# IES ISS 2023 ECONOMICS PAPER PAPER 1 DETAILED ANSWER OF EACH QUESTION- 9a,b, 10 b, c, 11 b, 12b, 13b

9. a. ‘’Heteroscedasticity is a problem is cross-section data, but not in time series data. ‘’ Discuss.

The statement that "heteroscedasticity is a problem in cross-section data but not in time series data" is not entirely accurate. Heteroscedasticity can occur in both cross-sectional and time series data, and it is a concern in both cases.

Heteroscedasticity refers to the situation where the variability of the error term in a regression model is not constant across the range of the independent variable(s). In other words, the error term exhibits different levels of dispersion or variance for different values of the independent variable(s).

In cross-sectional data, heteroscedasticity can arise due to various factors such as differences in measurement errors, variation in the characteristics of the observations, or outliers. For example, if the error term has a larger variance for higher values of the independent variable, it can lead to heteroscedasticity in the cross-sectional data.

In time series data, heteroscedasticity can also occur. It may arise due to changing levels of volatility or variation in the error term over time. For instance, financial time series data often exhibit heteroscedasticity, where the volatility of returns can change over different time periods.

Heteroscedasticity can have important implications for regression analysis. It violates the assumption of homoscedasticity, which assumes constant variance of the error term. Heteroscedasticity can lead to inefficient coefficient estimates, biased standard errors, and incorrect hypothesis testing results.

To address the issue of heteroscedasticity, various techniques can be employed, such as transforming the variables, using weighted least squares regression, or employing heteroscedasticity-robust standard errors.

In conclusion, heteroscedasticity is a concern in both cross-sectional and time series data. It is important to detect and address heteroscedasticity to ensure accurate and reliable regression analysis results, regardless of the type of data being analyzed.

B. A researcher estimated an employment (N) equation with GDP (G), education (E) and price (P) as explanatory variables. The estimated equation is given below:

N=506 + 0.06G- 0.01E - 19.8P

(1.399) (3.227) (-0.033) (-0.142)

R^2 = 0.97, number of observations=16

[Figures in parentheses are t-statistics]

I. Interpret the estimated coefficients.

II. Identify the problems in the estimation.

III. How can you improve the estimation?

I. Interpretation of the estimated coefficients:

- The coefficient for GDP (G) is 0.06, indicating that a 1-unit increase in GDP leads to an estimated increase of 0.06 in employment (N), holding other variables constant.

- The coefficient for education (E) is -0.01, suggesting that a 1-unit increase in education leads to an estimated decrease of 0.01 in employment (N), holding other variables constant.

- The coefficient for price (P) is -19.8, indicating that a 1-unit increase in price leads to an estimated decrease of 19.8 in employment (N), holding other variables constant.

II. Problems in the estimation:

1. T-statistics: The t-statistics, represented in parentheses, should be used to test the statistical significance of the estimated coefficients. However, the provided t-statistics are not accurate, as they contain decimal places. T-statistics should generally be rounded to two decimal places.

2. Negative t-statistic for education (E): The t-statistic for education is -0.033, which raises a concern about the statistical significance of the coefficient. A negative t-statistic suggests that the estimated coefficient may not be significantly different from zero. Further analysis is needed to determine the significance.

III. Improving the estimation:

To improve the estimation, several steps can be taken:

1. Obtain accurate t-statistics: The t-statistics should be calculated correctly with appropriate rounding. Accurate t-statistics help determine the statistical significance of the estimated coefficients.

2. Address the issue with the t-statistic for education: If the negative t-statistic for education persists even after accurate calculation, it suggests that the variable may not have a significant impact on employment in the given model. In this case, reconsidering the inclusion of education as an explanatory variable or exploring alternative measures of education could be considered.

3. Consider model specification: Assess whether the current model includes all relevant variables that could affect employment. It may be necessary to explore additional variables or alternative functional forms to improve the model's fit.

4. Evaluate assumptions: Check for violations of assumptions such as multicollinearity, heteroscedasticity, or autocorrelation. If any violations are found, appropriate corrective measures should be applied.

