# Microeconomics: What is the socially optimum price?

1. Feb 24, 2012

### lugita15

It is said in microeconomics that if there is a drought and you're the only one with a few spare water bottles, and you're feeling generous, you shouldn't just hand them out to people on a first-come, first-served basis. That's because you don't know how badly each person needs water, and if you want to maximize social welfare you'll want to give the water to the people who value the water more rather than those that value it less. So the solution to this problem is to sell the water bottles for a price. That way, people will automatically choose to buy the water if the water is worth more to them than the price, and they will choose not to buy the water if the price is worth more to them than the water. So you've classified people based on how much the water is worth to them, and you've given water to those for whom it is worth the most.

Microeconomic theory says that you should sell the water for a price even if you're not interested in making money; if you really don't care about the money you should just donate the sales proceeds to charity, but the imposition of a price increases social welfare. My question is, how is the right price determined? I know in the case where you have self-interested buyers and sellers, then the optimum price is the intersection of the supply and demand curves, i.e. the market clearing price. But how would you do it in this case, where the buyers are all self-interested, but the seller doesn't care about himself at all and just wants to help society?

The problem is, given a bunch of buyers, each with their own demand curve for water (let's assume for simplicity that they're all linear, but have different slopes), what is the right price to charge them, given a fixed supply of water bottles?

Any help would be greatly appreciated.

2. Feb 24, 2012

### 256bits

Microeconomic theory says that,,,,,
Really. I guess someone should tell all the people running the internet and that all the free stuff must now have a charge because social welfare is not being optimized, especially all those free programming people behind Linux and related software.

Optimum price for who? the seller or the buyer.

The right price.
Thus, if the seller's money has no use for society, then the optimum price is $0.00 The seller should sell his water FOR FREE! Optimally, the water would be "sold" to the people "willing" to pay most for them. In reality the only way to do this is through charging a price. I suppose what you could do is auction the water. The person who is willing to pay most for the water will receive the water, but he will get his money back as well as the water. This way you guarantee the water is actually going to the people who are willing to pay most for it, and yet you aren't charging them anything for it. But this transaction would have to be done out of sight of the other buyers, who may become angry if they find out what is going on. This would be a negative externality, which I won't consider here. You could also give away the water on a first-come first-serve basis. You can think of the early buyers as "willling to pay more" for the water. This argument is not that strong though. But in the end, what matters is that the price must be$0.00. A price of $0.00 guarantees greater consumer surplus than any other price. What may vary is who "buys" the water, which is irrelevant if you are asking only for the optimum price. EDIT: When you talk about supply and demand, you are actually talking about aggregate supply and demand, which is the sum of the supply and demand of each individual seller/buyer respectively. So whether each buyer has his own demand curve makes no difference to the problem. You care only about the aggregate demand at a given price. The individual demands add up to give the aggregate demand. BiP Last edited: Feb 25, 2012 7. Feb 25, 2012 ### lugita15 Let me rephrase my question as follows. Suppose we have N water bottles, and two people who want water, each with their own demand curve for water. Using these individual demand curves, how would you calculate the quantities Q1 and Q2 of water bottles that should be allocated to each person in order to maximize social welfare? And under what conditions does there exist a price P such that (Q1,P) and (Q2,P) are points on the respective demand curves of the two buyers? And finally, how would you generalize this to arbitrary numbers of buyers? 8. Feb 25, 2012 ### WhoWee How did you obtain the water bottles - did you pay a price? 9. Feb 25, 2012 ### lugita15 You just happen to have a fixed supply of water bottles, and no ability to get any more. You can assume they fell from the sky or something. 10. Feb 25, 2012 ### Physics Monkey I feel like this scenario depends a lot on your assumptions. How do you define social welfare? One can cook up all sorts of scenarios where the answer is anything you want. 11. Feb 25, 2012 ### Bipolarity For the purposes of answering his question, we can assume that social welfare is the same as the total surplus. That's how most microeconomics textbooks define social welfare, assuming no externalities. So it reduces to an optimization problem, whereby our goal is to maximize the total surplus. Terms in economics are generally not defined as vaguely as one might think; most of them are mathematically defined, so we need not argue over definitions. His problem is not really about subjective definitions or anything. It is just complicated mathematically because so many things have to be considered, and first principles come into question. Lugita, I am working on a solution as we speak. I think it may require a bit more mathematics than I had envisioned, so allow me to take my time. What I can assure you is that the price charged to EVERY buyer is definitely$0.00 no matter the number of bottles or the demand for them. Figuring out the quantities is a little more complicated though, but I'll get back to you on it.

BiP

Last edited: Feb 25, 2012
12. Feb 25, 2012

### Bipolarity

I think I have a solution but it assumes that all the demand curves are linear, which is only an approximation of the true general solution.

Let's suppose there are N water bottles and M buyers, each with their own individual demand curves.

