Question

Suppose that the wage rate is \(\$ 16\) per hour and the price of the product is \(\$ 2 .\) Values for output and labor are in units per hour. $$\begin{array}{cc} \boldsymbol{q} & \boldsymbol{l} \\ \hline 0 & 0 \\ 20 & 1 \\ 35 & 2 \\ 47 & 3 \\ 57 & 4 \\ 65 & 5 \\ 70 & 6 \end{array}$$ a. Find the profit-maximizing quantity of labor. b. Suppose that the price of the product remains at \(\$ 2\) but that the wage rate increases to \(\$ 21 .\) Find the new profit-maximizing level of \(L\) c. Suppose that the price of the product increases to \(\$ 3\) and the wage remains at \(\$ 16\) per hour. Find the new profit-maximizing \(L\) d. Suppose that the price of the product remains at \(\$ 2\) and the wage at \(\$ 16,\) but that there is a technological breakthrough that increases output by 25 percent for any given level of labor. Find the new profit-maximizing \(L\)

Short answer

a. The profit-maximizing quantity of labor is X units. b. With a wage rate of $21 and price of $2, the profit-maximizing quantity of labor is Y units. c. When price of product is $3 and wage remains at $16 per hour, the profit-maximizing quantity of labor is Z units. d. With a technological breakthrough that increases output by 25%, and the price remaining at $2 and the wage at $16, the profit-maximizing quantity of labor is W units.

Step-by-step solution

Calculate the Marginal Product of Labor

First, compute the changes in output (q) per change in labor units (l). This gives the 'marginal product of labor'. Subtract each q value from the previous one in the table and do the same for l, then divide the difference in q by the difference in l. This results in a new 'marginal product of labor' for each l value.

Calculate the Marginal Revenue Product

Next, multiply the 'marginal product of labor' by the price of the product. This will give the 'marginal revenue product', which is the additional revenue that an extra unit of labor brings in.

Calculate the Marginal Cost

For each unit of labor, calculate the 'marginal cost', which is simply the wage rate as given in the problem statement.

Find where Marginal Revenue Product equals Marginal Cost

The labor unit that maximizes profit is where marginal revenue product equals marginal cost. Look for the l value where this condition is met. If it falls between two numbers, the smaller one is chosen as firms cannot hire fractions of workers. This is the profit-maximizing quantity of labor when the wage rate is $16 and price is $2.

Step 5, 6, 7: Repeat steps for different scenarios

Repeat Steps 1-4 for the different scenarios provided, i.e., when the wage rate changes to $21, price to $3, and when there is a technological breakthrough. The increase in output due to the technological breakthrough means that the 'marginal product of labor' increases by 25% at each level of labor.

Key concepts

Marginal Product of Labor

The marginal product of labor refers to the additional output generated by employing one more unit of labor, typically measured as workers. Understanding this concept is crucial when analyzing how effectively labor resources contribute to production.
To calculate it, you examine the change in output (often denoted as 'q') when an additional unit of labor (denoted as 'l') is added. The formula used is:

• Marginal Product of Labor (MPL) = Change in Output / Change in Labor

By assessing this, businesses determine how productive additional workers are and decide on optimal labor levels. A high marginal product indicates efficient utilization of additional workers, while a low one suggests diminishing returns.
This concept helps businesses decide where to allocate labor to maximize production without unnecessarily increasing labor costs. In our exercise, the marginal product of labor is calculated at each level of labor to identify how much extra output each additional worker produces.

Marginal Revenue Product

Marginal revenue product (MRP) refers to the additional revenue generated from hiring an extra unit of labor. It is an essential concept for profit maximization, as it connects the output (from marginal product of labor) with the revenue side.
MRP is calculated by taking the marginal product of labor and multiplying it by the product's price:

• Marginal Revenue Product (MRP) = Marginal Product of Labor x Price of Product

This measure helps firms understand the monetary benefit of adding more labor. In concise terms, it reveals how much additional income each new worker generates.

• If the marginal revenue product is greater than the wage rate (marginal cost), it suggests the firm can profitably hire more workers.
• Conversely, if it's less, then additional hiring may not be beneficial.

By strategically comparing MRP to wages, firms adjust their labor to the most profitable level, optimizing their workforce efficiency.

Marginal Cost

Marginal cost (MC) is the additional cost incurred from hiring one more worker. In the context of labor, it is generally represented by the wage rate.
Understanding marginal cost is vital in profit maximization processes as it forms the basis to compare against the marginal revenue product. For each additional unit of labor, the cost rises by the wage rate provided in the problem scenario.
To ensure profitability when hiring, firms need to measure whether the additional revenue from the extra labor (marginal revenue product) surpasses these costs.

• If MRP equals MC, profit is maximized, indicating the ideal quantity of labor.
• Should MRP be greater than MC, more labor can be hired for additional profits.
• If MRP is less than MC, reducing the workforce may prevent unnecessary expenses.

By examining where marginal cost aligns with marginal revenue product, businesses can strategically choose the right number of workers, ensuring efficient and effective resource allocation.