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. $$\begin{array}{cc}9 & 1 \\\\\hline 0 & 0 \\\20 & 1 \\ 35 & 2 \\\47 & 3 \\\57 & 4 \\\65 & 5 \\\70 & 6\end{array}$$ 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

The profit-maximizing quantity of labor would be different under different scenarios. For (a) refer to the answer in Step 2, for (b) refer to the answer in Step 3, for (c) refer to the answer in Step 4, and for (d) refer to the answer in Step 5.

Step-by-step solution

Understanding the table and set up the profit function

The table is to be read as follows: 'If x units of labor are consumed, y units of the product are produced'. Therefore, assuming a linear relationship between number of units of labor consumed and output, one can find a function for Q(L) by dividing the number of units produced by number of units of labor consumed. The profit then can be calculated by subtracting the total cost from total revenue.

Calculating profit-maximizing quantity of labor (a)

Using the previously mentioned profit function, calculate the profits for each quantity level, by subtracting the total cost (wage rate times quantity of labor) from the total revenue (price of the product times output) and pick the quantity of labor where this difference is the maximum. This is the profit-maximizing quantity of labor.

Calculating the new profit-maximizing level of labor with an increased wage rate (b)

Repeat the process in Step 2, but use the new wage rate of \$21 per hour to calculate total cost. The profit-maximizing quantity of labor is again the one where profit is the maximum.

Calculating the new profit-maximizing quantity of labor with an increased product price (c)

Repeat the process in Step 2, but this time use the new product price of \$3 to calculate the total revenue. The profit-maximizing quantity of labor is again the one where the profit is the maximum.

Calculating the new profit-maximizing level of labor with a technological breakthrough (d)

With a 25 percent increase in the output due to technological advancement, it means that the output is 1.25 times what it was before for any given level of labor. So we will multiply the quantity of each output in the table by 1.25 and repeat the process of Step 2 with the new output-quantity pairs. The profit-maximizing quantity of labor is again the quantity where profit is the maximum.

Key concepts

Wage Rate

In the context of profit maximization, the **wage rate** is a crucial element that significantly affects a firm's decision regarding labor use. The wage rate is the cost per hour of labor that a firm needs to pay to employ workers. This cost influences how many workers a company can afford to hire while remaining profitable.

When calculating profits, a firm pays attention to its total costs, a substantial part of which is associated with wages. For instance, with a wage rate of \(\\(16\) per hour, total labor costs for employing 5 workers would be \(5 \times 16 = \\)80\). If the wage rate increases to \(\\(21\) per hour, the same labor costs jump to \(5 \times 21 = \\)105\), thereby affecting the profit margins.

A higher wage rate leads to increased costs, potentially reducing the number of workers employed unless these costs are offset by increased revenues or decreased other costs. Firms thus decide on their labor input by weighing additional wage costs against the additional revenue generated by employing extra workers.

Price of Product

The **price of a product** plays a pivotal role in determining total revenue, which directly influences profit levels. Total revenue is calculated by multiplying the price of the product by the output sold. Consequently, any changes in the product's price affect revenues and can alter profit-maximizing decisions.

For example, if a product sells for \(\\(2\) but later sees an increase to \(\\)3\), this rise means that any given level of output will generate more revenue. With a higher price per unit, a company can afford to pay more for inputs, such as labor, while still maintaining or escalating profits.

This change implies that the firm may consider employing more labor because the additional labor could result in more products that can now be sold at a higher price, boosting the firm's profit margins. Hence, understanding how shifts in the selling price impact revenue is vital for effective profit maximization.

Technological Breakthrough

A **technological breakthrough** can significantly enhance productivity without changing the amount of input, thereby allowing firms to achieve higher outputs at the same or reduced costs. If such a breakthrough occurs, it commonly involves an improvement that allows more to be produced with the same resources.

For instance, a 25% boost in productivity across all levels of labor means that each worker's output is magnified by 1.25 times. If previously a worker could produce 20 units, after a technological upgrade, the same worker can produce 25 units. This multiplication of productivity reflects directly on total output.

A technological breakthrough can thus shift the firm's output-quantity relationship, often enabling more significant profit potentials. With this improvement, firms might maintain current employment levels but increase their output and sales, or they might even adjust their labor distribution to further align with new production capabilities for maximum profitability.

Output and Labor Relation

The relationship between **output and labor** is foundational in understanding profit maximization as it captures how efficiently labor transforms into output. Typically illustrated in a table or a production function, this relationship showcases how different labor levels result in varying quantities of output. Understanding this link helps businesses identify optimal labor use for maximum production and profit.

The company observes diminishing returns over time—this occurs when each additional unit of labor produces less output than the previous one. Thus, a balance must be struck to determine the profit-maximizing labor quantity where the costs of labor begin exceeding their added value in production.

Analyzing and calculating these relations involve examining each labor point's output and consequently measuring which labor input maximizes profitability by aligning labor costs with resulting revenues.