Question

A firm is producing chocolate bars using only labour. The production function is \(Q=20 L-0.5 L^{2}\), where \(L\) denotes labour. The firm is selling its chocolate bars in a competitive market and the price of a chocolate bar is \(£ 1\). The firm hires workers in a competitive market. The wage paid to a worker is \(w\). Write down the profit function of the firm and find the labour demand function \((L\) as a function of \(w\) ) of the firm.

Short answer

The labour demand function is \( L = 20 - w \).

Step-by-step solution

Understanding the Production Function

The production function is given by \( Q = 20L - 0.5L^2 \), where \( Q \) represents the quantity of chocolate bars produced and \( L \) is the amount of labor employed. This function demonstrates the output level produced with varying levels of labor.

Define the Revenue Function

In a competitive market, the revenue (R) is the price per unit times the quantity of goods sold. Here, the price per chocolate bar is \( £1 \), so the revenue function can be written as \( R = Q \). Substituting in the production function, the revenue function becomes \( R = 20L - 0.5L^2 \).

Define the Cost Function

The cost of labor is given by \( wL \), where \( w \) is the wage rate per worker, and \( L \) is the amount of labor hired. This represents the total cost of hiring labor.

Write Down the Profit Function

Profit (\( \pi \)) is given by the difference between revenue and cost. Thus, the profit function is: \[ \pi = R - C = (20L - 0.5L^2) - wL \].

Simplify the Profit Function

Simplify the expression obtained for the profit function: \[ \pi = 20L - 0.5L^2 - wL \].

Find the First Derivative of the Profit Function

To find the labor demand, differentiate the profit function with respect to \( L \): \[ \frac{d\pi}{dL} = 20 - L - w \].

Solve the First Order Condition for Maximal Profit

Set the first derivative equal to zero to find the optimal amount of labor that maximizes profit: \[ 20 - L - w = 0 \]. Solve for \( L \) to get \( L = 20 - w \).

Use Second Derivative Test

To ensure that this is a maximum, compute the second derivative: \( \frac{d^2\pi}{dL^2} = -1 \). Since it is negative, the critical point is a maximum.

Key concepts

Labour Demand

Labour demand refers to the number of workers that firms are willing and able to hire at a given wage level. In our example involving chocolate bar production, the firm's decision on how much labour to hire revolves around maximizing profit. From the simplified profit function, we see that labour heavily influences production output and associated profits.
When determining labour demand, firms calculate the additional revenue generated by hiring an extra worker, known as the marginal revenue product of labour, which must be balanced against the wage paid. The resulting labour demand function, derived from maximizing the firm's profit, is given by the equation \( L = 20 - w \). This function implies that labour demand decreases as wages increase, reflecting the inverse relationship between wage levels and the number of workers a firm is willing to employ.

• Lower wages generally increase demand for labour.
• Higher wages may lead to firms hiring fewer workers due to cost considerations.

Competitive Market

In a competitive market, numerous buyers and sellers interact, with no single participant able to influence market prices. Firms in such markets are price takers, meaning they sell their products at the prevailing market price and must accept the wage set for labour.
For our chocolate bar firm, the competitive market context means it must sell each bar at the going rate of £1, regardless of production costs or strategies. Similarly, the labour market is competitive, setting standard wages that the firm must comply with when hiring workers.

• Price-taking behaviour leads firms to focus on efficient production.
• Firms compete primarily on cost management and output efficiency.

Profit Function

A profit function represents a firm's financial gain as a difference between total revenue and total costs. For the firm producing chocolate bars, this involves comparing the revenue from sales with the cost of labour.
The profit function provided is \( \pi = 20L - 0.5L^2 - wL \), where profit depends on output levels determined by labour and the cost associated with employing this labour. Simplifying this equation gives valuable insights into how profit varies with changes in labour and wage rates.
The nature of the profit function, with costs rising more than proportionally with labour due to the squared term, suggests diminishing returns as labour increases. This means adding more workers will eventually lead to smaller increases in output and profit.

• Firms seek optimal labour levels to maximize profits.
• Understanding the profit function aids in strategic decision-making.

Cost Function

The cost function represents the total expenses incurred by a firm in the production process. In our case, the primary cost is labour, expressed as \( wL \). The wage \( w \) indicates the cost per unit of labour, and the entire expression \( wL \) reflects the total labour cost.
The firm's aim is to manage these costs effectively while producing enough goods to maximize profits. Lowering costs or increasing productive efficiency allows the firm to improve profitability, especially in a competitive market.

• Efficient cost management gives firms a competitive edge.
• Minimizing waste and optimizing labour utilization are key strategies.