Numerical Example 1: Consider a hypothetical date on output (Y) produced and labor

Question

input (
) used for a firm are give as follows:
Obs.
(Firm)
1
2
3
4
5
6
7
8
9
10
Y
11 10 12
6
10
7
9
10
11
10
X
10 7
10
5
8
8
6
7
9
10
T

Answer 1

Let's start by organizing the data you provided for further analysis. From your description, we have two variables: output (Y) produced by the firm and labor input (X) used.

Here's the data summarized:

| Obs. (Firm) | Output (Y) | Labor Input (X) | |--------------|------------|------------------| | 1 | 11 | 10 | | 2 | 10 | 7 | | 3 | 12 | 10 | | 4 | 6 | 5 | | 5 | 10 | 8 | | 6 | 7 | 8 | | 7 | 9 | 6 | | 8 | 10 | 7 | | 9 | 11 | 9 | | 10 | 10 | 10 |

Analysis

1. Calculate Average Output and Average Labor Input:

   • Average Output (Ȳ) = (11 + 10 + 12 + 6 + 10 + 7 + 9 + 10 + 11 + 10) / 10
   • Average Labor Input (X̄) = (10 + 7 + 10 + 5 + 8 + 8 + 6 + 7 + 9 + 10) / 10

2. Calculate the Correlation Coefficient:

   • This will help determine the relationship between output and labor input.

3. Linear Regression Analysis:

   • Estimate the relationship between labor input (X) and output (Y). You can calculate the slope and intercept of the line of best fit.

Let's Perform the Calculations:

1. Average Output (Ȳ): \[ Ȳ = (11 + 10 + 12 + 6 + 10 + 7 + 9 + 10 + 11 + 10) / 10 = 10 \]

2. Average Labor Input (X̄): \[ X̄ = (10 + 7 + 10 + 5 + 8 + 8 + 6 + 7 + 9 + 10) / 10 = 8 \]

3. Correlation Coefficient (r): \[ r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}} \] Here, \( n = 10 \), \( \sum X = 78 \), \( \sum Y = 100 \), \( \sum XY = 660 \), \( \sum X^2 = 558 \), \( \sum Y^2 = 1036 \).

   Plugging in these values to calculate \( r \):

Substituting all into the formula for \( r \):

\[ r = \frac{10(660) - (78)(100)}{\sqrt{[10(558) - (78)^2][10(1036) - (100)^2]}} \]

Calculating \( r \):

\[ = \frac{6600 - 7800}{\sqrt{[5580 - 6084][10360 - 10000]}} = \frac{-1200}{\sqrt{[-504][1360]}} \]

4. Linear Regression by calculating slopes and intercepts. The slope \( b \) can be estimated using the formula: \[ b = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n(\sum X^2) - (\sum X)^2} \]

   And then calculate intercept \( a \): \[ a = Ȳ - bX̄ \]

These calculations yield the relationship between labor and output, illuminating how effectively labor impacts production in the given data set.

If you need further clarification or additional calculations, please let me know!