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# Ma economics entrance dse 2013

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### Ma economics entrance dse 2013

2. 2. PART I: One-mark questions Instructions: ~ First check that this booklet has pages numbered 1 through 24. Also check that the bottom of every page is marked AS02. Bring any missing pages to th attention ofthe invigilator. - This part ofthe examination consists of 20 rnultiple-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/} rnar/ <_/ E; each incorrect choice. lf you shade none ofthe bubbles, or more than one huhhlc, you will get 0 for that question. 0 You may begin now. Good luck. I. In 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,5“‘ P5, = l. S2: An index that satisﬁes the circularity condition P” * P5. = P, ,, r73 t, need not satisfy S l. S3: lfall prices change in the same proportion it, then the index should equal 1. S4: If we change the units of measurement ofthe prices, but not those ofthe 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) S1 and S3 2. Suppose that the probability that any particle emitted by a radioactive material will penetrate a given shield is .0l. If ten particles are emitted, what is the probability that exactly one of the particles will penetrate the shield? . l 7 V 7,‘ 9 (a) (. o1)(o.99)° /0" / O ” ’ Vets) (0. i)(o.99)" (c) 0.1 (d) 1/9 3. Consider a symmetric 90% conﬁdence interval for the population mean of a normal distribution with unknown variance, constructed using a random_sample of 400 observations. Which of tl? e"fUllowing_changcsE, ceteris paribus, would shorten the length of the conﬁdence interval b95!lteigr¢a: t__es£‘; 't_in"p_tln't’. >£l'. /.'t ; - «_, . (a) The conﬁdence level is cha'}tg‘e\$lgpf§€’{l>. R" : 45 3 "H90 . ,/(8) The sample mean is half its origina valuel8735"" :89 (c) The sample size is four times its original value. (d) The sample standard deviation is one third its original value. AS02
3. 3. ‘ Suppose a consumer has a utility function U = min. { - l, L‘l A and B be any tuo events. each ol‘hich has a positive probability of occurring ihiisidei‘ the lollm'ing statements‘ l ll‘. - and ll are independent. Ihe_ must he mutuztll} eclusie ll ll . - and B are mutually e. '<. ‘lusie, they must be independent. lll. ll} and B are independent, thc_' cannot be mutually e. :clusi e. l’ ll‘/ and l3 are mutually exclusive. they cannot be independent. l‘-l-lllCll0l. ll’lC(ll1IJC statements are true’? . .;i I Lind l' 4'1 ll and lll. '. ‘llll:1l1t. l l". all . one ofthc statements are true. Tansider a random variable X which can take on only nonzero iﬁger values from -—; (”J to *1’. and whose probability distribution is symmetric around 0. Suppose the function ’i’'/ . . .alled the / rm/ m/21'/ My niu. i.s'_/ unclimi of X, gives the probability that X = .r. V . ' —— ' *1 xi. 1.. ,20. Non considcrthe random variable Y = XI. Which ofthc l‘Oll()'ltlgi-'. ‘-li . :l. i he an appropriate deﬁnition lbrg('), the probability mass function for Y’? (a'lg(}') 1 A/ ([)“') V)/ '= 1,4, . . . , 4100 and g()') = 0 otherwise. lb" / I lhl;3t_') = [/ ‘(U)]3 Vy= 1,41. . . . .400 and go‘) = 0 otherwise. / ». / i; l5’l)') = 2f(/ T) V)‘: l, 4, . . . , 400 and g(}') = 0 otherwise. /i / A (d/ I_L'(1') = / _/‘l/ F—l V_i'= l, 4, . . . , 400 and gﬁv) = 0 otherwise. r W. consumer has a utility function U(. , 5‘) = l0(x2 + 4.'_ +4y3) 4 20. Which one olithe lnlhmiiig statements must be true’? « « (in The goods are imperfect substitutes / /. « lb) The goods are perfect substitutes net The goods are perfect complements (V I '/ id) None ofthc above. ,, ’ ( x+y. 2y}. llc ma. 'imizes his utility subject to his budget constraint and consumes (x", y‘) = (3, 3). Which one ofthe lllllt) ing statements must be true? (in price ofgootl . is necessarily equal to price of good _' (h) price of good x is double the price of good y (c) price ofgood X is less than or equal to price of good y Ltd) none olithe above. Suppose it monopolist sells his product in two separate markets. After the product is sold in one inurkct, there is no possibility ofit being resold in the other market. the lellou ing statements must be true‘? (a) Prices in both markets must be equal when the marginal cost ofoutput is constant. (b) Price must be liigher in the market with higher price elasticity of demand. / (L) l’rir: e must be higher in the market with lower price elasticity ofdcmand. (d) None ofthc above. Which one of Tonsider a firm producing a single good with the cost function [5. tfx = 0 C. _ 3 W l_1o+iox, in > 0 Rameshwari Photocopy Service Delhi School of Economics 9310867818, 9871975493 {V Z / AS02
