# Consumer surplus, producer surplus and Dead weight loss with inelastic supply curve

This is probably the question which is answered wrong most frequently by students when completing homework based on consumer and producer surplus. There are a few mistakes which are made which I will go over in this post.

The first mistake is labeling the producer surplus incorrectly.

The consumer and producer surplus are labelled above. The area "B" is the consumer surplus and there is nothing surprising about this. Most students get this part corect.

The producer surplus is the square area labelled "A" and this is typically what students get wrong. Sometimes they think there isn't any producer surplus.

The supply-and-demand diagram can be re-drawn to make it clearer to students. It also gives a better understanding of the nature of a good which has perfectly inelastic supply. This is what the supply curve should actually look like. You will notice that the supply curve is equal to 0 up until the point that it becomes vertical.

This is because a perfectly inelastic supply curve means the cost of producing a quantity of goods less than Q1 has no cost.

An example may be tickets to a sporting event. If the stadium isn't full, the cost of having another person in the stadium is essentially zero. Therefore, the supply curve is zero up until we reach the maximum capacity. And since it is practically impossible to add extra seats to a venue, you cannot supply a good past the point Q1.

Returning to the original problem. We first need to recall that the producer surplus is defined as the difference between the price paid and the price at which producers would have sold the good.

Up until the point Q1, producers would be willing to see the good for nothing, so the producer surplus is the entire square, which is shaded red in the above diagram.

Hopefully the addition of the supply curve at 0 clears up that problem.

The second problem encountered is the dead weight loss when supply is perfectly inelastic.

When the supply curve is perfectly inelastic, there is no dead weight loss when the government intervenes in the market-place. Dead weight loss can only occur when the quantity of goods is be produced and consumed decreases because of the intervention. The above diagram shows what happens when the government sets a price ceiling in a market with perfectly inelastic supply.

The thick black line is the price ceiling. When the government puts a price ceiling of Pc, the price of the good decreases to Pc. This increases the demand to Q2 - creating excess demand of Q2 - Q1. However, the supply remains the same and thus the equilibrium level of goods being sold in the market remains at Q1.

Since the price has decreased, the consumer surplus increases by the area "C". The lower price means suppliers get less for their good, so their producer surplus decreases by the area "C" - the same as the increase in consumer surplus.  Dead weight loss is the loss of consumer or producer surplus due to an intervention. In this case, there is no loss of consumer or producer surplus. There is only a transfer of producer surplus to consumer surplus. Even though there is now excess demand for the good, there will be no dead weight loss.

A common mistake for students to make is to draw the dead weight loss as it looks in the following diagram. This is incorrect as if there is no reduction in total surplus, there is no dead weight loss! ### 9 thoughts on “Consumer surplus, producer surplus and Dead weight loss with inelastic supply curve”

1. FSH says:

You are wrong. When there is an effective price ceiling, there is deadweight loss. The DWL is caused by the non-price competition. However, the cost caused by non-price competition cannot be shown in the diagram. thus, it seems that there is no DWL when supply curve is vertical.

1. econ101help says:

I am not "wrong". Non price competition, flow on effects (such as an queues caused by excess demand) are all potential consequences of binding price ceilings which might make such a policy a bad idea. However, the possibility for non-wage competition is abstracted away when we assume that there is one homogeneous good being sold within the market (It's implicitly assumed that this is a partial equilibrium model with one homogeneous good). If non-price competition occurred this would result in the good being sold being changed (e.g seat sizes being decreased in the stadium) which would mean we have a different good. Moreover, if there was a change to the good (e.g seat size decrease) this would also lead to a shift in the demand as peoples willingness to pay would be affected by the change to the good (I would be willing to pay less for a smaller seat, so the demand curve would shift leftwards) which would also affect our analysis.

All these factors are worth considering, but outside the scope of such a simplistic model. Hope that clears things up.

1. Warren Platts says:

Sir, you are "wrong", badly. It can be mathematically proven that there is a deadweight loss (DWL).

Consider a small town of 200 people, with an opera house with 100 seats. The demand curve D is P = 200 - Q. So maximum revenue is when sold out at \$100/seat.

