Then the consumers take the area above that price, and the suppliers get the value below. These are not necessarily equal areas; they depend on the econometrics shaping the curves.

In your example, the market-clearing price is $13. The consumer benefit is $3 to a and $0 to B, and the supplier benefit is $6. But no single party knows in advance what the market-clearing price is.

The easiest way to find it is for all 3 to engage in a reverse Dutch auction. A writes down "buy 1 @ $16". B writes down "buy 1 @ $13". C writes down "sell 10 @ $10". The auctioneer looks at all the slips and announces that the price is $13 and that C is the first (and only) supplier. Everyone leaves happy.

Trade only occurs in regions above the supply curve and below the demand curve. There is no way to slice that area up such that the pieces have greater total area than the whole.

Unless you get into odd situations like luxury goods or Geffen goods, the maximum benefit is always going to be obtained when the last good is sold at the market-clearing price. And the only way to really guarantee that--since consumers don't line up to buy in order of their positions on the demand curve--is to sell every good at that price.

There are many ways to divvy up that area such that suppliers get more or less than they would at one fixed price, but those ways do not make the trade worth more in absolute terms. In order for the suppliers to get more, the consumers must get less. Price discrimination is absolutely a zero-sum game that benefits the suppliers exactly as much as it hurts the consumers.

You are likely discounting the value of the consumer benefit as money not seen in the transaction. A dollar that a consumer would have spent on something, but didn't need to, can still be spent somewhere else.

I've heard examples where there are fixed costs for the producer/seller. Suppose to produce something it costs $1000 plus $1 per unit. Suppose there is a buyer A that values it at $800 and 500 buyers (B) that value it at $2.

What price should the seller charge for it?

If the seller charges a fixed price of $2 (or less), then he can make and sell 500 of them, which would cost him $1500, and he would be paid $1000 (or less). He would lose $500, so he would not do it and no value would be created.

If the seller charges a fixed price between $2 and $800, then he can make and sell 1 of them, which would cost him $1001 to produce, and we would be paid $800 (or less). He would lose money, so he would not do it and no value would be created.

There is no single fixed price where any value is created at all.

However, suppose he could charge a different amount to different people. He charges A $700 and charges all the B customers $1.80. He makes and sells 501 of the good at a cost of $1501. He is paid $1600. He profits $99. A gets $100 of surplus value. B buyers gets $0.20 of surplus value each for a total of $100.

By charging different customers different prices we have created $299 of value where a fixed price would have created none at all!

The suppliers do not know, a priori, the tastes of buyers and the shape of the demand curve. They only know that they can stay in business as long as their marginal cost per unit is less than the sale price on the open market. You don't know whether 500 people will buy at $2 or 50000 people will.

That $1000 is the entry ticket to the market. If the supplier produces just one unit, it is already paid and gone. As they say, sunk costs are sunk. That up-front cost is not recoverable through per-unit sales. (It is possibly recoverable through the sale of the business or its capital.)

So with respect to your example, the seller pays $1000, sells 501 units at $2, then sells the business assets while exiting the market. The buyer and seller benefit to trade is entirely unaffected.

The total amount buyers would pay is $1800. The total cost to sellers for a cleared market is $501. That makes a potential total benefit to trade for a cleared market of $1299. But the market clearing price only allows the seller to capture $501 of that benefit.

The seller probably does not know those exact numbers beforehand, and cannot use them to conclusively justify the $1000 investment. How does the seller know that one buyer is delusional enough to pay up to $800 for an item that costs $1 to make? If he does know, he simply charges that person that price and removes him from the equations. Then you're still stuck needing to know the next most demanding buyer, and how much they would spend. Each party has imperfect knowledge of the market, and there is no way for any one of them to make decisions based on knowing everything.

Another interesting example is kickstarter. It provides goals at different levels, with different rewards at each level. The different levels in many cases are effectively providing price discrimination even though they may have marginal benefits. Some projects reach their goals and get funded only because people at different levels of interest choose to fund those projects different amounts. If they could only set a single "price" for all backers they would never happen.

You don't need to subtract a penny to make it so. The value of the product for A must be greater than $13 or A won't buy it. The value of $13 cash must be greater than the value of not selling the product to the consumer or the seller won't sell. Everybody has more value than what they started with!

The only way this can possibly be true for all cases is via per-consumer price discrimination, anything less will always leave someone out.

  The value of the product for A must be 
  greater than $13 or A won't buy it.

For example, if I'm a trader and I know I can sell a widget on for $13 (after all costs are taken into account) and a supplier offers it to me for $13 I don't lose any money by taking the deal, but I don't make any money either. So I don't care if the transaction happens or not.

This, evidently, doesn't ever happen in the real world: in reality, you'd take into account whether you have time to buy it, whether you have the will to carry it around, etc., before arriving at a definite answer that implies (value - price) has collapsed to some non-zero value.

In my own mind, it's kind of irrelevant who gets it, because I don't recognize that buyer B has a right to pay only $10 or $13 or any other amount. I think it follows that if you allow sellers to discriminate among any classes of buyers, then there is no way to draw the line at how permissibly narrow those classes can be drawn.

And you think he's making loaded arguments? The value he's talking about is simply the sum of consumer surplus and producer surplus. They're both legitimate; a sale price of $10 allots $9 in gains from trade (or "value") entirely to consumer surplus, and perfect discriminatory pricing allots the same $9 of value to producer surplus, but the value is still produced despite the consumer surplus being $0.

Compare the $14-uniform-price scenario, where the consumer surplus is $2, the producer surplus is $4, and the gains from trade are $6 when they could be $9. That's $3 of lost value, and I don't see why you think that's a biased or controversial way of putting it.

The question of how to divide that surplus is very interesting and especially relevant to judgement of price discrimination.