10 Sept 2000

Demand Analysis and Estimation

In this unit of the course we examine the basics of consumer demand as a framework for the study of the elasticity concept, which itself is an analytic technique that quantifies the sensitivity or responsiveness of demand to changes in the market environment. The elasticity concept is useful to managers because it provides a way of evalating the effects of changes in important variables such as price, income, advertising, and the pricing activities of rival firms.

Lecture Outline

• The Basis for Consumer Demand
  • Utility Functions
  • Marginal Utility
• Consumer Choice
  • Indifference Curves
  • Marginal Rate of Substitution
  • Budget Lines
• Optimal Consumption
  • Utility Maximization
• Deriving a Demand Curve
• Demand Sensitivity Analysis: Elasticity
• Price Elasticity of Demand
  • Price Elasticity and Revenue Relationships
  • Price Elasticity and Optimal Pricing
  • Determinants and Uses of Price Elasticity
• Cross-Price Elasticity of Demand
• Income Elasticity of Demand
• Time and Elasticity
• Demand Estimation
  • Market Experiments
  • Regression Analysis

Lecture Notes

I. The Basis For Consumer Demand

Utility Functions

Early economists interested in comparing and aggregating utility levels across individuals hoped to find a way to develop a metric of comparison of utility across individuals. Such a process would require a cardinal utility in which a utility value of "13" to me is comparable to a value of "13" to you. You can see how this would be helpful in policy making, but it is fundamentally impossible, even when we use dollars as a proxy for cardinal utility (a dollar to a billionaire means less than a dollar to a family living in poverty).

Modern utility theory is based on observable rankings or orderings of various combinations of goods and services, referred to as ordinal utility. For example, on August 24, 1984 it could have been recorded that I preferred a bowl of hot curry to a hamburger, and a hamburger to a hot dog, and a hot dog to veal. This sort of ranking is the fundamental basis for modern utility theory. Moreover, we can use numerical representations for utility states as long as we recognize that those numbers are only useful in their ranking (ie, a bigger number implies more utility than a smaller number).

Take a moment and make up an example of your own utility function for lunch meals. You can make this a matrix where the rows consist of different meals, and the columns consist of quantity (e.g., number of pizza slizes, bowls of chile, etc). Rank from best to worst, and assign numbers to each meal alternative that transform that ranking to a numerical ranking. You may notice that total utility will eventually (or perhaps quickly if you have a modest appetite) decline with quantity. Thus, for example, you may find that 2 pizza slices yield the same total utility as one bowl of vegetarian curry (e.g., your hunger overcomes your sense of aesthetics), but eating a third slice would actually reduce your total utility (indigestion). In other words, there is likely to be different combinations of meals and quantities of meals that yield approximately the same total utility, over which you are indifferent. This interesting condition will be explored a bit later.

Marginal Utility

Overeating is perhaps one of the most universal experiences we have of the concept of diminishing marginal utility, meaning that the marginal utility of each additional unit consumed is smaller than the preceding unit. Marginal utility can become negative -- "I wish I hadn't eaten that last pizza slice."

The freedom to spend one's dollars as one sees fit in a market means that $ provides a metric of comparison for evaluating the utility of one thing (e.g., apples) to the utility of another (e.g., oranges) (in reference to the classical statement of irrelevance, 'thats apples and oranges', a famous economist is reputed to respond 'economics is about relating apples to oranges'; perhaps you can see why that's the case).

Consumer Choice

Indifference Curves

Consider the following example:

The numbers represent levels of total utility or satisfaction. Lets pretend we are in grammar school again and connect the numbers that are equal with a line, carefully making sure that the line connecting smaller numbers is always below the line that connects larger numbers -- they cannot cross. Once you are done you have something called an indifference map, which simply shows indifference curves for different levels of total utility.

One of the rules associated with indifference curves is that they cannot cross each other -- that would violate the very notion of 'indifference' used here. Another is that if both consumption objects are 'goods' (as opposed to 'bads' like IRS audits), then the indifference curves must be downward sloping. We will now turn to another property of indifference curves, namely their slope.

