Now that we have specified a cubic form for the short-run production function, we can discuss how to estimate this production function. As we see, only the simple techniques of regression analysis presented in Chapter 4 are needed to estimate the cubic production function in the short run when capital is fixed. We illustrate the process of estimating the production function with an example.
Now try Technical Problem 1.
F I G U R E 10.1 Marginal and Average Product Curves for the Short-Run Cubic Production Function:
Q 5 AL3 1 BL2
Average and marginal product of labor
Labor m
a
AP MP La = –2AB
Lm = –3AB
2Recall from Tables 8.2 and 8.3 in Chapter 8 that capital is held constant in any given column.
The entire marginal and average product schedules change when capital usage changes.
Suppose a small plant uses labor with a fixed amount of capital to assemble a product. There are 40 observations on labor usage (hours per day) and output (number of units assembled per day). The manager wishes to estimate the production function and the marginal product of labor. Figure 10.2 presents a scatter diagram of the 40 observations.
The scatter diagram suggests that a cubic specification of short-run produc- tion is appropriate because the scatter of data points appears to have an S-shape,
T A B L E 10.1
Summary of the Short- Run Cubic Production Function
Short-run cubic production function
Total product Q = AL3 + BL2
where A = a __ K 3
B = b __ K 2 Average product AP = AL2 + BL Marginal product MP = 3AL2 + 2BL Diminishing marginal returns Beginning at Lm = − ___3AB Diminishing average product Beginning at La = − ___2AB Restrictions on parameters A , 0
B . 0
F I G U R E 10.2 Scatter Diagram for a Cubic Production Function
Output
140 120 100 80 60 40 20
Labor
Lˆm = 19.36
10 15 20 25 30
Lˆa = 29.05 TP
0 5
similar to the theoretical total product curve set forth in Chapter 8. For such a curve, the slope first increases and then decreases, indicating that the marginal product of labor first increases, reaches a maximum, and then decreases. Both the marginal product and average product curves should take on the inverted-ứ-shape described in Chapter 8 and shown in Figure 8.3.
Because it seems appropriate to estimate a cubic production function in this case, we specify the following estimable form
Q = AL3 + BL2
Following the procedure discussed in Chapter 4, we transform the cubic equation into a linear form for estimation
Q = AX + BW
where X = L3 and W = L2. To correctly estimate the cubic equation, we must account for the fact that the cubic equation does not include an intercept term.
In other words, the estimated regression line must pass through the origin;
that is, when L = 0, Q = 0. Regression through the origin simply requires that the analyst specify in the computer routine that the intercept term be
“suppressed.” Most computer programs for regression analysis provide the user with a simple way to suppress the intercept term. After we use a regres- sion routine to estimate a cubic equation for the 40 observations on output and labor usage (and suppressing the intercept), the following computer output is forthcoming:
DEPENDENT VARIABLE: Q R-SQUARE F-RATIO P-VALUE ON F OBSERVATIONS: 40 0.9837 1148.83 0.0001
PARAMETER STANDARD
VARIABLE ESTIMATE ERROR T-RATIO P-VALUE L3 20.0047 0.0006 27.833 0.0001 L2 0.2731 0.0182 15.005 0.0001
The F-ratio and the R2 for the cubic specification are quite good.3 The critical value of F with k − 1 = 1 and n − k = 38 degrees of freedom is 4.1 at the 5 percent signif- icance level. The p-values for both estimates ˆ A and ˆ B are so small that there is less
regression through the origin
A regression in which the intercept term is forced to equal 0.
3For purposes of illustration, the hypothetical data used in this example were chosen to fit closely an S-shaped cubic equation. In most real-world applications, you will probably get smaller values for the F-ratio, R2, and t-statistics.
than a 0.01 percent chance of making a Type I error (mistakenly concluding that A ≠ 0 and B ≠ 0). The following parameter estimates are obtained from the printout
ˆ A = −0.0047 and ˆ B = 0.2731 The estimated short-run cubic production function is
ˆ
Q = −0.0047L3 + 0.2731L2
The parameters theoretically have the correct signs: ˆ A < 0 and ˆ B . 0. We must test to see if ˆ A and ˆ B are significantly negative and positive, respectively. The computed t-ratios allow us to test for statistical significance
t ˆ a = −7.83 and t ˆ b = 15.00
The absolute values of both t-statistics exceed the critical t-value for 38 degrees of freedom at a 5 percent level of significance (2.021). Hence, ˆ A is significantly nega- tive and ˆ B is significantly positive. Both estimates satisfy the theoretical character- istics of a cubic production function.
The estimated marginal product of labor is MP = 3 ˆ A L2 + 2 ˆ B L
= 3(−0.0047)L2 + 2(0.2731)L
= −0.0141L2 + 0.5462L
The level of labor usage beyond which diminishing returns set in (after MPL
reaches its maximum) is estimated as ˆ L m = − ˆ B ___
3 ˆ A = 0.2731 __________3(−0.0047) = 19.36
Note in Figure 10.2 that ˆ L m is at the point where total product no longer increases at an increasing rate but begins increasing at a decreasing rate. The estimated av- erage product of labor is
AP = ˆ A L2 + ˆ B L
= (−0.0047)L2 + (0.2731)L
The maximum average product is attained when AP = MP at the estimated level of labor usage
ˆ L a = − ˆ B ___
2 ˆ A = 0.2731 __________2(−0.0047) = 29.05
Maximum AP, as expected, occurs at a higher level of labor usage than maximum MP (see Figure 10.2). The evidence indicates that the cubic estimation of the production function from the data points in Figure 10.2 provides a good fit and has all the desired theoretical properties.
Now try Technical Problem 2.