#### Q. No. - 1 (a) - Show the conditions for a Cobb-Douglas production function under:
- (i) increasing returns to scale
- (ii) consant returns to scale
- (iii) dimishing returns to scale
Are the law of returns compatible?

(Comment for Solution.)

#### Q. No. 1 (b) - Define homothetic preferences. Explain the common characteristics of such preferences with the help of necessary diagrams.

Solution video:

#### Q. No. 1 (c) - What is monopoly power? What factors determine the amount of monopoly power?

(Comment for Solution.)

#### Q. No. 1 (d) - Explain the difference between Bandwagon effect and Snob effect.

(Comment for Solution.)

#### Q. No. 1 (e) - What is meant by deadweight loss? Why does a price ceiling usually result in a deadweight loss

(Comment for Solution.)

#### Q. No. 1 (f) - State the fundamental theorems of Welfare Economics.

(Comment for Solution.)

#### Q. No. 1 (g) - Public goods are non-rival and non-exclusive. Explain each of these terms and show clearly how they differ from each other.

(Comment for Solution.)

#### Q. No. 2 (a) - Explain the meaning of Nash Equilibrium. How does it differ from the equilibrium in dominant strategies? (8 Marks)

Nash Equilibrium is a strategy profile such that each player's equilibrium strategy is the best response to other players' equilibrium strategy and no player can benefit from deviating to other strategies.

While dominant strategy equilibrium is a strategy profile such that each player's equilibrium dominant strategy is the best response to other players' equilibrium dominant strategy and no player can benefit from deviating to other strategies. Dominant strategy equilibrium exists if only if both players have dominant strategies to play. All games do not hve dominant strategy equilibrium.

All dominant strategy equilibrium is a Nash Equilibrium but all Nash Equilibrium are not dominant strategy equilibrium.

#### Q. No. 2 (b) - Let market demand faced by duopolist be

\[P = 100 - 0.5Q\]
\[Q = {Q_1} + {Q_2}\]
and their respective cost function as:
\[{C_1} = 5{Q_1}\]
and
\[{C_2} = 5{Q_2}\]
Find out Cournot-Nash Equilibrium. (10 Marks)

Let profit function of the first duopolist be: \[{\pi _1} = P{Q_1} - {C_1}\] Substituting the value of P (inverse demand function): \[{\pi _1} = \left( {100 - 0.5Q} \right){Q_1} - 5{Q_1}\] Substituting the value of Q: \[{\pi _1} = \left( {100 - 0.5{Q_1} - 0.5{Q_2}} \right){Q_1} - 5{Q_1}\] \[{\pi _1} = 100{Q_1} - 0.5Q_1^2 - 0.5{Q_1}{Q_2} - 5{Q_1}\] \[{\pi _1} = 95{Q_1} - 0.5Q_1^2 - 0.5{Q_1}{Q_2}\] Applying the first order condition for maximization: \[\frac{{\partial {\pi _1}}}{{\partial {Q_1}}} = 0\] \[95 - {Q_1} - 0.5{Q_2} = 0\] \[{Q_1} = 95 - 0.5{Q_2}\] Similarly, let profit function of the second duopolist be: \[{\pi _2} = P{Q_2} - {C_2}\] Substituting the value of P (inverse demand function): \[{\pi _2} = \left( {100 - 0.5Q} \right){Q_2} - 5{Q_2}\] Substituting the value of Q: \[{\pi _2} = \left( {100 - 0.5{Q_1} - 0.5{Q_2}} \right){Q_2} - 5{Q_2}\] \[{\pi _2} = 100{Q_2} - 0.5{Q_1}{Q_2} - 0.5Q_2^2 - 5{Q_2}\] \[{\pi _1} = 95{Q_2} - 0.5{Q_1}{Q_2} - 0.5Q_2^2\] Applying the first order condition for maximization: \[\frac{{\partial {\pi _1}}}{{\partial {Q_1}}} = 0\] \[95 - 0.5{Q_1} - {Q_2} = 0\] \[{Q_2} = 95 - 0.5{Q_1}\]

Substituting the value of q_{2} in q_{1} and simplifying:
\[{Q_1} = 95 - 0.5\left( {95 - 0.5{Q_1}} \right)\]
\[{Q_1} = 95 - 47.5 + 0.25{Q_1}\]
\[{Q_1} - 0.25{Q_1} = 47.5\]
\[0.75{Q_1} = 47.5\]
\[{Q_1} = \frac{{47.5}}{{0.75}} = \frac{{190}}{3}\]
This is the quantity sold by the first duopolist at P.

