Question

Suppose that a firm's production function is given by \(Q=12 L-L^{2},\) for \(L=0\) to \(6,\) where \(L\) is labor input per day and \(Q\) is output per day. Derive and draw the firm's demand for labor curve if the firm's output sells for \(\$ 10\) in a competitive market. How many workers will the firm hire when the wage rate is \(\$ 30\) per day? \(\$ 60\) per day? (Hint: The marginal product of labor is \(12-2 L\).)

Short answer

The firm's demand for labor is given by the equation \(120-20L=W\), indicating that the wage rate is influenced by the amount of labor. When the wage rate is at $30 a day, the firm will hire 4 workers, and when the wage rate is at $60 a day, the firm will hire 3 workers.

Step-by-step solution

Derive Firm's Demand for Labor Curve

From the production function, we have the marginal product of labor (MPL) as \(12-2L\). The value of production of the last worker (or the marginal worker) is the product of MPL and the product price, which is \(10*(12-2L)=120-20L\). A profit-maximizing firm hires workers to the point where the value of the marginal product of labor (VMPL) equals the wage rate. So the demand function for labor by the firm is \(120-20L=W\), where \(W\) is the wage rate.

Number of Workers at Wage Rate $30 Per Day

To find out how many workers the firm will hire when the wage is $30 a day, we substitute \(W=30\) into the demand function and solve for \(L\): \(120-20L=30\), which gives \(L=(120-30)/20 = 4.5\). However, since the number of workers cannot be a fraction, the firm will hire 4 workers, because hiring the 5th worker would cost more than the $30 that the last unit of output can be sold for.

Number of Workers at Wage Rate $60 Per Day

Similarly, for the wage rate of $60 per day, we substitute \(W=60\) into the demand function and solve for \(L\): \(120-20L=60\), which gives \(L=(120-60)/20 = 3\). So when the wage rate is $60, the firm will hire 3 workers.

Key concepts

Understanding the Production Function

The production function is a key concept in economics that outlines the relationship between inputs and the resulting output. For this exercise, the production function given is \( Q = 12L - L^2 \), where \( L \) represents labor input, and \( Q \) is the output per day. This function describes how changes in labor input affect production. As a general rule, as labor input increases, output initially increases at a diminishing rate due to the nature of the squared term \( -L^2 \). This is because adding more workers eventually leads to overcrowding and inefficiencies, thereby causing a decrease in output. The production function is crucial for firms to understand how efficiently they are using their resources to produce goods or services.

Calculating the Marginal Product of Labor (MPL)

The Marginal Product of Labor (MPL) measures the additional output a firm receives from employing one more unit of labor. In this exercise, the MPL is calculated as \( 12 - 2L \). The MPL is derived from the production function by taking the derivative of \( Q \) with respect to \( L \). Understanding MPL helps businesses determine how much additional output a new worker can produce.

Important considerations include:

• Initially, each additional worker contributes significantly to output when few workers are employed.
• However, as more workers are added, the MPL declines due to poorer allocation of resources and potential congestion.
• Firms use MPL to decide the optimal number of workers to maximize output and efficiency.

Thus, recognizing the point at which additional labor results in decreased MPL is crucial for optimal staffing and productivity.

Firms in a Competitive Market

Being in a competitive market affects a firm's decision-making process. A competitive market is characterized by many sellers and buyers, none of whom can control the market price. Hence, firms are price takers, meaning they accept the market price as given. In this scenario, the firm's output sells for \( \$10 \) per unit contributing to consistent revenue per additional unit of output.

In this setup, firms maximize their profit by adjusting how many workers they hire based on the cost and production capacity. Since they cannot influence prices, firms focus on internal efficiency.

• They will employ workers up to the point where the cost of an additional worker (wage) equals the value of the additional products made by that worker (VMPL).
• This ensures they operate efficiently without overextending resources or incurring unmanageable costs.

Understanding how competitive markets operate aids firms in structuring operations to remain viable.

Determining the Optimal Wage Rate

The wage rate is the cost a firm pays for labor and is imperative in understanding labor demand. In this instance, we calculate the firm's demand for labor by considering the wage rate against the value of the marginal product of labor (VMPL), which equals \( 10(12-2L) \). A firm continues to hire more workers until the wage rate \( W \) equals VMPL.

For example:

• When the wage rate is \( \\(30 \), solving \( 120 - 20L = 30 \) shows that employing 4 workers is optimal.
• At \( \\)60 \), \( 120 - 20L = 60 \) leads to employing 3 workers as sufficient and cost-effective.

Understanding the interplay between wage rates and VMPL assists firms in maintaining budgetary constraints while maximizing productivity. This decision ensures that a firm does not incur excessive labor costs, especially in competitive markets where price control is minimal.