7/30/2019 MEC 003-J-11

1/12

No. of Printed Pages : 12 MEC-003MASTER OF ARTS (ECONOMICS)

Term-End ExaminationJune, 2011

MEC-003 : QUANTITATIVE TECHNIQUESTime : 3 hoursaximum Marks : 100SECTION - AAttempt any two questions from this section. 2x20=401.two product firm faces the following demandand cost functions :

demand functionQ2 =35-P 1-2c=Q12+2 , 2m , 2 2 +10 = cost function(a) Find the output levels that satisfy the first

order conditions for maximum profit.(b) Check the second order for sufficient

condition. Can you conclude that thisproblem possesses a unique absolutemaximum ?

:.Dc) What is the maximal profit ?2.a) Given the demand and supply for cobwebmodel as :Qdt =18 3PtQst = 3 + 4P t _ 1Find the intertemporal equilibrium price anddetermine whether the equilibrium is stable.

Q =40-2Pi P 2

MEC-003.T.O.

7/30/2019 MEC 003-J-11

2/12

(b) Establish the stability condition ofSamuelson's multiplier - acceleratorinteraction model.

3.a) Consider the Cobb Douglas productionfunction q = AL" k1 ' ; A, a > 0Prove that :(i )t is homogeneous of degree 1(ii) the marginal and averageproductivities of the 2 inputs L and kdepend on the ratio of the 2 inputs(iii) elasticity of substitution is unity.

(b) Determine whether the following functionis homogeneous. If so, to what degree ?

f (xy) =(x_y2

4.a) What is point estimation and how is itdifferent from interval estimation. What arethe characteristics of a good estimator.

(b) If x , x2, ..., xn is a random sample from aninfinite 'n' size population with variance o 2 ands the sample mean. Show-2(k X i X)2.1s a biased estimator of o- 2, buti=ithe bias becomes negligible for large n.

MEC-003

7/30/2019 MEC 003-J-11

3/12

(c)f x1 , x, and x3 is a random sample of size 3from a population with mean and variance2 and T i , T2 and T3 are the estimators usedto estimate the mean value p. whereT,x- . 1 1 3, T2 = 2X 1 4r3xandT3 = 3 [a xi +x2 + xl](i) Are T i and T2 unbiased estimator

of(ii) For what value of a will T3 be unbiased

estimator of R.(iii) With this value of a will T3 be a

consistent estimator.(iv) Which of the 3 is the best estimator.

MEC-003.T.O.

7/30/2019 MEC 003-J-11

4/12

SECTION - BAnswer any 5 questions from this section.x12=60

5. A subcommittee of 6 members is to be formed outof a group consisting of 7 men and 4 ladies.Calculate the probability that the sub-committeewill consist of (a) exactly 2 ladies and (b) atleast2 ladies.

6.onsider the matrices1 1 1 1 2 3

A = 2 3 4 and B= 6 1 2 63 2 3 5 1 0 5Find Rank of matrices A, B, [A + B], [A B] and[B A]

7.a) What is Karl Pearson's CorrelationCoefficient ? Prove that it lies between +1and 1.

(b) In a certain examination 10 studentsobtained the following marks in Maths andEconomics.

Roll No. 1 2 3 4 5 6 7 8 9 1 0Maths 90 30 82 45 32 6 5 40 88 73 66Economics 85 42 75 6 8 45 6 3 60 9 0 62 58

Find Spear man's rank correlation coefficient.

MEC-003

7/30/2019 MEC 003-J-11

5/12

8.arks obtained by 12 students in college test (x)and the university test (y) are as follows.x : 45 41 50 68 47 77 90 100 80 100 40 43y 63 60 60 48 85 56 53 91 '74 98 65 43What is your estimate of the marks a studentwould have got in the university test if he got 60in the college test but was ill at the time ofuniversity test ?

9. What is a Binomial Distribution ? Find the meanand standard deviation of a binomial distributionwith parameters n and P.

10. Solve the following problem.Max 10x1 +10x2 + 20x3 + 20x4

Subject to 12x1 + 8x2 + 6x,, + 4x4 5_2103x1 + 6x2 +12x3 +24x4 5..210x1 ' 2' Y:, 4

11. Given the input - output matrix and final demandvector as :

0.05 0.25 0.34 1 8 0 0A= 0.33 0.10 0 . 12 and d 20 0

0.19 0.38 0 .0 901)Find the solution the output levels of threeindustries.

MEC-003.T.O.

7/30/2019 MEC 003-J-11

6/12

12. (a) Let q =-3 + 2L2 +12 L is the production3function with L = labour employed. Findthe maximum L beyond which the averagereturn from labour starts diminishing.

(b) Find ddx whenY(i) it = log (ex + 3)

1(ii) YI X 22

MEC-003.

7/30/2019 MEC 003-J-11

7/12

01(11mil rI( 3TOFT17)'9', 2011

74.1.*-003 : 4ftHluIfcHch tqc)tiut fartirat7 7 777 : 3 E T T 41fg/W-Ti00vigrr -77 7777 4ch )- t 6 ,1x20=401.4 11 1 f T T -cr opicf ash:Q 1 =40-2P1P2} W IT conQ2=3 5 P1 P 2C Q1 2 +2 sh(a) 3 1 -N -*-a - 1 4P - T r i .0rradckilq-1 'f-c1. ;1'1 .0 I(b) frd17TrzilF r m0->k azir . 7 7 47 : 17RziT v h f 4 F a 9 T r iRItTr i(c) "a74 ""T t itr+ u ?2.a)4 4- 1 1 -0 1 1 5 \ - 7 1 1 ( 1T T 1T 33TrTM ast,Qdt =18 3 P tQst = 3+4 P 1 --13f -d-*ifFwh k if t

1 4 > = e a ] -#.199 T g r t 61 4n?MEC-003.T.O.

