14. Surplus Decomposition Proof
Why the DWL/Efficiency Gain Triangle is ALWAYS Between MSB & MSC
The Claim We Will Prove
When government corrects an externality with a tax or subsidy, each stakeholder's gain/loss is a geometric area on the graph. When you sum all areas, everything cancels except the triangle between MSB and MSC.
We prove this across 4 cases using actual course data. Each animation builds up the stakeholder areas one by one, then dramatically removes them to reveal the net triangle.
Case 1: Dr. Galal Negative Externality + Tax = $20
\(\text{MPB} = \text{MSB} = 90 - Q\) (no external benefit)
\(\text{MPC} = 10 + Q\)
\(\text{MEC} = 20\) (constant) → \(\text{MSC} = 30 + Q\)
Market (MPB=MPC): Q=40, P=$50 | After Tax (MSB=MSC): Q=30, Pc=$60, Pp=$40
CS change: Consumers now pay $60 instead of $50, and buy 30 instead of 40. CS loss = trapezoid area = ½(40+30)×10 = 350.
PS change: Producers now receive $40 instead of $50, and sell 30 instead of 40. PS loss = trapezoid area = ½(40+30)×10 = 350.
Gov revenue: Tax of $20 on 30 units = rectangle = 600.
3rd party gain: MEC×(Qold−Qnew) + reduction in total external cost = parallelogram = 200.
Net: −350 −350 + 600 + 200 = +100 = ½×10×20 = DWL triangle area.
Case 2: Vaccination Negative Externality + Tax = $20
\(\text{MPB} = \text{MSB} = 110 - 10Q\) (no external benefit)
\(\text{MPC} = 50 + 10Q\)
\(\text{MEC} = 20\) (constant) → \(\text{MSC} = 70 + 10Q\)
Market (MPB=MPC): Q=3, P=$80 | After Tax (MSB=MSC): Q=2, Pc=$90, Pp=$70
CS change: Trapezoid = ½(3+2)×10 = 25 loss.
PS change: Trapezoid = ½(3+2)×10 = 25 loss.
Gov revenue: $20 × 2 = 40.
3rd party gain: MEC reduction = 20.
Net: −25 −25 + 40 + 20 = +10 = ½×1×20 = DWL triangle area. (Matches sec9 crude oil!)
Case 3: Vaccination Positive Externality + Subsidy = $20
\(\text{MPC} = \text{MSC} = 50 + 10Q\) (no external cost)
\(\text{MPB} = 110 - 10Q\)
\(\text{MEB} = 20\) (constant) → \(\text{MSB} = 130 - 10Q\)
Market (MPB=MPC): Q=3, P=$80 | After Subsidy (MSB=MSC): Q=4, Pc=$70, Pp=$90
CS change: Consumers pay $70 instead of $80, buy 4 instead of 3. CS gain = trapezoid = ½(3+4)×10 = 35.
PS change: Producers receive $90 instead of $80, sell 4 instead of 3. PS gain = trapezoid = ½(3+4)×10 = 35.
Gov cost: Subsidy of $20 on 4 units = rectangle = 80.
3rd party gain: MEB × extra unit = 20.
Net: +35 +35 −80 + 20 = +10 = ½×1×20 = efficiency gain triangle area. (Matches sec5!)
Case 4: Dr. Galal Positive Externality + Subsidy = $20
\(\text{MPC} = \text{MSC} = 10 + Q\) (no external cost)
\(\text{MPB} = 90 - Q\)
\(\text{MEB} = 20\) (constant) → \(\text{MSB} = 110 - Q\)
Market (MPB=MPC): Q=40, P=$50 | After Subsidy (MSB=MSC): Q=50, Pc=$40, Pp=$60
CS change: Consumers pay $40 instead of $50, buy 50 instead of 40. CS gain = trapezoid = ½(40+50)×10 = 450.
PS change: Producers receive $60 instead of $50, sell 50 instead of 40. PS gain = trapezoid = ½(40+50)×10 = 450.
Gov cost: Subsidy of $20 on 50 units = rectangle = 1000.
3rd party gain: MEB × 10 extra units = 200.
Net: +450 +450 −1000 + 200 = +100 = ½×10×20 = efficiency gain triangle area.
Summary: All 4 Cases
| Case | Ext. Type | Policy | CS | PS | Gov | 3rd Party | Net | Triangle |
|---|---|---|---|---|---|---|---|---|
| 1 | Negative (MEC=20) | Tax $20 | −350 | −350 | +600 | +200 | +100 | ½×10×20 |
| 2 | Negative (MEC=20) | Tax $20 | −25 | −25 | +40 | +20 | +10 | ½×1×20 |
| 3 | Positive (MEB=20) | Subsidy $20 | +35 | +35 | −80 | +20 | +10 | ½×1×20 |
| 4 | Positive (MEB=20) | Subsidy $20 | +450 | +450 | −1000 | +200 | +100 | ½×10×20 |
Conclusion
In ALL 4 cases, the individual surplus changes (CS, PS, Gov, 3rd party) are different shapes and sizes, but they always sum to the triangle between MSB and MSC.
This is why the DWL/efficiency gain is always \(\frac{1}{2} \times \Delta Q \times \text{MEC or MEB}\) — it's the area between MSB and MSC from Qmarket to Qefficient.