Marginal Rate of Substitution

• We are consuming more of one good, and less of another
• We are keeping total utility constant

The marginal rate of subsitution (MRS) refers to the slope of the indifference curve. The slope of the indifference curve tells us, as we increase consumption of one good (e.g., one more orange on the 'x' axis), how much of the other (e.g., how many apples on the 'y' axis) must we give up to keep total utility constant and thus stay on the same indifference curve. Let's use the symbol '@' to mean 'change in':

MRS = [(@ good on y axis)/(@ good on x axis)],

given we remain on the same indifference curve, and thus that total utility is constant. In fact, as we move along an indifference curve, we know that

(@ total utility) = 0 = [MU(oranges) x @ oranges] + [MU(apples) x @ apples].

If we re-arrange the right-hand-side of the equation above, we get:

[MU(oranges)/MU(apples)] = -(@ apples)/(@ oranges) = MRS.

From this last equation we can see that the slope of the indifference curve, the MRS, is determined by the ratio of marginal utilities of the two goods in question.

Budget Lines

A consumer's budget constraint is made up of:

• A budget, or an amount of money available per unit of time for the relevant purchase choices. For example, it may be your weekly lunch budget.
• A set of prices for each of the goods the consumer is considering. For example, the price of a bowl of chile, a pizza slice, or a hamburger.

Note that in this simple model we do not account for saving for later. We could, by introducing the discounted value of future consumption as yet another choice, but that would complicate the analysis without adding much in terms of understanding the basics. Below I shall give an example of a simple budget constraint for Freida. Lets assume that Frieda has a well-defined budget for her weekly consumption of fruit, and in particular, apples and oranges. Then we get:

Budget = P(oranges)xOranges + P(apples)xApples

Recall that apples were on the 'y' axis. Thus solving for this budget constraint as an equation of the line we get:

Quantity of Apples = Budget/P(apples) - [P(oranges)/P(apples)]x(Quantity of Oranges)

Note that if Frieda only bought apples, then the number of apples she can buy each week is simply her fruit budget divided by the price of an apple. If, however, she wants to add a few oranges to her diet, to stay within her budget constraint she must give up some oranges. In particular, for every extra apple Frieda buys, she must give up

[P(oranges)/P(apples)]

units of apples to keep within her budget constraint. For example, if the price of an apple is $0.50, and the price of an orange is $0.25, then for every extra orange Frieda buys, she must give up 1/2 an apple to keep within her budget constraint.

Thus the slope of the budget line is the ratio of prices given above:

Note that every combination of apples and oranges that are on or inside of the budget line are feasible combinations that Frieda can afford to buy. Thus this area on and inside the budget line is referred to as the feasible set.

Optimal Consumption

Utility Maximization

Thus MRS = -(@ apples/@ oranges) = MU(oranges)/MU(apples) = P(oranges)/P(apples)

At this point of tangency the slope of the indifference curve [-(@ apples)/(@ oranges)] is equal to the slope of the budget line [P(oranges)/P(apples)].

Note that rearranging this equality yields:

MU(oranges)/P(oranges) = MU(apples)/P(apples).

Thus a consumer's constrained-optimal consumption occurs where the marginal utility per dollar is equal across both (or all) consumption alternatives.

Remember that apples are the "y" variable and oranges are the "x" variable

Summary:

• Note in this example that Frieda finds that her constrained optimal weekly consumption corresponds to 6 oranges and 8 apples per week.
• At this point of tangency, the MRS = price ratio (the budget line has the same slope as the indifference curve).
• This will remain her weekly consumption of fruit, and thus is an equilibrium, unless some part of the market environment changes, such as her budget, preferences, or the product's prices.
• Finally, note that we can derive Frieda's demand curve for one of these goods, say apples, by observing how her equilibrium consumption bundle (combination of oranges and apples) changes if we were to observe a change in the price of apples. We turn to this issue below.

