Question

Labour supply \(L\), job acceptances \(J\) and labour demand \(D\) are, respectively, related to the real wage \(W\) by: $$ L=10+W, J=b W, 00 $$ (a) Find equilibrium unemployment. (b) If there is now an income tax at rate \(t\) on wages, what happens to equilibrium unemployment?

Short answer

Equilibrium unemployment is \( 10 + \frac{50(1-b)}{b+d} \). With tax, unemployment tends to increase as net wages decrease.

Step-by-step solution

Set Equilibrium Condition

In equilibrium, the number of job acceptances \( J \) equals the number of job demand \( D \). Therefore, we set the expressions for \( J \) and \( D \) equal to each other: \( bW = 50 - dW \). To find \( W \), we solve for \( W \) in this equation.

Solve for Equilibrium Wage \( W \)

From the equilibrium condition \( bW = 50 - dW \), rearrange terms to isolate \( W \):\[ bW + dW = 50 \]Factor out \( W \):\[ W(b + d) = 50 \]Solve for \( W \):\[ W = \frac{50}{b + d} \]

Calculate Equilibrium Labour Supply \( L_e \)

Substitute \( W = \frac{50}{b + d} \) back into the labor supply equation \( L = 10 + W \):\[ L_e = 10 + \frac{50}{b + d} \]

Calculate Equilibrium Employment

From \( J = bW \) with \( W = \frac{50}{b + d} \), the equilibrium employment job acceptances \( J_e \) is:\[ J_e = b \times \frac{50}{b + d} = \frac{50b}{b + d} \]

Determine Equilibrium Unemployment

The equilibrium unemployment is the difference between labor supply and job acceptances in equilibrium, which is \( U_e = L_e - J_e \):\[ U_e = \left(10 + \frac{50}{b + d}\right) - \frac{50b}{b + d} \]Simplify the expression:\[ U_e = 10 + \frac{50}{b+d} - \frac{50b}{b+d} = 10 + \frac{50 - 50b}{b+d} \]

Consider Effect of Income Tax

If an income tax \( t \) is implemented on wages, the net wage becomes \( W(1-t) \). Substitute \( W(1-t) \) into the equilibrium condition:\[ bW(1-t) = 50 - dW(1-t) \]Solve for the new net wage and follow similar steps to find the new equilibrium unemployment. The substitution and simplification will show that unemployment levels change due to reduced effective wages affecting \( J \) and \( D \).

Key concepts

Labour Supply

The concept of labour supply revolves around the willingness and ability of individuals to work at various real wage levels. In our exercise, the labour supply equation is given by \( L = 10 + W \), where \( L \) represents the quantity of labor supplied and \( W \) stands for the real wage. This relationship indicates that as the real wage \( W \) increases, the labour supply \( L \) also increases. Why does this happen? It's quite simple—higher wages attract more workers because the opportunity cost of not working becomes greater. People are more likely to offer their labor when they are compensated better. Hence, the labour supply curve typically slopes upwards, reflecting the direct relationship between wages and labour supply. When we consider taxes, this concept can change slightly. If there is a tax on wages, the effective wage decreases, potentially reducing the overall labour supply, as individuals may not find work as attractive compared to their leisure time. This interplay is crucial when analyzing how policies can affect workforce participation.

Labour Demand

Labour demand refers to how many workers employers are willing to hire at various real wage levels. In the exercise, labour demand is expressed as \( D = 50 - dW \), with \( D \) being the labor demanded and \( d \) as a positive constant. This relationship exhibits that as wages rise, the demand for labour decreases. This inverse relationship is because higher wages increase the cost of hiring for employers. Companies will demand less labor if wages rise because it becomes more expensive to employ the same number of workers, leading them to either hire fewer employees or to automate tasks. This forms a downward-sloping labor demand curve. Additionally, when income tax is introduced, it can further impact labor demand. If taxes cause the net wages to fall, this might reduce the overhead for companies, altering how much labor they wish to employ, depending on the economic elasticity of the labor market. Understanding this mechanism helps in grasping how changes in wage-related policies might affect unemployment rates.

Equilibrium Unemployment

Equilibrium unemployment arises when the quantity of labor supplied equals the quantity of labor demanded, but some workers still remain unemployed. This scenario occurs due to various reasons such as job friction or mismatches in skills. From our solution steps, equilibrium unemployment \( U_e \) is calculated as the difference between the labour supply and job acceptances: \[ U_e = \left(10 + \frac{50}{b+d}\right) - \frac{50b}{b+d} = 10 + \frac{50(1-b)}{b+d} \]In this equation, it becomes clear that even with equilibrium wages and employment, there might still be people who are willing to work but remain unemployed for reasons not related to wage rates. When a tax is imposed, the effective or net wage decreases, impacting both labor supply and demand. As a result, equilibrium unemployment might increase because the effective diminishing of wages discourages labour supply while possibly altering labor demand. Recognizing this dynamic is essential for understanding how policy changes can affect real-world labor market outcomes and overall societal employment levels.