Utility and Demand

UtilityMax Manual

Purpose of the Module

This module lets you see how individual demand for goods depends on individual preferences, income, and prices. In this module there are two goods, beer and (music) CDs. The module starts with the determination of the individual preferences. Once those are set you have to try to determine what “available” combination of the goods a person with these preferences would like “best”. “Available” means combinations that can be bought with current income, given current prices. “Best” refers to that combination of the two goods that will maximize the consumer’s utility, or satisfaction, given current income and prices. (A synonym for “best” often used by economists is “optimal.”) Once you complete this part of the module you can change the price of one good and see what effect this price change has on this person’s “best” decision. Subsequently, you can change the price of the other good and see what effects that price change has.

By the time you have finished with this module you should understand why a rise in the price of one good always reduces the amount bought of that good, and why such a price increase for one good can either increase or decrease the amount of another good a person buys. In other words, you should understand the “Law of Demand” and what it means for goods to be substitutes or complements.

Nature of the Problem

Economists, businesses, and sometimes governments are interested in being able to predict the effects of price and preference changes on consumer demand for goods. They are also interested in how the demand for one good is related to the demands for other goods. There are some simple rules that allow for general statements about these things. First, the “Law of Demand” states that if the price of a product goes up the quantity demanded of that product goes down (and vice versa for price reductions). Secondly, if the price of one good goes up the demand for similar goods will increase (substitutes), and the demand for things that are used together (complements) will decrease. It is easy to come up with examples of for the latter statement: goods that most people consider to be substitutes, i.e. ice cream and frozen yogurt, or complements, i.e. hamburgers and French fries.

Even with the obvious examples above, knowing what everyone knows won’t tell you how much of an effect a price increase for ice cream would have on the quantity of ice cream demanded, or on the demand for frozen yogurt, or how large an effect a cut in the price of hamburgers would have on the quantity of hamburgers bought, or on the demand for French fries. Knowing how much of an effect there will be is as important as knowing whether the effect will be positive or negative. To find that out it is necessary to have a more precise analysis than can be obtained using general verbal statements.

To get more precise answers economists developed mathematical and graphical tools to describe individual reactions to these kinds of changes, determine how they depend on things like income and personal preferences. This module lets you use those tools, without having to do any real mathematics on your own, make changes and see results and arrive at a deeper understanding of the ways economists analyze demand by consumers.

Utility Maximization Model

Preferences

Economists summarize a person’s preferences using a “utility function.” The basic ideas embedded in the utility function are “more is better” and “declining marginal rate of substitution (MRS).” The first means people always want more of everything. The second is a little more complicated. Suppose a person is currently willing to trade one ounce of French fries for one ounce of steak. If he/she now has a lot more steak and a lot fewer French fries he/she would be reluctant to give up any more French fries for a mere ounce of steak. It might take a pound of steak to get him/her to give up even an ounce of French fries. In essence, the more you have of something, the easier it is to get you to give up some of it, the less you have the harder it is to get you to give up any of it.

The same information can also be shown in graphical form. The graphical presentation involves indifference curves. An indifference curve shows all combinations of goods that give this person the same level of utility, or equal satisfaction. Different curves indicate different levels of utility with curves above and to the right indicating higher levels of total utility. A typical indifference curve graph is shown in Figure 1.

Figure 1 Indifference Curves

Beer

(quantity)

I2

I1

0 CDs (quantity)

In Figure 1, I1 is all the combinations of beer and CDs that provide the lowest level of total utility, or satisfaction (among the levels shown); I2is all the combinations that provide a given level of total utility that is higher than I1, and I3 has combinations that are still higher in the total utility provided. There are many other indifference curves, above, below, and between these three. Notice these curves do not cross. They can’t cross. Crossing would mean the same combination of goods provides two different levels of total utility. It would also mean that some combinations on one curve (i.e. I1) were better than those on another (i.e. I2) and some were worse, even though all the points on I1 give the same total utility, and all the ones on I2 give the same total utility. The shape of the curves (steep downward slope starting at the upper left, flattening as you go toward the lower right) reflects the declining marginal rate of substitution.

Choice and the Budget Constraint

What actual combination of Beer and CDs will this person buy? Any point on I3 would be preferred over any point on I2, and any point on I2 (if possible) would be preferred over any point on I1. However, no information has been offered on what is possible. The indifference curves only show what is preferred or desirable to the consumer in the sense of utility, or satisfaction. What is possible is set by the person’s budget constraint.

