You are planning to estimate a short- run production function for your firm, and you have collected the following data on labor usage (L) and output (Q):
Labor usage Output
3 1
7 2
9 3
11 5
17 8
17 10
20 15
24 18
26 22
28 21
30 23
To estimate the short-run production function, we need to first determine the relationship between labor usage (L) and output (Q). One common approach is to use regression analysis to estimate a linear or non-linear equation.
In this case, let's start by graphing the data to visually analyze the relationship between labor usage and output. You can use a scatter plot with labor (L) on the x-axis and output (Q) on the y-axis.
By examining the scatter plot, you can get a sense of whether there is a linear or non-linear relationship between the variables. If the plot shows a clear pattern (such as a straight line), it suggests a linear relationship. However, if the plot does not show a clear pattern, it indicates a non-linear relationship.
After analyzing the scatter plot, you can proceed with estimating a production function. Let's assume that the relationship seems to be non-linear in this case.
One commonly used non-linear production function is the Cobb-Douglas production function, which has the following general form:
Q = A * L^α * K^β
where Q is output, L is labor usage, K is capital, A is a constant, and α and β are parameters representing the elasticity of output with respect to labor and capital.
Since we only have data on labor usage and output, we would need to estimate a simplified version of the Cobb-Douglas production function, which is known as a single-input production function:
Q = A * L^α
To estimate this simplified production function, we can use a regression analysis to calculate the values of A and α. Regression analysis helps us find the best-fitting equation that represents the relationship between L and Q in the given data set.
You can use statistical software, such as Excel, Python, or R, to perform the regression analysis. The software will calculate the coefficients A and α, as well as provide statistical measures of the fit, like R-squared.
Once you have the estimated coefficients, you can use them to predict output (Q) for any given level of labor usage (L) in the short run.