4. 4. This lirm's sunk cost and lixed cost are respectively (a) l0 and I0 (b) l0 and 0 (c) 0 and l0 (d) 5 and l0 l0. The elasticity of substitu (a) C/ ab (b) ab/ c (c) a + b / (3) 1 tion ofthe production functionj(x. y) = c. "’y" is l l. In the standard [S-LM framework if you introduce endogenous money supply such that money supply depends positively on the nominal rate of interest, the corresponding LM < ' - - , / curve (a) becomes steeper . / (,b')’ becomes ﬂatter (c) becomes horizontal (d) remains unchanged r l'l. Consider a simple Keynesian model where equilibrium output is determined by aggregate demand. investment is autonomous and a constant proportion of the income is saved. In this framework an increase in the savings propensity has the following effect: (a) it leads to higher level of output in the new equilibrium . (.l5) it leads to lower level ofoutput 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 the degree of increase in the savings propensity. 13. In the Solow model of growth, an increase in the s impact: (a) it leads to a higher steady state rate of growth (b) it leads to a lower steady state rate of growth the steady state rate of growth remains unchanged ' (d) the steady state rate of growth may increase increase in the savings propensity. avings propensity has the following or decrease depending on the degree of 14. if covered 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 ex ected currency de reciation must e ual the interest rate differential lus the risk 9 _ P at P premium. (d) interest rate differential must equal the risk premium. 15. Which of the following constitute the “impossibl simultaneously ensured in an open economy? 1. Fixed exchange rate 2. Balance in the balance of payments 3. Free intemational capital mobility 4. Independent monetary policy (a) l, 2 and 3 (b) 1, 2 and 4 Xe) 1, 3 and 4 (d) 2, 3 and 4 e trinity" whiqh cannot be AS02
5. 5. 6. Consider the l0llHVlllg two functions mapping plane. (i) f(x, ,. ’2 ) = (xl -t- l, .‘2 + l) and (ii) g(x, ,x2) = (x2,x, ). Which ofthe above functions is a linear function? m Both functions are linear. -(b) Neither function is linear. (t: )fis a linear function. g is not linear. (d)g is a linear liinction, f is not linear. points on the plane back to points on the iConsider the equation xy + yz + 2"‘ = /r , delined for all positive values of. ' and); and I Where It is a given positive constant. The partial derivative 52/ax then equals (3) —(xz‘"’ +y" lny)/ (yx"' + 2' ln z) , S(b) —(yx’"' +2‘ lnz)/ (xz"' +y‘ lny) '. (c) - (zy‘"‘ +. '-’ ln x)/ (xz"" + y‘ In y) : (d) —(zy"' + 2‘ ln z)/ (.t'z"' + z" lnz) w he derivative ofthe function deﬁned by f(X) = sin2 .1‘ + COS2 x is (a) an increasing function ofx ~(b) an oscillating function ofx ' ) a constant ‘(d) a decreasing function ofx Rameshwari Photocopy Service Delhi School of Economics 9810867818, 9871975493 7 Consider an n x :1 real matrix A with n > ' . will change the sign ofdet/1 '(b) will not change the sign ofdet/1 . (c) may or may not change the sign of det A, depending on the value ofn (d) may or may not change the sign ofdet A, depending on the positions ofthc two columns 4. lnterchanging the positions oftwo columns Suppose we have a chair with n legs and it stands with all its legs touching the r, regardless ofthe floor quality, i. e., evenness, smoothness. etc. Then, :1 is (8) 2 tb)3 (6)4 QM; /as; -jLa47,[: : 7'! * 0 . (d)5 AS02
