A bi-lateral monopoly in a labour market exists when a single employer - a monopsonist - and a trade union - a monopolist - control the demand for and supply of labour in a labour market.
The monopsonist and the trade union have conflicting objectives - the monopsonist would like to pay a lower wage than the free market equilibrium, and the trade union would like to negotiate as high a wage as possible.
If a labour market is competitive in nature, with many buyers and sellers of labour, the competitive equilibrium wage and employment levels are determined by the interaction of the demand for labour D=(MRP)[1] and the supply of labour S (the average cost of labour[2]), with the equilibrium at 'e', and the wage and employment at W and Q respectively.
However, if there is a single buyer of labour - a monopsonist - it will employ up to the point where the marginal cost of labour (MCL) equals the marginal revenue product (MRP), at Qm (fewer employed than at Q). The monopsonist will only pay the average wage, at Wm (less than at W).
It we now add a union as a single supplier of labour (with all employees union members) the union can negotiate for a higher wage, at Wu in the diagram, with employment at Qu. It could actually set a wage at W, or even higher than W, but in this case employment would fall.
The actual outcome of negotiations depends upon the relative bargaining power of the employer and the union. The more powerful the union, the higher the wage rate negotiated. Employment would also increase, up to a maximum of Q, but if the wage rate is set above W, employment would fall.
Relative bargaining power is determined by several factors, including the elasticity of demand for labour, and the elasticity of supply. For example, if the elasticity of demand for labour is low, the union will have more power to influence wages and employment.
The power of a union also depends upon whether the monopsonist in the labour market is also a monopolist in the product market, or has significant monopoly power. If the employer is also a monopolist it will be easier to pass on any wage increase to the consumer in terms of increased prices.
[1] MRP determines the demand curve for labour and is determined by multiplying the physical productivity of workers by the price of the products they produce.
[2] When the supply of labour is competitive, with homogeneous units of labour, the average and marginal cost of labour is the same. Only when the supply is less than perfectly competitive will the marginal cost of labour rise above the average cost. The monopsonist does not have to pay all workers the same wage.