Similarly, quantity sold by the second duopolist will be:
\[{Q_2} = \frac{{190}}{3}\]

Note that \(\frac{{190}}{3}\) is not the competitive market equilibrium quantity in the given market. We can find the competitive equilibrium quantity using
D = S. We know the demand curve and supply curve which is also the Average Revenue (AR) curve is the upward sloping portion portion of Marginal Cost (MC) curve. Since the demand curve is linear, AR = P. Thus, we can say P = MC is the equilibrium condition in a competitive market. Now, we can find the equilibrium quantity.

We first need to calculate the MC. We know that:
\[TC = {C_1} + {C_2}\]
\[TC = 5{Q_1} + 5{Q_2}\]
\[TC = 5\left( {{Q_1} + {Q_2}} \right)\]
\[TC = 5Q\]
Taking the first derivative:
\[MC = \frac{{d\left( {TC} \right)}}{{dQ}} = 5\]
Now, equilibrium condition is:
\[D = S\]
\[P = MC\]
\[100 - 0.5Q = 5\]
\[0.5Q = 100 - 5\]
\[Q = \frac{{95}}{{0.5}} = 190\]
The market equilibrium quantity (Q) is 190. Cournot Nash equilibrium quantity of duopolists is \(\frac{{190}}{3}\), that is, each firms sell one-third of total output which can be sold in a perfectly competitive market.

#### Q. No. 3 (a) - Suppose the utility function for the consumer take one of the following forms: \[\left( i \right)U = 50x + 20y\] \[\left( {ii} \right)U = 20x + 50x\] \[\left( {iii} \right)U = 80x + 40x\] The budget of the consumer is ₹10,000. The price of good X and good Y are ₹50 and ₹20 per unit respectively. Determine the possibility of determination of the equilibrium basket in each case using diagram and comment on the nature of the solution

Solution: Video:

#### Q. No. 3 (b) - Outline how the production possibility frontier can be used to explain the concept of opportunity cost. Why is the production possibility frontier concave to the origin?

(Comment for solution.)

#### Q. No. 4 - Suppose that a firm's production function is given by the Cobb-Douglas function:\(Q = {K^\alpha }{L^\beta }(where\,\,\alpha ,\beta > 0)\).The firm can purchase all the K and L it wants in competitive input markets at rental rates of r and w respectively.

(i) Show that cost minimisation requires \(\frac{{rK}}{\alpha } = \frac{{wL}}{\beta }\). What is the slope of the expansion path for this firm?

(ii) Assuming cost minimisation, show that total costs can be expressed as a function of Q, r and w of the form
\[TC = B{Q^{\frac{1}{{\alpha + \beta }}}}.{w^{\frac{\beta }{{\alpha + \beta }}}}.{r^{\frac{\alpha }{{\alpha + \beta }}}}\]
where B is a constant depending on \(\alpha \) and \(\beta \).

(iii) Show that if \(\alpha + \beta = 1\), total cost (TC) is proportional to Q.

(iv) Calculate the firm's marginal cost curve.

(Comment for solution.)

#### Q. No. 5 (a) - Distinguish between economic rent and transfer earnings. Can economic rent exist in the long run? Justify your answer.

(Comment for solution.)

#### Q. No. 5(b) - Explain graphically the role of elasticity of supply of a factor determining the economic rent.

(Comment for solution.)

#### Q. No. 6 (a) - Why do externalities prevent markets from being efficient? How does Coase theorem correct an externality?

