How to Calculate Deadweight Loss from a Tax on a Competitive Market
Master welfare economics with this complete walkthrough of calculating deadweight loss from a per-unit tax. Learn how to find the new equilibrium, calculate consumer and producer surplus changes, and understand why taxes create efficiency losses even when they raise revenue.
📹 Video Walkthrough: This Exact Problem
Watch the full solution for calculating deadweight loss from a tax step-by-step with graph analysis and numerical calculations.
Table of Contents
The Problem
Calculate the deadweight loss:
In a competitive market, the demand curve is given by Qd = 100 - 2P and the supply curve is Qs = 20 + 3P, where P is price in dollars and Q is quantity.
The government imposes a per-unit tax of $6 on producers. Calculate the deadweight loss created by this tax.
Express your answer in dollars and show your work step-by-step.
This is one of those classic microeconomics problems that shows up on every exam. You've got supply and demand curves, a tax gets thrown in, and now you need to figure out how much efficiency is lost. It looks straightforward, but there are a lot of places to mess up.
The trick here is understanding that the tax creates a wedge between what consumers pay and what producers receive. That wedge shrinks the market, and the lost transactions create deadweight loss. If you're stuck on a similar problem, you can always generate a custom video solution on Torial.
Understanding Deadweight Loss
Deadweight loss (DWL) is the loss of economic efficiency that occurs when the equilibrium quantity is not at the socially optimal level. When a tax is imposed, it drives a wedge between the price consumers pay (Pd) and the price producers receive (Ps).
Tax = Pd - Ps
The tax creates a gap between consumer price and producer price
Important things to remember:
Key points about deadweight loss:
- Deadweight loss is a triangle on the supply and demand graph. It's the area between the curves from the new quantity to the old quantity.
- It represents lost transactions that would have been mutually beneficial but don't happen because of the tax.
- It's separate from tax revenue. The government collects tax revenue, but DWL is the efficiency loss on top of that.
- It increases with the tax size, but not linearly. A bigger tax creates disproportionately more DWL.
- It depends on elasticity. More elastic demand or supply means bigger deadweight loss for the same tax.
For our problem, we need to find the area of the triangle that represents the transactions that would have happened without the tax but don't happen with it. If you need more help visualizing welfare economics, check out other economics videos in our library.
Finding the Pre-Tax Equilibrium
First, we need to know where the market was before the tax. This gives us our baseline for comparison.
Step 1: Set demand equal to supply
At equilibrium, quantity demanded equals quantity supplied:
Qd = Qs
100 - 2P = 20 + 3P
100 - 20 = 3P + 2P
80 = 5P
P* = $16
Step 2: Find the equilibrium quantity
Plug P* = $16 into either the demand or supply equation:
Q* = 100 - 2(16) = 100 - 32 = 68 units
Or: Q* = 20 + 3(16) = 20 + 48 = 68 units ✓
💡 Key Insight: Before the tax, the market clears at P* = $16 and Q* = 68 units. This is the efficient outcome where total surplus (consumer + producer surplus) is maximized.
After the tax, we'll see this quantity shrink, and that shrinkage is what creates the deadweight loss.
This is where a lot of students get tripped up. They forget to find the pre-tax equilibrium first and jump straight to the tax calculation. But you need this baseline to calculate how much surplus is lost. Always start with the no-tax case.
How the Tax Changes the Market
When a $6 per-unit tax is imposed on producers, they now receive $6 less than what consumers pay. This shifts the supply curve upward by the amount of the tax.
Understanding the tax wedge:
If consumers pay Pd and producers receive Ps, then:
Pd = Ps + Tax
$6 = Pd - Ps
The supply curve shifts up because producers need to receive Ps, but consumers must pay Pd = Ps + $6 to get that quantity supplied.
New supply curve with tax:
The original supply is Qs = 20 + 3P. But now, if producers receive Ps, they supply:
Qs = 20 + 3Ps
Since Pd = Ps + 6, we can write:
Qs = 20 + 3(Ps)
Qs = 20 + 3(Pd - 6)
Qs = 20 + 3Pd - 18 = 2 + 3Pd
This is the supply curve as seen by consumers. It's shifted up by $6.
Notice how the supply curve effectively shifts left (or up). For any given quantity, producers now need a higher price from consumers to be willing to supply it, because they're paying the tax. This is the key insight that makes the tax calculation work.
Finding the New Equilibrium with Tax
Now we find where the new supply curve (with tax) intersects demand. This gives us the new quantity and prices.