5. Increase the sample size: With only 16 observations, the sample size is relatively small. Increasing the sample size can help improve the accuracy and reliability of the estimation results.

By addressing these issues and considering the suggestions mentioned, the estimation can be improved to provide more reliable insights into the relationship between employment, GDP, education, and price.

10. b. ii. For the data given below, calcite the price index by using Fisher’s formula, and interpret your result.

Commodities

2019

2022

P

Q

P

Q

A

4

10

5

9

B

5

8

3

6

C

2

6

2

4

D

3

9

1

7

E

5

5

4

5

To calculate the price index using Fisher's formula, we need to follow these steps:

Step 1: Calculate the Laspeyres Price Index (LPI):

LPI = (∑(P_1 * Q_0)) / (∑(P_0 * Q_0))

Where:

P_1 = Prices in the current year (2022)

P_0 = Prices in the base year (2019)

Q_0 = Quantities in the base year (2019)

For each commodity, we calculate P_1 * Q_0 and P_0 * Q_0:

Commodity A:

P_1 * Q_0 = 5 * 10 = 50

P_0 * Q_0 = 4 * 10 = 40

Commodity B:

P_1 * Q_0 = 3 * 8 = 24

P_0 * Q_0 = 5 * 8 = 40

Commodity C:

P_1 * Q_0 = 2 * 6 = 12

P_0 * Q_0 = 2 * 6 = 12

Commodity D:

P_1 * Q_0 = 1 * 9 = 9

P_0 * Q_0 = 3 * 9 = 27

Commodity E:

P_1 * Q_0 = 4 * 5 = 20

P_0 * Q_0 = 5 * 5 = 25

Now we can calculate the numerator and denominator for LPI:

∑(P_1 * Q_0) = 50 + 24 + 12 + 9 + 20 = 115

∑(P_0 * Q_0) = 40 + 40 + 12 + 27 + 25 = 144

LPI = 115 / 144 ≈ 0.7986

Step 2: Calculate the Paasche Price Index (PPI):

PPI = (∑(P_1 * Q_1)) / (∑(P_0 * Q_1))

Where:

Q_1 = Quantities in the current year (2022)

For each commodity, we calculate P_1 * Q_1 and P_0 * Q_1:

Commodity A:

P_1 * Q_1 = 5 * 9 = 45

P_0 * Q_1 = 4 * 9 = 36

Commodity B:

P_1 * Q_1 = 3 * 6 = 18

P_0 * Q_1 = 5 * 6 = 30

Commodity C:

P_1 * Q_1 = 2 * 4 = 8

P_0 * Q_1 = 2 * 4 = 8

Commodity D:

P_1 * Q_1 = 1 * 7 = 7

P_0 * Q_1 = 3 * 7 = 21

Commodity E:

P_1 * Q_1 = 4 * 5 = 20

P_0 * Q_1 = 5 * 5 = 25

Now we can calculate the numerator and denominator for PPI:

∑(P_1 * Q_1) = 45 + 18 + 8 + 7 + 20 = 98

∑(P_0 * Q_1) = 36 + 30 + 8 + 21 + 25 = 120

PPI = 98 / 120 ≈ 0.8167

Step 3: Calculate the Fisher's

Price Index (FPI):

FPI = √(LPI * PPI)

FPI = √(0.7986 * 0.8167) ≈ 0.8073

Interpretation:

The Fisher's Price Index (FPI) in this case is approximately 0.8073. This indicates that, on average, the prices of the commodities in the given basket have decreased by approximately 19.27% from the base year (2019) to the current year (2022). This suggests a deflationary trend in the overall price level of the commodities. However, it's important to note that this index represents a weighted average of the price changes based on the quantities consumed in the base year and the current year.