At this equilibrium, the consumer surplus is \$5000 (triangle "B" in your 1st figure), and the producer surplus is is \$10,000 (box "A"). Net welfare is B + A = \$15,000

Government comes along and mandates a free show. Q: What, if any, is the DWL?

Since A (in your diagram above) goes to zero, the consumer surplus, determined graphically, is B + A, and producer surplus is zero, so net welfare is (apparently) still \$15,000.

It is there. Because the performance is free, all 200 people in town want to go, but there are only 100 seats. To avoid waiting in line losses, a lottery is held, and individual winners are required to show ID upon entrance to the theatre.

Therefore, of the people on D to the left of S, only half of them get to go: their consumer surplus is thus \$15K/2 = \$7.5K. Meanwhile, of the people on the right of S, only half of them get to go as well: half the area of that triangle is \$5K/2 = \$2.5K

Thus the actual realized consumer surplus is \$7.5K + \$2.5K = \$10K implying a DWL of \$5K--which just so happens to equal the area of your green triangle in your last figure. Call it "D". In the end, net welfare is B + C + A - D. QED

1. econ101help says:

Now, when you said “non-price competition” I thought you were referring to firms differentiating their product based on quality etc. hence why I was talking about homogenous goods. However,since it appears you were talking about non-price rationing, e.g lotteries. I better understand your point now.

Let’s firstly not suppose that these are purely my ideas. Here are a list of other people who got it “badly wrong” too:

In this solutions file here: https://msu.edu/course/ec/301/Matraves/H6Answers.doc in question 2 it says DWL is zero when supply is inelastic.

From wikipedia, we can see here: https://en.wikipedia.org/wiki/Deadweight_loss#Hicks_vs._Marshall that it also says that DWL is zero when the supply curve is inelastic.

We also see here (From question 3A in chapter 8: http://www.geneseo.edu/~stone/SOLUTIONS.Chapters6-13.doc) that

“The statement, ‘A tax that has no deadweight loss cannot raise any revenue for the government,’ is incorrect. An example is the case of a tax when either supply or demand is perfectly inelastic. The tax has neither an effect on quantity nor any deadweight loss, but it does raise revenue.”

Why are all these people “wrong”? Because it’s implicitly assumed in all these models that there is perfecting sorting, i.e, those who value the good the most are the ones who attain it after the introduction of a price ceiling. When you suggest that a lottery could be used, this violates that assumption. Understandable, this assumption might not make sense. However, without it, it’s hard to “peg down” what is the consumer surplus. Moreover, your issue arises for all price ceiling problems, not just when there is inelastic supply - see here:

In the above diagram, it is assumed that after the introduction of the price ceiling all the people with the highest willingness to pay are the ones who get to consume the good, based on the perfect sorting assumption . However, we could introduce a lottery in this case and get similar results to you.

For example, suppose that we have the same inverse demand function P = 200 - Q and an upward sloping inverse supply function P = Q. Equilibrium in this case will again be 100 units at a price of \$100. Now suppose that we set a price ceiling of \$50. In this case, we have demand being 150 units; supply being 50 units and an excess demand of 100 units.

Before the introduction of the price ceiling, consumer surplus would be 0.5*(200-100)*100 = \$5,000.

If there was perfect sorting, the consumer surplus would be \$3750 after the introduction of a price ceiling (this is in the area shaded green labelled A).

Let’s now suppose that the lottery allocates the tickets to the 100 consumers with the lowest willingness to pay (this is in the area shaded pink labelled B in the diagram attached). In this case, the DWL is only \$1,250. Thus your criticism is more aimed at the sorting assumption and not this particular example. However, as we can see, the consumer surplus would depend on who consumes the goods.

Finally, if you do wish to show the deadweight loss associated with the case of non-price competition among buyers, the worst case assumption is that the people with the highest willingness to pay use up all their surplus acquiring the good, in which case you get a diagram looking like such:

1. Warren Platts says:

Thank you for your detailed and thoughtful response. This is a great website!

