2/17/12

Calculating equilibrium and surplus with a tax, a question and answer



This intensive economics question goes over calculating equilibrium price and quantity, then using those numbers to get consumer and producer surplus, and finally implementing a tax to see how that will change the previous results:

1. The inverse demand curve (or average revenue curve) for the product of a perfectly competitive industry is give by p=80-0.5Q where p is the price and Q is the quantity. The short-run industry marginal cost function is MC=50+0.25Q

a) Calculate the equilibrium price and quantity assuming perfect competition and profit maximization and hence calculate the consumer and producers' surplus.


b) A tax of 15 per unit sold is now imposed on every unit sold. Calculate the new equilibrium price (including tax) and quantity, the tax quantity raised and the dead weight loss caused by the tax.

To solve part a) we need to follow the steps in calculating equilibrium price and quantity.  We can set p=MC and solve for Q which will be our equilibrium Q.

P=MC=80-0.5Q=50+0.25Q, subtract 50 and add .5Q to both sides to get:
30 = .75Q or Q=40

So our equilibrium Q is 40, and we can plug this into our p or MC equation to solve for equilibrium price which will be 60 (50+10 or 80-20).

To get consumer surplus, we want to find the area above the price, but below the demand curve.  To do this we need to find where the demand curve intercepts the Y axis, or what Q will be when p is zero.  If Q=0, then p will be 80, using the inverse demand curve given above.  Since the price is 60, we know that the height of the triangle is 20 (80-60), and the base is 40 (our quantity from above).  Plugging this into our area of a triangle equation and we get:

½*(20*40) = 400

So our consumer surplus is 400, now to get producer surplus we want the area below the price but above the MC/supply curve.  Our supply curve intersects the Y axis at a value of 50, so the height of the triangle is 10, and the base is again 40.  Plugging this into our area of a triangle equation we get:

½*(10*40) = 200

So our producer surplus is 200.



Now if each unit is taxed $15, we need to modify either the inverse demand function, or the MC function (depending on whether the tax affected suppliers or consumers).  It doesn’t really matter which we shift as the result will be the same. 

Let’s add it to the inverse demand function.  We can add the tax on the left hand side of the equation, representing the new amount to be paid by the consumer:
p + T = 80 – 0.5Q, where T is the amount of the tax or in this case $15.

We can subtract the tax from both sides to get:
P = 65 – 0.5Q, which shows us that the demand curve has shifted down/left. 

We can set p and MC equal to each other and solve for equilibrium quantity which will be:

P = MC = 65 – 0.5Q = 50 + 0.25Q, subtract 50 and add 0.5Q to both sides to get:

15 = 0.75Q or Q = 20

And this means that equilibrium price will be 55.

You can follow the steps outlined above to get the new equilibrium consumer and producer surplus which will be 100 and 50 respectively.

Finally, tax revenue in this situation will be the amount of the tax times the quantity sold so:

Tax revenue = 15*20 = 300, and the deadweight loss is the difference in total surplus between the two scenarios (in this case, tax revenue counts as a surplus for the government).

Before total surplus was 600, and now total surplus is 450 so our deadweight loss in this situation is 150.

You can also calculate the area of the triangle representing deadweight loss, which will result in a triangle with a height of the tax amount and a base of the difference between equilibrium quantities so ½*(15*20) which is also 150.




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