Tax Incidence and Deadweight Loss

Econ 502: Advanced Microeconomics

Setup

Consider a competitive market in equilibrium. Let \(Q^D(p)\) denote market demand and \(Q^S(p)\) denote market supply. In equilibrium:

\[Q^D(p^*) = Q^S(p^*) = Q^*\]

Now impose a per-unit tax \(t\) on sellers. This creates a wedge between the price buyers pay (\(p^D\)) and the price sellers receive (\(p^S\)):

\[p^D = p^S + t\]

The new equilibrium requires:

\[Q^D(p^D) = Q^S(p^S)\]

Deriving the Tax Incidence Formula

Step 1: Differentiate the equilibrium condition

Starting from \(Q^D(p^D) = Q^S(p^S)\) and \(p^D = p^S + t\), totally differentiate both sides with respect to \(t\):

\[\frac{dQ^D}{dp^D} \cdot \frac{dp^D}{dt} = \frac{dQ^S}{dp^S} \cdot \frac{dp^S}{dt}\]

Step 2: Use the tax wedge

Differentiating \(p^D = p^S + t\) with respect to \(t\):

\[\frac{dp^D}{dt} = \frac{dp^S}{dt} + 1\]

Substitute into Step 1:

\[\frac{dQ^D}{dp^D} \left(\frac{dp^S}{dt} + 1\right) = \frac{dQ^S}{dp^S} \cdot \frac{dp^S}{dt}\]

Step 3: Solve for \(dp^S/dt\)

Expand:

\[\frac{dQ^D}{dp^D} \cdot \frac{dp^S}{dt} + \frac{dQ^D}{dp^D} = \frac{dQ^S}{dp^S} \cdot \frac{dp^S}{dt}\]

Rearrange:

\[\frac{dp^S}{dt}\left(\frac{dQ^S}{dp^S} - \frac{dQ^D}{dp^D}\right) = \frac{dQ^D}{dp^D}\]

\[\frac{dp^S}{dt} = \frac{dQ^D/dp^D}{dQ^S/dp^S - dQ^D/dp^D}\]

Step 4: Convert to elasticities

Define the elasticities at the initial equilibrium:

\[e_{D,p} = \frac{dQ^D}{dp} \cdot \frac{p^*}{Q^*} \quad \text{(negative)} \qquad e_{S,p} = \frac{dQ^S}{dp} \cdot \frac{p^*}{Q^*} \quad \text{(positive)}\]

So \(dQ^D/dp = e_{D,p} \cdot Q^*/p^*\) and \(dQ^S/dp = e_{S,p} \cdot Q^*/p^*\). Substitute:

\[\frac{dp^S}{dt} = \frac{e_{D,p} \cdot Q^*/p^*}{(e_{S,p} - e_{D,p}) \cdot Q^*/p^*} = \frac{e_{D,p}}{e_{S,p} - e_{D,p}}\]

Similarly, using \(dp^D/dt = dp^S/dt + 1\):

\[\frac{dp^D}{dt} = \frac{e_{D,p}}{e_{S,p} - e_{D,p}} + 1 = \frac{e_{S,p}}{e_{S,p} - e_{D,p}}\]

Step 5: Take the ratio

\[\boxed{\frac{dp^D/dt}{dp^S/dt} = \frac{e_{S,p}}{e_{D,p}}}\]

Since \(e_{D,p} < 0\), this ratio is negative. Taking absolute values, the side with the smaller elasticity (more inelastic) experiences the larger price change — and therefore bears more of the tax burden.

Deriving the Deadweight Loss Formula

Step 1: The Harberger triangle

The deadweight loss from the tax is the triangle between the supply and demand curves over the range of lost quantity:

\[DWL = \frac{1}{2} \cdot t \cdot |\Delta Q|\]

where \(\Delta Q = Q_{\text{tax}} - Q^*\) is the reduction in equilibrium quantity caused by the tax.

Step 2: Find \(\Delta Q\) in terms of elasticities

From the equilibrium condition \(Q^D(p^D) = Q^S(p^S)\), the change in quantity can be expressed through either the demand or supply side. Using supply:

\[\Delta Q = \frac{dQ^S}{dp^S} \cdot \Delta p^S = \frac{dQ^S}{dp^S} \cdot \frac{dp^S}{dt} \cdot t\]

Substituting \(dQ^S/dp = e_{S,p} \cdot Q^*/p^*\) and the result from the incidence derivation:

\[\Delta Q = \frac{e_{S,p} \cdot Q^*}{p^*} \cdot \frac{e_{D,p}}{e_{S,p} - e_{D,p}} \cdot t\]

Step 3: Substitute into the Harberger triangle

\[DWL = \frac{1}{2} \cdot t \cdot \left|\frac{e_{S,p} \cdot e_{D,p}}{e_{S,p} - e_{D,p}} \cdot \frac{Q^*}{p^*} \cdot t\right|\]

Since \(e_{D,p} < 0\), the product \(e_{S,p} \cdot e_{D,p} < 0\), and \(e_{S,p} - e_{D,p} > 0\), so the fraction is negative. Taking the absolute value and pulling the negative through:

\[\boxed{DWL \approx -\frac{1}{2} \cdot \frac{e_{S,p} \cdot e_{D,p}}{e_{S,p} - e_{D,p}} \cdot \frac{t^2}{p^*} \cdot Q^*}\]

This expression is positive because the numerator \(e_{S,p} \cdot e_{D,p} < 0\) and the leading negative sign flips it.

Key Takeaways

DWL grows with the square of the tax rate. The \(t^2\) term means that doubling the tax rate quadruples the deadweight loss. This is the core intuition behind the Ramsey rule for optimal taxation: it is more efficient to levy small taxes across many goods than a large tax on one good.

Higher elasticities mean larger DWL. When buyers and sellers are more responsive to price changes, a tax causes a larger quantity distortion, generating more welfare loss.

Perfectly inelastic demand or supply implies zero DWL. If either \(e_{D,p} = 0\) or \(e_{S,p} = 0\), the numerator is zero and \(DWL = 0\). Intuitively, if quantities don’t change, there is no allocative distortion.