The use of budgeting categories creates isolated choice problems. The standard rational model supposes that all goods are considered together, implying that an overall optimum can be achieved. If a person does not have the cognitive resources to conduct the types of complicated optimization this might imply, he or she might reduce the problem into various budgets by category. So, for example, a person might have one budget for food, another for clothing, another for utilities, and so forth. In the past, people often kept separate envelopes of money for each of these budgets. More recently, people tend to use software to keep track of spending in each category.
The consumer decision problem could now be written as max
x1, , xn
v x1, , xnk 3 4
subject to a set of budget constraints
p1x1+ + pixi< y1, pi+1xi +1+ + pjxj< y2,
⋮
pkxk+ + pnxn< yl
3 5
where xs is the amount of good s the person chooses to consume, ps is the price the person pays per unit of good s, and ymis the allotted budget for category m, and where the consumer has allotted budgets to l categories over n goods.
In any budget, m, the problem functions much like the standard consumer choice problem. In the category there is a budget constraint. If this constraint is binding, then the consumer chooses to consume at the point of tangency between the budget constraint and an indifference curve, given the consumption level of all other goods as pictured in Figure 3.4. Note that all of the consumption choices now depend on the reference point. We will ignore this reference point for now; however, later discussions develop this point further.
If instead we compared across budgets, we would see that the overall optimum consumption bundle is not necessarily achieved, because the budget process imposes artificial constraints. Suppose we plot two goods from separate budgets, good 1 from budget 1 and good 3 from budget 2 as pictured in Figure 3.5. With the same level of expenditure between these two goods, p1x1* y1k + p2x2* y2k , the consumer could purchase any bundle such that p1x1+ p3x3< p1x1* y1k + p3x3* y2k , where we now FIGURE 3.4 The Consumption Problem within a Single Budget Category (Holding Consumption of All Other Goods Constant) x1 * (ym, p1, p2 ǀ k) x1 x2 * (ym, p1, p2 ǀ k) v(x1, x2 ǀ k) x2 x2 = (ym –p1x1)/p2 = v(x1 * (ym, p1, p2 ǀ k), x2 * (ym, p1, p2 ǀ k) ǀ k) FIGURE 3.5 Nonoptimality between Budgets x1 x3 x1 * (y1 ǀ k) x3 * (y2 ǀ k) v(x1, x3 ǀ k) x3 = ( p1x1 * + p2x2 * –p1x1)/p3 = v(x1 * (y1 ǀ k), x3 * (y2 ǀ k) ǀ k)
suppress the prices of all goods in the arguments of the consumption functions. However, the consumer did not compare these possible bundles because of the artificial budget category. Instead, the consumer found the tangency of the indifference curve to budget 1 for all items in budget 1 and then found the tangency to the indifference curve for all items in budget 3. The budget may be set such that the rational optimum is excluded. For example, the consumer might allocate less money to budget 1 than would be required to purchase the unconditional optimal bundle suggested by the standard choice problem depicted in equations 3.1 and 3.2. Further, the consumer might allocate more to budget 2 than would be suggested by the unconditional optimal bundle. In this case, the consumer will purchase less of good 1 than would be optimal and more of good 3 than would be optimal. This is the condition displayed in Figure 3.5. The indifference curve crosses the budget curve so that there are many points along the budget curve that lie to the northeast of the indifference curve, constituting the dashed portion of the budget constraint. The consumer would be better off by choosing any of these consumption points. Each of these points consists of consuming more of good 1 and less of good 3.
Thus, budgeting leads to misallocation of wealth so that the consumer could be made better off without having access to any more resources. Except in the case where the budget allocations happen to line up exactly with the amount that would be spent in the unconditional optimum, this will be the case. If particular income sources are con- nected with particular budgets, any variability in income leads to a further shifting of funds. For example, if money that is received as a gift is only budgeted for entertainment or for items that are considered fun, a particularly large influx of gift money will lead to overconsumption of entertainment and fun, relative to all other items. The consumer who optimizes unconditionally could instead spend much of this money on more practical items for which he or she will receive a higher marginal utility.
The consumer problem in equations 3.4 and 3.5 is solved much the same way as the standard consumer problem from Chapter 1. Now, in each budget, the consumer will consume each good until marginal utility divided by the price for each good in a budget is equal. Where v x1, , xnk xi is the marginal utility of good i, this requires 1 ps xsv x 1*, , xn* k = 1 pr xrv x 1*, , xn* k , 3 6
which is the standard condition for tangency of the budget constraint with the indifference curve. Additionally, we can use the budget constraint for this particular budget to determine the amount of each good in the budget. However, the solution does not imply equality of the marginal utility divided by price for goods appearing in different budgets because they are associated with a different budget constraint. This necessarily implies that the indifference curve will cross the overall budget constraint except in the rare case that the budget is set so that the unconditional optimum described is attainable.