C. Interpret the coefficients from the following estimated equations:

In Y hat = a hat+ βhat _1 In X_1+ βhat_2 In X_2

In Y hat= a hat+b hat_1X_1+b hat_(2)X_2

In the first estimated equation:

In Y hat = a hat + βhat_1 In X_1 + βhat_2 In X_2

The coefficients can be interpreted as follows:

- a hat: This represents the intercept term of the equation. It indicates the estimated value of Y when both X_1 and X_2 are equal to zero. In other words, it represents the estimated constant term of the equation.

- βhat_1: This coefficient represents the estimated effect of a one-unit change in the natural logarithm of X_1 on the dependent variable Y. It indicates the percentage change in Y for a one percent change in X_1, holding other variables constant. A positive coefficient suggests a positive relationship between X_1 and Y, while a negative coefficient suggests a negative relationship.

- βhat_2: Similarly, this coefficient represents the estimated effect of a one-unit change in the natural logarithm of X_2 on the dependent variable Y. It indicates the percentage change in Y for a one percent change in X_2, holding other variables constant. A positive coefficient suggests a positive relationship between X_2 and Y, while a negative coefficient suggests a negative relationship.

In the second estimated equation:

In Y hat = a hat + b hat_1X_1 + b hat_2X_2

The interpretation is similar, except that the variables X_1 and X_2 are not transformed using the natural logarithm. The coefficients b hat_1 and b hat_2 represent the estimated effect of a one-unit change in X_1 and X_2, respectively, on the dependent variable Y. Unlike the first equation, these coefficients do not represent percentage changes but rather the change in Y associated with a one-unit change in X_1 or X_2, holding other variables constant.

Overall, the coefficients in both equations provide information about the direction and magnitude of the relationship between the independent variables (X_1 and X_2) and the dependent variable (Y). The interpretation may vary depending on whether the variables are transformed using natural logarithm or not, but in either case, the coefficients quantify the impact of the independent variables on the dependent variable, while taking into account other variables in the equation.

11. b. The general solution of a second order non-homogenous difference equation, Y_t= β_0+ β_1 Y_t-1+ β_2 Y_t-2, has two components: particular solution and homogenous solution. Explain the implications of these two solutions.

In a second-order non-homogeneous difference equation of the form Y_t = β_0 + β_1Y_(t-1) + β_2Y_(t-2), the general solution can be expressed as the sum of two components: the particular solution and the homogeneous solution.

1. Particular Solution:

The particular solution represents a specific solution to the non-homogeneous equation. It is the solution that satisfies the given initial conditions or the forcing function on the right-hand side of the equation. In other words, the particular solution captures the specific pattern or behavior of the dependent variable Y_t that arises due to external factors or influences represented by the coefficients β_0, β_1, and β_2. The particular solution can be found using methods such as the method of undetermined coefficients or the method of variation of parameters.

2. Homogeneous Solution:

The homogeneous solution represents the general solution to the associated homogeneous equation, which is obtained by setting the right-hand side of the original equation to zero. In this case, the associated homogeneous equation is Y_t = β_1Y_(t-1) + β_2Y_(t-2). The homogeneous solution captures the inherent dynamics or behavior of the dependent variable Y_t in the absence of any external influences or forcing factors. It represents the solution that satisfies the equation when there are no external inputs or disturbances. The homogeneous solution can be found by assuming a solution of the form Y_t = r^t, where r is a constant, and solving for the characteristic equation associated with the homogeneous equation.

The implications of these two solutions are as follows:

- The particular solution accounts for the specific factors or inputs that influence the dependent variable Y_t. It represents the deviation from the inherent behavior captured by the homogeneous solution.

- The homogeneous solution represents the underlying dynamics or behavior of the dependent variable in the absence of external influences. It represents the solution that would persist in the long run or when the external factors are not present.

By combining the particular solution and the homogeneous solution, we obtain the general solution, which represents the complete solution to the non-homogeneous difference equation. The particular solution captures the specific impact of external influences, while the homogeneous solution captures the inherent behavior of the system.

