DSE-Entrance Solutions

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1 2010 DSE-Entrance Solutions Economics This document contains solutions of past year exam problems asked in Delhi School of Economics entrance exams held for admissions to their masters program( ). Disclaimer: The solutions offered in the document can be incorrect, the likelihood of which is very small though, so use at your own risk. The possible reasons for mistakes could be my ignorance or laziness with former being more likely of the two. Amit Kumar Goyal economicsentrance.weebly.com, beaneconomist.com 2/15/2010

2 Prepared by: Amit Kumar Goyal DSE: Year 2004 Part (b) only proposition (i) is true 2. (b) there exists a solar system in the milky way galaxy such that every planet in it has a moon revolving around it without any life forms 3. (c) x 1=y 1, for x 1Є[0,1000] 4. (c) x 1=y 1, for x 1Є[0,1000] 5. (d) both α and β are pareto efficient 6. (b) x 1=999 and y 1Є[999,1000] 7. (a) (1,0) 8. (c) s 1 = (3-4c)/(3-2c) and s 2 = 2c/(3-2c) 9. (b) no 10. (a) ½ 11. (d) produce nothing in Plant (c) ¼ 13. (a) (-5/2, 0) 14. (d) (b) ((10-200)/4, (10-200)/6) 16. (a) deficit decreases by (d) removing the water subsidy and providing a lump-sum subsidy. 18. (b) (µβ+α)/p[µ(1-c)+αλ] 19. µ/[µ(1-c)+αλ] 20. (d) Y = f(½) 21. (b) max {W 0, P/2} 22. (a) min{½, 1- W 0/P} 23. (c) f(½) 24. (c) Y does not decrease and P increases 25. (b) Y does not increase and P decreases 26. (c) 2/3 [2]

3 27. (d) 3/ (a) 2/3 29. (c) p 2 Part 2 1. (A) Effect of a marginal increase in G on AD: Y = C(Y-T) + I(r) + G, M/P=L(Y, r) Differentiating totally, (1-C Y)dY - I rdr = dg L YdY + L rdr = dm/p Solving, dy/dg = L r/(l r(1-c Y)+ I rl Y) Refer book for explanation (B) Refer book for explanantion 2. (A) (a) np(1-p)(p n-2 +(1-p) n-2 ) (b) (1- np(1-p)(p n-2 +(1-p) n-2 )) k-1 (np(1-p)(p n-2 +(1-p) n-2 )) (B) (a) a=1/48 (b) 7/24 (c) 3/4 3. (A) f(x 1, x 2)= x 2g(x 1/x 2) ( BY CRS ) MP 1= g (x 1/x 2) MP 2= g(x 1/x 2)-(x 1/x 2) g (x 1/x 2) MP 11= (1/x 2)g (x 1/x 2) < 0, (given in (b)) Hence, MP 12=- (x 1/(x 2) 2 )g (x 1/x 2) > 0 Similarly, MP 21 > 0. (B) For a 33 3 Economic interpretation : For a 33 3, the three securities when combined appropriately can generate any return. (C) To show : f is convex if and only if {(x,r) f(x) r} is convex. Suppose f is convex. [3]

4 To show: {(x,r) f(x) r} is convex Let (x 1,r 1), (x 2,r 2) belongs to {(x,r) f(x) r}. So, f(x 1) r 1 and f(x 2) r 2 Take 0<t <1, we want to show that f(tx 1+(1-t)x 2) tr 1+(1-t)r 2 By convexity of function, f(tx 1+(1-t)x 2) tf(x 1)+(1-t)f(x 2) tr 1+(1-t)r 2. Now suppose {(x,r) f(x) r} is convex To show: f is convex. (x 1,f(x 1)) and (x 2,f(x 2)) belongs to {(x,r) f(x) r}. Since the set is convex, (tx 1+(1-t)x 2, tf(x 1)+(1-t)f(x 2)) belongs to {(x,r) f(x) r}. This implies f is convex. 4. (A) (a) SC =p xx+(25p y)/x. (b) LC =10(p xp y) 1/2 (c) x = 5(p y/px) 1/2 (d) x = (3.5) (2p y/px) 1/2 (B) Member 1 will veto b. Member 2 will veto d. Member 3 will veto c. [4]

5 Prepared by: Amit Kumar Goyal DSE: Year 2005 Part (d) (x 1,y 1)=(100,100) and (x 2,y 2)=(50,0) 2. (a) (x 1,y 1)=(50,50) and (x 2,y 2)=(100,50) 3. (c) (1/3, 2/3) 4. (b) (x 1,y 1)=(50,50) and (x 2,y 2)=(100,50) 5. (b) a Tribe 6. (d) a Tribe 7. (b) There are at least two sets of individuals that are both a Club and a Tribe. 8. (a) a Club 9. (a) q 1=a/3b and q 2=(a-3)/3b 10. (b) The market share of the more efficient firm decreases. 11. (c) q 1 will increase and q 2 will decrease. 12. (d) produce nothing in plant (d) Every book by Bourbaki contains a chapter such that the validity of all the theorems in it is independent of the reader s gender. 14. (c) (c) 1/3 16. (a) 5/8 17. (b) 2/9 18. (b) 1/2 19. (c) 2p (a) 2/3 21. (b) n-n(mean(y)) 22. (c) 48/ (c) (c) right if the price level falls and/or the quantity of money rises. 25. (d) None of the above choices is correct [5]

