# Consumer theory

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Consumer theory is a theory of economics. It relates preferences (through indifference curves and budget constraints) to consumer demand curves. The models that make up consumer theory are used to represent prospectively observable demand patterns for an individual buyer on the hypothesis of constrained optimization.

## Indifference curves and budget constraintsEdit

For an individual, indifference curves and an assumption of constant prices and a fixed income in a two-good world will give the following diagram. The consumer can choose any point on or below the budget constraint line BC. This line is diagonal since it comes from the equation $xp_X + y p_Y \leq \mathrm{income}$. In other words, the amount spent on both goods together is less than or equal to the income of the consumer. The consumer will choose the indifference curve with the highest utility that is within the budget constraint. I3 has all the points outside of their budget constraint so the best that the consumer can do is I2. This will result in them purchasing X* of good X and Y* of good Y.

Income effect and price effect deal with how the change in price of a commodity changes the consumption of the good. The theory of consumer choice examines the trade-offs and decisions people make in their role as consumers as prices and their income changes.

## Price effectsEdit

These curves can be used to predict the effect of changes to the budget constraint. The graphic below shows the effect of a price shift for good Y. If the price of Y increases, the budget constraint will shift from BC2 to BC1. Notice that because the price of X does not change, the consumer can still buy the same amount of X if he or she chooses to buy only good X. On the other hand, if the consumer chooses to buy only good Y, he or she will be able to buy less of good Y because its price has increased.

To maximize the utility with the reduced budget constraint, BC1, the consumer will re-allocate consumption to reach the highest available indifference curve which BC1 is tangent to. As shown on the diagram below, that curve is I1, and therefore the amount of good Y bought will shift from Y2 to Y1, and the amount of good X bought to shift from X2 to X1. The opposite effect will occur if the price of Y decreases causing the shift from BC2 to BC3, and I2 to I3.

If these curves are plotted for many different prices of good Y, a demand curve for good Y can be constructed. The diagram below shows the demand curve for good Y as its price varies. Alternatively, if the price for good Y is fixed and the price for good X is varied, a demand curve for good X can be constructed.

## Income effectEdit

Another important item that can change is the income of the consumer. As long as the prices remain constant, changing the income will create a parallel shift of the budget constraint. Increasing the income will shift the budget constraint right since more of both can be bought, and decreasing income will shift it left.

Depending on the indifference curves the amount of a good bought can either increase, decrease or stay the same when income increases. In the diagram below, good Y is a normal good since the amount purchased increased as the budget constraint shifted from BC1 to the higher income BC2. Good X is an inferior good since the amount bought decreased as the income increases.

$\Delta y_1^n$ is the change in the demand for good 1 when we change income from $m'$ to $m$, holding the price of good 1 fixed at $p_1'$:

$\Delta y_1^n = y_1(p_1', m) - y_1(p_1',m').$

## Substitution effectEdit

Every price change can be decomposed into an income effect and a substitution effect. The substitution effect is a price change that changes the slope of the budget constraint, but leaves the consumer on the same indifference curve. This effect will always cause the consumer to substitute away from the good that is becoming comparatively more expensive. If the good in question is a normal good, then the income effect will reinforce the substitution effect. If the good is inferior, then the income effect will lessen the substitution effect. If the income effect is opposite and stronger than the substitution effect, the consumer will buy more of the good when it becomes more expensive. An example of this might be a Giffen good.

Substitution effect, $\Delta y_1^s$, is the change in the demand for good 1 when the price of good 1 changes to $p_1'$ and, at the same time, the money income changes to $m'$:

$\Delta y_1^s = y_1(p_1', m') - y_1(p_1,m).$

Consumer theory can also be used to analyze a consumer's choice between leisure and labor. Leisure is considered one good (often put on the horizontal-axis) and consumption is considered the other good. Since a consumer has a finite and scarce amount of time, she must make a choice between leisure (which earns no income for consumption) and labor (which does earn income for consumption).

The previous model of consumer choice theory is applicable with only slight modifications. First, the total amount of time that an individual has to allocate is known as her time endowment, and is often denoted as T. The amount an individual allocates to labor (denoted L) and leisure (l) is constrained by T such that:

$l + L = T$

or

$l + (T-l) = T$

A person's consumption is the amount of labor they choose multiplied by the amount they are paid per hour of labor (their wage, often denoted w). Thus, the amount that a person consumes is:

$C = w(T-l)$

When a consumer chooses no leisure $(l=0)$ then $T-l = T$ and $C = wT$.

From this labor-leisure tradeoff model, the substitution and income effects of various changes in price caused by welfare benefits, labor taxation, or tax credits can be analyzed.