A firm estimates its cubic production function of the following form:

A firm estimates its cubic production function of the following form: Q = AL^3 +BL^2  and obtains the following results:
Dependent VARIABLE Q R-SQUARE F-RATIO P-VALUE ON F
OBSERVATIONS 32 0.7547 92.31 0.0001
VARIABLE PARAMETER ESTIMATE STANDARD ERROR T-RATIO P-VALUE
L3 -0.0016 0.0005 -3.2 0.0032
L2 0.4 0.095 4.21 0.0002
a. Based on the estimation results, write thee equations for total product, average product, and marginal product curves:
b.The estimated values of A and B are statistically significant at exact what level of significance? (use the p-value)
c. Find the level of labor at which the average product of labor reaches its maximum value.

2. Suppose the short-run production function for a particular product is represented by:
Q = 60,000L2 – 1,000L3
a. Find the marginal product and show mathematically that labor is subject to diminishing returns.
b. Beyond what labor usage, the marginal product becomes negative.
c. Show mathematically that when the average product is at maximum, marginal product is equal to it?

Assume Caspian is a firm operating in a competitive market. The estimated its average variable cost function is shown below: AVC = A + BQ + CQ2
The total fixed cost (TFC) = $1,000.
Dependent VARIABLE AVC R-SQUARE F-RATIO P-VALUE ON F
OBSERVATIONS 35 0.8713 108.3 0.0001
VARIABLE PARAMETER ESTIMATE STANDARD ERROR T-RATIO P-VALUE
INTERCEPT 43.4 13.8 3.14 0.0036
Q -2.8 0.9 -3.11 0.0039
Q2 0.2 0.05 4 0.0004
a. Are the estimated coefficients statistically significant at the 5% level of significance?
b. Find the total cost function.
c. Find the equation for the ATC. Is it U-shaped? A simple yes/no is not enough. Use math.
d. Find the shut-down price.
e. How much will the firm produce at the market price of $53.4?