DSE 2018 question paper

All 49 questions from the DSE 2018 Delhi School of Economics MA entrance paper. Open any question for the worked solution.

1. Q1Let ℝ be the set of real numbers and f: ℝ → ℝ be a continuous and concave function. Which of the following statements is correct?
2. Q2The maximum value of f(x,y) = (xy)^(1/2), subject to |x| ≥ |y| and |x| + |y| ≤ 1, is
3. Q3Consider an exchange economy with two agents, 1 and 2, and two goods, X and Y. Each agent's consumption set is ℝ²₊. The endowments of agents 1 and 2…
4. Q4Let ℝ be the set of real numbers and let D be the set of functions d: ℝ × ℝ → ℝ that satisfy, for all x, y, z ∈ ℝ: (i) d(x,y) ≥ 0; (ii) d(x,y) = 0…
5. Q5Let f: [0,1] → ℝ be twice differentiable. Suppose that the line segment joining the points (0, f(0)) and (1, f(1)) intersects the graph of f at a…
6. Q6Consider the matrix A = [cos θ -sin θ; sin θ cos θ] where θ ∈ [0, 2π). The inner product of vectors v = (v₁, v₂) and w = (w₁, w₂) in ℝ² is defined…
7. Q7The set [0,1]
8. Q8A sequence of real numbers (xₙ) converges to x. Consider the following claims: (I) The sequence (xₙ₊₁/xₙ) converges to 1. (II) The sequence (xₙ +…
9. Q9Persons 1, 2 and 3 divide 12 indivisible chocolates; each person's preference is strictly increasing in chocolates. Person 1 proposes a division and…
10. Q10The set {f₁, …, fₙ}, where each fₖ is a real-valued function defined on ℝ, is said to be linearly independent if c₁, …, cₙ ∈ ℝ and Σₖ₌₁ⁿ cₖ fₖ(x) =…
11. Q11Let f: [0,1] → ℝ be differentiable and suppose that |f′(x)| < 1 for every x ∈ [0,1]. Then, there
12. Q12Consider a closed macroeconomy with demand side Y = C₀ + c(Y - τ Y) - α r + G₀ and M₀ = KPY - β r, where C₀, G₀, M₀, K, c, τ, α, β are positive…
13. Q13Consider a closed macroeconomy with Y = C₀ + c(Y - τ Y) - α r + G₀ and M₀ = KPY - β r, where all constants are positive and c, τ ∈ (0,1). The AD…
14. Q14Consider a closed macroeconomy with Y = C₀ + c(Y - τ Y) - α r + G₀ and M₀ = KPY - β r, all constants positive and c, τ ∈ (0,1). Suppose the interest…
15. Q15In a multiple regression model with three right-hand-side variables and 105 observations estimated by OLS, the researcher must decide whether to…
16. Q16For a variable x the standard error of the sample mean is 20 when samples of size 25 are taken and 10 when samples of size 100 are taken. A…
17. Q17A consumer lives for two periods and chooses consumptions C₁ and C₂ to maximise utility u(Cₜ) = σ/(σ - 1)(Cₜ^((σ - 1)/σ) - 1). Future consumption is…
18. Q18Consider the game with row-player payoff first, column-player second: for rows U, M, D and columns L, M, R, the payoffs are U: (2,0),(3,3),(0,0); M…
19. Q19In a multiple regression model, the Durbin-Watson test statistic is 1.3, while the critical lower and upper values are 1.5 and 1.7 respectively.…
20. Q20Consider the model y = β₀ + β₁ X₁ + β₂ X₂ + ϵ, estimated using OLS. Which of the following will lead to a higher Var(β̂₂)? (i) Smaller sample size…
21. Q22Consider a country with two citizens, 1 and 2. The government is considering a scheme that will cost 100. It does not know the citizens' true…
22. Q23Consider a Solovian economy with aggregate production function Yₜ = Kₜ^(1/2) Nₜ^(1/2). The population is 100 (no growth) and the initial capital…
23. Q24Consider a Solovian economy with Yₜ = Kₜ^(1/2) Nₜ^(1/2), population 100 (no growth), initial capital 9, saving half of income, and 100% depreciation…
24. Q25Suppose X₁, …, Xₙ are observed completion times of an experiment, each uniformly distributed on [0,1] and independent. If Y is the maximum observed…