6. 6. - Each correct choice will iH('()I‘I'cL‘I c/ mice. l l‘ you 5 get 0 for that question. earn you 2 marks. However, you will lose 2/3 mark for cat '1 hade none ofthc bubbles. or more than one bubble, you wl The median monthly family expenditure is in the range: (rt) Rs 2000 to 3000 (b) Rs 3000 to 4000 (c) Rs 4000 to 5000 (d) Rs 5000 to 6000 (2 places, the mean (a) 0.95 and 0.50 9 l (b)().50 and 0.95 ' A tun’/ V I"-45 — 0 °°”' ’(,2ro.95 and 0.55 ) 0.95 and 0.45 ndYwehavethedata zxY=3so, 2x=5o, >: v=5o, X=5, of, i I A} 7 4, and 0', — 9 Which ofthc following holds” ’ , I ~ (a)A one unit chan ' Pw’ - 7' l l - n 6 unit change in Y and a one unit change in Y is associated with a L25 unit change in X , 9 g _ .7 Al , _ ; (c7 The covariance between X and Y exceeds l7 359.: 2?: él . ’ (d) The regression ofY on X passes through the origin -9 >1 3 ' 5; 9 24. The life 0 a cycle tyre is normally distributed with mean 350 days and variance 64. ll is ‘ true that: y 7 ,3, : ’ (Z1) lhe probability that the life ofthc tyre will be less than 336 84 clays is greater than 5% / (b) The probability that the life ofthc tyre will be greater than 363 I6 days is greater than . . " ” '% ~ * ’ ) K‘, ) The probability that the life ofthe tyre will be between 336 84 and 363.16 days is 1 1‘, ‘ )"L’dyp 90% ":7 ‘ l , ,)’V (d) The probability that the life ofthc tyre will be less than 334 32 days is greater than / / , ‘ “ ‘AL ‘l AS02
7. 7. In 11 random sample of400 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%. It is true that: ta)Given HA: /r : ‘— 0.1, we can‘t reject the null at the We level ofsigniticancc, and the probability of error type I ofthis test is 0.005 lb)Given HA: 7r < O. l . we reject the null at the 5% level ofsigniﬁcartee. and the probability oferrer type I ofthis test is 0.05 IC) Given HA. 7%’ : t 0.1, we reject the null at the 10% level ofsignilicance. and the probability olierror type I ol‘ this test is (). l Id) IFHA: II > (H , the power of the associated test is higher than il‘l—l, t: / I < 0.1 I In a sample of I000 mangoes, the mean weight ofa mango is 2l0 g, and the standard deviation 9.5 g. Iii‘ another sample of I200 mangoes, the mean is I80 g and the standard deviation l 1.5 g. Assume that the respective populations from which these samples are drawn have the same variances. Given the null hypothesis H. ,: p, — p2 = 0 , where pl 6. and [~12 are the population means, it is true that: (at) lfH, ,: pl ~ p: at 0 , we reject the null hypothesis at the 15%, I3%, 12% and 8% V ~72 ‘ levels of signiﬁcance. ‘ (b) If HA: p, ~ / .12 at 0 , we do not reject the null hypothesis at the I% level of signiﬁcance. (c) The appropriate estimator for testing whether the samples are from essentially the same population is pl ~ pl. I (d) lfl‘l, .: pl — pi i 0 , and we conduct a test at the 7% level ofsigniﬁcance, the probability oferror type I of this test is 0.035. 27. Data on lndia’s exports oflute and Tea for the years 2000-2003 are as follows: 2000 2001 2002 2003 lute Q‘la“‘“y (000 0 871 706 724 627 Rameshwari Photocopy Service ljnce (RS ‘C0000/i000 I) 202 320 311 302 schooj of Economics ea Quantity (‘000 t) 199 179 214 288 9310867818’ 9871975493 Price (RS ‘Q0000/i000 K) 577 767 834 799 Wliich of the following are true? (a) The chain—base price indices with 2000 as base year are P0, = l46.3, ‘Tm = 152.2, and Pm= 144.9 (b) The chain-base price indices with 2000 as base year are E“ = I483, 702 = 152.2. and 170, = 146.9 (c) The chain—base price indices with 2000 as base year are F0, = I483, E2 = 154.2, and F0, = 144.9 (d) The chain—base price indices with 2000 as base year are E“ = l48.3, F02 = l56.2, and P, ,,= 146.9 2't‘. 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 AS02 i 7
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) s/9 / (9)6) ; 40:: go 2 Erv (b) 0.8 _’; '”? /F7? cw r/ ma (C)-03 _/ jx’5': >.; 2_oA’; ;_' - _§. ,'} 1‘ l/ (d)80/215 my 7.9 / av / av , ,,, -~—__, ,1 ; D, 3 29. Each cell of the following table provides the probability ofthejoint occurrence ofthc corresponding pair of values ofthc random variables X and Y. Consider tltecfolloxving statements about X and Y : ‘*‘l. l‘r(Y=2)>Pr(. X=l) 0'’ - F/ ll. l’r(Y= l|X=2)= Pr(Y= nx=1) ' V. lll. The events X = 3 and Y = 3 are mutually éxclusiv W. X and Y are independent. Which of the above statements are true? (a) only I and II. (X/ la’) only it and Ill. (c) only Ill and IV. (d) only ll, lll and IV. 0 ; _,l. _ 0 (“l V 30. A survey ofasset ownership in poor households in rural UP and Bihar found that 40% of P (Kl 1 “/7 W the households own a radio, l5% own a television and 60% own a bicycle. lt also found ((7.