Question

Suppose that the labour supply of workers is \(L^{S}=w-5\) and the labour demand is \(L=20-w\). Derive the labour market equilibrium.

Short answer

The equilibrium wage is 12.5 and the equilibrium labour quantity is 7.5.

Step-by-step solution

Understand the Supply and Demand Equations

The labour supply function is given as \( L^S = w - 5 \) and the labour demand function is given as \( L^D = 20 - w \). In equilibrium, supply equals demand, so \( L^S = L^D \).

Set Supply Equal to Demand

To find the equilibrium, set the supply function equal to the demand function:\[ \begin{align*} w - 5 &= 20 - w. \end{align*} \]

Solve for Equilibrium Wage (w)

Add \( w \) to both sides to eliminate the \( w \) on the right-hand side:\[ \begin{align*} 2w - 5 &= 20. \end{align*} \]Add 5 to both sides to solve for \( w \):\[ \begin{align*} 2w &= 25. \end{align*} \]Divide both sides by 2:\[ \begin{align*} w &= 12.5. \end{align*} \]

Calculate Equilibrium Labour Quantity (L)

Substitute the equilibrium wage \( w = 12.5 \) back into either the supply or demand equation.Using the supply equation \( L^S = w - 5 \):\[ \begin{align*} L^S &= 12.5 - 5 = 7.5. \end{align*} \]This is also the equilibrium labour quantity.

Key concepts

Supply and Demand

In economics, supply and demand form the foundation of understanding how markets operate, including those for labor. Supply refers to how much of a particular good, service, or resource people are willing to offer at varying price levels. In our case, the labor supply is represented by the function \(L^{S} = w - 5\). This equation tells us that the number of workers willing to work increases as wages (\(w\)) increase. Conversely, demand represents how much of that good, service, or resource people want to purchase at different prices. The labor demand function \(L^D = 20 - w\) suggests that employers want more labor as wages decrease. To understand a market, we need to look at both supply and demand together, observing how they interact to determine prices and quantities in a given market.

Equilibrium Wage

The equilibrium wage is a crucial component in the labor market, representing the wage at which the quantity of labor supplied equals the quantity of labor demanded. This balance is essential for a stable market.
The equilibrium wage is found by setting the supply and demand equations equal:

• Start with \(w - 5 = 20 - w\).
• By solving, add \(w\) to both sides to get \(2w - 5 = 20\), then add 5 to each side resulting in \(2w = 25\).
• Divide both sides by 2 to find \(w = 12.5\).

At this equilibrium wage of \(12.5\), the market clears, meaning there are no surpluses or shortages of labor.

Labour Supply Function

The labor supply function \(L^{S} = w - 5\) describes how the supply of labor responds to changes in wage levels. This function shows a direct relationship between wages and the number of workers willing to work.
As wages rise, the willingness of workers to supply more labor typically increases. The constant term "5" indicates that when the wage is low enough, there are no workers willing to work, as shown with the condition \(w > 5\) for any positive labor supply.Several factors might shift the labor supply function:

• Population growth can increase \(L^{S}\) since more people could potentially enter the workforce.
• Changes in worker preferences or in non-work income sources might also impact how labor is supplied at various wage levels.
• Policy changes, such as tax benefits or social benefits, can influence people's willingness to supply labor.

Understanding how the labor supply function operates is essential for analyzing labor market dynamics and ensuing equilibrium outcomes.