Step 1: Set new supply equal to demand
The new supply curve (from consumer's perspective) is Qs = 2 + 3Pd. Set it equal to demand:
Qd = Qs (new)
100 - 2Pd = 2 + 3Pd
100 - 2 = 3Pd + 2Pd
98 = 5Pd
Pd = $19.60
Step 2: Find producer price
Producers receive $6 less than consumers pay:
Ps = Pd - Tax = $19.60 - $6 = $13.60
Step 3: Find the new quantity
Plug Pd = $19.60 into the demand equation:
Qnew = 100 - 2(19.60) = 100 - 39.20 = 60.8 units
Or verify with supply: Qnew = 2 + 3(19.60) = 2 + 58.80 = 60.8 units ✓
⚠️ Check your work: The new quantity (60.8) should be less than the original quantity (68). The consumer price ($19.60) should be higher than the original ($16), and the producer price ($13.60) should be lower.
This confirms the tax is working as expected: it reduces quantity and creates a price gap.
Notice how the quantity dropped from 68 to 60.8 units. That's 7.2 units of lost transactions. Those lost transactions are what create the deadweight loss. Want to see more worked examples? Browse through hundreds of economics solutions on Torial.
Calculating Deadweight Loss
Deadweight loss is the area of the triangle between the supply and demand curves, from the new quantity to the old quantity. This represents the lost consumer and producer surplus from transactions that no longer happen.
The deadweight loss triangle:
DWL is a triangle with:
- Base: Change in quantity = Q* - Qnew = 68 - 60.8 = 7.2 units
- Height: The tax amount = $6
DWL = ½ × Base × Height
DWL = ½ × (Q* - Qnew) × Tax
DWL = ½ × 7.2 × $6
DWL = ½ × $43.20 = $21.60
Alternative method: Using the formula
For a linear supply and demand with a per-unit tax, the general formula is:
DWL = ½ × |ΔQ| × Tax
Where |ΔQ| is the absolute change in quantity. This works because the triangle's height is the tax wedge.
Final Answer:
Deadweight Loss = $21.60
The tax creates an efficiency loss of $21.60. This is the value of the mutually beneficial transactions that no longer occur because of the tax.
What this tells us:
- The tax reduces market quantity from 68 to 60.8 units
- 7.2 units of transactions that would have been beneficial no longer occur
- The deadweight loss of $21.60 is separate from the tax revenue collected
- This represents a real loss to society, not just a transfer
Notice how we used the simple triangle formula. The deadweight loss is always a triangle when supply and demand are linear, which makes the calculation straightforward. The key is finding the right base (quantity change) and height (tax amount).
Graphical Interpretation
Understanding the graph helps you see why deadweight loss exists and how to calculate it visually.
On the graph, you'll see:
- Original equilibrium: Where supply and demand intersect (P* = $16, Q* = 68)
- New supply curve: Shifted up by $6 from the original supply
- New equilibrium: Where new supply intersects demand (Pd = $19.60, Qnew = 60.8)
- Tax revenue rectangle: The area between Pd and Ps, from Q = 0 to Qnew
- Deadweight loss triangle: The area between supply and demand, from Qnew to Q*
💡 Visual check: The DWL triangle should be to the right of the new equilibrium quantity. It's the "missing" surplus from transactions that don't happen.
The triangle's left edge is at Qnew, its right edge is at Q*, and its top is bounded by the supply and demand curves.
Being able to identify the deadweight loss triangle on a graph is crucial for exams. Professors love to ask you to shade it in or calculate its area. If you can visualize it, the math becomes much easier.
Common Mistakes to Avoid
Here are the mistakes that cost students the most points. Learn them now so you don't make them on test day. If you want personalized help avoiding these errors, create a custom study video for your specific problem.
❌ Mistake #1: Using the Wrong Quantity Change
Calculating DWL using the new quantity instead of the change in quantity. The base of the triangle is Q* - Qnew, not just Qnew.
Fix: Always use the difference between the original and new quantities. DWL = ½ × (Q* - Qnew) × Tax.
❌ Mistake #2: Confusing Tax Revenue with Deadweight Loss
Calculating the area of the tax revenue rectangle instead of the DWL triangle. Tax revenue is Qnew × Tax, which is different from DWL.
Fix: Tax revenue is a rectangle. DWL is a triangle. They're separate concepts. DWL is always smaller than tax revenue for small taxes.
❌ Mistake #3: Not Shifting the Supply Curve Correctly
Adding the tax to the price instead of shifting the supply curve. If tax is on producers, supply shifts up by the tax amount.