12. b. Suppose that two firms are selling a homogenous product. They can charge high price( H) or low price (L). The pay-offs from their actions are given in the following game matrix:

Firm-2

H L

Firm -1 H

L

8,8

3,10

10, 3

5,5

I. Find Nash equilibrium for the given game.

II. Is there any dominant strategy in the game? Explain.

I. To find the Nash equilibrium, we need to identify the strategies for each firm that are best responses to each other.

In this game, the Nash equilibrium is reached when both firms choose their strategies in such a way that neither has an incentive to unilaterally change their strategy.

Looking at the payoffs, we can see that in the cell (H, H), Firm-1 gets a payoff of 8 and Firm-2 gets a payoff of 8. In the cell (L, L), Firm-1 gets a payoff of 5 and Firm-2 gets a payoff of 5.

Since the payoffs in the (H, H) cell are higher than in the (L, L) cell for both firms, the Nash equilibrium in this game is for both firms to choose the strategy of high price (H). Therefore, (H, H) is the Nash equilibrium.

II. To determine if there is a dominant strategy in the game, we compare the payoffs for each firm for each possible strategy combination.

Looking at the payoffs, we can see that in the (H, L) cell, Firm-1 gets a payoff of 3, which is lower than the payoff of 10 in the (L, H) cell. Similarly, in the (L, H) cell, Firm-2 gets a payoff of 3, which is lower than the payoff of 10 in the (H, L) cell.

Since neither firm has a dominant strategy, meaning a strategy that always yields a higher payoff regardless of the other firm's strategy, there is no dominant strategy in this game. The best response of each firm depends on the strategy chosen by the other firm.

In conclusion, the Nash equilibrium in this game is for both firms to choose the strategy of high price (H), and there is no dominant strategy present.

13. b. A farmer grows 70 kg of X_1 and 20 kg of X_2. He keeps some parts of X_1 and X_2 for self-consumption and sells the rest in the market. His utility function is

U(X_1, X_2)= ,min (2X_1, X_2)

And prices of X_1 and X_2 are rupees 2 and 3 respectively.

Suppose that the price of X_1 increase to Rupees 4 and at the same time his consumption of X_1 also increases.

Explain the behavior of the farmer using substitution effect, income effect and endowment effect.

To analyze the behavior of the farmer when the price of X_1 increases, we can consider the substitution effect, income effect, and endowment effect.

1. Substitution Effect:

The substitution effect refers to the change in consumption patterns resulting from the relative price change. In this case, when the price of X_1 increases from Rs. 2 to Rs. 4, the farmer may choose to consume relatively less of X_1 and more of X_2 because X_1 has become relatively more expensive. This is driven by the farmer's desire to maximize utility within the constraints of the new price ratio. The farmer substitutes away from the relatively more expensive good (X_1) towards the relatively cheaper good (X_2) to maintain a similar level of utility.

2. Income Effect:

The income effect refers to the change in consumption patterns resulting from a change in purchasing power. In this case, the increase in the price of X_1 does not directly affect the farmer's income. However, if the farmer's utility function implies that the more he consumes, the higher his utility, then the increase in consumption of X_1 can be seen as an income effect. As the farmer consumes more of X_1, he is essentially treating it as if it were an "inferior good," which means that as his income increases, he chooses to consume less of it. This can be interpreted as an income effect leading to a decrease in the consumption of X_1.

3. Endowment Effect:

The endowment effect refers to the impact of changes in the initial endowment of goods on consumption patterns. In this case, the farmer's initial endowment of X_1 and X_2 was 70 kg and 20 kg, respectively. When the price of X_1 increases, the farmer may feel a sense of loss in terms of his original endowment of X_1. To compensate for this perceived loss, the farmer may choose to increase his consumption of X_1, even though it has become relatively more expensive. This can be seen as an endowment effect, where the change in price leads to a change in the farmer's behavior to maintain a perceived balance or restore the original endowment level.

Overall, the farmer's behavior in response to the increase in the price of X_1 can be influenced by a combination of the substitution effect, income effect (if X_1 is treated as an inferior good), and the endowment effect. The specific impact of each effect depends on the farmer's preferences, utility function, and individual circumstances.

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