25. Q26Consider the function f(x,y) = xy/(x² + y²) if (x,y) ≠ (0,0), and f(0,0) = 0. Which of the following is correct?
26. Q27Suppose we have estimated y = 10 + 1.5x + 4D, where y is earnings, x is experience, and D = 0 for females, 1 for males. If we had coded the dummy as…
27. Q28Which of the following is/are the consequence(s) of including an irrelevant variable in a multiple linear regression model? (i) The variances of the…
28. Q29Suppose Y is a random variable with uniform distribution on the interval [-π/2, π/2]. The value of the (cumulative) distribution function of the…
29. Q30A student answers a multiple-choice question with m possible answers. The student knows the correct answer with probability p; if she knows it she…
30. Q31Consider the game with row payoff first, column second: for rows U, D and columns L, R, the payoffs are U: (x,x),(z,y) and D: (y,z),(y,y). This game…
31. Q32Consider an economy with no capital. The final good is produced with a linear labour technology Yₜ = Aₜ(1 - λₜ)L₀, where L₀ is the constant labour…
32. Q33Consider the economy of the previous problem, with final-good technology Yₜ = Aₜ(1 - λₜ)L₀ and technology evolving as dA/dt = λₜ L₀ Aₜ. For any…
33. Q34Duopolist firms 1 and 2 sell a homogeneous good with demand Q = 100 - 2P, where Q is quantity demanded at price P. Firms 1 and 2 have constant…
34. Q35Voters arrive at a social ranking of alternatives by consulting a 'holy book': the social ranking is the ranking found in this book. Which of…
35. Q36A consumer weakly prefers a basket (a₁, a₂, a₃) to a basket (b₁, b₂, b₃) if a₁ ≥ b₁, a₁ + a₂ ≥ b₁ + b₂, and a₁ + a₂ + a₃ ≥ b₁ + b₂ + b₃. Which of…
36. Q37Suppose the economy-wide union sets the wage for employed workers by W = P^e(Z - α u) with unemployment rate u and labour force L. The producer sets…
37. Q38Consider the regression model yᵢ = α₀ + α₁ x₁ᵢ + α₂ x₂ᵢ + uᵢ. A researcher wants to conduct White's heteroscedasticity test using the residuals from…
38. Q39Suppose you have run the regression y = α + β x + γ Urban + θ Immigrant + δ Urban×Immigrant + ϵ, where Urban indicates living in a city and…
39. Q40Consider the Solovian economy with Yₜ = Kₜ^(1/2) Nₜ^(1/2), population 100 (no growth), initial capital 9, saving half of income, and 100%…
40. Q41Consider the Solovian economy with Yₜ = Kₜ^(1/2) Nₜ^(1/2), population 100 (no growth), and 100% depreciation within one period. Suppose households…
41. Q42Consider an exchange economy with two agents, 1 and 2, and two goods X and Y. Given bundles (a,b),(c,d) ∈ ℝ²₊ with (a,b) ≥ (c,d) and (a,b) ≠ (c,d)…
42. Q43Consider the Solovian economy with Yₜ = Kₜ^(1/2) Nₜ^(1/2), population 100 (no growth), and 100% depreciation within one period. In the previous…
43. Q44A monopolist faces demand D(p) = α - p and cost function C(q) = cq. She can advertise to increase demand: advertisement level θ costs θ²/2 and…
44. Q45Persons 1, 2 and 3 divide 12 indivisible chocolates; each person's preference is strictly increasing in chocolates. Person 1 proposes a division and…
45. Q46Player 1 moves first and chooses L or R. If she plays R, the game ends with payoffs (10, 0). If she plays L, player 2 chooses L or R; if R, the game…
46. Q47If the function f: ℝ² → ℝ is concave, then
47. Q48Let ℝ be the set of real numbers. A subset E of ℝ is said to be open if, for every x ∈ E, there exists r > 0 such that (x - r, x + r) is a subset of…
48. Q49You have 100 observations on y, with average value 15, and on x, with average value 8. From an OLS regression, you have estimated the slope on x to…
49. Q50Suppose your data produces the regression result y = 10 + 3x. Scale y by multiplying observations by 0.9 and do not scale x. The new intercept and…