The efficient allocation of a consumers budget

We have just seen how a utility maximising consumer allocates a budget between two goods. It is possible to generate a simple rule from this analysis that enables us to get a good intuitive understanding of consumer behaviour. Before we derive this result, another concept is needed – the concept of marginal utility.

Let us return to the notion of an individual’s utility function; we wrote this earlier as

Suppose that, in some period of time, an additional unit of good x is consumed without any change taking place in the amount of y which is consumed. This additional consumption of good x will lead to an increase in the consumer’s total utility. The marginal utility of good x is the change in total utility resulting from the consumption of an additional unit of that good. It can be represented by:

where the symbol [read as “delta”] means “a small change in” some variable of interest, Qx is the quantity of X consumed per period, and MUx is the marginal utility derived from good X. In a similar fashion, MUy = U/Qy, is the change in total utility resulting from the consumption of an additional unit of good y (with consumption of x unchanged).

Now let us return to the tangency condition we noted above. When the consumer is maximising utility, the slopes of the budget line and the indifference curve are equal. The slope of the budget line is given by (the negative of) the price of x divided by the price of y. That is

Slope of budget constraint =

On the other hand, the slope of an indifference curve at any point along it is given by (the negative of) the marginal utility of x divided by the marginal utility of y. That is

Slope of indifference curve =

The derivation of these two results about the slopes is given separately in Box 3.1. If you would prefer to avoid the maths, you will have to take the results on trust.

Finally, as these two slopes are equal at a point of utility maximisation, the following must be true when an individual is optimising his or her expenditure pattern:

=

which after multiplying through by -1 and then rearranging gives

=

What does this result tell us? It shows that maximum utility is achieved by allocating total income between different goods in such a way that the marginal utility derived from each good consumed per unit of money spent is equal over all goods. If we interpret MUx and MUy as the “values” derived at the margin from small, additional amounts of x and y respectively, then we can say the following. A utility maximising individual purchases goods in quantities such that the “value-for-money” derived from each good at the margin is equalised.

To obtain an intuitive understanding of the condition, it may be helpful to ask what would happen if the consumer’s pattern of purchases did not equalise ‘value for money’ at the margin for each good purchased. If these values for money ratios are not equal, then some goods must offer more value for money than others. A rational individual would switch expenditure between goods, away from those with low value for money and into those giving high value for money. This reallocation would enable the individual to get more total utility from the same income level, demonstrating that the original allocation was not an efficient one.