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Exercise: How to Measure Welfare

You are tasked by your city council to find the economic surplus the market for good X creates. Furthermore, the council members want to know how the economic value is divided between the producers and the consumers. Lastly, calculate the tax burden for each party ,the dead weight loss, and the revenue that would go to the city's budget if a per unit tax of $15 is imposed on the producers.

You know that the demand schedule is,
and the supply curve can be modeled by the following equation;

In order to find the consumer and producer surpluses, one need to first solve for the equilibrium price and quantity:

Now that we know the equilibrium price and quantity we can find the consumer surplus and the producer surplus. In this problem I will demonstrate two ways to solve this exercise. The first method uses algebra, which is well suited for this type of question, since the demand and supply curves are linear. The second method is more general and involves the use of Calculus.
  • 1st method:

Notice that both areas are triangles. Therefore, we can use the formula for the area of a triangle to find both surpluses.

The total economic surplus is equal to $902.5.
  • 2nd method:
Since we know that for the consumer surplus, the area is bounded by demand curve from above and by the equilibrium price from below. We can set up a double integral to find the area:

The same idea can be applied for the producer surplus, the area is bounded above by the equilibrium price and by the supply curve from below;
If the city introduces a per unit tax of $15 on good X, then supply curve shifts to the left and is now equal to:
 
Solving for the equilibrium price and quantity we see that the market clearing price is $103 and 16 units are sold at that price.

The tax incidence on the producers and consumers is calculated by looking at the ratio of the change in price to the per unit tax.
One nice way to check if you have the correct answer is if the sum of the tax burden for both parties add up to 1. Furthermore, we see that 40% of the tax is 'payed' by the consumers while the suppliers take on most of the tax (60%).

 To calculate the dead-weight loss, realize that this area is a triangle. We can use the same formula utilized to find the consumer and producer surpluses.

This means that the new total economic surplus is equal to $880 after the tax.

The revenue that will go the city's budget  corresponds to the number of units produced times the tax.


let's visualize the problem:

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