Fix: For a tax on producers, the new supply curve (from consumer perspective) is Qs = f(P - Tax). Substitute and solve.
❌ Mistake #4: Using the Wrong Price in Calculations
Using the original equilibrium price instead of recognizing that consumers pay Pd and producers receive Ps after the tax.
Fix: After a tax, there are two prices: Pd (what consumers pay) and Ps (what producers receive). They differ by the tax amount.
❌ Mistake #5: Forgetting the ½ in the Triangle Formula
Calculating DWL as (Q* - Qnew) × Tax instead of ½ × (Q* - Qnew) × Tax. This gives you double the correct answer.
Fix: It's a triangle, so you need the ½. Always remember: area of triangle = ½ × base × height.
❌ Mistake #6: Not Finding Pre-Tax Equilibrium First
Jumping straight to the tax calculation without establishing the baseline. You need Q* to calculate the quantity change.
Fix: Always solve for the original equilibrium first. This gives you Q* which you'll need for the DWL calculation.
❌ Mistake #7: Mixing Up Tax on Producers vs. Tax on Consumers
Shifting the wrong curve. A tax on producers shifts supply; a tax on consumers shifts demand. The DWL is the same either way, but the curves shift differently.
Fix: Read the problem carefully. Tax on producers → shift supply up. Tax on consumers → shift demand down. The economic burden is the same, but the math looks different.
Practice Problems with Video Solutions
Best way to get good at this? Practice. Try these similar problems and check your work with video solutions. You can also generate instant video explanations for any economics problem you're working on.
Practice Problem 1: Tax on Consumers
Demand: Qd = 80 - P, Supply: Qs = 2P - 10
A $5 per-unit tax is imposed on consumers. Calculate the deadweight loss.
Hint: This time the tax is on consumers, so demand shifts down. Your answer should be around $8.33.
Get instant video solution on Torial →Practice Problem 2: Larger Tax
Demand: Qd = 120 - 3P, Supply: Qs = 2P + 20
A $10 per-unit tax is imposed on producers. Calculate the deadweight loss.
Hint: Bigger tax means bigger DWL, but not proportionally bigger. DWL should be around $30.
Get instant video solution on Torial →Practice Problem 3: Finding Tax Revenue Too
Demand: Qd = 100 - 2P, Supply: Qs = 3P - 20
A $4 per-unit tax is imposed. Calculate both the deadweight loss and the tax revenue.
Hint: Tax revenue = Qnew × Tax. Make sure you can distinguish between the rectangle (revenue) and triangle (DWL). DWL ≈ $4.80, Revenue = $96.
Get instant video solution on Torial →Practice Problem 4: Percentage Tax
Demand: Qd = 200 - 4P, Supply: Qs = 2P + 40
A 20% ad valorem tax is imposed on producers (tax = 0.20 × price). Calculate the deadweight loss.
Hint: This is trickier. The tax depends on price, so Pd = Ps(1 + 0.20). You'll need to solve this system. DWL ≈ $13.33.
Get instant video solution on Torial →When to Use Triangle Formula vs. Integration
Should you always use the triangle formula, or do you sometimes need integration?
✓ Use Triangle Formula When:
- Supply and demand curves are linear
- You have a per-unit tax (not percentage)
- The problem gives you equations
- You want a quick calculation
✓ Use Integration When:
- Curves are nonlinear (exponential, logarithmic, etc.)
- You're given a graph and need to estimate
- The problem explicitly asks for integration
- You need the exact area under a curve
For this problem? Triangle formula is perfect. Linear curves mean the DWL is always a triangle, and the formula DWL = ½ × |ΔQ| × Tax works every time. When you're juggling multiple concepts, it helps to have a step-by-step video walkthrough that shows exactly which method to use.
Other Videos
No Action Found
Improving Goal Setting with AI Implementation
Explaining Droplet Communication in Microchannels
Explaining Microfluidics for Preserving Fragile Biological Molecules
Stuck on a Different Problem?
Get instant, personalized video explanations for your homework
Create Your First VideoRelated Articles
How to Calculate Consumer and Producer Surplus from Market Equilibrium
Complete walkthrough of calculating consumer and producer surplus. Learn how to find equilibrium, set up the integrals, and interpret what these measures mean.
How to Calculate Price Elasticity of Demand Using the Midpoint Formula
Understand microeconomics with this complete walkthrough. Learn the midpoint formula, understand what elasticity values mean, and avoid common mistakes.