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Basically, I'm trying to understand why the total surplus is maximized at the equilibrium and what happens if the price isn't at the equilibrium.

Say the price of a good is the equilibrium price. Now make it more expensive. I know that after a while the price will adjust to the equilibrium again, but what will happen before that? I'm trying to use an example but there are things I can't fill in:

The equilibrium price of the good is \$20 and the equilibrium quantity is 20. Given a straight demand and supply curve and a maximum buyer's imposed value of \$40, the consumer surplus is $\frac{1}{2}(40-20)\cdot20=200$ and the producer surplus is $\frac{1}{2}(20-0)\cdot20=200$.

Now make the good \$30. The quantity demanded falls to 15 and the quantity supplied rises to 25. Now, the consumer's surplus is the triangle I marked in blue, $\frac{1}{2}(40-30)\cdot15=75$ (because only 15 items will be sold), but what about the producer's surplus? Is this the triangle I marked in red? I think not, because that seems to imply that the price is only \$10. But if not, what is the producer's surplus, and why?

enter image description here

Thanks in advance and my apologies for the possibly confusing phrasing.

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    $\begingroup$ This question boils down to "what is producer's surplus?" which, without the homework element is a much better question in my opinion. $\endgroup$
    – Giskard
    Apr 18 '18 at 9:38
  • $\begingroup$ Maybe you could think about who ends up with the rectangular chunk between p=10 and p=30. $\endgroup$
    – Dan
    Apr 18 '18 at 11:12
  • $\begingroup$ @Dan I don't really know. If the price rise were due to a tax, that area would be the tax revenue, but I don't know what the area means when there is just a price change not due to taxes. $\endgroup$
    – Sudera
    Apr 18 '18 at 11:20
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    $\begingroup$ @Sudera It seems like you're close. If the price rise were due to a tax, that chunk of money would go to the government. If it's just because the producer put the price up, where does that money go to? $\endgroup$
    – Dan
    Apr 18 '18 at 11:26
  • $\begingroup$ @Dan Oh I guess it just goes to the producer then. Thanks for your help! $\endgroup$
    – Sudera
    Apr 18 '18 at 11:45
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What is CS and PS out-of-equilibrium at $P=30$:

I think that answer to this question lies in misunderstanding the definition's of consumer and producer surplus.

The consumer and producer surplus are actually (following Mankiw Principles of Economics, 8th ed, pp 135 and 140

consumer surplus: the amount a buyer is willing to pay for a good minus the amount the buyer actually pays for it

producer surplus the amount a seller is paid for a good minus the seller’s cost of providing it [where Mankiw later in chapter explicitly states cost is here to be interpreted broadly as "the value of everything a seller must give up to produce a good", including the opportunity cost (or more broadly s wilingness to sell)].

You will note that the definitions above talk about price people pay (i.e. actual market price) not 'equilibrium price'. Some lecture notes or textbooks will include the word equilibrium in the definitions but that is only because virtually all analysis using Supply-Demand is an analysis of some equilibrium.

More rigorously, since the maximum willingness to buy is bounded by the demand curve and willingness to sell by supply curve we can define consumer surplus as the area below the demand curve and above price paid by consumers and

$$CS = \int^{q_m}_0 D(q) dq − p_mq_m $$

where $D(q)$ is arbitrary demand curve, $p_m$ is price consumers have to pay in the market, and $q_m$ quantity sold at the market.

$$PS = p_mq_m - \int^{q_m}_0 S(q) dq $$ wher $S(q)$ is some arbitrary supply curve (also if we would want to be even more general we could allow $p$ to be different for suppliers and consumers e.g. situations with taxation where there would be wedge between price consumers pay and price per quantity suppliers get due to the tax wedge).

Yes, in some chapters or lecture notes the above formula will be defined in terms of equilibrium price but that is only because the market price/quantity is the equilibrium price/quantity in any equilibrium analysis (i.e. in equilibrium $p_e = p_m, q_m=q_e$). However, in an out-of-equilibrium analysis you have to use actual market prices (this would be equivalent of analyzing price floor or price celling (e.g. see Mankiw Principles of Microeconomics, 8th ed, pp. 116).

Consequently, in your case the consumer and producer surplus can be properly visualized as shown on the tikz graph below I made:

enter image description here

where the area of consumer surplus (out-of-equilibrium) at price $P_m=30$ is the blue shaded region below demand and above market price bounded by market quantity, and the producer surplus (out-of-equilibrium) at price $P_m=30$ is the red trapezoid (area under the market price and above supply bounded by market quantity).

We can also calculate the area of that trapezoid, although to do that we need to know what is the price at which $Q_S=0$ (or explicitly know what the supply and demand equations are). In your question you don't state what the price at which supply is equals zero is but if you would provide it we could calculate it (also you seem to have made some mistake when copying down the problem as your demand does not lie on a line since it is given by points $(0,40);(15,30);(20,20)$ - which simply mathematically cannot lie on a single line - I originally planned to go all the way deriving the linear demand and supply equations based on the points but this mistake in problem set up makes it impossible ).

Why is total welfare maximized in equilibrium?

Well the reason for this is that under all the background assumptions of the stylized partial equilibrium model (e.g. no market failures and other 'shenanigans' that would affect PS and CS in way not shown in the partial equilibrium analysis here), the total combined area of PS and CS is maximized at the equilibrium price and quantity.

