I build a dynamic stochastic general equilibrium model with search and matching frictions and two sectors in order to study the labour market effects of public sector employment and wages. Public sector wages plays an important role in achieving the efficient allocation. High wages induce too many unemployed to queue for public sector jobs, while if they are low, the government faces recruitment problems. The optimal steady-state wage premium depends mainly on the labour market friction parameters. In response to technology shocks, it is optimal to have procyclical public sector wages. Deviations from the optimal policy can increase the volatility of unemployment significantly. Public sector wage and employment shocks have mixed effects on unemployment. A wage shock raises the unemployment rate, while a reduction in the separations lowers it. Hiring more people can increase or decrease the unemployment rate. All shocks raise the wage and crowd out employment in the private sector. In the empirical part, I employ Bayesian methods to estimate the parameters of the model for the United States. I find that the direct search mechanism between the two sectors is an important element to explain business cycle fluctuations of the labour market variables.