Thoughtful Thursday, Lindahl Equilibrium

We use the concept of marginal benefit and total benefit from a public good. The classical situation in public goods analysis is security guards in a coop. (Or on a national level, the number of Poseidon missiles, Buchanan) Each cooperator is asked how much would you pay, or how much is it worth to you, if there were one security guard, two security guards, three security guards, etc. the question then, is how do we take this information and decide:

1. how many security guards to hire
2. how much to charge each cooperator.

Some coooperators might want more security guards and pay for all of them. Yet all cooperators will have the benefit, even if they don't actually care, of whatever protection the security guards provide.

In the table below we show what each cooperator says is ther total benefit. Dr. James M. Buchanan talk about this type of schedule in Chapter Two and Three of his book, Public Finance in Democratic Process, (on which I hope to have one of my next Thoughtful Thursday series).

Number Security Guards	Citizen 1	Citizen 2	Citizen 3
0	7	8	9
1	12	15	11
2	17	21	13
3	22	27	14
4	28	33	15
5	32	38	16
6	36	43	17
7	40	48	18
8	43	52	0
9	45	55	0
10	47	58	0
11	48	0	0

We compute the marginal benefit, which is the benefit from each new security guard. Which is simply the amount of benefit they get from each additional security guard. Or how much they would pay to have one more security guard in their cooperative.

Number Security Guards	Citizen One	Citizen Two	Citizen Three
Security Guards	1	2	3
0	5	7	2
1	5	6	2
2	5	6	1
3	6	6	1
4	4	5	1
5	4	5	1
6	4	5	1
7	3	4
8	2	3
9	2	3
10	1
11
12
13
14
15
16
17
18
19

For example, when citizen three goes from paying nine dollars to eleven dollars, that is a difference of two dollars.

My program reads in these tables and computes the second. It reads from the keyboard the cost of a secuirty guard. When I entered the cost of five dollars, the program says that we should buy eight security guards. If we add the marginal benefits for eight security guards (2, 3,0) adds up to five. That is the answer to the first question. Under Lindahl Equilibrium we divide the total cost for eight security guards proportional to the marginal equilibruium at that ponit so Citizen One pays sixteen dollars and Citzien Two pays twenty-four.

If we entered twelve for the cost of one security guards, the program tells us that the Citizen One pays ten dollars, Citizen Two pays twelve dollars and two dollars for the Third Citizen. (Notice that this doesn't quite add up to fourty-eight. Our program does not divide the integer to the cost.

We have two citizens, whose benefits from the public good can be read from the two figures below. That is citizen A gets ten units of benefits for one additional item going from zero to one item down to zero units of benefit after the fiftieth item and intermediate five units of benefit after twenty five item.

Citizen B gets 30 units of benefit for the first item, but gets no more benefit after twenty units.

Assume that it costs twenty dollars per unit. then, we read from the curve that the city shoudl get twelve units of benefit. At twelve units of benefit, we read back from the individual citizen curves that Citizen B should pay 10.5 per unit of benefit and citizen A should pay eight units of benefit.

Calculating this out, that meens that citizen Two would pay $126.00 in tax for this public good and citizen One would pay ninety six dollars in tax.

Within margin of error from reading things graphically, that is close to the $240.00 that it actually costs.

I like to use Maple to do my economics calculations. It becomes much clearer and more precise than diagrams. More importantly, I make so many stupid algebraic/arithmetic errors I cannot get the right answer on problems in engineering, physics, etc. I found the solution, at least for homework, is to do the calculations in a symbolic math system like Maple. Last century, my research area was symbolic math in Mechanical Engineering Computer AIDed Design. It was the topic of my Ph.D. dissertation. When I had an engineering course, I used symbolic math to do my homeworks.

One can plot the marginal benefit of each additonal unit as a function of quantity for a particular user. Or, how much would they pay if that was sufficient to get the government, the public at large, to supply one more acre of farm land. For user One, we have (blogf.jpg) -Q + 40. BlogF.jpg

This idea came from Hyman's book but was far better explained by my Public Fianance Professor, Dr. Warren Jones, who arranged for a Summer videotape class. To keep things simple, we assume each user has a linear curve that ends when it crosses the x-axis, or in the computerese we say that the demand for User One that the max (-Q + 40, 0). This example has two other citizens. The formula for Citizen Two is max (0, -Q/2 + 30) and the formula for the Citizen Three is max (0, -Q/3+30). These three graphs are shown in (blogg.jpg). BlogG.jpg BLog H.jpg BlogI In Lindahl equilibrium, we sum these marginal curves (blogh.jpg) and (blogi.jpg). Assume our good costs twenty dollars per unit. I asked Maple to solve for this function (blogi.jpg) and twenty, and we get fourty-eight units purchased. At this number of units, Citizen One, function F1 pays nothing-- substitute Q=48 into the equation for Citizen One (max(-Q+40,0)). Plugging into F2 and F3, we see that Citizen Two pays six dollars and Citizen Three pays fourteen dollars per unit.

Lindahl Equilibrium is a metric of success for an algorithm and voting procedure to determine the tax rate for each group of citizens. I described

And blogG.jpg shows the curves for Citizens One, Two and Three.) Assume that is the netire public. The total demand for the units of parkland is given by BLOGH.JPG as a sum of these curves, and alone s blogI.jpg

If we assume the cost of an acre of parkland in our city is always twenty units, we can ask Maple to solve for us to find out how much we will buy. (48 units)