DSE 2015 question paper

All 58 questions from the DSE 2015 Delhi School of Economics MA entrance paper. Open any question for the worked solution.

1. Q1Two individuals, 1 and 2, are each offered a lottery paying Rs 160 or Rs 80, each with probability 1/2; the alternative is a fixed sum of money.…
2. Q2In an exchange economy with agents 1,2 and goods x,y, the agents' preferences are fixed. If it rains, 1 's endowment is (10,0) and 2 's is (0,10)…
3. Q3Deadweight loss is a measure of
4. Q4To regulate a natural monopolist with cost function C(q)=a+bq (with a>0), the government has to subsidize the monopolist under
5. Q5Suppose an economic agent lexicographically prefers x to y. Then her indifference curves are
6. Q6For f:ℝ²→ℝ with f(0,0)=0 and f(x,y)=xy/(x²+y²) for (x,y)≠(0,0), which statement is correct?
7. Q7For the same f (f(0,0)=0, f(x,y)=xy/(x²+y²) otherwise), which statement is correct?
8. Q8Consider the system x+2y+2z-s+2t=0, x+2y+3z+s+t=0, 3x+6y+8z+s+4t=0. The dimension of the solution space of this system is
9. Q9Vectors v₀,v₁,…,vₙ in ℝ^m are affinely independent if Σᵢ₌₀ⁿ cᵢvᵢ=0 and Σᵢ₌₀ⁿ cᵢ=0 imply cᵢ=0 for all i. For such a set, which is implied? I. v₀,…,vₙ…
10. Q10Let F:ℝ²→ℝ² be linear with F(1,2)=(2,3) and F(0,1)=(1,4). Then in general F(x₁,x₂) equals
11. Q11A correlation coefficient of 0.2 between Savings and Investment implies that
12. Q12In a simple regression model estimated using OLS, the covariance between the estimated errors (residuals) and the regressors is zero by…
13. Q13Consider the uniform distribution over the interval [a,b]. Then
14. Q14Let F:ℝ→ℝ be a distribution function and define b(c)=inf F⁻¹([c,1]) for c∈(0,1] (and b(0)=0). If F has a jump at x, say c=F(x)>a≥ F(x-), then
15. Q16Suppose agents have static expectations. Static expectation implies that
16. Q17With Cₜ=α Y^eₜ (0<α<1) and Iₜ=γ(Y^eₜ)² (γ>0), output is demand-determined (Yₜ=Cₜ+Iₜ). Under static expectations, starting from any Y₀≠(1-α)/γ, in…
17. Q18Suppose now agents have rational expectations. Rational expectation implies that
18. Q19Under rational expectations, in the long run actual output in this economy (Yₜ=α Y^eₜ+γ(Y^eₜ)²)
19. Q20Suppose we conduct n independent Bernoulli trials, each with success probability p. If k is such that the probability of k successes equals the…
20. Q21A pure exchange economy has three persons 1,2,3 and goods x,y, with utilities u¹=xy, u²=x³y, u³=xy². If the endowments are (2,0),(0,12),(12,0)…
21. Q22For the same three agents (u¹=xy, u²=x³y, u³=xy²), if the endowments are (0,2),(12,0),(0,12) respectively, then
22. Q23A monopoly electricity supplier has unit cost c. Type- i utility is uᵢ(q,t)=θᵢln (1+q)-t (q consumption, t tariff), with θ_H>θ_L>c. If the supplier…
23. Q24Now the supplier cannot observe types but offers both packages from the observable-types case; consumers freely choose. Then
24. Q25Buyers of ice-cream are uniformly distributed on [0,1]. Two sellers simultaneously choose locations to maximize market share (each buyer goes to the…
25. Q26In the previous setting, suppose all players understand that a third seller will locate on [0,1] AFTER observing sellers 1 and 2, and seller 3…
26. Q27A government and two citizens: a park costs Rs 100; its value is Rs 30 to citizen 1 and Rs 60 to citizen 2 (private). Each reports r₁,r₂; the park…
27. Q28In the same mechanism, suppose citizen 1 's valuation rises to 50 (citizen 2 's stays 60) and the procedure is unchanged. Then
28. Q29Consider two games (player 1 chooses a row, player 2 a column). Game A: player 2 has only Hawk, with payoffs Enter (-1,1), Not enter (0,6). Game B…
29. Q30An exchange economy has agents 1,2 and goods x,y. Agent 1 lexicographically prefers x to y; agent 2 's utility is min {x,y}. Endowments are (0,10)…
30. Q31For a strictly increasing differentiable u:ℝ²→ℝ, consider (D₁u(x₁,x₂))/(D₂u(x₁,x₂))=p₁/p₂ and p₁x₁+p₂x₂=w. What guarantees existence of continuously…
31. Q32As n↑∞, the sequence (-1)ⁿ(1+n⁻¹)
32. Q33The set (0,1) can be expressed as
33. Q34The set [0,1] can be expressed as
34. Q35For a linear transformation P:ℝⁿ→ℝⁿ that is a projector (every x uniquely x=y+z with y∈R(P),z∈N(P), and P(y+z)=y), which holds?
