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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?

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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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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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