Deriving a Demand Curve

First, note that with the price of apples changing, the slope of the budget line (equal to the price ratio) MUST also change. If Frieda only ate oranges, this price change would not matter, so the budget line rotates from the fixed-point intercept on the 'x' axis (oranges) and allows Frieda to consume more apples and oranges than below -- unless, of course, she only ate oranges! Observe how the budget constraint rotates outward, and how this changes the Frieda's consumption of apples and oranges:

From the diagram above we can see that as the price of apples drops from $.5 to $.25, consumption of apples rises from 8 to 19. If we then plot these price-quantity demanded points (p=$0.5, QD=8), (p=$0.25, QD=19) we will have a simple linear estimate of Frieda's demand for apples in the range from 50 to 25 cents per apple:

Demand Sensitivity Analysis: Elasticity

Question: if we are computing price elasticity of demand, for example, why not simply compute the slope of the demand curve to measure demand sensitivity to price changes? (Hint: Think about what happens to the slope for someone's weekly demand for coffee if the quantity of coffee per week is measured in ounces rather than in pounds...do we want a sensitivity measure that changes when we use different units of measure?)

In the most general terms, an elasticity is computed as follows:

elasticity = (% change in variable Y)/(% change in variable X)

where Y is the variable being acted upon by a change in X, meaning that Y is dependent on X, while X is independent of Y.

We can calculate elasticities for very, very small changes (where calculus techniques are relevant), referred to as point elasticity, or for more discrete changes, referred to as arc elasticity. The difference in these two elasticity measures has to do with the amount of change that has occurred in the variables. If the changes are small, point elasticity can be used (where again, @ refers to 'change in'):

point elasticity = (@Y/Y)/(@X/X)

As the changes in X and Y grow larger, however, then we will get increasingly different elasticity numbers depending on whether we use the old or the new values for X and Y.

Example: Suppose that when X grows from 10 to 20, Y grows from 10 to 30. If we use the old X and Y values, the point elasticity is 2, while if we use the new X and Y values, the point elasticity is 1.333. To resolve this problem, we use an approximation or averaging technique embedded in the arc elasticity of demand:

arc elasticity = [@Y/(avg Y)]/[@X/(avg X)]

where (avg Y) refers to (old Y + new Y)/2 (parallel for X).

Price Elasticity of Demand

Ed = abs[(% change in QD)/(% change in P)]

When Ed is greater than 1, we say that demand is price elastic in that price range, meaning that a 10 percent increase in price will result in a more than 10 percent reduction in quantity demanded, or that a 10 percent decrease in price will result in a more than 10 percent increase in quantity demanded. Demand is responsive to price, meaning that there are likely to be lots of available substitutes, or perhaps the good is quite expensive to people, or we are evaluating elasticity over a longer time horizon allowing for more change to occur. In this case, raising price will lower a firm's total revenue, while lowering price will raise a firm's total revenue.

When Ed equals 1 we say that demand is unitary elastic, meaning that a given percentage change in price always results in an inversely proportionate change in quantity demanded. Total revenue is independent of price changes when demand is unitary elastic.

When Ed is less than 1, we say that demand is inelastic, meaning that a 10 percent change in price results in a less than 10 percent change in quantity demanded. Demand is inelastic when quantity demanded is not very responsive to changes in price. This occurs when people are very brand loyal, when there are few substitutes, when there is a very small time horizon for adjustment, or when the good is relatively inexpensive.

The price elasticity of demand can be calculated using either the point or the arc method, whichever is most applicable. Suppose that regression analysis indicates the following reduced-form demand equation:

QD = 48,129 - 16P

Then the '16' refers to the change in QD caused by a change in P. We can then use the point elasticity measure (since regression analysis is based on small changes) to get the following price elasticity of demand expression:

Ed = 16x[P/Q].

So, for example, if price is $1000, then QD = 32,129, and Ed = .498. If, on the other hand, price is $2000, then QD = 16,129, and Ed = 1.98. Thus as price rises, elasticity rises when demand is linear. Consumers become more price sensitive as price rises, and thus for a 1% increase in price when price is high, quantity demanded falls by more than 1%.

