### Previous Year Paper Solution | Indian Economic Service Exam 2009 | General Economics - I

#### Q. No. 1 (a) - How do you draw a Lorenz curve? Explain its use.

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#### Q. No. 1 (b) - What is 'shadow price'? Why are shadow prices used in project analysis as against market prices?

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#### Q. No. 1 (c) - What is Engel's law? Which sector/product(s) of an economy operate under this law?

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#### Q. No. 1 (d) - State the first and second fundamental theorems of welfare economics, and comment on their usefulnesses.

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#### Q. No. 1 (e) - State the Kuhn - Tucker conditions.

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#### Q. No. 1(f) - Explain how Pareto's law of distribution is useful in measuring income distribution.

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#### Q. No. 1 (g) - Explain total factor productivity and mention any two popular measures of the same.

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#### Q. No. 2 - Show how a demand function may be derived from the Cobb-Douglas utility function.

The Cobb-Douglas utility function can be written as follows: $U = {x^\alpha }{y^\beta }$ Where x and y are quantity demanded of the two goods, say, Good X and Good Y. A consumer's objective is to maximize utility with the given budget constraints which can be written as: $I = {p_x}x + {p_y}y$ Equlibrium Condition in this two commodity is given by: $MR{S_{x,y}} = \frac{{{p_x}}}{{{p_y}}}$ Where, $MR{S_{x,y}} = - \frac{{dy}}{{dx}} = \frac{{M{U_x}}}{{M{U_y}}}$ We can derive the MU of x and y from the utility function as follows: $M{U_x} = \frac{{\partial U}}{{\partial x}}$ $M{U_x} = \alpha {x^{\alpha - 1}}{y^\beta }$ $M{U_x} = \frac{{\alpha {x^\alpha }{y^\beta }}}{x}$ $M{U_x} = \frac{{\alpha U}}{x}$ Similarly, $M{U_y} = \frac{{\beta U}}{y}$ Now, we can derive MRS from the above result: $MR{S_{x,y}} = \frac{{M{U_x}}}{{M{U_y}}} = \frac{{\alpha U}}{x} \times \frac{y}{{\beta U}}$ $MR{S_{x,y}} = \frac{{\alpha y}}{{\beta x}}$ From the results derived above, we can write equlibrium condition for Cobb-Douglas as: $\frac{{\alpha y}}{{\beta x}} = \frac{{{p_x}}}{{{p_y}}}$ Rearranging the equation above, we can write: $y = \frac{\beta }{\alpha }\frac{{{p_x}x}}{{{p_y}}}$ Substituting this value of y in the budget constraint, we get: $I = {p_x}x + {p_y}\frac{\beta }{\alpha }\frac{{{p_x}x}}{{{p_y}}}$ We can remove py from the equation above: $I = {p_x}x + \frac{\beta }{\alpha }{p_x}x$ Rearranging the equation above, we get: $I = {p_x}x\left( {\frac{{\alpha + \beta }}{\alpha }} \right)$ Let $${\alpha + \beta = 1}$$, $x = \frac{{\alpha I}}{{{p_x}}}$ this is the demand function for Good X, which shows a negtaive relationship between quantity demanded and the price of the commodity. Similarly, by substituting the value of this demand function in the budget constraint, we can derive the demand function for Good Y as follows: $y = \frac{{\beta I}}{{{p_y}}}$

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#### Q. No. 3 Formulate a translog cost function and show how the elasticity of input substitution may be obtained.

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#### Q. No. 4 - What is "moral hazard" in economic theory? Discuss a situation that would describe a moral hazard problem.

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#### Q. No. 5 - What is "free-rider" problem? Discuss the possible solutions ot this problem.

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#### Q. No. 6 - State and explain the Coase theorem in the context of pollution control.

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#### Q. No 7 - Explain Leontief's static input-outut model and describe its limitations.

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Q. No. 8 - What is "prisoner's dilemma"? How is it related to strictly dominant strategy?

