### Section - A

1. Answer any ** ten** parts of the following questios. Answers to each part should be of approximatel 50 words.

Marks: 5 × 10 = 50

#### (a) If the law of demand is \(x = a{e^{ - bp}}\), where p is price and x is quantity demanded. Express price elasticity of demand, total revenue and marginal revenue as function of x.

(Comment for solution.)

#### (b) Explain 'Leontief Inverse' in the input-output model suggested by W.W. Leontief.

(Comment for solution.)

#### (c) Graphically explain the expansion path of a firm taking labour and capital as inputs.

(Comment for solution.)

#### (d) What is adverse selection in insurance markets? How the problem can be solved?

(Comment for solution.)

#### (e) Describe Gini's coefficient as a measure of inequality.

(Comment for solution.)

#### (f) Show that Cobb-Douglas production function \(Q = A{L^\alpha }{K^{1 - \alpha }}\), where symbols have usual meaning, exhibits constant returns to scale but diminishing returns to a factor of production.

(Comment for solution.)

#### (g) What is monopoly power? Give an expression for measuring it.

(Comment for solution.)

#### (h) Why does a perfectly competitive firm keep on producing in the short-run even when it is incurring losses? Explain also when the firm will shut down. Use suitable diagram.

(Comment for solution.)

#### (i) WHat are type I and type II errors in testing of a hypothesis?

(Comment for solution.)

#### (j) Given utility function U = q_{1}q_{2} and budget constraint

Y = p_{1}q_{1} + p_{2}q_{2}

derive the indirect utility function. (Marks - 5)

Solution video:

#### (k) State the causes of market failure.

(Comment for solution.)

### Section - B

Answer any **SIX** of the following questions in about 150 words.

Marks: 15 × 6 = 90

#### 2. Cardinal utility approach and ordinal utility approach to demand suggest the same decision rule for the optimizing consumer (which one?). Yet, the latter approach is preferred over the former. Why?

Solution Video:

#### 3. Describe Von Neuman and Morgenstern utility index. Is this index unique? Explain.

(Comment for solution.)

#### 4. Define elasticity of goods substitution and distinguish it from cross-price elasticity of demand. Which one is a better measure of substitution and why?

(Comment for solution.)

#### 5. Write dual of the following linear programme problem and solve the obtained dual graphically:

Minimize : Z = 3x_{1} + 3x_{2}

subject to:

x_{1} + 2x_{2} ⋝ 1

2x_{1} + x_{2} ⋝ 1

x_{1} ⋝ 0, x_{2} ⋝ 0

(Marks - 15)

Solution video:

#### 6. Critically examine Hicks-Kaldor criterion of compensation. Give Scitovsky's improvement over this criterion.

(Comment for solution.)

#### 7. State and explain the assumptions applying ordinary least square method (OLS) method to two variable linear regression model: \[{Y_t} = {b_0} + {b_1} + {u_t}\] \[t = 1,2,...,n\]

Following assumptions apply to ordinary least square (OLS) method to two variable linear regression model:

**Linearity in parameters:**The regression model is**linear in theparameters**. In the regression model in question, b_{0}and b_{1}are parameters to be estimated, they need to be linear else OLS will not be applicable. However, the regression model need not to be linear in variables, that is, X and Y here.- Values of X are fixed: When values of X are fixed, the disturbance term, u
_{t}is not correlated with the independent variable, X_{t}. It means X_{t}and u_{t}are independent. Symbolically: \[{\mathop{\rm cov}} \left( {{u_t},{X_t}} \right) = 0\] **u**This means the values of u_{t}is a random variable:_{t}in any one period depends on chance and each value has a probability of occurance. It values may be positive, negative or zero.**Zero Mean of the Disturbance term:**The disturbance term u_{t}has zero mean in any particular period given the values of X_{t}. For each value of X, u has a random value. Some are positive, some negative or 0. Positive and negative values cancel each other and the sum of u_{t}is equal to zero. As a result, the mean is also zero. Symbolically, we can write: \[E\left( {{u_t}|{X_t}} \right) = 0\] Since, X is not a random variable (stochastic), we can also write: \[E\left( {{u_t}} \right) = 0\]**Homoscedasticity:**Homoscedasticity means that the variance of the disturbance term u_{t}is constant in each period regardless of the values of X_{t}. We can derive this as follows: \[{\mathop{\rm var}} \left( {{u_t}} \right) = E{\left[ {{u_t} - E\left( {{u_t}} \right)} \right]^2}\] Since, \(E\left( {{u_t}} \right) = 0\) \[{\mathop{\rm var}} \left( {{u_t}} \right) = E\left[ {u_t^2} \right] = \sigma _u^2\]**The disturbance term, u**This assumption and the above assumption can be sumarized as follows: \[u \sim N\left( {0,\sigma _u^2} \right)\] This means that u is normally distributed with zero mean and constant variance._{t}is normally distributed.**No Autocorrelation:**Correlation between any two series, u_{t}and u_{s}is zero given that the corresponding series of X_{t}and X_{s}are non-stochastic. This is possible when observations are sampled independently. Symbolically: \[{\mathop{\rm cov}} \left( {{u_t},{u_s}} \right) = 0\] Covariance is zero means that correlation is also zero and zero correlation means that there no linear relationship between concerned variables.- The number of observations (n) must be greater than the number of explanatory variables. It is not possible to estimate parameters with a single pair or observation.
- Nature of Independent variable: Here, all values of the independent variable, X, must not be same. In technical terms,
the variance of X
_{t}must be a positive number. The should also not be outliers, that, very small or large values of X compared to other data set.

#### 8. "In the long-run competitive equilibrium rewarding each input according to its marginal physical product precisely exhausts the total physical product." Critically examine the above statement.

(Comment for solution.)

### Section - C

Attempt any ** two** of the following questios in about 300 each.

Marks: 30 × 2 = 60

#### 9. Consider the following duopoly. Demand is given by \(P = 10 - Q\), where \(Q = {Q_1} + {Q_2}\). The firm's cost functions are:
\[{C_1}({Q_1}) = 4 + 2{Q_1}\,\,{\rm{and}}\,\,{C_2}({Q_2}) = 3 + 3{Q_2}\,\,\]

(a) Suppose both firms have entered the industry. What is joint profit maximising level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry?

(b) What is each firm's equilibrium output and profit if they behave non-co-operatively?

(Comment for solution.)

#### 10. Can the threat of a price war deter entry by potential competitors? What actions might a firm take to make this threat credible? Give example.

(Comment for solution.)

#### 11. For statistically estimated demand function for the commodity X,
\[{D_x} = \frac{{1547P_x^{0 \cdot 2}P_y^{0 \cdot 3}{A^{0 \cdot 4}}}}{{P_z^{0 \cdot 5}{B^{0 \cdot 3}}}}\]
(where x, y, z are goods, A stands for advertisement outlay, B for budget of the consumer and \({P_x},\,{P_y},\,{P_z}\) are prices of goods x, y, z respectively).

Answer the following:

(a) How are x, y and z related?

(b) Whether x is an inferior, normal or Giffen type good?

(c) What would be the percentage change in demand for \(x\,\,(i.e.\,\,\,\,{D_x})\) and in which direction if advertisement outlay increases by 50 percent?

(Comment for solution.)

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