### See how deep the rabbit hole goes…

In our first appendix, we wandered a little into the math behind public goods. That post has less algebra and more graphs. If what you read here is moving too fast, that might be a good place to start.

Public goods, in brief, are goods which you don’t have to own to enjoy. These are things like public parks, an army, or your neighbor’s front lawn. In that post, we assumed a particular form for a utility function for public goods.

[latex] \mbox{Utility} = \sqrt{\alpha S_0 + \sum\beta S_i}[/latex]

where $latex S_0$ is the amount I spend on the good, $latex S_i$ are my neighbors’ spending. The parameters, $latex \alpha$ and $latex \beta$ determine how much benefit I get from my own spending and my neighbors’ spending, respectively. There’s more on why this specific functional form was chosen here. Other forms are certainly valid.

In this model, the profit is the utility minus my expenditure, $latex S_0$

[latex] \mbox{Profit} = \sqrt{\alpha S_0 + \sum\beta S_i} – S_0[/latex].

If an overbearing government (or home owners association, or business) enforced the same spending on public goods by all beneficiaries, we can determine the optimal spending by equating $latex S_0$ and $latex S_i$ and taking the derivative of profit with respect to this single variable:

[latex] \frac{dP}{dS} = \frac{\alpha+(n-1)\beta}{2\sqrt{S(\alpha+(n-1)\beta)}} – 1[/latex]

Here we’ve introduced n, the total number of participants. Equating this derivative to zero, we find that the optimum level of spending is

[latex] S = \frac{\alpha + (n-1)\beta}{4} [/latex]

for a profit of

[latex] P = \frac{\alpha+(n-1)\beta}{4} [/latex].

per person.

If, on the other hand, we leave it up to each individual to contribute what she thinks is best for herself, we differentiate the profit equation with respect to changes in $latex S_0$ alone.

[latex] \frac{\partial P}{\partial S_0} = \frac{\alpha}{2\sqrt{\alpha S_0+\beta \sum S_i}} – 1[/latex]

Then, since all the participants are assumed to be 100% rational (and will therefore make the same decision), we equate $latex S_0$ and $latex S_i$, set the derivative equal to zero and discover that spending becomes

[latex] \frac{\alpha^2}{4(\alpha + (n-1)\beta)}[/latex]

and profit is

[latex] \frac{\alpha}{2}-\frac{\alpha^2}{4(\alpha+(n-1)\beta)}[/latex].

In the other post on public goods,we reported that, for our simple model of paving the street leading to a cul-de-sac with 5 houses, our homeowners would only be willing, on their own, to spend $2 on paving rather than the ideal $50, which reduced their profit from $50 to $18 each.

Now, let’s introduce a “Galt factor,” G, representing the increased productivity of workers in John Galt’s mountain enclave due to the absence of moochers. How high would the Galt factor have to be for the Galt society to have more efficient lawn care than us regular shmoes? Both $latex S_0$ and $latex S_i$ should be multiplied by G since the labor and other input in the Galt society is that much more potent. So, we multiply all $latex S_0$ and $latex S_i$ by G in the profit for the non-cooperative profit and equate that to the (unmodified) profit for the cooperative system.

[latex] \frac{G\alpha}{2} – \frac{G\alpha^2}{4(\alpha+(n-1)\beta)} = \frac{\alpha+(n-1)\beta}{4} [/latex]

Solving for G, we find that the Galt factor necessary for the Galt society to operate more efficiently than a “moocher” society which recognizes and subsidizes the public good in question is

[latex]\frac{(\alpha+(n-1)\beta)^2}{\alpha(\alpha+2(n-1)\beta)}[/latex]

Here are some sample results using numbers from the beekeeping and road paving examples in the last post. In the gardening example, most of the benefit of keeping up my lawn goes to me. Only a little goes to my neighbors which is why $latex \alpha$ is greater than $latex \beta$ in the first two rows. The final 3 rows are based on a good that’s completely public–everyone benefits equally no matter who paid. Last post we used the example of paving the road that leads to the cul-de-sac. Since everyone enjoys the benefits of the spending equally, $latex \alpha=\beta$.

$latex \alpha$ $latex \beta$ $latex n$ $latex G$

80 10 5 1.17

80 10 100 7.0

40 40 5 3.27

40 40 20 10.7

40 40 100 50.7

We could even simplify the above expression by constructing the “public-ness” of a good, P.

[latex] P = \frac{(n-1)\beta}{\alpha+(n-1)\beta} [/latex]

Public-ness ($latex P$) is the fraction of the total benefit that goes to the community as whole. The Galt factor in terms of $latex P$ is

[latex] G= \frac{1}{1-P^2}[/latex]

Now, if we had any reason to believe our model this would be a very useful formula. Should a particular service be left to the market or paid for with public funds? Just figure P, make a guess as to how much more efficient your private market would be and compare. For goods that are completely shared, P is just $latex 1- 1/n$ with $latex n$ being the size of the community. For a cul-de-sac of 5, the Galt factor must be 25/9 = 2.78. For a small town of 10,000, P = .9999 and G = 100,000,000. So, for this adorably naive model the Galt society could likely manage to outdo us in gardening, as long as their cul-de-sacs didn’t get much bigger than 5 houses, but when it comes to truly public goods consumed by hundreds of people (like a public park or a security guard at your apartment complex) they’ll be hard pressed to accomplish in one hour what takes the rest of us a full work week. For goods shared by hundreds of thousands of people, the Galters might as well not try.

In many cases, at least some of the benefit can be secured exclusively for the benefit of the producer and her customers. For example, a road can be built with toll booths or the radio station can interrupt your music with commercials. These actions are effectively changing $latex \alpha$ and $latex \beta$. These efforts almost always lead to a reduction in both (i.e. it’s less useful to everyone), but also reduces $latex P$ making the private option more attractive and enticing people to invest in production. If we assume that private systems are more efficient than public ones, we can even reach the conclusion that adding obstacles (like commercials, patents and toll booths) can boost efficiency.

It’s fair to say that a model this crude isn’t likely to be accurate to any degree, but it gives an idea of some of the qualitative factors that effect evaluation of public goods. Too often our political discussion never gets past pithy platitudes. When we say, “private markets are more efficient,” it’s important to specify, even with a crude estimate, just *how much* more efficient. Liberals should not pretend the Galt factor is 1, but conservatives should not pretend it’s infinite. And, we should recognize that neither private nor public production is the right solution for all good in all markets. Math matters.

Do you have corrections or comments? How would you improve the model? Can you think of some goods for which the utility functions could be measured? Send me an email.