Question

2. Consider the following model of the labour market. where w is the wage rate, Ld is labour demanded by the firms and Ls is labour supplied by workers What condition should δ satisfy in order for the second equation to be a reasonable labour supply function (i) What condition should satisfy in order for this system to have a unique equilibrium. (iii) Assume that δ = 1, express the systemin matrix form and use matrix algebra to find the equilibrium values of L and w.

Answer 1

i)

Labor supply should be an increasing function of wage. Thus the slope of L-w graph must be positive. Hence for a reasonable labor supply, $\delta > 0$.

ii)

In order for system to have a unique equilibrium -

$L_d = L_s \Rightarrow 2 - w^* = 1 + \delta w^*$

$\Rightarrow \delta (w^*+1) = 1 \Rightarrow \delta = \frac{1}{1+w^*}$ where w* is the equilibrium wage.

c)

If $\delta = 1$, equations become -

$L_d = 2 -w \Rightarrow L_d + w = 2$

$L_s = 1 + w \Rightarrow L_s - w = 1$

In the equilibrium, L_d = L_s = L

Hence, these equations can be written in the matrix form as -

$\begin{bmatrix} 1 & 1\\ 1& -1 \end{bmatrix}$$\begin{bmatrix} L\\ w \end{bmatrix}$$=\begin{bmatrix} 2\\ 1 \end{bmatrix}$

$\Rightarrow\begin{bmatrix} L+w\\ L-w \end{bmatrix}$$=\begin{bmatrix} 2\\ 1 \end{bmatrix}$

Solving them, we get:

Adding both rows : 2L = 3 => L = 3/2

and thus w = 2 - 3/2 = 1/2