-J7 = l; ' that 5% ofthc households own both a radio and a television, 26% own both a radio and : . F ( 5) , c, /:0 bicycle, 5°/ o own both a television and a bicycle, and I% own all three. lfa randomly N . 5/we selected poor householdin these areas is found to own exactly one ofthcsc three assets, Flkrx-r 26/Hm what is the probability that it is :1 bicycle? ,_ mremr S/6°‘ (a)2()/23 : .j= .gn‘ ﬁn)” f_OE°+—, ’;, __, ._ g '3 ﬂ. .—. .,r or l_/ I,-e(b)l7/23 _. . . .-‘--_- 4.; 2;. (‘(‘_f. ()tb)- JQIS/23 51+ 32+ Q lll’ (d) 12/23 In In) ten: 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‘1"2"2”'. Person l's endowment is (1, 0) and person 2's endowment is (0, I). Denote person l's allocation by (x. , y. ) and person 2's allocation by (x2,y; ). The set ofefficient allocations ((x. , y. ), (X2, y; )) is such that (L3) (-1’z, )’2)= (l ‘XI, l -)’1) for allxhxz 5 [or ll I (b) (x2,y2)= (l —x. , l —y. ) for all x. , x; e (0, I) (<3) (Xt. .VI) = (V2, 1/2) = (X2, Y2) (d) (. '. ,y. ) = (e‘', e”) and (x; ,y; ) = (I — e", l — e”) for all a, b E (—<>o,0) 32. Consider an exchange economy with persons l and 2 and goods xand y. Person l‘s utilit function is u. (x, y) = xzyz and person 2's utility function is u; (x, )2) = e"". The total endowment ofthe economy is (2, I). The allocation (x. , y. ) = (I, I) and (x2, yz) = (I, 0) i (a) Pareto inefficient P" (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 (, (d)_ Parctoxcfficient AS02
9. 9. Consider the situation ofthe preceding question. lfperson l's endowment is (1, l) and 2's , ndowment is (l. 0), then the following allocation is a competitive equilibrium: (a) (x. , y‘) = (3/2, H2) and (x2, )‘2) = (l/2, 1/2) ‘ ' t ' )(Xi. ,Vi)= (l» l)3nd(X2-Y2) = (l.0) ‘ (C) (Xi. ,Vi) ‘Q. 0) i1“d(«'z. )’2)= (0u l) / " - ', (d) (it. y. ) = (2. I)-and(x2.y1) = (0. 0) TO ‘ ; Consider the situation ofthe preceding question. Which ofthc following is an equilibrium price vector? ta) (ﬂiiﬂz) = (l. 0) (b)(Pi, l’2)= (0. l) (7)(Pi. P2) = (l7 l) '(d) none ofthe above ' . Consider an exchange economy with the same two agents and utility functions as the last thiee questions, but now the endowments ofthc two persons are (0, l) and (2, 0). Which ofthe following pre-trade lump-sum transfers ofwealth will lead to allocations (in, y. ) = (l, l) and (x2, y2) = (1, 0) and prices ([2,, P2) = (l, I) being a competitive equilibrium? (a) subsidies of l to both persons (1), : l . Mzt -(1 (b) taxes of 1 on both persons _ . _ subsidy of l to person I and a tax of 1 on person 2 i ‘ (d) a subsidy of l to person 2 and a tax of l on person l - . There are 3 items of choice, 2:, 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, 2} and {x, 2}. She chooses the items x, y, and 2 respectively ll these 3 situations. In 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 ofthc following is correct? 7 —(a) Ms A’s choices violate the weak axiom ofrevealed preference. , (b) 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 {x, _y, z}. (d) We can't say (a) or (b), even though we can deduce her choice from the set -‘. .t'. )2 2}- Qucstions 37-39 use the following information: A chemicalfactory produces a cliemical K rid an cfﬂiwnl E which it dumps iii a river. A downstrcaniﬁshcrjv [)f()dtlC(’. ‘_ﬁ. S‘/ I F and its‘ éu. it. i are affected by the level of efﬂiienl in the river. The twaﬁrmsrare competitive anu'fu('c unit prices PK = 10 and l’, .~ = 20 for the cliemieal and thcjish respectively. The costfinictimi of the chemical factory is C(K, E) = K I [(5 — E): + l]. The cost function of the ﬁsliery is E(1~‘,1:)= I-“E2. - P 37. If the 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 of the chemical, effluent and ﬁsh (i. e. K_E. F) are respectively (a)S,0,S/8 7,6 _ ,0/(_ - / :7,/5. U31] (b)3,0,2/5 ' _, 1F , _ ‘Mr. .10» .4’. -:1 wt”/ “° * . / ’ 0 gays, 5, 2/5 if‘, .7 / V (d)S,5,5/8 » , g _ aria - +KZ(a°«[6—€)‘ 0 9