(Comment for solution.)

#### Q. No. 6(b) - Using a particular industry, explain what is meant by economies of scale and economies of scope. How do these affect the industry you have identified?

(Comment for solution.)

#### Q. No. 7 (a) - In a contest, two judges ranked eight condidates A, B, C, D, E, F, G and H in order of their preference as shown in the following table. Find the rank correlation coefficient.

A | B | C | D | E | F | G | H | |
---|---|---|---|---|---|---|---|---|

First Judge | 5 | 2 | 8 | 1 | 4 | 6 | 3 | 7 |

Second Judge | 4 | 5 | 7 | 3 | 2 | 8 | 1 | 6 |

#### Q. No. 7 (b) - The regression equation of the variables x and y are \[8x - 10y + 66 = 0\] and \[40x - 18y = 214\] The variance of x is 9. Identify the two regression lines. Find the simple correlation coefficient between the two variables and variance of y.

Solution Video

#### Q. No. 8 - Discuss Social Choice theory in Economics. Distinguish between the views of Amartya Sen and Kenneth Arrow in making choices for social welfare.

(Comment for solution.)

#### Q. No. 9(a) - Consider a two-variable linear regression model \[\begin{array}{l}{Y_t} = \alpha + \beta {X_t} + {U_t}\\and\,\,{U_t} = {}_\rho {U_{t - 1}} + {\varepsilon _t};\,\left| \rho \right| < 1\end{array}\]Find Mean, Variance and Covariance of random disturbance term \(\left( {\,{U_t}} \right)\).

(Comment for solution.)

#### Q. No. 9(b) - Consider the model of wage determination:
\[{Y_t} = {\beta _1} + {\beta _2}{X_t} + {\beta _3}{Y_{t - 1}} + {U_t}\]

where

Y = wages

X = productivity

\({U_t} = {}_\rho {U_{t - 1}} + {\varepsilon _t};\,\,\, - 1 < \rho < 1\)

Discuss the method of testing with the help of appropriate test statistic.

(Comment for solution.)

#### Q. No. 9(c) - Cosider a model
\[\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,{Y_t} = {\beta _1} + {\beta _2}{X_t} + {U_t}\\and\,\,\,{U_t} = {}_\rho {U_{t - 1}} + {\varepsilon _t}\end{array}\]

Discuss the process of the removal of autocorrelation when

(i) \(\rho \) is known

(ii) \(\rho \) is unknown (using Cochrane-Orcutt iterative method)

(Comment for solution.)

#### Q. No. 10(a) - An economy produces only coal and steel. The two commodities serve as intermediate inputs in each other's production. 0.4 tonne of steel. Similarlly, 0.1 tonne of steel and 0.6 tonne of coal are required to produce a tonne of coal. No capital inputs are needed. 2 and 5 labour days are required to produce a tonne of coal and steel respectively. If the economy needs 100 tonnes of coal and 50 tonnes of steel,

(i) Calculate the gross output of the two commodities and the total labour required.

(ii) Write down technology matrix.

(iii) Do you think that the system is viable?

(iv) Determine the equilibrium prices, if the wage rate is ₹ 10 per man-day.

(Comment for solution.)

#### Q. No. 10(b) - Mohan is paid ₹ 8 if two coins turn both heads and ₹ 1 if two coins turn both tails. Ram is paid ₹ 3 when the two coins do not match.

(i) Write down the pay-off matrix of the above problem.

(ii) Whom do you consider in the better situation?

(Comment for solution.)

#### Q. No. 11(a) - Compare the distribution theory of Marx with that of Ricardo.

(Comment for solution.)

#### Q. No. 11(b) - Explain when Kaldor's theory of distribution beomes more appropriate.

(Comment for solution.)

#### Q. No. 11(c) - Narrate the areas where Kaldor's distribution model fails.

(Comment for solution.)

#### Q. No. 12(a) - The kinked demand curve describes price rigidity. Explain how the model works. Why does price rigidity occur in oligopolistic market

(Comment for solution.)

