Who Really Pays a Tax showed that the side of the market that responds less to price bears more of a tax, whoever pays it legally. With Point Elasticity with Calculus we can say exactly how much.
Key idea
For a small per-unit tax $t$, the price buyers pay rises by $$\Delta P_b = \frac{\varepsilon_S}{\varepsilon_S + |\varepsilon_D|}\,t,$$ and the price sellers keep falls by $\Delta P_s = \dfrac{|\varepsilon_D|}{\varepsilon_S + |\varepsilon_D|}\,t$. The less elastic side bears the larger share. It doesn't matter which side sends the money to the government.
Why it works
Start at equilibrium $(P, Q)$. After the tax, buyers pay $P + \Delta P_b$ and sellers keep $P - \Delta P_s$, with $\Delta P_b + \Delta P_s = t$. Quantity demanded falls by about $|\varepsilon_D|\,Q\,\dfrac{\Delta P_b}{P}$; quantity supplied falls by about $\varepsilon_S\,Q\,\dfrac{\Delta P_s}{P}$. The market still clears, so the two falls are equal: $|\varepsilon_D|\,\Delta P_b = \varepsilon_S\,\Delta P_s$. Solving with $\Delta P_b + \Delta P_s = t$ gives the formula. Notice the "other side's" elasticity on top: buyers bear more when supply is very elastic, because sellers can easily walk away.
Worked example: From elasticities
At equilibrium, $\varepsilon_D = -0.5$ and $\varepsilon_S = 1.5$. What share of a small tax do buyers bear?
- $\dfrac{1.5}{1.5 + 0.5} = 0.75$.
- Buyers bear 75%, sellers 25%: demand is the less elastic side.
Answer: 75%.
Worked example: From straight lines
$Q_d = 100 - 2P$, $Q_s = 3P - 20$, and a tax of \$5 per unit on sellers.
- Equilibrium: $100 - 2P = 3P - 20$, so $P = 24$, $Q = 52$.
- With the tax: $100 - 2P_b = 3(P_b - 5) - 20$, so $5P_b = 135$ and $P_b = 27$; $P_s = 22$.
- Buyers' share: $3/5 = 0.6$. Elasticity check at (24, 52): $\varepsilon_D = -\tfrac{12}{13}$, $\varepsilon_S = \tfrac{18}{13}$, and $\dfrac{18/13}{18/13 + 12/13} = 0.6$.
Answer: $P_b = 27$, $P_s = 22$; buyers bear 60%.
Common mistake
Giving the larger share to the more elastic side, which turns the formula upside down. The side that can easily change how much it buys or sells escapes the tax; the side that can't, pays. Check: if demand were perfectly inelastic ($\varepsilon_D = 0$), your formula should give buyers the whole tax: $\varepsilon_S/(\varepsilon_S + 0) = 1$.
For straight-line curves the result is exact for any tax size: buyers bear $d/(b + d)$ when $Q_d = a - bP$ and $Q_s = c + dP$. For curved demand and supply it holds for small taxes, using the elasticities at the starting point.