I won't be providing a rigorous proof of this as that would require a lot of calculus and this problem is long as it is, but I encourage you to take a look at proofs provided in MWG, Microeconomic analysis pp 328 or Varian, Microeconomic Analysis pp 224 if interested.

However, the reason why the welfare be easily seen graphically by the following thought experiment: would the total surplus be larger if $P>P_e$ or if $P<P_e$? The answer is no as you can see visualized in the two figures below. As in both figures below the total surplus area will always fit within the big triangle given by area below demand and above supply clearly the total welfare is in both situations lower than the total welfare in a situation where $P_m=P_e$.

enter image description here


PS: An important caveat to note is that the above applies when you want to analyze what would be the total welfare outside equilibrium vs the welfare in equilibrium. If you want to compare two situations which are both equilibrium e.g. supply shifts to the right due to technological progress, you have to compare the total welfare in the market in the old equilibrium vs the new one. You should not confuse such situations with market not being in equilibrium.

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I see a lot of these posts are very complete, and I think a more direct answer might be beneficial here.

The supply curve represents the amount that producers are willing and able to sell at. Producer surplus is the space between the existing price and the supply curve.

The demand represents the amount that buyers are willing and able to buy at. Consumer surplus is the space between the existing price and the demand curve.

The CS and PS are bound to be below the observed Q (that's to the left!), since no one receives anything for units that have not been exchanged in the market. They also must stay above 0 also.

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This four page reference shows how producer surplus is defined and calculated using integral calculus:

https://www.math.ubc.ca/~malabika/teaching/ubc/spring11/math105/surplus.pdf

As quantity increases from q = 0 to q = qe (equilibrium quantity) the price rises at each point on the supply curve S(q).

The equilibrium point (pe, qe) is taken independently by finding the intersection supply curve S(q) and demand curve D(q).

The producer surplus is defined as the area in the rectangle pe x qe minus the area under the rising supply curve AS(q) as quantity goes up from q = 0 to q = qe. When you get to qe you stop doing the calculation so there is no meaning associated with data points to the right of the equilibrium point with respect to producer surplus definition.

I don't know why the terms consumer surplus and producer surplus are used in this context. In a counterfactual world the limits of the calculation might be less than or greater than the equilibrium point, and then consumers would spend less or spend more, and producers would sell less or sell more, but in those counterfactual models one must move the supply and demand curves to a different point of intersection to recognize sales at the same supply and demand price.

In reality goods sell at price times quantity. If the producer drives down the price sufficiently consumers tend to buy in greater quantity. Early adopters of high definition TV or electric cars are willing to pay more so sales occur at higher prices and lower quantities. Producers use the learning curve and invest in fixed costs to drive down prices and increases quantities produced, and demand goes up as prices come down over time. All during this process the supply and demand curves are moving around and do not represent a static calculus solution.

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  • $\begingroup$ the second part of this A is incorrect and the first half is misleading at the best. Consumer and producer surplus are not necessarily defined by equilibrium price - only when we are doing analysis where market is at an equilibrium they are defined in that way (you can see this explained in any undergraduate 101 economics textbook for example Principles of Economics by Mankiw). Second, it is completely valid to make this sort of analysis without any movements of supply and demand - in fact this is how effects of price roofs/floors are analyzed in any 101 textbook $\endgroup$
    – 1muflon1
    Oct 31 '20 at 13:57
  • $\begingroup$ When innovators introduce new goods, such as electric cars or high definition televisions, the initial price of goods is relatively high and the initial quantity sold is relatively low. Does the 101 textbook model map these actual sales data to points on a supply and demand curve that is static at the time of such sales? Or is the supply and demand curves dynamic such that supply intersects demand at every price where an actual sale occurs? So don't the calculations for consumer and producer surplus change over time as conditions change in the actual market where transactions are data points? $\endgroup$ Oct 31 '20 at 14:26
  • $\begingroup$ A situation where there is a new good that is scarce (or limited because supplier is monopolist due to patent etc) would not be out of equilibrium situation. That would have its own equilibrium price and quantity here the question specifically asks about how to calculate consumer/producer surplus when price is not in the equilibrium, the question does not ask what is the CS and PS in a particular equilibrium with new products. It is actually completely valid to do such out of equilibrium analysis and it is done when evaluation some policies $\endgroup$
    – 1muflon1
    Oct 31 '20 at 14:34
  • $\begingroup$ - hence the answer above is incorrect or if you want it could be said it is answer to some another Q not question given here. Here the author of the question is literary stating in the first paragraph: "I'm trying to understand why the total surplus is maximized at the equilibrium and what happens if the price isn't at the equilibrium." $\endgroup$
    – 1muflon1
    Oct 31 '20 at 14:34
  • $\begingroup$ If the price changes in a market transaction then I would take that as a new equilibrium price which invalidates my model of the equilibrium point causing me do draw either another supply curve, or another demand curve, or another supply and demand curve to account for the new data point. If the change in stated price results in no sales then I would infer that there is no demand at that stated price. If it results in sales I would then know the quantity demanded at that price and this would be a new equilibrium model. $\endgroup$ Oct 31 '20 at 14:50

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