-zf(a, b) < 1 EEE 2008 B 3 iskysoft
19. 19. Question 8. Suppose X and Y are independent ranrloin variables with stanrlai-<l . *«, :zv ‘ distributimis. The probaliility of X > 1 q E (0,1). What is the probability of the event: X3 > 2 and Y2 > 1? (a) M (1)) p21] (6) P02 (d) 2120 Question 9. Suppose each of ll players tosses a coin that has probability 1/2 :2; L}. ijIi: ; Heads and probability 1/2 of falling Tails. A player wins if her coin toss is Heads and the other players‘ coin tosses are Tails. This game is played until someone wins Tlu: probability of finding a winner in the / c-th iteration of the game is 2l—n(1 _ 2l—n)k—l (b) n2“"(1 — n2“")k" (C) n2‘"(1 — n2"‘)"“ ((1) 2—n(l _ 2—n)k—l a winner in up to two iterations is ) 7 16 . ‘ - (a / Rameshwari PhotocoPY 5°_”"°° (bl 15/16 Delhi School of Economics (c) 39/64 9810867818. 9871975493 (d) I5/64 Section II Instructions Answer an; -;I"‘1‘"our 0} the §oIlovlIirz': grﬂV§(quest1ons in the Answer booklet ‘ ‘ : giapor Question 11. (A) Prove that d: 82" X 9%" —-9 3R, deﬁned by d(: r:, y) = ( a distance function, where (. , denotes an inner product on ER". 1 — y, .r — 3/)1/2, (B) Prove that, if (X, d) is a compact metric space and f : X —» X is such thal d(: c,y) = d(f(1:), f(y)) for all 3:, y E X, then f is surjective. I (8,112) Question 12. (A) Prove that if (X, :1) is a metric space, then the distance function (1 is continuous. EEE 2008 B -I is 1) 6 ((1,1) and the probability of Y ‘J »
20. 20. ric space and I : X ~+ X is ,6d(I. y) for (’. '(‘l_' (Hint: Start with (B) Suppose (X, d) is a nonempty and complete met in. there exists 5 6 (0,1) such that d(f(1:), f(y)) § one I E X such that f(r) = :r. X. Now use the c a contraction, 1'. y E X. Show that ther X and use | iI'(I| l(‘. l'i. l(‘S. ) (3, W. e exists exactly f to construct a Cauchy sequence in ompleteness some point in zilirl (-mitl‘: i<: tinIi lT,1‘2l, let A(t) be an n x It and for every t E d only one 3?", there exists one an = A(t)d>(t) for every t 6 l7‘1,7’-_. i) 3. Let I‘; < 1'2, 1' E l1‘1,T‘2l, Suppose, for every 111 6 l - ER” such that Dq5(t) Question 1 x and let b(t) E R". real matri lTi . T2 differentiable function (15 : and 0(7) = L’). (A) Show that the spa diﬁerential equation Dx(t) A(t): c(t) is an n-dimensional vector space. (B) A collection {q5‘ , . . . , d>"‘) of solutions 0 if and only if (d)‘(t), . . . , d)'"(t)) C 9?" is linearly in ce of solutions of the ordinary l DJ: (t) '—= -/l(C).1lt(l. ) is linearly independent dependent [or every t E ll'[,1"; l. y E X and I. (, ((), l) imply zi (‘. ()l'(‘. X set ll I, ll? is said to be c. nnve. ’ Questimi 14. X C lll" is S‘«li(l to lll‘. ta‘ + (l l. )y (—_ X. (liven zi (‘. OI'(‘. X set X C lll", a function f : X - ‘S tf(I) + (1 ~ t)f(U)- C: X and t. E (0,1) imply f(tJ: + (L — t)y, is convex and f : X ~’ 3? is continuous, where TR" Show that, if [(1/‘2+y/2) 3 f(a: )/‘2+ f(y)/ '2 for a and ? 'R are given if 3:, 1; ll ; :.y E X, then Suppose X C lll" the Euclidean metrics. f is a convex function. OC « : ‘ rmt = }: ,___, or’). (Hint: Use the fact t is 0 or I. ) hat every t E (0. 1) can he represented in the lo wliere each 01,. o ~‘~ (. ~.lJ, ' an unknown distribution P Question 15. (A) Suppose an observation X is drawn from and that the following simple hypotheses are to be tested: II" 2 I’ is :1 uniform ilistrilintion on the interval ((3, l) H1 : P is a standard normal distribution [)r. :tcrniin<: the niost powerful test of size 0.01 and calculate the power of the test when 1;’, is true. (B) Suppose X1, . . the parameter 0 is unknown. n sample from the uniform di od estimator of 9. (l0.i‘ . , X" are a randox Derive the maximum likeliho where stribution on [0, /7).