Consumer choice

THE BUDGET CONSTRAINT

Consumers can not buy everything they want but they are limited in their expenses by a budget. The set of bundles which are available to the consumer make up his budget constraint. The combinations of two goods x1 and x2 with prices respectively p1 and p2 that the consumer can purchase are represented graphically by the budget line expressed as:

p1x1 + p2x2 = I

the value of what the consumers buys = the income he has

A consumer can buy any basket which lies on his budget line (thus consuming all of his income) or any bundle inside the budget line (thus spending less than he has). The vertical intercept of the budget line is I/p2 which represents the maximum quantity of good 2 the consumer can buy without consuming good 1, the horizontal intercept of the budget line is I/p1 which represents the maximum quantity of good 1 the consumer can buy without consuming good 2. The slope of the budget line is -p1/p2 and it represents how the market (which sets the prices for the goods) allows the consumer to exchange good 2 with good 1 and vice versa.

HOW A CHANGE IN INCOME AFFECTS THE BUDGET LINE

As said before the budget line depends on the income of the consumer we are analysing. If the income of our consumer suddenly decreases, the line intuitively shifts inward because now the consumer can buy bundles which have a lower value, if the income instead increases the line shifts outward because now the consumer can afford bundles which are worth more. Because the slope of the line does not depend on the income of the consumer but on the prices of the goods, it does not change and therefore the line shifts in a parallel fashion.

HOW A CHANGE IN THE PRICES OF THE GOODS AFFECTS THE BUDGET LINE

If the prices change the slope of the budget line must change as well and the line becomes:

• Flatter when the ratio -p1/p2 becomes closer to 0

• Steeper when the ratio -p1/p2 becomes more negative

When the price of good 1 increases the horizontal intercept moves closer to 0 because, after the price increase, the consumer can afford to buy less of it. If the price of good 2 increases the horizontal intercept moves farther from the origin because, after the price has decreased, the consumer can buy more of it. The same reasoning holds for good 2 on the vertical axis.

WHAT THE CONSUMER ENDS UP BUYING

The bundle that maximizes the satisfaction of a consumer while allowing him to remain on the budget line is called the optimal choice. In order to do so the consumer will choose the point on the budget line which allows him to reach the highest indifference curve. Bundles which are outside the budget line can not be purchased by the consumer and therefore can not be optimal. Baskets inside the budget line do not use all the resources available to the consumer and therefore there are surely other points that reach higher utilities. This is because, by the assumption that more is better, consumers prefer bundles that are valued more to the ones that are valued less. The optimal choice must therefore lie on the budget line and, more precisely, where the indifference curves and the budget line are tangent because to every other point on the budget line corresponds a lower level of utility because the corresponding indifference curve is closer to 0. The tangency condition at the optimal choice requires that:

So:

The optimal choice in the picture is called an interior optimum: a choice where the consumer purchases positive quantities of both commodities. The tangency condition can also be rewritten as:

Expressed in that form the tangency condition means that the consumer gets the same satisfaction per dollar spent by consuming good 1 or good 2.

LOOKING AT THE PROBLEM FROM ANOTHER PERSPECTIVE

Until now we have considered the problem of maximizing utility given the budget constraint. We can consider the problem in an opposite way and we could ask ourselves which bundle will the consumer choose in order to have a certain level of utility while trying to minimizing costs to do so. This kind of problems is called the expenditure minimization problem. Graphically we have to look at the bundle that minimizes costs while remaining on the same indifference curve. If the curve is too close to the origin the consumer is not spending enough money to reach the indifference curve regardless of which bundle on that line he chooses. If the budget line crosses twice the indifference curve, the consumer can reach the level of utility desired in the points where the two intersect but, by doing so, the consumer is not minimizing costs because there other budget lines that touch the indifference curve and that make the consumer spend less. The optimal choice also in this problem is the point where the budget line is tangent to the indifference curve. As said a lower budget line would not reach the desired level of utility while an higher one would not minimize costs. Because both the problem of maximizing utility given a budget and minimizing costs given a specific level of utility to reach have the same solution, they are said to be dual to one another. This means that, no matter how you look at the problem, the optimal choice of the consumer stays the same.

CORNER POINTS

We defined an interior optimum as an optimal choice where the consumer possesses both goods but, in reality, consumer might also choose to consume only one good. If the consumer can not find an interior optimum that suites his needs he will choose to consume a basket where only one good is present. This kind of optimal choice is named as a corner point and it is a point along the horizontal or vertical axis. We find corner solutions when the consumer is willing to substitute a good for the other and vice versa and therefore in the case of perfect substitutes. Take as an example butter and margarine: you will buy and use only one of them, there is no need for you to buy both. Remember that the consumer is always trying to maximize his own utility. If the indifference curves representing the preferences of the consumer are always steeper than the budget line, the consumer will always consume a bundle which has only good 1 because, by doing so, he reaches the highest possible indifference curve. If instead the indifference curves are always flatter than the budget line, in order to maximize utility, the consumer will choose a bundle lying on the vertical axis. If you have linear indifference curves described by the general function U=ax1+bx2 the problem of finding the optimal choice can be summarized in:

• The consumer will choose to consume only good 1 if:

• The consumer will choose to consume only good 2 if:

CHOICE WITH PERFECT COMPLEMENTS

If we are considering goods which are considered as perfect complements the preferences of the consumer will be represented by L-shaped indifference curves. In this case the problem of finding the optimal choice is simply the one of finding the highest indifference curve touching the budget line so the curve which has his kink lying on the budget line.