10. 10. 38. Suppose the socially optimal levels ofthc chemical, effluent and ﬁsh, denoted by E, E. F respectively, are the levels that maximize the joint proﬁt ofan integrated ﬁrm consisting ofthe chemical factory and the ﬁshery. Then if is equal to (.1) 5/? /(1172 — F) (b) 5.73 / (E2 + F) (c) sf! /(I? -‘ — F? ) ya 5/? ! / (E1 +rT1) 39. Suppose a government knows the ﬁrms‘ production functions and their output prices. and can costlessly monitor the amount ofeflluent released by the chemical factory. If it sets . : tax I per unit ofeffluent produced by the chemical factory, the level ofthis tax that will result in the socially optimal effluent level E is equal to (a): =io/ ?rT2/(1?’+rT1) (b): =1o1?’F’/ (1?’—rT') , ge; /=1oI?1F’/ (K7‘+rT2) ” (.1) 1 =51?? ? / (752 +F’) 40. Consider a (inn with one output and two possible choices ofcapital stock, say I and z The associated cost functions are C(x. l) = 2 + 2x and C(x. 2) = 4 + x. Bcforc clioosing I’ capital stock, the ﬁrm's cost function is 2+2x, ifxe[O,2] V’ t -/ "‘)C*(x)= {4+x, ifx>2 A K (b)C"(x)=2+2x (c)C‘(x)=4+x _ 4+x, ifxe[O,2] (d) C*(x)_{2+2x, ifx>2 ’ = ALI} (l7)l_a, where L is the total employment of labour; V K7 is the total capital stock (which is ﬁxed in the short run); and A > 0 and 0 < a < l are parameters of the system. Let P and W denote the aggregate price level and the money wage rate respectively. Assume that the producers in the economy maximize proﬁt in a perfectly competitive set up. . by: I I I I (a) L, ;=[ —Yl_a]a 7*. //M 15; A“) ix /4ir‘< "/ {iii ; 9< /9 (Elm! _? :9 l U . F . r - he U r, '/1“ L ’ "i/ ’ , “ i "i. ‘L N01 /9 l< ' l AS02 ’ I0
11. 11. l (G) La 2 1 0 ([j'iTa' Aa(t? )'“f’ P (d) None ofthc above . ll‘ there is 21 one-sltot increase in the stock of capital (K). the demand for labour schedule, a‘s derived above, , R h ‘ Ph t Service . “3"; :.; ":; ;o. :t°; ::t. .m I 0 'n M does no‘ SM“ 9810867818, 9871975493 (d) the information available so far is not adequate to answer this question pose the above economy is characterized by a single household which takes the aggregate _‘: ice level and the money wage rate as given and decides on its consumption and labour j_ ply by maximizing its utility subject to its budget constraint. The household has a total ‘ owment of Z units of labour time, of which it supplies L5 units to the market at the ney wage rate W, and enjoys the rest as leisure. Its utility depends on its consumption and 1’ we in the following way: U = (Cy; +(l. ——L5)ﬂ, ' 0 < ,8 <1. The only source of ome of the household is the wage income and it ‘ ‘ ' ‘ ' umplion goods at the price P. The corresponding supply of labour schedule as a function of the real wage rate is given A CI = (C) ’g+ /1) ’g T ‘ l‘W“l''’’ h At tk) 1;‘ = _—”—— 5 W 1/ . « L . ﬂ——g « "’ t+[K]'~/3 / =1/= <.>r"’ I, _ 5_ Z W ‘ l+[. K)"/ l P er (c)I. ‘= ‘D — _ W / ’ ~l~J P - (d) None of the above 7 . If there is an exogenous increase in the total endowment of labour time (Z ), the supply of labour schedule, as derived above, AS02 l l 4.; .-
12. 12. (zi) shifts up / (,b’) shiﬁs down '(c) does not shift '/ __. ._. 45 Let a - = —3% K =10; L =10, A = 4 Given these parameter values, the uniui noii-negative equilibrium value of real wage rate that clears the labour market is um by: ) H, ’. 1 H2 3/3 P 2 2 ' 1/2 W i+ a/5 (b) —= - 2 / (c) L= l2—2«/ fl/2 l’ W /2 (d) ? = [2 + 2‘/3] supply curve (oiitpu (a) upward sloping (b) downward sloping (c) vertical (d) horizontal ml is e, mgeriau. i'ly ﬁxed at F 7'/ ii’ Y= C+1+G, where the COILYIIIII/ )// ii‘ disposable income (Y—T) such (r) such that I = I — d. r, d > 0." total tax re aggregate real income such that T ‘ I I GIIIOIIOIIIOILY such that G = G . M The money market clearing condition is given by: f : — L. where supply of money (A0 is exogenous such that M = Ill‘; and the demand for i-mu’ balances (L) is a function of the real income (Y) and the nominal interest rate (1) such i/ mi L = (LY — b. i, a, b > 0. Finally. the nominal interest rate , of irylatian (7re)such that i nomy starts from an equilibrium . S‘lIIltlIl0I1 where and the money market clear. 47. In ceteris paribus, a unit increase in govemment expenditure has the following impact on the equilibrium income level: 1 . (a) it increases by —————>———— units b[1— c(I ~r)]+ ad AS02 l2