To know what is possible, we need to know the person’s income and the prices of the goods. Assume that this person cannot spend more than their income (no credit) and will not spend less (no saving). Neither of these assumptions is required for the theory of demand, but they simplify the analysis, and both assumptions are incorporated into this module. The consequences of these assumptions can be stated mathematically as:

B = PBeer x Beer + PCDs x CD

B is the consumer’s budget constraint, or the money income available to spend, PBeeris the price of beer, PCDs is the price of a CD, and Beer and CD are the quantities bought. The equality sign means that the amount spent on these two goods adds up to the available income. The budget equation can be is added to the content of Figure 1 and is shown as Figure 2.

Figure 2 Indifference Curves and the Budget Constraint

Beer

(quantity)Point D

Point C

Point A

Point A’

CDs (quantity)

In Figure 2, the downward sloping, straight line shows the combinations of goods that can be purchased, given current income and prices. This person would prefer to buy at any point on I3, i.e. Point A or Point A’, but cannot afford those combinations. Point D could be chosen, since it is on the budget line, but it is not as desirable Point C because PointD is on I1. Point C is the “best” choice for this person because any other available point would be on a lower indifference curve and yield less total utility. Any more desirable point, such as Point A, is not affordable. Also notice that at Point C, the slope of the budget line matches the slope of the indifference curve I2. The significance of this equality of slopes is discussed below.

Utility Maximization

Indifference Curve Slope

The slope of an indifference curve is called the “Marginal Rate of Substitution (MRS).” For example, if you move from Point A on I3 to Point A’ also on I3, the level of satisfaction is the same at both points. To get from Point A to Point A’ the person gets less Beer but more CDs. The slope of the indifference curve is the decrease in Beer divided by the offsetting increase in CDs that brings the person back to the original (I3) level of utility. This is the proportion at which the person would be just barely willing to give up Beer in exchange for CDs (substitute one for the other). Getting any fewer CDs in exchange for the given amount of Beer would be unacceptable since such an exchange would involve a loss of utility, or a lower indifference curve. Getting any more CDs would more than make up for the loss of Beer and move the person to a higher indifference curve.

The slope of the indifference curve, or MRS, is represented as:

MRSCDs,Beer = - MUCDs

MUBeer

What all that means is that the Marginal Rate of Substitution is the ratio between the marginal utility of CDs (MUCDs) and marginal utility of beer (MUBeer). If you wish to consume more CDs, you must give up a little Beer (how much depends on the marginal utility of Beer). To retain the original level of utility (same indifference curve) you need to balance the utility gained with additional CD consumption against the utility lost from less beer consumption. The MRS may also be expressed as:

MRSCDs,Beer = - Beer

CDs

where Beer is the amount of beer given up to obtain an additional unit of CDs while retaining the same level of total utility. Similarly, CDs is the change in the amount of CDs associated with that amount of beer given up that maintains the same level of total utility. The minus sign denotes the fact that the consumer is decreasing the quantity of beer (in the numerator) while simultaneously increasing the quantity of CDs (in the denominator).

(Note: The MUCDs =  TU and the MUBeer =  TU )

 CDs  Beer

One can show the equivalency of the two MRS expressions by dividing the MUCDs by the MUBeer:

MRSCDs,Beer = - MUCDs =  TU/  QCDs = - QBeer ).

MUBeer  TU/  QBeer QCDs

Budget Line Slope

The budget line represents all combinations of Beer and CDs for which total money spent is equal to the available money income. The budget line (B) determines what combinations of the two goods are attainable. On the right hand side, the total expenditure on beer, or PBeer x Beer, plus the total expenditure on CDs, or PCDs x CDs equals the total budget of the consumer.

B = PBeer x Beer + PCDs x CDs

If you bought one more Beer that would take up dollars that you had previously been spending on CDs. Exactly how many depends on the price of beer. And how many CDs you give up depends on the price of CDs. The slope of the budget line is the negative of the ratio of one price to the other, or

- PCDs

PBeer

The magnitude of this slope indicates the rate at which the two goods can be substituted for each other without changing the total amount of money income spent.

(Note: One can find the slope of the budget line by dividing B/PBeer by B/PCDs).

Total Utility Maximization

At Point C the two slopes are equal -- that is required for this to be the “best” point, or the combination of the two goods that maximizes the consumer’s utility or satisfaction. If the slopes are not equal, then it is possible to find another, higher, indifference curve that reached the budget line. At Point C:

MRS CDs,Beer = PCDs

PBeer

Income and Substitution Effects

When a price changes there are two ways in which this affects the demand for goods. One way involves the change in opportunity cost, the rate at which one good is exchanged for another. The other involves the overall purchasing power of the (fixed) available money income, or budget.