Price Elasticity and Revenue Relationships

Price Elasticity and Optimal Pricing

MR = d(PxQ)/dQ = P + QxdP/dQ,

if we multiply the top and the bottom of the right-hand-side of the above equation by 'P', we get

MR = P[1 + 1/Ed],

where Ed refers to the price elasticity of demand, and is less than zero by the law of demand. Thus a firm's optimal pricing decision can be derived from the following:

MC = MR = P[1 + 1/Ed], implying

P = MC/[1 + 1/Ed].

Note that this optimal pricing rule has an implicit constraint, namely that the firm NOT operate on the inelastic portion of its demand curve (the lower part), because if it did, the optimal price would be negative, which is absurd. This requirement is equivalent to requiring that MR be nonnegative, which can be easily seen in the equations above (you can't set MR = MC and have a negative MC).

Thus the price above satisfies profit maximization (it is consistent with MR = MC), and relates this optimal price to two observables: MC and Ed. Read the example on pages 209 and 210, and Table 5.7 of the Hirschey and Pappas text.

Determinants and Uses of Price Elasticity

Cross-Price Elasticity of Demand

Managers use cross-price elasticities to help them make pricing and product design decisions. If cross-price elasticities are large, then it is difficult for a firm to charge a price that is very much higher than its rivals. Thus a firm developing a new product or service may want to try different versions in test cases in order to find a niche in which cross-price elasticities are small.

Cross-price elasticities between complements indicate how strong a complementary relationship exists, which may have a big impact on marketing decisions in the firm, and may suggest cooperative marketing strategies.

Cross price elasticity = [% change in quantity demanded of x]/[% change in price of y].

Cross-price elasticity is positive between substitutes and is negative between complements. The larger is the cross-price elasticity in absolute value, the stronger is the substitute or complement relationship.

Income Elasticity of Demand

Income elasticity = [% change in quantity demanded of x]/[% change in consumer incomes].

When income elasticity is positive, we say that the good is normal, meaning that as consumer incomes rise consumption also rises. This would be the case for brand-name goods and services. Within the class of normal goods we also have a subcategory of luxury goods, in which case the income elasticity of demand is greater than 1, meaning that a 10% increase in consumer income results in more than a 10% increase in consumption. Some economists refer to luxury goods as cyclical goods because during the 'upswing' in the business cycle (when the whole economy is growing), the demand for luxury goods rises relatively rapidly. Examples include restaurant meals, vacations, etc. Consumption of luxury goods is highly sensitive to income, and are the first things given up when income falls (such as when people are laid off), and which are added on when incomes rise.

Inferior goods have a negative income elasticity of demand, meaning that as incomes rise, quantity consumed falls. These are sometimes referred to as countercyclical goods, because the demand for them rises when the economy is in a downturn (recession). Examples include generic goods, used clothing and cars, haircuts at beauty colleges, etc. Inferior goods are those goods that low-income people are forced by their conditions to consume, and which are dropped when their incomes rise.

Time and Elasticity

Demand Estimation

• Consumer Interviews: Survey consumers, or potential consumers, to get their anticipated responses to questions regarding how their purchase decision may be influenced by variation in the product's price, the price of rival (substitute) goods and complementary goods, as well as in their own incomes. Such surveys may also include questions designed to determine the effectiveness of advertising campaigns. Survey methods are highly developed in terms of the statistics of a representative sample, and in terms of the way questions are asked.
• Market Experiments: Test markets may be actual, such as 'representative' cities like Peoria Illinois or Spokane Washington, or 'laboratory' markets in which representative consumers are brought in under simulated market conditions. An advantage of market experiments is that actual 'dollar votes' are measured, rather than responses to hypothetical questions. Market experiments can be expensive, and the results may not generalize for situations outside the test market.
• Regression Analysis: One begins by constructing the estimating equation, such as a demand function, then gathers data and estimates the coefficients of the equation that tell us the marginal effects of each individual variable on sales quantity. Regression analysis can be used in conjunction with data-gathering market experiments.

   

  All pages copyright Steve Hackett unless otherwise noted.