Yes, my criticism was aimed at the assumption of perfect sorting where all quantity is allocated to those with the highest willingness to pay. Willingness to pay may not translate 1-for-1 into willingness to wait in line. Also, such perfect sorting defeats the very idea of a price ceiling, which is to make the product affordable to more people. The perfect sorting merely gives a bonus to those who can already afford the product!

Thus, a random allocation model should be the default assumption IMHO (and in any case, there are some genuine lottery allocation systems, such as when limited hunting licenses are distributed).

What this would look like in practice would be something along these lines:

http://i.imgur.com/QzbgsKd.png

The result is a rather messy looking chart, and it's not clear what the DWL really should be. As you say, it would be desirable to know the exact DWL. One could run a Monte Carlo simulation of course, but it would be desirable to have an exact, visually attractive graphic solution. I think I've finally figured it out.

Here are pictures of the "textbook" price ceiling models where the product is perfectly allocated to those with the highest willingness to pay, both inelastic and elastic supply:

The demand curve D is P = 200 - Q and S is P = Q = 100 for the inelastic supply case (total net welfare = 15,000), P = Q for the elastic supply, where under free market conditions, consumer surplus (CS) and producer surplus (PS) are equal (7500) and total net welfare is 10,000.

In the inelastic supply curve, there is no DWL, just a transfer of PS to CS. In the elastic supply, DWL = 2,500, but as you point out, this underestimates the DWL compared to random allocation.

The way to think about the random allocation model IMO is to calculate the average individual CS, and then multiply by the quantity actually supplied Qs. In our examples, individual CS's range from 150 to zero; since D is a straight line, the average CS is 150/2 = 75.

Doing this calculation is practically identical to increasing the negative slope of the demand curve by the proportion Qd/Qs (where Qd is the quantity demanded):

D' ==> P = 200 - (Qd/Qs)Q

Thus the new "demand" curves for the inelastic and elastic cases would now be respectively:

P = 200 - (3/2)Q
P = 200 - 3Q

Doing it this way gives the precise CS expected with random allocation, and neatly shows the DWL on the S&D chart.

Since the point of a price ceiling is to make the product affordable to the poor, and if we assume one's willingness to pay is directly proportional to one's ability to pay, if there was a means test to perfectly sort consumers at the right end of the demand curve, we get the following charts:

Here, the new "demand" curve has the same slope as the original, but the curve has moved to the left (or equivalently, the P intercept moves down). The new P intercept can be precisely specified:

P(intercept) = Pc + (Qs/Qd)CSmax

where Pc is the price ceiling, and CSmax is the maximum consumer surplus, given the price ceiling.

2. Warren Platts says:

PS here's the source code of a Monte Carlo simulation that calculates the total consumer surplus with random allocation:

Module Module1
Sub Main()
Dim Q As Double 'Quantity
Dim CS As Double 'Consumer Surplus
Dim PC = 50 'Price Ceiling
Dim b = 200, m = -1 'Demand Curve Parameters
Dim Qs = 100 'Quantity Supplied
Dim Qd = 150 'Quantity Demanded
Randomize()
For Q = 0 To Qd Step 0.000001
If Rnd() < (Qs / Qd) Then
CS = CS + ((b + m * Q) - PC) / 1000000
End If
Next
Console.WriteLine("Total Consumer Surplus = " & CS)
End Sub
End Module

2. Warren Platts says:

Using your chart above, it would look something like this:

http://i.imgur.com/KZRBnIX.png

3. FSH says:

Thank you very much for both of your discussion.

I think the deadweight loss caused by an effective price ceiling under a vertical supply curve can be shown by the area C in the last diagram. This area represent the rent dissipation caused by the non-price competition. When price is set below the equilibrium due to government control, there is a shortage. More people can afford to buy the product at this lowered price. Hence, beside paying the money price, people have to use some non-money way to compete for the products, say the 100 seats. The maximum non-price payment is represented by area C. Thus, this part of consumer surplus, in fact, is dissipated due to non-price competition.

4. Cosette Daymore says:

This is absolutely right. A great resource for my Econ 101 students!