13. 13. b Z[T: units b (d) it decreases by —%————— units bll — c(l — r)l+ ad ' it increases by .4 hi celeris purilm. s', a unit increase in the expected rate of inﬂation has the liollowiiig impact on the equilibrium value of the real interest rate: [1 _ C0 — T)]—— units Z? ____j bll — c(l - r)]+ ud (a) it increases by (b) it decreases by T)]+ ad units bk — Cu _ TH units ‘(c) it increases by b [1— CU — r)]+ ad . _ b[1 — c(l — r)l y) it decreases by ——-—————-—-b[1 _ C(1_ T)]+ ad :49. in ceieris paribus, a unit increase in the general price level has the following im the equilibrium value ofthc real interest rate: [V / (Fﬂ units (a) it increases by units b[1— c(l — r)l+ ad Rameshwari Ph t ~ — Delhis ° °°°Py Se , - M/ P } Ch r‘ ‘C [ ( )2 units 931086780103’ o9f8Ec°"°mics e - 71975493 (b) it decreases by b[l _ C0 _ 1)] + ad }, c) it increases by &? —————— (cl) it decreases by al price level to be ﬂexible. Given the above 50. Suppose we now allow the gener the corresponding aggregate demand curve (output characteristics of the macro—economy, demanded as a function of the general price level) is (a) upward sloping Law) downward sloping (c) vertical _ (d) horizontal ‘,1. Fix an m x n matrix A and an mtvector b. A condition that ensures the existence of a solution (an n—vector x) of the equation Ax = b is ‘(Q det A ; t 0 (b) m = n (c) the columns of A are linearly independent (d) the rows of/1 are linearly independent l3 AS02
14. 14. S2. The real—valued functionj'(. ‘) = x’ is i/ ﬁt) strictly convex . (b) strictly concave r ﬂ ~) « — '. (c) neither strictly concave nor strictly convex / , l J _ / * V ‘ J l . , o «J t ' , .‘ (d) convex but not strictly convex L ’ J / / ‘( L J t- V I 53. Consider the equation A): = 0 where A is an n x n matrix such that (1,, ¢ 0 For every is {l, ... .,r1} and a. , = 0 whenever i >j. This equation has (a) n distinct solutions (b) an n e l dimensional vector space ofsolutions . K/ (E) exactly one solution (d) an n dimensional vector space ofsolutions. /, Z// 9, /9 54. There is a pile of l7 matchsticks on a table. Players 1 and 2 take turns in removing matchsticks from the pile, starting with Player l. On each turn, a player has to remove a number ol‘ sticks that equals the square ofa positive integer, such that the number of matchsticks 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? . (»a') ll‘ Player 2 plays appropriately, he can win regardless ofhow 1 actually plays. v l (b) lfP| aycr l plays appropriately, she can win regardless of how 2 actually plays. ,, (c) Both players have a chance to win, ifthcy play correctly. y_, ,,', ,,, .,a (d) The outcome of the game cannot be predicted on the basis of the data given. / ' I ~/ . . ' Yglf/1 is a set of real numbers, let g, be the function such that g, .(x) = l if x <-: A , and g, .(x) 77.. = 0 if x ea .4. With this notation, consider the infinite sequence of functionsﬁ, where I" _/ ,,(x) = n g[°_, ,,, ‘(x) (i. e., n multiplied by gmml (x) ) for all real numbers x and for each It . I = l,2,3.. ..Then ; (a) For every x, the‘ sequence of numbers (fn (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) | im, __m f" (X) exists for all but a ﬁnite set ofreal numbers x. "SE§ContintIing with the sequence of functions above, we consider the sequence of real numbers (If, _): °=, (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 ‘ 57. For each positive integera = 1,2, 3, . .., let S, be the set of points lying on the curvey= (l/ 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 Xb) 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 AS02
15. 15. 8. Which ofthe followin ‘M e4‘? g two numbers is larger: 0" or Ire’? (b) Ir" (c) they are equal (d) it depends on the value ofe Ra”m__l_ 50 sCh0Dl U1 ' l/ ,". 0 818,981 9. The function f(x)= {Sm( x) ll l > 9810867 a, If x = 0 (a) is continuous or discontinuous at x = (b) is continuous at x = 0 [\$12) is discontinuous :1 x = 0 (d) is upper hemicontinuous at x = 0 0, depending on the value a 0. Suppose v. , v; and y; are three vectors in 3 Then the vectors v. + v; , v; + v; , and v, +v_; (:1) are linearly independent - (b) may be linearly dependent J1) are linearly dependent (d) are linearly independent, except when one ofthe vectors is zero. ‘The remaining pages of this book let are for your rough work, a lziclz will not be checked. . . fter you finish, hand in this booklet along with your bubble heat to the in vig ilators before you leave the examination hall. 2 ' f" Z 517 1 lv ‘ '33 Jt . I I, ,, 5| ' ‘ 1/: .e: ‘ ' J, AS02 I5 -dimensional space, and are linearly dependent.