Substitution Effect

The first effect is found by looking at the reaction to the change in the price of one good compared to the price(s) of other good(s), holding a measure of the consumer’s total satisfaction, or total utility, constant. We look at the quantities of the goods that the individual would buy if relative prices change, but there is only enough “real income” to reach the level of total utility achieved before the price change.

Another way of looking at this is that the substitution effect, via the change in relative prices, means a change in the consumer’s opportunity costs. For example, if the price of a CD rises then the opportunity cost of a beer is lower -- you give up fewer CDs to buy one beer than before. This leads you to buy fewer CDs and more beer. In Figure 3, when the price of a CD rises, there is a movement from Point C to Point C’ in Figure 3 below. B1is the original budget line, B2 (dotted line) the one with the same purchasing power, but a higher relative price for CDs.

Figure 3 Substitution Effect of A Price Change

Beer

(quantity) B2

Point C’

Point C

0 CDs (quantity)

At Point C’ this person buys fewer CDs and more beer than at Point C, this is the substitution effect of the price change. Points C and C’ are on the same indifference curve—the same level of utility. However, at the two points the slope of the indifference curve is different, matching the two different slopes of the budget line (due to the different relative prices). Points C and C’ have the same “real income”, defining that as the level of total utility reached with the same purchasing power. They differ only by the different slopes and different prices.

Income Effect

When there is a price change the purchasing power, or “real income”, of the fixed money income changes. In the case of a price increase for CDs, this means the individual can reach a lower indifference curve than the one reached in Figure 3. The movement from C’ to Point E in Figure 4 is the result of this “real income” change. There is no change in opportunity cost (relative prices) between these two points, just a change in purchasing power. In Figure 4, B1 is the original budget line, B2 (dotted line) is the budget line with the original purchasing power but new, lower CD price, and B3 is the budget line with the original money income, the new CD price, and therefore a lower purchasing power, buying less utility.

Figure 4 Income and Substitution Effects of a Price Change

Beer

(quantity) Point C’

Point C

Point E

B3I1

CDs (quantity)

Notice the B1 and B3 budget lines reach the beer axis at the same point. With no change in money income and no change in the beer price, the maximum amount of beer that could be bought has not changed. At Point C the Marginal Rate of Substitution equals the original price ratio. At Points C’ and E the Marginal Rate of Substitution equals the new price ratio. The move from Point C’ to E is the income effect of the price change. The move from Point C to C’ is the substitution effect.

Utility Maximization and the Demand Curve

This module does not generate a demand curve for you. However, you should be aware that when you change, for example, the price of Beer you get two values for the quantity of beer that will be bought in equilibrium (i.e., when the consumer has maximized utility) one at each price. These are two points on the consumer demand curve. If you changed the price a few more times you would have even more of the demand curve. If you changed the price of CDs you would get points on the demand curve for CDs.

Running the Module Step by Step

1)The first stage in the module involves determining the consumer’s preferences. This means setting the parameters “a” the relative importance of Beer and CDs to the module’s hypothetical consumer, and “b,” the degree to which the two goods are substitutable

Parameter “a” involves how much weight this person puts on having Beer versus CDs. If a = 0, this person does not like Beer at all (gets no utility from it), all satisfaction is from buying CDs. If a =1, all that matters in terms of the consumer’s satisfaction is Beer, getting CDs does not yield any satisfaction. Obviously, parameter “a” cannot be less than zero or greater than 1. Since this module is meant to involve choices, values of exactly 0 and exactly 1 are ruled out since either way the person would buy only one good no matter what.

The value of parameter “b” has to do with how easily this person can be induced to switch from one good to the other -- to what extent the two goods are substitutes for one another. (See the “Mathematical Presentation of the Module” section for a more detailed discussion of parameter “b”.)

Thus, in this module parameter “a” has to be set between zero and one, “b” must be less than one, greater than negative 1 (in terms of the theory even these restrictions are not necessary but they are convenient). You have a choice, you can either set “a” and “b” yourself, or you can let the program set them for you. If you let the program set them they will be chosen at random, within the range of permissible choices.

The permissible choices do not let you see certain extreme cases. You cannot make the two good perfect substitutes (which would create straight line indifference curves). However, if two goods are perfect substitutes they are, in effect, the same product so they do not offer an interesting case for switching between them. You cannot make the two goods perfect complements (which would make the indifference curves look like sets of axes—a vertical line which ends at a horizontal line). However, two goods that are perfect complements for everyone in the market (or even almost everyone) would usually be sold as one joint product—like left and right shoes. Since they would always be boughttogether in the same proportions, they do not make for a very interesting case either.