Prisoner's Dilemma is a hypothetical situation used in Game Theory to explain Nash Equilibrium and Dominance Rule. The hypothetical situation is:
Two suspects, involved in a crime together, are taken into police custody. The police do not have sufficient evidence to convict them so they can be convicted if only if they confess. The police interrogate them in separate cells so that they couldn't communicate and the police put some conditions before them which can be explained with the following pay-off matrix:

Suspect B
ConfessNot Confess
Suspect AConfess (-3, -3)(0, -6)
Not Confess (-6, 0)(-1, -1)

The payoffs in the payoff matrix represent years of jail as punishment. Minus sign implies that jail is a loss, not profit. So, suspects would like to minimize year of jail. It exhibits the following conditions:

• Payoffs (-3, -3) implies that both will get three year jail if both confess.
• Payoffs (0, -6) implies that Suspect A will get zero year jail, that is, no punishment and Suspect B will get six years of jail if the former confesses and the latter does not confess.
• Similarly, payoffs (-6, 0) implies that Suspect B will get zero year jail, that is, no punishment and Suspect A will get 6 years of jail if the former confesses and the latter does not confess.
• Lastly, payoffs (-1, -1) implies that both will get only one year jail if both do not confess.

Let us first look at the matrix from A's point of view. Given the conditions above, it is beneficial for A to confess. If he confesses, he will get either three year jail (in case B also confesses) or zero year jail(in case B does not confess). If he does not confess, he will get either six year jail (in case of B confesses) or one year jail (in case B does also not confess). A would choose to confess as zero to three year jail is better than one to six years jail.
Similarly, if you look at the payoffs from B's point of view, you will find the same result, that is, B will also confess. It means that the optimum strategy for both is to confess which represented payoffs (-3, -3).
The situation described above can be found in many type of real life situation where two persons are interdependent for optimization of their respective payoffs. This is what we call Prisoner's Dilemma.

Strictly Dominant Strategy: It is called a dominant strategy if a player in a game can play its optimal strategy regardless of what other player will play. Here, A and B both have the dominant strategy. A's optimal stategy is to confess where he can expect that he will get either 0 or 3 year jail which is better than the the alternative strategy, that is, not to confess where he can get 1 or 6 year jail. He can easily choose his best strategywithout worrying about what B will play. Similar situation is faced by B. The best stategy for B is also to confess without worrying about what A will play.
It is called strictly dominant strategy when the alternative stategy always makes worse off. Here, we can see that alternative strategy (Not Confess) of A makes him better off than the optimal strategy if B does not confess. Same situation is faced by B. Therefore, neither of them has strictly dominant strategy.

#### Q. No. 9 - What is a superlative index number? How is it related to the theory of aggregation?

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#### Q. No. 10 - Define a quadratic form, and state the conditions under which it is (i) positive definite, (ii) positive sem-definite, and (iii) negative definite.

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#### Q. No. 11 - Explain the principl of least squares as a basis for multiple regression analysis. Also state the underlying standard assumptions of ordinary least Squares estimation. Explain further the consequences of violation of such assumptions.

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#### Q. No. 12 - In the case of a pure exchange econom how do you characterize Walrasian equilibrium? Also establish the conditions under which such an equilibrium exists.

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#### Q. No. 13 (a) - State and prove Euler's theorem. Is i relevant in the context of a firm?

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#### Q. No. 13 (b) - Distinguish between technical and allocative efficiency in the context of a firm. Give an example.

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#### Q. No. 13 (c) - Distinguish between parametric and non - parametric tests in testing of hypotheses.

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1. Sir, when I substitute the demand function for good x in equation of budget constraint, i am unable to obtain the demand function of y. There is no Beta value while rearranging the equation..

2. But i got the correct demand function of Good Y when i substituted the value of demand function for good x in the equation where you found value of Y first and also used it in the budget constraint.

1. $x = \frac{{\alpha I}}{{{p_x}}}$
Substituting:
$I = {p_x} \times \frac{{\alpha I}}{{{p_x}}} + {p_y}y$
$I = \alpha I + {p_y}y$
${p_y}y = I - \alpha I$
$y = \frac{{I(1 - \alpha )}}{{{p_y}}}$
We know that:
$\alpha + \beta = 1$
$\beta = 1 - \alpha$
Therefore,
$y = \frac{{\beta I}}{{{p_y}}}$

3. Thank you sir