2012-phd.pdf

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Time. 3 hours Maximum marks. 100 General instructions: Please read the following instructions carefully

• Check that you have a bubble-sheet and an answerbook accompanying this examination booklet. Do not break the seal on this booklet until instructed to do so by the invigilator.

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• Only after the invigilator announces the start of the examination, break the seal of the booklet. Check that this booklet has pages 1 through 12. Bring any missing pages to the attention of the invigilator.

Signature Name

Roll number

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Part I

• This part of the examination will be checked by a machine. Therefore, it is very important that you follow the instructions on the bubble- sheet.

• Fill in the required information in Boxes 1, 2, 4 and 5 on the bubble-sheet.

The invigilator will sign in Box 3. You do not need to fill in Box 6.

• Part I of the examination consists of 20 multiple-choice questions. Each question is followed by four possible answers, at least one of which is correct.

If more than one choice is correct, choose only the ‘best one’. Among the correct answers, the ‘best answer’ is the one that implies (or includes) the other correct answer(s). Indicate your chosen best answer on the bubble- sheet by shading the appropriate bubble.

• For each question, you will get 2 mark if you choose only the best answer.

If you choose none of the answers, then you will get 0 for that question.

However, if you choose something other than the best answer or multiple answers, then you will get −²/3 mark for that question.

QUESTION 1. Consider the following model, estimated using OLS Y_i = βX_i +ε_i; i = 1,2, ..., n

where there is no intercept, and V ar(εi) = σ². Which of the following statements is not true?

(a) The R² from this regression can be large even if X and Y have low correlation.

(b) The least squares residuals need not sum to zero.

(c) The mean square error is given by ^P(Y_i −Yˆ)²/(n−1).

(d) The least squares estimator of the slope coefficient is given by:

n^PX_iY_i−^PX_i^PY_i n^PX_i² −(^PX_i)² Answer: (d)

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QUESTION 2. An analyst trying to estimate the demand for rice has estimated the following two models

Model 1: D = 50 + 0.3Y + 0.1P + 12N; R² = 0.7.

Where D is the demand for rice per household, Y is income per household, P is the price of rice andN is household size. The standard errors associated with the coefficient Y is 0.1, that associated with P is 0.05, and of N is 3.

Model 2: D/N = 50 + 0.2Y /N −0.5P; R² = 0.9

where the standard errors of the estimated slope coefficients are 0.1 and 0.2 respectively. Which of the following statements is true?

(a) Model 2 is preferred to Model 1 because it has a higher R².

(b) Model 1 is preferred to Model 2 because the coefficients are all signif- icant

(c) The two models are not comparable in terms of fit (d) None of the above

Answer: (c)

QUESTION 3. Consider the regressions Y_i^∗ = ˆβ₁^∗ + ˆβ₂^∗X_i^∗ + ˆu_i^∗, and Y_i = βˆ₁ + ˆβ₂X_i+ ˆu_i, where Y_i^∗ = w₁Y_i and X_i^∗ = w₂X_i; w₁, w₂ are constants. Is it true that

(a) ˆβ1

∗ = w2βˆ1

(b) ˆβ₂^∗ = ^w_w²

1

βˆ₂ (c) V ar( ˆβ2

∗) =

w1

w2

2

V ar( ˆβ2)

(d) r_xy² 6= r_x²∗y^∗, where r denotes correlation coefficient Answer: (c)

QUESTION 4. For variables X and Y we have the data

PXY = 350, ^PX = 50, ^PY = 60, X = 5, σ_X² = 4, σ²_Y = 9

where X denotes the mean of X and σ_X² denotes the variance of X. Which of the following holds

(a) A one unit change in X causes a 1.25 unit change in Y, and a one unit change in Y causes a 0.6 unit change in X

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(b) A one unit change in X causes a 0.6 unit change in Y, and a one unit change in Y causes a 1.25 unit change in X

(c) A 10% change in X causes a 15% change in Y

(d) The regression of Y on X passes through the origin Answer: (a)

QUESTION 5. X is a normally distributed random variable with unknown mean µ and standard deviation equal to 2. The value of the sample mean from a random sample of size 25 is 10. Which of the following values lie within the 95% confidence interval for µ?