16. 16. Delhi School of Economics Department of Economics Entrance Examination for M. A. Economics Option B June 28, 2008 Time 3 hours! Maximum marks mo General instructions. Please read the following instructions carefully. n Check that your examination llllfi pages 1 to 5 and you liave been given :1 blank Answer booklet. Do not start writing until instructed to do so by the invigilator. o Fill in your Name and Roll Number on the small slip attached to the Answer booklet Do not write this information anywhere else in this booklet. 0 When you iinish, hand in this Examination along with the Answer booklet to the invigilator. o Do not disturb your neighbors at any time. Anyone engaging in illegal exam- ination practices will be immediately evicted and that person’s czmdidaturc will be cancelled. - The examination has two sections. Follow the instructions given at the beginning of tfiltill section. Do not write below this line. This space is for ollicial use only Fictitious Roll N umber Section/ Question Marks I II/11 Rameshwari Photocopy Service " I 1 Economics :7: °; ;:'o: ::: :s°W493 1 I n/14 II/15 Total EEE 2008 B 1 F177‘
17. 17. Section I Instructions. Answer Questions 1 to 10. Each question is followed by four asset (a) to (cl), one of which is correct. Indicate the correct choice by circling it on Examination. Each correct choice will earn you 2 marks. However, you will lose 2/3 for each incorrect choice. Please ensure that you hand in this Examinati on {- with the Answer booklet. Question 1. There is a pile of 17 matchsticks on a table. Players 1 and 2 take tur ? removing matchsticks from the pile, starting with Player I. On each turn, a players to remove :1 number of sticks that equals the square of a positive intc number of matchsticks that remain on the table equals some non-ne player who cannot do so, when it is his turn, loses. ger, such that gative integer. ‘» (a) ll Player 2 plays appropriately, he can win regardless of how I actually play. . (b) ll" Player 1 plays appropriately, he can win regardless of how 2 actually plays (C) Both players have a chance to win, if they play correctly. (d) Tliegoutcome of the game cannot be predicted on the basis of the data given. Question 2. Given .4 C fit, let IA be the function deﬁned on 9? by: and 1,4(z) = 0 if :1: ¢ .4. Consider the sequence of functions (f, ,) where f, ,(z) = 7ll[0'l/ nI(I) for :1: E 9?. l (n) For every 1:, the sequence of numbers ([, ,(1')) has :1 limit in lll. (b) For every 1.‘, lim, ,_. ,,° f, ,(z) does not exist. (c) If lim, ,_. m f, .(x) exists, then the limit depends on 1:. (d) lini, ._. <x, f, .(z) exists for all but a ﬁnite set of 2: 6 ER. lA(: ::) = 1 in for positive intege a: uéstion 3. Ther pr thr e identical boxes, each with two drawers. Box A contai iv -5;‘ ' Pfstvrll -- Eold c<3:irli'_ln eacli‘d1"awer. .ﬁ9ﬁ0 Jlcontains a silver coin in each drawer. Box C cont‘ - ‘ : ; 1- - a gold coin in <ii'%}hn<: lgg silver coin in another drawer. A box is chosen, a d : opened and a gold coin is found. What is the probability that the chosen box is C? (8) 2/3 (bl 1/3 l (C) 1/2 ((l) :5/.1 Question 4. A random variable has outcomes Success and Failure with probabilities 7; and 1/4 respectively. A gambler observes the sequence of outcomes of this variable. 2. 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 0‘ambler‘s rize? D Question 5. A number of iiiatlieiiiaticizuis in the middle of the 20th century 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 reader's gender. “ If every assertion by this critic is false, which of the following assertions must be true? (21) 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 readers gender. (c) There exists a book by Bourbaki such that every chapter in it contains a theorem whose validity is independent of the reader's gentler. (rl) Every book by Bourbaki contains a cliaptei‘ such that the validity ol all the thee» reins in it is independent of the reader's gender. Question 6. A family has 55 children. Suppose the probability that a child will he a girl is 1/2 and that all births are independent. ll the family has at least one girl, what is the probability that the family has at least one boy? (a) 5/7 Service ' PY (b) 6/ 7 Rameshgva; :°l: )l: ‘:: f°Ec: °n°mics (C) 5/6 D; l8l'1lo8%7 18 9871975493 (cl) 4/6 ’ Question 7. Suppose f 2 5R2 —~ ER 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(: r, d>(: r)) = 0 for all :1; sufficiently close to a is (a) D1f(a. b) > 1 (b) D1f(a, b) <1 (9) D2f("-ub) > 1 (d) D2f(a, b) < 1 EEE 2008 B 3