6 26. (b) one; zero 27. (a) IS curve was vertical and stuck at a low level of income. 28. (c) ½; -½ 29. (b) decreasing; decreasing 30. (a) increasing; increasing Part 2 1. (A)(a) Required Variance=25/49+36/49=61/49 (b) Required Variance = 50 (c) Null Hypothesis: H 0 = µ A-µ B=0 Test statistic value: z = (mean(x A)-mean(x B))/ [(25/36)+(36/49)] = -25/1.2-20, Null is rejected. (B) (a) True (b) True (c) False (d) False (e) True 2. Given the information we can derive that Aggregate supply curve will be vertical, hence an increase in supply of money will shift AD curve outwards and can only lead to rise in prices without affecting the output. 3. (A) (a) SC =p xx+(25p y)/x. (b) LC =10(p xp y) 1/2 (c) x = 5(p y/px) 1/2 (d) x = (3.5) (2p y/px) 1/2 (B) (q 1,q 2,q 3)=(1/2,1/4,1/8) 4. (A) Refer Textbook (B) (a) a33-3; (b) Price = 61/6 [6]

7 Prepared by: Amit Kumar Goyal DSE: Year 2006 Part (b) pareto efficient 2. (d) all of the above 3. (b) a private good 4. (b) monopsony 5. (d) C = Q (wr) 6. (d) complete crowding out and no increase in equilibrium income 7. (b) the IS curve is vertical and the LM curve is upward sloping. 8. (b) (d) 3/8 10. (d) A consumer s budget share for an inferior good decreases with an increase in money income. 11. (b) Income elasticity of good x is (c) 1/3 13. (c) remains constant 14. (d) any of the above three conditions is satisfied 15. (c) f is quasi-convex 16. (a) does not change the value of the determinant of A 17. 1/ (d) (c) under flexible exchange rates when the exchange rate rises (depreciates) above and then falls down to equilibrium after a monetary expansion 20. (c) fiscal policy is an effective tool for stabilizing the economy Part (b) not a competitive equilibrium allocation but is pareto efficient 22. (a) a competitive equilibrium allocation and is pareto efficient [7]

8 23. (b) (0,1) 24. (c) 50 units 25. (a) equilibrium output increases by 160/3 units 26. (b) Current Employment: 400; Employment tomorrow: (c) Rs (b) (b) q 1=2 and q 2=2 30. (b) A has x=2, y=0 and B has x=0, y=2 31. (a) 5/ (d) 2/7 33. (b) (b) (b) has a greater impact on income than in a closed economy 36. (d) an increase in the interest sensitivity of money demand 37. (c) The line defined by y = x + 2 and the vertical axis. 38. (a) the smaller the induced change in interest rates and smaller the responsiveness of investment to these changes 39. (d) the level of profits 40. (b) (x 1, x 2) = (2/18, 5/18) 41. (c) x 1 increases and x 2 decreases 42. (c) (15, 7) 43. (b) Y = r 44. (d) does not change, ever increasing 45. (c) leaves unchanged 46. (d) 7/ (b) 121/ (c) 4/ (a) 28/ (d) Both countries produce both goods 51. (½)t 2 ((1+b)/b) 52. (c) is greater than or equal to 0 at every x Є R 53. (d) In every election, every party has at least one dishonest candidate. [8]

9 54. (b) a Gang 55. (d) a Gang 56. (b) There are at least two sets of individuals that are both a Family and a Gang. 57. (a) a Family 58. (a) (b) 3/8 60. (d) f(x) = x, where x Є R and x 0 [9]

10 Prepared by: Amit Kumar Goyal DSE: Year 2007 Part (c) (20, 0) 2. (b) a vertical line 3. (b) K = (3/2)L 4. (c) MR < 0 in such a case 5. (a) A, B 6. (b) inflation is constant 7. (d) change the composition of monetary base 8. (a) always reduce borrowing 9. (a) improve it in the short run 10. (a) only raises the price level and the interest rate 11. (a) f(x) = x (c) det A = det B 13. (b) Df(x) = 0 for some x Є R 14. (d) In every legislature and every party, there exists a legislator who does not pay taxes 15. (c) singular 16. (b) 2/5 17. (c) increases steadily from 0 to (d) 4/7 19. (b) (i) 0.16; (ii) (c) 7.5 Part (c) L+72L 2-4L (d) (i) Rs. 4,80,000 and (ii) risk-averse 23. (a) (i) 1 km and (ii) Rs (a) Some firms will exit the industry, but the others will remain [10]