(a) 9.3 (b) 9.8 (c) 10.6

(d) All of the above Answer: (d)

QUESTION 6. An urn contains equal number of green and red balls. Sup- pose you are playing the following game. You draw one ball at random from the urn and note its colour. The ball is then placed back in the urn, and the selection process is repeated. Each time a green ball is picked you get 1 Rupee. The first time you pick a red ball, you pay 1 Rupee and the game ends. Your expected income from this game is

(a) ∞

(b) Positive but finite (c) Zero

(d) Negative Answer: (c)

QUESTION 7. Consider the system of equations αx+βy = 0

µx+ νy = 0

α, β, µ and ν are i.i.d random variable. Each of them takes value 1 and 0 with equal probability.

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Statement A: The probability that the system of equations has a unique solution is ³₈.

Statement B: The probability that the system of equations has at least one solution is 1.

(a) Both the statements are correct (b) Both the statements are false

(c) Statement A is correct but B is false (d) Statement B is correct but A is false Answer: (a)

QUESTION 8. What is the total number of local maxima and local minima of the following function

f(x) =







(2 +x)³ if −3 < x ≤ −1 x²^/³ if −1 < x < 2 (a) 1

(b) 2 (c) 3 (d) 4

Answer: (b)

QUESTION 9. A rectangle has its lower left hand corner at the origin and its upper right hand corner on the graph of f(x) =x² + (1/x²). For which x is the area of the rectangle minimized?

(a) x = 0 (b) x = ∞ (c) x = ¹₃¹^/⁴ (d) x = 2¹^/³ Answer: (c)

QUESTION 10. Let x = (x₁, x₂, . . . , x_n) and y = (y₁, y₂, . . . , y_n) be two vectors in <ⁿ. Define, x ⊗ y = ^Pn_i=1x_i|y_i|, where |y_i| stands for absolute value. How many of the following statements are correct

Statement 1: x⊗y = y ⊗x

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Statement 2: x⊗x = 0 implies x = (0,0, . . . ,0)

Statement 3: x⊗(c.y) = c(x⊗y), where c ∈ <₊₊ andc.y = (cy₁, cy₂, . . . , cy_n) (a) 0

(b) 1 (c) 2 (d) 3

Answer: (b)

QUESTION 11.A firm producing hockey sticks in Punjab has a production function given byQ = 2√

KLwhereK stands for capital, L stands for labour and Q stands for output. The rental rate of capital is Rs 1 and the wage rate is Rs 4. What will be the firm’s cost function?

(a) 2Q (b) Q (c) Q² (d) 2Q² Answer: (a)

QUESTION 12. A and B live in an exchange economy. There are two goods X and Y. utility functions of A and B are U^A(x, y) = x + y and U^B(x, y) = x² +y², respectively. A’s endowment is 2 units of X and 1 unit of Y, while B’s endowment is 1 unit of X and 2 units of Y. Consider the following allocations

Allocation 1: A gets 1.5 units of X and zero unit of Y, B gets 1.5 units of X and 3 units of Y

Allocation 2: A gets 1.5 units of X and 1.5 unit of Y, B gets 1.5 units of X and 1.5 units of Y

(a) Both the allocations are Pareto optimal (b) None of the allocations is Pareto optimal (c) Only allocation 1 is Pareto optimal

(d) Only allocation 2 is Pareto optimal Answer: (c)

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QUESTION 13. Fill in the blanks: Given a downward sloping linear demand curve and constant marginal cost curves, if a per unit tax is imposed on a monopoly, a monopoly will the quantity of its good and its revenue after tax will

(a) Increase; Decrease (b) Increase; Increase (c) Decrease; Increase (d) Decrease; Decrease Answer: (d)

QUESTION 14. Consider the following two-player game. The players simultaneously draw one sample each from a continuous random variable X, which follows U nif orm[0,100]. After observing the value of her own sample, which is private information (that is, opponent does not observe it), players simultaneously and independently choose one of the following: SW AP, RET AIN. If both the players choose SW AP then they exchange their initially drawn numbers. Otherwise, if at least one person chooses RET AIN, both of them retain their numbers. A player earns as many Rupees as the number she is holding at the end of the game.

Probability that the players will exchange their initially drawn numbers is (a) 1

(b) ¹/² (c) ¹/³ (d) 0

Answer: (d)

QUESTION 15. The productivity of a labourer depends on his daily wage.

