Both can happen. A producer can increase prices unilaterally (that is without an increase in the costs of production). In this case the producer can be undercut

Both can happen. A producer can increase prices unilaterally (that is without an increase in the costs of production). In this case the producer can be undercut by those accepting a greater volume of sales for lower margins. When an increase in the costs of production happens, for a sufficiently spherical cow, the price level will rise across all producers depending on several factors including the elasticity of demand and consumer indifference. In the case of the tax, it is not entirely clear as to which specific case applies. Privately held companies may be more willing to maintain market share by eating a portion of the profits where a publicly held company will have a driving force to maintain its total margin.