19. 19. Question 8. Suppose X and Y are independent ranrlmn variables with stanclarrl fJ. .i: x.u if distriluitioiis. The probability of X > i is 1) E (U, 1) and the probability of i’ -i '3' q 6 (0,1). What is the probability of the event: X3 > 2 and Y2 > 1? (a) M (l)) p21] (C) P42 (d) 2120 Question 9. Suppose each of ll players tosses a coin that has probability 1/2 vn iniljilz; Heads and probability 1/2 of falling Tails. A player wins if her coin toss is Heads mul the other players‘ coin tosses are Tails. This game is played until someone win»: Tin probability of ﬁnding a winner in the k-th iteration of the game is 2l—n(1 __ 2]-—n)k—l (b) n2""(1 —— n2"")"'l (c) n2‘”(1 — n2"")"“ (d) 2—n(1 _ 2—n)k—l a winner in up to two iterations is ) 7 16 . ‘ - (3 / Rameshwari PhotocoPY 5°_""'°° (bl 15/16 Delhi School of Economics (c) 39/64 9810867818. 9871975493 (d) 15/64 Section II Instructions Answer any-‘(our at, the §OllOVW11glﬂY§(‘qUeSClOﬂS in the Answer booklet I tiia. -'7" Question 11. (A) Prove that d: 82"‘ X 92" —-9 3?, deﬁned by d(: c,y) = (1 — y, J: — y)‘/2, is a distance function, where (. , denotes an inner product on 9?". (B) Prove that, if (X, d) is a compact metric space and f : X —+ X is such that d(: c,y) = d(f(1:), f(y)) for all :5, y E X, then f is surjective. ' (8,12) Question 12. (A) Prove that if (X, d) is a metric space, then the distance function at is continuous. EEE 2008 B -I
20. 20. tric space and I 2 X ~J- X is (Il. f(yl) S .0d(I. yl for e'vI. v (B) Suppose (X, d) is a nonempty and complete me (Hint: Start with i. e.. there exists 8 E (0.1) such that d(f one I E X such that HI) = I. X. Now use the c a contraction, 1‘. y E X. Show that ther X and use f to (tons iu'u| u-_i'ti(-s_) (S, '7 e exists exactly truct a Cauchy sequence in ompleteness some point in and uiritrwu-t. iuii [r, r2l, let AU) be an n x I1 and for every t E d only onw YR". there exists one an )d>(t) for every t E il'| ,l-H 3. Let 1‘, <1‘-3,-r e it-1_rg, Suppose. for every 11; E l ~+ ER” such that Dq5(t) ; A(t Question 1 x and let b(t) E ? R". real rnatri [Ti . T2 differentiable function (1) ‘ and 0(7) : L’). differential equation D.1;(t) (A) Show that the spa ional vector space. . . , d>"‘} of solutions 0 ce of solutions of the ordinary A(I. ):c(t) is an n—dimens independent (B) A collection {(1;‘ , . if and only if {d>‘(t), . . . , dJ'"(t)} C 33" is linearly in l DI(t) = 'A(t)J: (L) is linearly dependent for every t E [r, ,r2i. : i ('. ()l‘A"X . ~:t-. t it my C-_ X and I. <' ((l, ll imply X C ii? " is S'rti(i to in: El? is S'r. i( Question l4. I, )y (: X. (liven u (‘. ()llVt‘. X set X L 372", it function f : X - E X and t. E (0,1) imply f(tI + (1 — Qty) 3 tf(1;) + (1 ~ t)f(y). - nuous, where iii“ l to be cmivcx ta‘ + (l and 37? are given if 3;, y Suppose X C iii" is convex and f : X ~~ 31 is conti the Euclidean metrics. Show that, if f(I/ '2+ y/2) 3 f(1:)/ '2 + f(y)/ '2 for all : :.y E X. then f is a convex function. (Hint: Use the fact that every t E (0, 1) can be represented in the form! = 0,2‘ ’ wliere each oz, is () or l. ) (‘-“"7 an unknown distribution P bservation X is drawn from stion 15. (A) Suppose an o hoses are to be tested: Que e hypot and that the following simpl II“ 2 IV’ is :1 unilurni (li. ~'. l.riliution on the interval (0, l) alculate the power of the test xx-hen J. -', U'.2t()1'ni!1<: the most powr-: rful test of size 0.01 and c is true. (B) Suppose X1.. . he parameter 0 is unknown. n sample from the uniform d od estimator of 9. (l0,: ‘ . , X" are a randox Derive the maximum likeliho where t istribution on [0, ’7i.