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Although the tax incidence is generally dependent on the nature of the elasticity of the demand or supply, in special scenarios can the producer take up the complete burden of the tax so that the price paid by the consumers does not change? If this is possible does this mean that the supply curve is shifted in order to change the tax incidence?

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There are two possible answers here, depending on whether you want to take a partial or general equilibrium setting. Note that the partial equilibrium analysis of tax incidence is only taught in undergraduate micro textbooks but is not considered a sensible approach in current research.

Partial Equilibrium: Suppose your supply is perfectly inelastic (for example a non-rival but excludable good which you produced last period) and demand is perfectly elastic (let's say another firm produces a perfect substitute with perfectly elastic supply sold at some price p). Then the producer will bear the full burden of the tax.

General Equilibrium: In a general equilibrium setting, anything can happen. In response to a tax, both supply and demand curves may shift. However, in this case the standard definition of tax incidence is no longer meaningful. In such cases, researchers either calculate some loss function for each consumer or simply look at price changes to analyze tax incidence.

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