A firm has the following short-run production function:
Q = 100L + 10L2 - 0.7L3
Where Q = quantity of output per week; L = Labor (number of workers)
a. When does the law of diminishing returns take effect?
b. Calculate the value for Stage I in the production process?
c. Assume the firm hires 10 workers, each worker is paid $800 per week, and the price of output is $15 per unit sold. Determine whether the firm is maximizing profit, or in other words, using the optimal amount of their input (labor).
d. Given the answer to part c, what is your recommendation to the firm if their objective is profit maximization?
I assume L2 means l squared, and L3 means L cubed.
Q=100L + 10L^2 - 0.7L^3
dQ/dL= 100-20 L -3.1L^2
to maximize profit, set to zero, and solve for L. If the 10 workers are not equal to L, the firm is not maximizing.
Q=7L+10 L2-L3 DETERMINE
Q=F(L) DETERMINE
Q=F(L)
a. The law of diminishing returns takes effect when the marginal product of an additional unit of labor starts to decrease. In this case, we can determine when this occurs by examining the derivative of the production function with respect to labor. When this derivative becomes negative, it indicates that the law of diminishing returns has taken effect.
b. To calculate the value for Stage I in the production process, we need to find the range of labor values where the marginal product of labor is positive. Stage I occurs when the marginal product of labor is positive, meaning that each additional unit of labor adds more output to the total. To determine this range, we need to look for the values of labor where the derivative of the production function (dQ/dL) is positive. This would indicate the range of labor values where the marginal product of labor is positive.
c. To determine whether the firm is maximizing profit, we need to consider the relationship between their inputs (labor) and their outputs (quantity of output). In this case, the firm hires 10 workers, each worker is paid $800 per week, and the price of output is $15 per unit sold. We can calculate the total cost of labor by multiplying the number of workers (10) by the wage rate ($800). We can also calculate the total revenue by multiplying the quantity of output (Q) by the price per unit ($15). If the total revenue exceeds the total cost, then the firm is maximizing profit. However, if the total cost exceeds the total revenue, then the firm is not maximizing profit and may need to adjust their input (labor) levels.
d. Based on the answer to part c, if the firm's objective is profit maximization and they are not currently maximizing profit, the recommendation would be to adjust their input (labor) levels. This could involve hiring more workers if the total revenue exceeds the total cost or reducing the number of workers if the total cost exceeds the total revenue. The specific adjustment would depend on the labor and production constraints of the firm and would require further analysis.