The demand curves for the M buyers can be notated as follows:
$$D_{1} = c_{1} - m_{1}P$$
$$D_{2} = c_{2} - m_{2}P$$
$$D_{3} = c_{3} - m_{3}P$$
$$...$$
$$D_{M} = c_{M} - m_{M}P$$

The price charged to all these buyers will be $0 because any price higher than that results in reduced consumer surplus, as explained in my previous posts. We have to decide on the quantity. Who will receive the N water bottles? Well, when the price is zero, the demand for each of the following buyers can easily be solved by plugging in $P = 0$: Quantity Demanded by each particular buyer when the price is zero: $$D_{1} = c_{1}$$ $$D_{2} = c_{2}$$ $$D_{3} = c_{3}$$ $$...$$ $$D_{M} = c_{M}$$ Thus, the respective quantities that will be sold to each buyer is $(c_{1},c_{2},c_{3}...c_{M})$. This is basically the x-intercept of each buyer's demand curve. This is only possible if $\sum c < N$. If $\sum c > N$, then we do not have enough bottles to feed everyone and so the bottles must be rationed more wisely. I will work on a solution for that case. Last edited: Feb 25, 2012 13. Feb 25, 2012 ### lugita15 . In economics, welfare is defined as utility measured in dollars (or any currency). If you'd be willing to buy something at$9.99, but willing to sell it at $10.01, then we say that the benefit you get from it is worth$10 in welfare. Social welfare is defined to be the sum of everyone's individual welfare. Why do economists care about social welfare? Because it is a measure of economic efficiency. A situation is called economically inefficient if someone can be made better off without making anyone worse off, and it's called efficient if all opportunities to mae people better off without hurting anyone have already been taken advantage of. In order to achieve that, we have to maximize social welfare.

14. Feb 25, 2012

### Bipolarity

IF it so happens that If $\sum c > N$, then we must ration the bottles to the buyers willing to pay most for the water (though we will not charge them anything). This we can do through auctioning and looking for the highest bidder. Once we have depleted all our water, we will return the money to them, effectively making the price $0.00 But the actual quantities that each buyer receives can be mathematically derived. For each buyer, we calculate his consumer surplus by integrating the whole area under his demand curve: $$Surplus \ of \ 1st \ buyer = \int^{c_{1}}_{0}D_{1}\ dQ$$ $$Surplus \ of \ 2nd \ buyer = \int^{c_{2}}_{0}D_{2}\ dQ$$ $$Surplus \ of \ 3rd \ buyer = \int^{c_{3}}_{0}D_{3}\ dQ$$ $$...$$ $$Surplus \ of \ last \ buyer = \int^{c_{M}}_{0}D_{M}\ dQ$$ We start by "selling" the bottles for$0.00 to those buyers for whom the surplus is highest, and then continue until we have either run out of bottles, or run out of buyers.

In reality, we will not know the demand curves for anybody. We must simply auction them and then refund the money (to prevent confusion, this must be done after all the bottles/buyers have been depleted).

I think for now this solution is adequate, but I wonder what we would do for a demand curve that has a more general shape.

BiP

15. Feb 25, 2012

### lugita15

Thanks for all this. I think we can fairly assume that everyone demands all the water bottles at a price of 0 dollars. Once we have integrated the surpluses by integrating each demand curve with respect to quantity, how do we calculate the quantities we should give each person? Is the quantity we should give linearly proportional to the surplus? Or is the relationship more complicated?

16. Feb 25, 2012

### Physics Monkey

I understand the very narrow definitions in classical microeconomics texts. It just seemed to me that the original question might have been after a more interesting answer. After all, these kinds of scenarios are often used to suggest the serious inadequacies of very simplistic microeconomic models.

17. Feb 25, 2012

### Bipolarity

I made a small mistake regarding the case where you don't have enough bottles to satisfy all the buyers.

If you have enough bottles, the problem is very easy. Just calculate the x-intercept for each buyer, and sell each buyer that many bottles.

If you don't have enough bottles, which is likely the case in a drought, the problem becomes quite complex. I think you would have to employ computational techniques to solve it. Mere calculus/algebra won't be sufficient, or at least I think a solution would take me (or anyone) a good while to find.

You would have to have a program that calculates which buyer is willing to pay the most for the next bottle. You would then sell a single bottle to him. Then the amount he's willing to pay would decrease due to the law of demand. The program would then repeat the process until all the bottles have been sold or until all the buyers have been satisfied.

I can't think of any non-computational solution out of the top of my head.

BiP

18. Feb 25, 2012

### Bipolarity

My knowledge of economics does not extend much beyond the simple microeconomic or macroeconomic models. There are many economic theories, all with their own axioms and postulates. But this scenario I think was invented by the OP perhaps as more a mathematical challenge than as a question of fundamental economic principles. The conditions in the problem may suggest that the simple models are flawed, but I think the math behind it proves otherwise. It's just that the math itself is frustratingly complicated.

I do not think there any other economic models that would provide a different answer to this question. It's just that the answer itself involves some fairly advanced techniques to solve. I do not have much experience in computational math, but I don't think this problem can be solved otherwise. Of course I may be wrong.

19. Feb 25, 2012

### WhoWee

Even if they fell from the sky - a fixed supply means you'll need to first calculate your own needs - the amount left over is the quantity to price.

20. Feb 25, 2012

### lugita15

Assume you don't need any.