7/30/2019 MEC 003-J-11

8/12

(b )- f-- r w z r r=q-rfz ct1- Q T 7 1 -9- r3.a)1 - 1 1 - 1 7-{ fdq T {' c.f.) :q= AL(' kl a.A, cx > 0

rct :(i) L I@ 7 1 114 t * ;(ii) - 1 1 ' 3771 9'f, l a2 . T T k*=Ti ff 37Ta9k 3-97Tff fff*t(iii) -5 r t h - T - 2. 1 -rcrir r i t-*rU

(b ) ch f * - 1 1- ) ( 1 - 1 tf A-y2)/24.a) fq3 3 T-jF T r f-d1 t1ifuf 9 - T 5 1c r T 3 u 31711icRqc t iI(b ) -c r w 1.17LP:it,wui if 2 t 31t644 '1 1 / 3 - T r w R c h i Tit0n - 1 ( 1 1 1 0 1N - 1 1 t1737 F-- -0 cni er-7n x)vI- 2 -" W i1f1-9 39 614 11 -i = 1

, R - R T T - 61 4 -c T-TRT9t- 1 4 1 u 1 .771crt IMEC-003

7/30/2019 MEC 003-J-11

9/12

(c)E R z i p t ,a-q-tur u 2 tf fq r01 1rfd0x, x2 3Rx3 Tir7741---Frrfur r z rT1 = x1+ x2 x3, T2 = 2x1 4x 2 + 3x3

1T=3 [a xi +x, +x3](i)E { T T 1 AT T2 p, t(ii) T3- 1 1 1 - 11 9-ff +-iNch qffT4tt7atl4 - 1 1 - 1 c 1-11 "'f 4 1 1 ?(iii) tatti -1 k T IM T 3 Bch{4 - 1 1 4 0 1 1 141. 4 1 1 I(iv) 74 1- c r-Tit t71 7 ?

MEC-003.T.O.

7/30/2019 MEC 003-J-11

10/12

%T M -1 F T/ 7 -a. 4 - ; Ix12=605.f - ) T i17 if 7 77 - 4:Tt11 6 7774 ci;) -)64*-114-i rd a-d11 X111 tiqrff a - )1 m i R i c b c t i,3-11 01 1(-1 cf-;44 :(a) q(--1 ci Ti 6) 4 1(b) c h H14-1t 1 : 1F-6 - T-71 X 1 4 ?6.Thalefil7 fay or) :1 1 1 1 2 3A= 2 3 4 AB= 6 1 2 6

3 2 3 5 10 5.71 1-ichla 34MTRITTiT7e A, B, [A + B],[A B] A [B A] t c11 re.9t-o-lich1 T 3-TTT-07 01-) :

7.a) irf Lilqtt-1-1IT-44q TITfT qzfT 101 ?fi: r w r HI-1 +1 1 tt(b) f-711 4k-T i if 10 W tif i 3i 3-147 1 1 T q3 : 4 5 6 7 8 9 10511flic1 H gcbR14 , 4 - 4 1 . 1 ,Tifurff 90 30 82 45 32 65 40 88 73 663 1 2 4 - T r - - - q 85 42 75 68 45 63 60 90 62 58T chR A-1 .wr 9 4 , 4 - 11 to chMEC-0030

7/30/2019 MEC 003-J-11

11/12

8.4R7 x) afty)fdriti A mm icb i:x : 45 41 50 68 47 77 90 100 80 1 0 0 40 43y : 63 60 60 48 85 56 53 91 74 98 65 43

3-17IF r i+ It" tT47 14 60 3ft 7 1 "FoRcircmic-N aiT 7:1771 11 71 1.-14aft(49. fg---Eg air479 T Z F T pla t t? N I -c 1 C 4 n a ft P q f c,* fg-trq3-TT4ait: 1 T - z i-nlcb F IT - 1 0 1T IT -( 9 7 < 4 1- I10. T I T I T z I TITIT NT9' ch :3 T R T-*-7r AA :0x1 +iox2 +20x3 +20x4T i t A T P q -17 :2x1 + 8x2 + 6x3 + 4x4 .2103x1 + 6x2 +12x3 + 24x4 _210x2, x3, x411. 3779.-3- T r q 3 1- r -&7 A 3-fr-T r Hi4fTrfr.7cr Trtg :

0.05 0.25 0.34 1800A= 0.33 0 .10 0 .12 d 20 00 .19 0.38 0 .0 9 007R:T*N-3E 1 111&fq79-TT t 6c-tug-I tit44MEC-0031.T .O.

7/30/2019 MEC 003-J-11

12/12

12. (a) '7f 6c41q 797 It f-d17L3q L2 +12 L , qiL= 11: f "1- cili v-Ii"t IWI 397'-. 3f9-----14 -c-R 7.110 4) - NT A3 1-r4 *m 7-T r q -4 f r rua z v tr 7 rcn- ft I3-11 "W ff i "e qk :dx( 1 )= log (ex +3)1(ii)/- = Vx2 + a2

MEC-0032