9/25/11

Calculating consumer and producer surplus for a competitive, social, and monopoly scenario. A rare earth product example, part 2



This post picks up from a previous post located here. 

In this post we will work on filling out the rest of this table:

Equilibrium
Q
P
PS
CS
Ext. Cost
Eco. Surplus
Competitive
70
25




Social
52.5
33.75




Monopoly
44.2
37.9






Competitive outcome:

To calculate consumer and producer surplus, we are going to have to find some areas.  For the competitive outcome, producer surplus is going to be the area below the equilibrium price, and above the supply curve.  Since this area is a triangle, we can use the formula for finding the area of a triangle (1/2 base * height).  The height of the triangle is the price (25) and the base of the triangle quantity sold (70).  So ½(25*70) is 875, which is our producer surplus. 

For consumer surplus, we need to find the area above the price level, but below the demand curve.  Again we are calculating the area of a triangle, where the height is now 60 – 25 = 35 (this is the intercept on the price axis minus the equilibrium price level), and the base is again equilibrium quantity (70).  So ½(35*70) is 1225, which is our consumer surplus.

Calculating the external cost is a little simply.  Since the external cost is equal to 15 for each unit of output, the total external cost is 15*70 (our equilibrium quantity).  So the total external cost is 1050. 

Adding producer and consumer surplus together will give us total economic surplus (or total welfare, a good thing) but we also have to subtract the external cost.  Summing these three components together gives us:

875 + 1225 -1050 = 1050
Which is our total economic surplus accounting for negative externalities. 




 Social optimum outcome:

Under this scenario, we do pretty much the same thing for the competitive outcome, but we use the MESC line instead of the S line, so our equilibrium point is point C. 

Our social equilibrium quantity is 52.5 and our social equilibrium price is 33.75.  So the height of the producer surplus triangle is 33.75 – 15 (our included social cost) which is 18.75 while the base is 52.5.  So ½(52.5*18.75) is 492.1875. 

{NOTE:  there are two ways to calculate producer surplus here.  The way demonstrated above INCLUDES the externality, and is easier.  The other way to do it is to calculate the area above the true S curve but below the price, and then subtract out the externality.  This would involve a little more math because you would have to calculate the area of a different triangle and then a rectangle to get producer surplus. 

To do this, find the area of the triangle where the price is 18.75 (the price the producer receives without the external cost) and the base is again 52.5.  Then calculate the area of the rectangle, where the price is 15 and the base is 52.5.  Add these areas up (492.1875 + 787.5) to get a total producer surplus of 12.79.6875.  But since external costs equal 15 * 52.5, we have to subtract this from our producer surplus, so we end up with the same value of 492.1875 shown above.}

Consumer surplus is going to be the triangle on top, so our height is 60 – 33.75, and our base is 52.5.  So our consumer surplus will be ½(26.25*52.5) which is 689.0625.

To get our total economic surplus we add these two up to get (689.0625+492.1875) which is 1181.25.  And this total economic surplus is larger than our competitive outcome surplus, which makes sense because we are now accounting for the externality.  By including the externality, we have made society better off.



Monopoly outcome:
To calculate producer surplus for a monopoly we need to measure the area between the firm’s cost curve (the supply curve) and the price they are receiving for the product.  This involves calculating the area of a rectangle and a triangle.

The area of the triangle will be the area lower than the price paid by the monopoly (point A) and higher than the supply curve.  Here the height of the triangle will be 15.8 (where MR intersects S), and the base will be 44.2 (equilibrium quantity for the monopoly outcome).  So ½(15.8*44.2) is 349.18. 

The height of the rectangle will be the difference between the price the item is sold for (37.9) minus the cost of the item (15.8) or 37.9-15.8 which is 22.1.  The base of the rectangle is equilibrium quantity (44.2).  To find the area of a rectangle we multiply height times base which gives us: 22.1*44.2 = 976.82. 

To get total producer surplus, we then add up the areas of these shapes to get: 349.18+976.82 = 1326.

To find consumer surplus we need to find the area of the triangle below the demand curve, but above the price paid by the consumers.  The height of this triangle will be the intercept of the P axis (60) minus the price paid (37.9) which is 22.1.  The base is equilibrium quantity (44.2).  So the area will be ½(22.1*44.2) which is 488.41.

Now we need to calculate the total external cost for this economy which is 15 * 44.2 (output quantity).  This gives us 663.

When we add these components together, we get:
1326+488.41-663 = 1151.41


 Notice that the economic surplus under the monopoly scenario is higher than the competitive scenario.  This is because monopolies restrict output in order to charge a higher price.  Because the competitive scenario doesn’t include the external cost, it results in a lower economy wide surplus level.  This doesn’t mean that monopolies care about economic surplus, it is just coincidence that in order to fix a negative externality we need to restrict quantity and raise price, which is what monopolies do.

Also note that total economic surplus is lower in the monopoly scenario than the social outcome.  We would expect this because the social outcome is a competitive outcome accounting for the externality.  There is no other price or quantity level that would give us a greater total economic surplus than the one achieved from the social scenario.



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