#### Q. No. 12(b) - State and prove Product Exhaustion Theorem. How does it differ from Clark-Wicksteed-Walras Theorem?

(Comment for solution.)

#### Q. No. 13 (a): Consider the two variable regression model: \[{Y_i} = \alpha + \beta {X_i} + {U_i}\] and \[Var\left( {{U_i}} \right) = E\left( {U_i^2} \right) = \sigma _i^2\] Show that \({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } }\) is unbiased and inefficient estimator of \(\beta \).

1. \({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } }\) is unbiased estimator
of \(\beta \).

Proof: We can use deviation form formula for the estimator which is given as follows:
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } = \frac{{\sum {{x_i}{y_i}} }}{{\sum {x_i^2} }}\]
Let:
\[\frac{{{x_i}}}{{\sum {x_i^2} }} = {k_i}\]
Now, we can write:
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } = \sum {{k_i}{y_i}} \]
Since,
\[{y_i} = {Y_i} - \bar Y\]
We can also write:
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } = \sum {{k_i}\left( {{Y_i} - \bar Y} \right)} \]
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } = \sum {{k_i}{Y_i} - \bar Y\sum {{k_i}} } \]
Let us find the value of \(\sum {{k_i}} \)
\[\sum {{k_i}} = \frac{{\sum {{x_i}} }}{{\sum {x_i^2} }}\]
The sum of deviation from mean is always equals zero. So,
\[\sum {{x_i}} = \sum {\left( {{X_i} - \bar X} \right)} = 0\]
It means:
\[\sum {{k_i}} = 0\]
Now, we can write:
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } = \sum {{k_i}{Y_i}} \]
Substituting the regression model in the above equation:
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } = \sum {{k_i}\left( {\alpha + \beta {X_i} + {U_i}} \right)} \]
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } = \alpha \sum {{k_i}} + \beta \sum {{k_i}{X_i}} + \sum {{k_i}{U_i}} \]
We have already proved that:
\[\sum {{k_i}} = 0\]
Now, let us find the value of \(\sum {{k_i}{X_i}} \):
\[\sum {{k_i}{X_i}} = \frac{{\sum {{x_i}{X_i}} }}{{\sum {x_i^2} }}\]
\[\sum {{k_i}{X_i}} = \frac{{\sum {{X_i}\left( {{X_i} - \bar X} \right)} }}{{\sum {x_i^2} }}\]
\[\sum {{k_i}{X_i}} = \frac{{\sum {X_i^2 - \bar X\sum {{X_i}} } }}{{\sum {x_i^2} }}\]
\[\sum {{k_i}{X_i}} = \frac{{\sum {X_i^2 - \bar X\sum {{X_i}} - \bar X\sum {{X_i}} + \bar X\sum {{X_i}} } }}{{\sum {x_i^2} }}\]
\[\sum {{k_i}{X_i}} = \frac{{\sum {X_i^2 - 2\bar X\sum {{X_i}} + n{{\bar X}^2}} }}{{\sum {x_i^2} }}\]
\[\sum {{k_i}{X_i}} = \frac{{\sum {\left( {X_i^2 - 2\bar X{X_i} + {{\bar X}^2}} \right)} }}{{\sum {x_i^2} }}\]
\[\sum {{k_i}{X_i}} = \frac{{\sum {{{\left( {{X_i} - \bar X} \right)}^2}} }}{{\sum {x_i^2} }}\]
\[\sum {{k_i}{X_i}} = \frac{{\sum {x_i^2} }}{{\sum {x_i^2} }} = 1\]
Now, we can write:
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } = \beta + \sum {{k_i}{U_i}} \]
Taking expectation keeping in mind that X_{i} are fixed values:
\[E\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = E\left( \beta \right) + \sum {{k_i}E\left( {{U_i}} \right)} \]
Expectation of a constant, here \(\beta \), is the constant itself and by assumption, it is know that:
\[E\left( {{U_i}} \right) = 0\]
Therefore,
\[E\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = \beta \]
The mean (expectation) of the estimator is equal to the population parameter. Therefore, \({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } }\)
is the unbiased estimator of \(\beta \)

2. \({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } }\) is inefficient estimator
of \(\beta \).

