PART ONE: THE HOUSEHOLD
Chapter 2 Consumer Choice
2.1 Basic Setup
To understand consumer choice, there are three steps:
1. We need to know what the consumer wants to do.
2. We need to know what the consumer can do, given her income and the prices she faces (her constraints).
3. We put together the consumer’s preferences and constraints to determine feasible choices that will maximize well-being
This is all done, given her particular tastes. Economist cannot judge whether a person’s tastes are sensible, only whether they are attempting to achieve their goals in a rational fashion.
To specify consumer tastes, we need to make a number of assumptions:
1. Completeness Assumption: A consumer, when confronted with any two bundles, can tell us which one she prefers, or whether she is indifferent between them.
2. Transitivity Assumption: Preferences are such that if bundle x is preferred to bundle y, and bundle y is preferred to bundle z, then x is preferred to z.
3. Nonsatiation Assumption: More is better. For all feasible quantities, the consumer is never satisfied. [Image pg. 25] Without this assumption, the indifference curve could slope upward.
4. Diminishing Marginal Rate of Substitution Assumption: When the marginal rate of substitutions falls as we move along an indifference curve. This ensures that the indifference curve is bowed inward – i.e. convex to the origin.
The marginal rate of substitution (MRS) is the negative of the slope of an indifference curve; it measures the rate at which the consumer is willing to trade one good for the other. It is the rate at which the consumer would be willing to substitute an item of type a for one additional item of type b, and still be as happy as she was before.
Deriving an Indifference Curve
An indifference curve Ui is generated by connecting bundles of the commodities among which the consumer is indifferent. A bundle above the indifference curve is preferred to any bundle on the indifference curve. [The axes on this graph are quantity of item a and quantity of item b.] This could be done by starting at any point and asking how many item a’s she would be willing to give up for one additional item b.
Deriving an Indifference Map
An indifference curve can be drawn through each point in the quadrant. The entire collection of indifference curves is called an indifference map. Indifference curves cannot intersect – this would violate the assumption of transitivity. However, they do not have to be parallel.
Properties of “typical” convex indifference curves:
1. Negative Slope
2. Marginal Rate of Substitution (MRS) = negative of slope
3. Diminishing MRS
4. Indifference curves lying to the northeast represent higher levels of satisfaction
5. Indifference curves cannot cross.
Other Types of Indifference Curves
Perfect Substitutes: Goods that can be substituted for each other at a constant rate, that is, that have a constant marginal rate of substitution. Their indifference curves are straight lines (now bowed curves).
Perfect Compliments: Perfect compliments are goods that have to be consumed in fixed proportions – having more of one without having more of the other does not provide any benefit. The indifference map for a pair of perfect complements is a series of right angles. The right angles are placed on a ray from the origin where the slope is the ratio in which the goods are consumed.
Pollution is an example of an economic “bad.” We can model preferences, even if one of the goods is a “bad.” These indifference curves do not satisfy the assumption of nonsatiation. Therefore, the indifference curves with one good and one bad slope upward. The more of the “bad” a person consumes, the more he dislikes it.
Utility Theory: Assigning Numbers to Indifference Curves
It is possible to attach numerical “scores” to each bundle. This can be convenient particularly in deciding between bundles with many different goods. The numerical score associated with each commodity bundle is called its total utility. A utility function is a formula that shows the total utility associated with each bundle. These numbers are ordinal – they can tell us that one indifference curve is better than another, but not by how much. An ordinal utility function is a utility function allowing the ranking of bundles by their amount or utility, but not precise comparisons of how various bundles are valued relative to each other.
2.3 Budget Constraints
A price taker is a consumer whose price per unit of a commodity is not affected by the number of units purchased. She has no control over the prices she faces.
A budget constraint is the representation of the bundles among which a consumer may choose, given her income and the prices she faces. It is drawn as a straight line on a graph with axes that are quantity of item a and quantity of item b. The slope of the budget constraint represents the relation between the price of the two quantities. (i.e. A slope of -2 indicates that the price of the commodity on the horizontal axis is twice the price of the commodity on the vertical axis.)
(Equation 2.2) Budget Constraint: pxx + pyy = I
*px is price per unit of x; py is price per unit of y; I is income.
*The x intercept is I/px; The y intercept is I/py
The feasible set is the collection of bundles satisfying the budget constraints. These are the sets on or below the budget constraint.
Changes in Income: When income changes but relative prices do not, a parallel shift in the budget constraint is induced. If income decreases, the constraint shifts in toward the origin; if income increases, it shifts out from the origin.
Changes in Price: When the price of once commodity changes and other things stay the same, the budget line moves along the axis of the good whose price changes. (The slope and one of the intercepts change.) If the price goes up, the line pivots in; if the price goes down, the line pivots out.
Summary of Properties of Linear Budget Constraints
1. Slope = -px/py
2. Horizontal intercept = Income/px
3. Vertical intercept = Income/py
4. Change in income: parallel shift of budget constraint
5. Change in price of one commodity: rotation of budget constraint
Nonlinear Budget Constraints
Quantity Rationing: A budget constraint with quantity rationing changes the feasible bundles. The budget constraint line looks like the normal diagonal budget constraint line up to the ration quantity, and then a vertical line, since it is not possible to buy more of the rationed quantity.
Quantity Discounts: When the price per unit of a commodity depends on the number of units purchased, the budget constraint is nonlinear. For example, if the price falls as more units are consumed, the constraint becomes flatter as more units are consumed.
2.4 The Consumer’s Equilibrium
The consumer equilibrium is the point where the indifference curve is tangent to the budget constraint. This represents the highest indifference curve possible, given the budget constraint.
The interior solution is an equilibrium bundle that contains some amount of each good. This is the most common type of solution. The interior solution is the point at which the budget constraint line is tangent to the indifference curve. By definition, the slope of the indifference curve is equal to the marginal rate of substitution, MRSyx. Also, the negative of the slope of the budget line is px/py. At equilibrium these two slopes are equal: MRSyx = px/py
The marginal rate of substitution shows the rate at which the consumer is willing to trade one good for the other; the slope of the budget constraint is the rate at which she is able to trade one good for another. In equilibrium these must be equal.
The corner solution is an equilibrium bundle in which the consumption of some commodity is zero. At this point, the highest indifference curve hits the budget constraint at the intercept. At this point, the MRS may not be equal to the slope of the budget constraint MRSxy != px/py.
Equilibrium with Composite Commodities
In reality, we want to compare more than two commodities at a time. This can be done by having item x on one axis and having a composite of all goods other than x on the other axis. For convenience, we can define a unit of all other goods as the amount that you could purchase by spending $1.
Using Utility to Characterize the Consumer’s Equilibrium
The consumer’s objective is to maximize the value of her utility function subject to her budget constraint. Marginal utility is the change in total utility associated with consumption of one additional unit of a good. The marginal utility from gaining (del)x of item x is (del)x*MUy. Also, moving along an indifference curve, the utility gain from getting additional units of item x must be equal to the utility loss from giving up units of item y. From this we can derive the following equation:
(2.6) MUx/MUy = px/py
(2.7) MUx/px = MUy/py
The second equation shows that a bundle maximizes total utility only if the marginal utility of the last dollar spent on each good is the same. When marginal utility of the last dollar is the same for each commodity, there is no way that income can be reallocated between commodities so as to increase total utility.