11 25. (b) 9 apples 26. (a) 3.5 kg 27. (d) ((0, 0),( 8, 2),(3-8, 1)) 28. (c) Pareto efficient, because there is a bound on Agent 1 s ability to consume. 29. (b) (c) ((0, 0), (0, 3-3), (3, 3)) 31. λ/[α(1+λ)] 32. (b) P/P e = 1+λ 33. (a) an increase in the actual price level 34. (b) a decrease in the actual price level 35. (a) β 2/[1-(α 1+β 1)] units 36. (c) 1/[1-(α 1+β 1)+(Υ 1β 2)/Υ 2] 37. (c) (1+δ)w 2 > (1+r)w (a) goes up 39. (d) Country 1 has a lower equilibrium level of output than country 2 and it runs a trade surplus vis-à-vis Country (b) (i) and (ii) remain unchanged; (iii) depreciates 41. (a) {(x, r) f(x) r} is convex 42. (a) 2xe sqr(x) -e x 43. (d) converges to neither 1 nor (d) [-1, 1] 45. (c) (-1, 1) 46. (d) d(x, z) d(x, y) +d(y, z) 47. (b) (A-B) X C = (A X C)-(B X C) 48. (c) Attains a saddle point 49. (c) (d) During warm weather, my crops suffer from pests more than during cooler weather. Therefore, a warm environment must help pests to multiply. 51. (d) None of the above 52. (3!3!)/6! [11]

12 53. (b) 5/ (b) (½, ¾, ½) 55. (b) 37.6 years 56. (a) (d) 5, 0.5, (d) (b) There is a significant difference in the performance of the classes at the 5% level but no significant difference at the 1% level of significance 60. (b) There is no significant difference between the districts and the candidate is preferred in district A [12]

13 Prepared by: Amit Kumar Goyal DSE: Year 2008 Part (d) S1 and S3 (Fischer s Index is ideal., it satisfies only these two) 2. (b) (0.1)(0.99) 9 3. (d) The sample standard deviation is one third its original value 4. (c) III and IV 5. (c) g(y) = 2f( y) for any y = 1, 4,..,400 and g(y) = 0 otherwise. 6. (b) The goods are perfect substitutes 7. (c) price of good x is less than or equal to price of good y 8. (c) price must be higher in the market with lower price elasticity of demand 9. (d) 5 and (d) (b) becomes flatter 12. (b) it leads to lower level of output in the equilibrium 13. (a) it leads to a higher steady state rate of growth 14. (c) expected currency depreciation must equal the currency differential plus the risk premium. 15. (c) 1, 3 and (d) g is a linear function, f is not linear 17. (b)-(yx y-1 +z x In(z))/ (xz x-1 +y z In(y)) 18. (c) a constant 19. (a) will change the sign of det A 20. (b) 3 Part (b) Rs to (c) 0.95 and 0.55 [13]

14 23. (a) A unit change in X is associated with a 1.25 unit change in Y, and a one unit change in Y is associated with a 0.6 unit change in X 24. (c) The probability that the life of the tyre will be between and days is 90% 25. (c) Given H A: π 0.1, we reject the null at the 10% level of significance, and the probability of error type I of this test is (a) If H A: µ 1-µ 2 0 we reject the null hypothesis at the 15%, 13%, 12% and 8% level of significance 27. (c) The chain base price indices with 2000 as base year are P 01 = 148.3, P 02 =154.2, and P 03 = (d) 80/ (b) only II and III 30. (c) 15/ (a) (x 2, y 2) = (1-x 1, 1-y 1) for all x 1, y 1 Є [0, 1] 32. (d) Pareto Efficient 33. (b) (x 1, y 1) = (1, 1) and (x 2, y 2) = (1, 0) 34. (c) (p x, p y) = (1, 1) 35. (c) a subsidy of 1 to person 1 and a tax of 1 on person (a) Ms A s choices violate the weak axiom of revealed preference 37. (c) 5, 5, 2/5 38. (d) 5K 2 /(K 2 +F 2 ) 39. (c) 10K 2 F 2 /(K 2 +F 2 ) 40. (a) C*(x) = 2+2x, if x Є [0, 2] 4+x, if x > (b) L d = K(Aα) 1/(1-α) (W/P) 1/(α-1) 42. (a) shifts up 43. (a) L s = L(W/P) β/(1-β) /(1+(W/P) β/(1-β) ) 44. (b) shifts down 45. (d) W/P = [2+2 2] 1/2 46. (c) vertical 47. (c) it increases by b/[b{1-c(1-τ)}+ad] units 48. (d) it decreases by b{1-c(1-τ)}/ [b{1-c(1-τ)}+ad] units [14]

15 49. (c) it increases by {M/(P) 2 }{1-c(1-τ)}/ [b{1-c(1-τ)}+ad] units 50. (b) downward sloping 51. (d) the rows of A are linearly independent 52. (a) strictly convex 53. (c) exactly one solution 54. (a) If Player 2 plays appropriately, he can win regardless of how 1 actually plays. 55. (d) lim n f n(x) exists for all but a finite set of real numbers 56. (c) a constant sequence 57. (b) a set having a single point 58. (a) e π 59. (c) is discontinuous at x = (c) are linearly dependent [15]