35. Q36If P is a projector and Q:ℝ^m→ℝⁿ is a linear transformation with R(P)=R(Q), then
36. Q37Let a<b be two consecutive, non-repeated roots of a polynomial f, so f(x)=(x-a)(x-b)g(x). Consider: I. g(a) and g(b) have opposite signs. II.…
37. Q38Suppose f:[0,1]→ℝ is twice differentiable with f′′(x)+f′(x)=1 for all x∈(0,1) and f(0)=0=f(1). Then
38. Q39Suppose x₁,…,xₙ are positive and λ₁,…,λₙ are non-negative with Σᵢ₌₁ⁿλᵢ=1. Then
39. Q40Let N={1,2,3,…}. Suppose there is a bijection between N and a set X, and also a bijection between N and a set Y. Then
40. Q41A regression of wages on gender on 200 individuals (150 men) yields Wᵢ=300-50Dᵢ+uᵢ (Dᵢ=1 if male). The average wage in the sample is
41. Q42For the same wage-gender regression, the most precise estimate of the male-female wage difference would have been obtained if, among the 200…
42. Q43For the same regression (Wᵢ=300-50Dᵢ+uᵢ; 150 men, 50 women), the explained (regression) sum of squares is
43. Q44A researcher estimates Model A: yᵢ=β₀+β₁Sᵢ+β₂Aᵢ+εᵢ and Model B: yᵢ=β₀+β₁Sᵢ+εᵢ, where S is study hours and A is ability. β̂₁=7.1 in A but 2.1 in B…
44. Q45An analyst estimates Y=β₀+β₁X₁+β₂X₂+β₃X₃+u by OLS, but the true β₃=0. By including the irrelevant X₃,
45. Q46Let β̂ be the OLS slope on X₁ from regressing Y on X₁ alone, and β̃ the coefficient on X₁ from regressing Y on X₁ and X₂. Then
46. Q47In Y=a+b₁X₁+b₂X₂+u (large sample), let t₁,t₂ test b₁=0 and b₂=0; treat them as independent. Testing the joint null b₁=b₂=0 by the rule 'reject if…
47. Q48Four testers independently rank three chocolates A,B,C (best =1, then 2, 3); ranks are summed per chocolate across testers. If each tester ranks at…
48. Q49Suppose 0.1% of a town has tuberculosis. A test is positive with probability 0.999 if a person has TB and 0.002 if not. Given a positive test, the…
49. Q50There exists a random variable X with mean μ_X and variance σ_X² for which P[μ_X-2σ_X≤ X≤μ_X+2σ_X]=0.6. This statement is
50. Q51A sample of size 2 is drawn without replacement from an urn with balls numbered 1,2,3. Let X be the smaller and Y the larger of the two drawn. The…
51. Q52From the square with vertices (0,0),(0,2),(2,0),(2,2), five points are chosen independently at random. If a point (x,y) satisfies x+2y≤2 a pair of…
52. Q53Let S have n>1 elements and A₁,…,Aₘ⊆ S satisfy: for any x≠ y in S, some Aᵢ contains exactly one of them. Then necessarily
53. Q54Solow model Y=K^αL^(1-α) with class savings: households save s_w of wage earnings and sᵣ of interest earnings; labour grows at n, no depreciation.…
54. Q55In the same class-savings Solow model, now with sᵣ=0 and 0<s_w<1, an increase in the labour-force growth rate n
55. Q56In the same model, let both s_w,sᵣ be positive with s_w<sᵣ. In the long run the capital-labour ratio approaches
56. Q57Now the government taxes wage earnings at rate τ and transfers the revenue to capital-owners; households save s_w of net wages and sᵣ of net…
57. Q59In the setting of the previous (endogenous labour-growth) question, an increase in the parameter A
58. Q60A profit-maximizing firm owns two plants with cost functions c₁(q)=q²/2 and c₂(q)=q², and may use either or both to produce any output. For this…