In some range of wage, the more he is paid, the better his health and the more work he is able to do in a given day. Suppose that the relationship between productivity and daily wage is as follows. No work is done for wage below Rs. 20 per day. Each rupee earned above 20 increases productivity by 5 units until the daily wage is Rs. 140 per day. Beyond this level of wage, productivity is constant. If a farmer needs to hire labour for a total of 6000 units of work per day, how many labourers is he likely to hire?

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(a) 4 (b) 5 (c) 6 (d) 10

Answer: (d)

The following set of information is relevant for the next five questions. Read the information carefully before answering the questions below.

Consider an economy where the nominal wage rate is set by a process of wage bargaining between the workers and the producers before actual production takes place. Thus at any period t, the nominal wage rate (W_t) is a function of the expected price level (P_e^t), the rate of unemployment (ut) and the average productivity of the workers (At). The exact functional relationship is given below:

W_t = P_t^eF(u_t, A_t); F_u < 0;F_A > 0

Once the nominal wage is determined, the producers set the actual price level (P_t) as a constant mark up (µ) over the nominal wage rate: P_t = (1 +µ)W_t. Define the actual rate of inflation as π_t = ^P^t_P^−P^t−1

t−1 and the expected rate of inflation as π_t^e = ^P^t^e_P^−P^t−1

t−1

QUESTION 16. Given the above wage and price setting equations, derive the relationship between expected rate of inflation and actual rate of inflation.

Which of the following equations represents this relationship?

(a) π_t = (1 +π_tê)[F(u_t, A_t)−µ]−1 (b) π_t = (1 +πê_t)(1 +µ)F(u_t, A_t)−1 (c) π_t = πê_t(1 +µ)F(u_t, A_t)

(d) None of the above Answer: (b)

QUESTION 17. Suppose the average productivity of workers remains constant at a level A. Given the relationship in previous question, which of the following equations defines the ‘natural rate of unemployment’ ?

(a) F(u_t, A) = µ

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(b) F(u_t, A) = _1+µ¹ (c) F(ut, A) = _1+µ^µ (d) None of the above Answer: (b)

QUESTION 18. A one-shot increase in the average productivity of worker, ceteris paribus, leads to

(a) An increase in the natural rate of unemployment (b) A decrease in the natural rate of unemployment (c) No change in the natural rate of unemployment

(d) Some change in the natural rate of unemployment but the direction is ambiguous

Answer: (a)

QUESTION 19. Go back to the relationship between π_t and π_tê above (first question of this list). Suppose expectations are static, i.e., π_tê = π_t−1. Also let F(u_t, A_t) = Â_u^t

t −1 and A_t = A (a constant). The corresponding value of the non-accelerating inflation rate of unemployment is given by

(a) u_t =

µ 1+µ

A (d) u_t =

µ 2+µ

A (c) ut =

1+µ 2+µ

A (d) None of the above Answer: (c)

QUESTION 20. Now suppose the workers productivity increases at a constant rate γ, that is, ^A^t_A^−A^t−1

t−1 = γ . To maintain a non-accelerating inflation rate, rate of unemployment has to increase at the rate

(a) γ (b) µ (c) µ+γ (d)

µ 1+µ

γ Answer: (a)

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End of Part I. Proceed to Part II of the examination on the next page.

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Part II

• Attempt any three questions, in the answerbook provided to you. If you attempt more than three questions, only the first three will be counted.

Each question carries 20 marks. The allocation of marks for each part is given alongside.

• Fill in your Name and Roll Number on the detachable slip of the answerbook.

QUESTION 21. Suppose that a single observation X is taken from the uniform distribution on the interval [0, θ], where the value of θ is unknown, and the following simple hypotheses are to be tested:

H₀ : θ = 1 H₁ : θ = 2

Let α(δ) and β(δ) denote the Type I and Type II errors associated with this problem of testing hypotheses.

(a) Find a test procedure for which α(δ) = 0 and β(δ) < 1. [2]

(b) Among all test procedures for whichα(δ) = 0, find the one for which β(δ)

is a minimum. [2]

(c) If instead that we had a sample of sizendrawn from the same distribution.