Proof: An estimator is called inefficient when it does not has minimum variance property.
We can calculate variance as follows:
\[Var\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = E{\left[ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } - E\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right)} \right]^2}\]
\[Var\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = E{\left[ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } - \beta } \right]^2}\]
From:
\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } = \beta + \sum {{k_i}{U_i}} \]
We can write:
\[Var\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = E{\left[ {\sum {{k_i}{U_i}} } \right]^2}\]
Removing summation operatou:
\[Var\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = E{\left[ {{k_1}{U_1} + {k_2}{U_2} + ... + {k_n}{U_n}} \right]^2}\]
\[Var\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = E\left[ {k_1^2U_{_1}^2 + k_{_2}^2U_{_2}^2 + ... + k_{_n}^2U_{_n}^2 + 2{k_1}{k_2}{u_1}{u_2} + .... + 2{k_{n - 1}}{k_n}{u_{n - 1}}{u_n}} \right]\]
\[Var\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = k_1^2E\left( {U_{_1}^2} \right) + k_{_2}^2E\left( {U_{_2}^2} \right) + ... + k_{_n}^2E\left( {U_{_n}^2} \right) + 2{k_1}{k_2}E\left( {{u_1}{u_2}} \right) + .... + 2{k_{n - 1}}{k_n}E\left( {{u_{n - 1}}{u_n}} \right)\]
From the assumption of no serial correlation:
\[E\left( {{u_{i}}{u_j}} \right) = 0\]
Now, we can write:
\[Var\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = k_1^2E\left( {U_{_1}^2} \right) + k_{_2}^2E\left( {U_{_2}^2} \right) + ... + k_{_n}^2E\left( {U_{_n}^2} \right)\]
From the question:
\[E\left( {U_i^2} \right) = \sigma _i^2\]
So, we can write:
\[Var\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = k_1^2\sigma _1^2 + k_2^2\sigma _2^2 + ... + k_n^2\sigma _n^2\]
\[Var\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = \sum {k_i^2\sigma _i^2} \]
Substituting the value of k:
\[Var\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } } \right) = \frac{{\sum {{x_i}\sigma _i^2} }}{{{{\left( {\sum {x_i^2} } \right)}^2}}}\]
This variance of estimator \({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \beta } }\) does not follows minimum variance
property because there are different values of variance for different samples. Therefore, it is an inefficient estimator.

#### Q. No. 13(b) Consider a three variable linear regression model
\[{Y_i} = {\beta _1} + {\beta _2}{X_{2i}} + {\beta _3}{X_{3i}} + {\varepsilon _i}\]

and suppose that

(i) \(\sigma _i^2 = {\sigma ^2}Z_i^2\)

(ii) \(\sigma _i^2 = {\sigma ^2}{X_{1i}}\)

(iii) \(\sigma _i^2 = {\sigma ^2}X_{_{1i}}^2\)

Discuss Generalised Least Squares (GLS) method to overcome the heteroscedasticity problem under three cases (i, ii and iii).

(Comment for solution.)

Sir, in the cournot nash equilibrium question I could't understand the last a few steps. When you use the profit maximisation condition, how did 190 come after derivation? After derivation wont 95 be left and the market equilibrium quantity therefor be 95? pls help! Thank you.

ReplyDeleteYou're right. There is something wrong with this. Let me check it.

DeleteThe answer has been corrected. Do you still have any confusion?

Deleteno confusion! thank you so much sir!

DeleteSir any source/reading material through which I can solve more quesitons like these and also understand the theoretical foundation behind this?

ReplyDeleteWhich question are you talking about?

DeleteFor Microeconomics - Nicholson and for statistics SC Gupta. I have told about the references in the video.

Microeconomics is available in the Google Drive folder.