Bluematter.

Or: Are private contributions offset by a greater decrease in public spending?

A reader asks me to place the claim I made in the concluding paragraph of my previous post in a formal setting in order to make clear what I meant (apologies to the readers who submitted requests for posts earlier - coming soon guys, promise). The paragraph in question is this:

'If the median voter knows that the higher the size of the deficit the more some citizens will voluntarily chip in, how do you think his decision on the size of the deficit will be affected? Yes, you conscientious citizens, you just made running a larger deficit the most appealing proposition.' This statement of course, depending on the extent my assumptions hold, can equally well apply to charitable giving, corporate social responsibility and the amount of household chores each of a group of housemates carry out.

While I am temporarily diverging from the customary style of this blog here, I have tried my best to keep this accessible to anyone with a year or two of economics education under their belts.

The simple (even simplistic) model below builds on standard models of public good provision. I examine what the equilibrium level of provision of the public good (call it 'clean air' if you like) is when it is provided both by government and the (perfectly competitive) private sector. Standard expositions of public good models focus on the equilibrium amount of the public good that would be provided by either voting or in a free market economy with no government under general conditions, while the aim here is to attempt to evaluate the implications of concurrent public provision and voluntary private purchases of the public good in a more restrictive setting.

I assume a closed economy with no growth. There is a finite number, n=3, of consumers/citizens. I label the three consumers as ‘left-wing’ (l), ‘median’ (m) and ‘right-wing’ (r). There are only two goods produced in this economy, a private good Yprv and a public good Ypub and there is no saving. For simplicity, I assume that there are no fixed costs of production and that the marginal cost of producing one unit of Yprv is the same as producing one unit of Ypub, with both being equal to one. This is not restrictive since the ‘units’ used are arbitrary. Also, there is perfect competition so that the price of goods equals their marginal cost.

Each individual has an endowment, Wi, and I further assume that there is perfect equality. One unit of this endowment can be used to produce either one unit of the public good or one unit of the private good.

Consumers are assigned simple Cobb-Douglas utility functions:

, i = l,m,r (1)


and I further assume that αi+bi = 1, αi, bi>0.

I define the median consumer/citizen simply to be the one that has the median α in the population of αi’s. Also, for ease of expression, I refer to the consumer/citizen that has a higher value of α, i.e a ceteris paribus higher preference for the public good, as being to ‘the left’ or being ‘left-wing’ and the consumer/citizen with the lower value of α as being to ‘the right’ or being ‘right-wing’.

The government provides Ypub with revenues obtained through taxation, and it selects the tax level in order to maximize the utility of the median voter.

Now, let’s suppose that the private sector is restricted to producing the private good only. The public good is provided solely by the government via taxation, which is required to be levied equally on every citizen. The level of tax (and thus the public good) is chosen to maximize the median voter’s utility.

All consumers (except the median voter) do not in any way choose their consumption bundle in this setup. The median voter (via the government) chooses the tax rate to maximize his own consumption and all other consumers simply contribute the amount of tax prescribed and spend the rest of their endowment on the private good.

Since this post is already unduly long, I will leave the equations determining the amount of the public good/taxation and each voter's utility as an exercise to the reader - for anyone interested, please email me and I will be happy to provide the answers.

To help get an intuitive feel for the results, the table below shows the values our variables take when we set w=100, αr=0.3, αm=0.5 and αl=0.7. The variable g refers to the amount of the public good purchased by the various individuals directly in the market (rather than provided by government via taxation), which in this case is zero for every consumer.

I will now turn to an alternative scenario, in which the private and the public good are both provided in a perfectly competitive environment, with consumers able to purchase goods at marginal cost. I model this as a sequential game of perfect information, in which the median voter selects the tax rate (which has to be equal for all citizens) and then all consumers decide the amount of the public good to purchase privately, on top of that provided via taxation.

Setting the values of the variables at the same level as in the previous example we have:








The tax rate, and the level of the public good, are now lower. The median voter and the ‘right-wing’ consumer are now better off, while the ‘left-wing’ consumer is worse off. In fact, even though we now have private contributions (by the ‘left-wing’ consumer) to the public good, the equilibrium amount of the public good is now lower than in the case with no voluntary contributions. Given our assumptions and perfect competition, introducing voluntary contributions increases the utility of the majority of the population but leads to a lower equilibrium amount of the public good.

This seemingly counterintuitive result makes perfect sense: knowing that the ‘left-wing’ consumer will find it beneficial to contribute more when the tax rate is lower, the median voter can gain by lowering the tax rate (and thus increasing his consumption of the private good and, incidentally, the ‘right-wing’ consumer’s consumption) and ‘free-riding’ on the ‘left-wing’ consumer’s contribution to the public good.

I have a hunch this may also help explain the pattern of charitable giving observed in the US: Americans have a very high level of private contributions per capita coupled with extremely lousy public funding of 'good causes' (look, for example, at donations for victims of the Tsunami). The relatively excess generosity of the private citizens is lower than the relative stinginess of the state compared to European countries, and America consistently comes out as a laggard when it comes to contributing to 'good causes'.