Consider once again all test procedures for which α(δ) = 0 and find the one

for which β(δ) is a minimum. [8]

(d) Suppose now that we are trying to test a simple null hypothesis that H0 : θ = θ0 against the composite hypothesis that θ > θ0. Based on a sample of size 1, derive and graph the power function of the optimal test

with α(δ) = 0. [8]

QUESTION 22. This question has two parts.

(i). Consider the following simple regression model Y = α+βX + e

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where E(e) = 0 and V ar(e) = σ², where X takes on value 0 for the first n1 observations and 1 for the next n2 observations (the total sample size is n = n1 +n2).

Derive the least squares estimator of β and interpret it in this specific case.

Note that no marks will be given for a general least squares formula or inter-

pretation. [5 + 5]

(ii). Consider the following two-variable regression model:

Y = α +β₁X1 +β₂X2 +e

where E(e) = 0 and V ar(e) = σ². This model is estimated using ordinary least squares.

Now run three separate regressions:

Regression 1: Y on X1 alone with estimated residual ˜Y Regression 2: X2 on X1 with estimated residual ˜X2 Regression 3: ˜Y on ˜X2

(a) Show that the slope coefficient in Regression 3 above is the same as the estimator for the least squares slope coefficient β₂ in the two-variable regression model. This is part of the Frisch-Waugh-Lovell theorem. [5]

(b) Interpret the implications of your result. [3]

(c) What would happen if X1 and X2 were perfectly uncorrelated? [2]

QUESTION 23. This question has two parts.

(i). Consider the following game

L R

T (x, x) (b, y) B (y, b) (a, a)

Find all Nash equilibria of this game (including mixed strategy Nash equilibria) if

(a) y > x > a > b [4]

(b) x > y > a > b [4]

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(c) x = y > a > b [4]

(ii). Consider a two period game consisting of two firms. Firm 1 is an in- cumbent having an installed capacity of ¹₆ unit and Firm 1 cannot change its capacity level due to financial constraint. However, firm 2 can choose a capacity q₂^∗ at a per unit capacity cost ¹₂. Thus, the capacity is a choice variable only for firm 2. In period 1, firm 2 enters the market and chooses a capacity while in period 2 firms compete with each other in prices. Both the firms produce output at zero cost upto their capacity levels and beyond their capacity levels they cannot produce any output at a finite cost. The market demand is given by p = 1−q. Derive the optimal choice of capacity for firm

2 and the equilibrium market price. [8]

QUESTION 24. Two individuals, 1 and 2, live next door to each other.

They derive utility from the beauty of both their own garden and that of their neighbour, as well as their respective consumption of other goods. The utility functions are as follows

u₁ = 8 logx₁ + 2 logx₂ +y₁ u₁ = 4 logx₁ + 6 logx₂ +y₂

wherexi is the number of hours of gardening work applied to agenti’s garden, and y_i is agent i’s spending on other goods (whose price is one). Each hour of gardening work costs 2 (the cost of raw materials, foregone income, etc.).

Each individual has initial income I, which must be divided between beauti- fying his own garden and purchase of other goods. Money can be transferred costlessly from one individual to another.

(a) Find the amount of gardening each agent will undertake if they acted

rationally and selfishly. [4]

(b) Is the resultant allocation Pareto efficient? Derive the values of x1 and x2 that every Pareto efficient allocation must satisfy. [4]

(c) Suppose the government can provide a per unit subsidy s towards the cost of gardening, which is financed by an equal lump sum tax on the two individuals. Find the optimal subsidy which maximizes social surplus. [4]

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(d) Does the optimal subsidy lead to a Pareto improvement relative to no subsidies and selfish behaviour? Does it lead to a Pareto efficient allocation?

Discuss why or why not. [4]

(e) In the light of this example, discuss whether there can be a non paternal- istic argument for laws that prohibit the demolition of historic buildings or

impose regulations on new construction. [4]

QUESTION 25.

(a) Show that in a neoclassical growth model without technical progress the growth rate of per capita income in the long run is zero. [10]

(b) ‘The long run constancy of per capita income in this model is entirely due to the assumption of diminishing marginal productivity of capital’ - do

you agree? Elaborate. [10]

QUESTION 26.

(a) Explain the concept of rational expectations. How is it different from

adaptive expectations? [8]

(b) Describe the policy ineffectiveness proposition. Does it always hold?

Explain intuitively. [12]

End of Part II