A modification of the primitive model of a polyelectrolyte solution, containing macroions and counterions, is presented. The macroions are assumed to carry 20 (or in some calculations 10) negative charges and the counterions are monovalent. Asymmetry in size between the two species is 15:2. In addition to the Coulombic forces, the macroions also interact via a short-range directional attractive force. The parameters of this attraction are chosen to result in the formation of dimers, but no trimers or higher clusters Can be formed. This model of a polyelectrolyte solution is studied for a range of concentrations and for several values of the depth of the potential well, using the statistical-mechanical theory based on the two-density formalism for associating fluids. The results indicate that the short-range attraction between macroions and consequently the formation of dimers, yields a significant decrease of the osmotic coefficient for the model polyelectrolyte solution. Part of the effect is due to the strong attraction between the doubly charged aggregates and counterions in the solution. The effect of dimerization is also reflected in the excess internal energies and in all three pair distribution functions.