Price elasticity of supply is important for government because it can tell the government something about what the incidence of taxes will be.
When governments impose taxes, they can be felt more or less by different groups of people. For example, if the government imposes an excise tax on goods, will the consumers pay more of that tax or will the producers pay it? This issue of who pays the tax is called the incidence of the tax.
Price elasticity of supply helps to determine the incidence of the tax. Whenever the price elasticity of supply is higher than the price elasticity of demand, the burden of the tax falls more on the consumer. Whenever the elasticity of demand is higher than the elasticity of supply, the burden of the tax falls on the producer.
When governments impose taxes, they want to know who those taxes will affect. If the government knows the price elasticity of supply (and that of demand) it can know what the incidence of the tax will be.