Budgeting and the Slutsky Matrix

I've been trying with not much success over the last few years to come up with a workable budget for my family. I should find my lack of success surprising. After all, I'm an economist. I teach about the "budget constraint": you can't spend more than what you have. It should be simple, add up all revenues, then pick commensurate expenditures. But it is not. I've been wondering why.

Despite the timeless models routinely used by economists, income and expenditures are flows, not stocks. A household's finances are like a river: water (income) flows in more or less regularly and at a more or less predictable rate, and water (expenditures) flows out more or less quickly for various uses. At any given moment, there is water in the river, sometimes more, sometimes less, it does not usually get all used up. Of course, if one was to dam the river at both ends, one could measure the stock of water available, and then exactly parcel out that amount to various uses such that all the water would be used. In that sense, the quantity of water available for various purposes would be constrained by the quantity available in the same way as a consumer's expenditures are constrained by his available wealth. But does anyone face a budget constraint of this type in their daily lives? i.e., dividing a fixed stock of wealth among competing expenditures?

People who live paycheck to paycheck may come closest to that model of behavior: one gets a thousand dollar paycheck and must decide how much of it to apply to rent, utilities, groceries, transportation, entertainment. Until the next paycheck, that is all the money available, and spending it one way means that it's not available for another purpose. This is what is meant by a binding budget constraint.

But on further examination, even households living hand to mouth do not always have to give up something to get something else. Since life is a flow, some purchases can be postponed. Not buying an item now does not necessarily imply doing without, it may just mean waiting a little longer to buy it. Credit also allows people to transfer future income into the present and avoid having to wait. In other words, people are aware that their income is a flow and that more will flow in soon. We don't normally think of our expenditures as being subtracted from a stock of depletable wealth.

Life's pervasive uncertainty also complicates budgeting. Many people, such as those paid commission or tips and those with occasional jobs have an uncertain stream of income. Even for those with a predictable income, many expenditures are not. Who could have predicted the toothache that necessitated an emergency trip to the dentist? or the pothole that forced a trip to the auto repair shop to fix a flat tire and bent wheel rim? People may set money aside for these unforeseen events, but who knows whether that would be too much or not enough? The flow of our lives includes an unpredictable future.

My point here is not that budgeting is impossible, but that budgeting is at best approximate. A family earning $2,000 per month would be foolish to plan as much expenditures as one earning $6,000. Its river of income is just not large enough. When people choose an expenditure pattern, they are not usually spending a budget (i.e., planning how to spend every last dime of their stock of wealth), but are instead vaguely aware of having to live within their means. For example, the family earning $2,000 a month will be aware that its flow of income is not large enough to sustain expenditures that would be reasonable if it earned $6,000 a month but would be extravagant at its current level.

The metaphor of a budget constraint is a useful metaphor in economics. It points to the reality that even large rivers do not have an infinite amount of water, and that diverting some to irrigate an area may mean that there would not be enough left to irrigate another. This is certainly a useful lesson to understand and apply in real life. Further, using a budget constraint and the notion that people optimize, one can derive properties of demand functions such as the Slutsky equation, which breaks down the impact of a change in prices into income and substitution effects. To a certain extent, it is also useful and enlightening to understand that an increase in the price of something people would like to have makes them poorer.

But here's where some of modern economics goes awry: given that it's difficult to even think of any non contrived example where people clearly choose a consumption bundle that uses up their entire budget, is there any point, other than vain bragging about one's mathematical dexterity, in demonstrating that the Slutsky matrix of substitution effects is symmetric and negative semi-definite? That some mathematician some years ago thought to verify this ought, I suppose, to be recorded for posterity in the pages of an obscure journal, but it should be left on the shelves of a library to gather dust, not be required reading for graduate students in economics. (You can find example of this here, here and here.)

Economics ought to be about understanding the kinds of choices people make and the factors that affect their choices, so that we could better understand and improve the societies that these choices produce. It should not be a game among a handful of Illuminati to come up with the nth esoteric footnote to what was in the beginning a pretty simple, common sense idea.

I'm glad to see that this point of view is beginning to gain traction among at least a few economists. See in particular this post by Karl Smith: What Kind of Papers Should We Celebrate: Math Free Essays and the Baby-Sitting Economy and this one by Paul Krugman: Models, Plain and Fancy.