Source: Exam 1 Objectives (Section 3) | Varghese, Texas A&M
Tags: demand analysis, elasticity, price elasticity of demand, income elasticity, cross-price elasticity, normal good, inferior good, substitutes, complements, demand shifts, demand curve, price discrimination, hurdle model, first-degree price discrimination, second-degree price discrimination, third-degree price discrimination, rule of thumb pricing, ECON 323
This chapter builds out everything about demand: what shifts the curve vs. movement along it, how to measure responsiveness with elasticity, the link between elasticity and revenue, and how firms use elasticity to set prices (including price discrimination). The exam tests both calculation and intuition, especially around elasticity changing along a linear demand curve.
Demand vs quantity demanded
Demand refers to the entire relationship between price and quantity (the whole curve). Quantity demanded is a specific number at a specific price (a point on the curve). A change in price causes a change in quantity demanded (movement along the curve). A change in any other factor shifts demand (the whole curve moves).
Normal good
A good for which demand increases when consumer income rises. Positive income elasticity.
Inferior good
A good for which demand decreases when consumer income rises. Negative income elasticity. Also described as counter-cyclical, because demand for these goods rises in recessions.
Superior good
A normal good with income elasticity greater than 1. Demand grows proportionally faster than income.
Substitutes
Two goods where an increase in the price of one raises demand for the other. Positive cross-price elasticity.
Complements
Two goods where an increase in the price of one lowers demand for the other. Negative cross-price elasticity.
Price elasticity of demand (Ep)
A measure of how responsive quantity demanded is to a change in price. Defined as the percentage change in quantity demanded divided by the percentage change in price. Usually negative, but often discussed in absolute value.
Elastic demand
|Ep| > 1. Quantity is highly responsive to price. A price increase causes total revenue to fall.
Inelastic demand
|Ep| < 1. Quantity is relatively unresponsive to price. A price increase causes total revenue to rise.
Unitary elastic demand
|Ep| = 1. A price change causes total revenue to remain unchanged.
Point elasticity (point-slope method)
Elasticity calculated at a specific point on the demand curve: Ep = (dQ/dP) × (P/Q). Uses the slope of the demand function and the coordinates of the point.
Direct elasticity method
Elasticity calculated from observed changes: Ep = (%ΔQ) / (%ΔP). Uses actual data rather than a known functional form.
Rule of thumb pricing
A pricing formula derived from the margin-elasticity relationship. Given MC and Ep, the profit-maximising price is P = MC / (1 + 1/Ep). When |Ep| = 2, for example, P = 2 × MC.
Price discrimination
Charging different prices to different consumers (or for different units) for the same good, where the price difference does not reflect a cost difference.
Hurdle model
A price discrimination strategy where the firm sets a "hurdle" (coupon, rebate, waiting period) that only price-sensitive consumers will clear, effectively segmenting the market by willingness to pay.
Consumer surplus
The difference between what a consumer is willing to pay and what they actually pay. Price discrimination captures some of this surplus for the firm.
Non-price factors shift the entire demand curve: income, prices of related goods, tastes and preferences, expectations, number of buyers.
A price change causes movement along the existing demand curve, not a shift.
In the demand function Q = a + bP + cI + dP_other, the coefficients on I and P_other determine how those variables shift demand.
You should be able to explain, in words and on a graph, the direction and size of the shift when a non-price factor changes.
Tuition has changed over time, but so have many other factors (income, population, alternatives, financial aid).
The demand function was not stable across those years; it shifted.
To calculate elasticity from data, you need to know that the demand curve itself stayed put and only price moved. If the curve shifted between observations, the two data points are not on the same curve.
Direct method (from observed data):
Ep = (ΔQ / Q) / (ΔP / P) = (%ΔQ) / (%ΔP)
Point-slope method (from a known demand function):
Ep = (dQ/dP) × (P / Q)
where dQ/dP is the slope of the demand function (not the inverse demand function).
Both methods should give the same answer when applied correctly at the same point.
On a straight-line demand curve, elasticity is not constant. It varies from point to point.
At the top of the curve (high price, low quantity), demand is elastic.
At the midpoint, demand is unitary elastic.
At the bottom of the curve (low price, high quantity), demand is inelastic.
The slope is constant along a line, but elasticity depends on the ratio P/Q, which changes.
Two exceptions where elasticity is constant everywhere:
A perfectly elastic demand curve (horizontal line): |Ep| = ∞ at every point.
A perfectly inelastic demand curve (vertical line): |Ep| = 0 at every point.
There is also the constant-elasticity demand function Q = aP^b, where elasticity equals b everywhere, but the key exam distinction is between "line elasticity" (varies along a straight line) and "overall elasticity" (a single number summarising responsiveness for the good as a whole).
If demand is elastic (|Ep| > 1): a price decrease raises total revenue.
If demand is inelastic (|Ep| < 1): a price decrease lowers total revenue.
If demand is unitary elastic (|Ep| = 1): a small price change leaves total revenue roughly unchanged.
This means a profit-maximising firm should never price on the inelastic portion of a linear demand curve. On that portion, the firm could raise price, sell fewer units, and earn more revenue while also reducing costs.
Availability of substitutes (more substitutes = more elastic)
Proportion of budget spent on the good (larger share = more elastic)
Time horizon (longer time = more elastic, as consumers find alternatives)
Necessity vs luxury (necessities tend to be inelastic)
Income elasticity:
E_I = (%ΔQ) / (%ΔI)
Positive: normal good
Greater than 1: superior good (cyclical, demand rises faster than income)
Between 0 and 1: necessity
Negative: inferior good (counter-cyclical, demand rises when income falls)
Cross-price elasticity:
E_xy = (%ΔQ_x) / (%ΔP_y)
Positive: x and y are substitutes
Negative: x and y are complements
The profit-maximising condition MR = MC can be rewritten as:
(P − MC) / P = −1 / Ep
This links the price-cost margin directly to the price elasticity of demand. A firm with more elastic demand has a thinner margin; a firm with less elastic demand can charge a wider margin.
Rearranging the margin-elasticity formula:
P = MC / (1 + 1/Ep)
Since Ep is negative, this produces a price above MC. For example, if |Ep| = 4 and MC = $2:
P = 2 / (1 + 1/(−4)) = 2 / (1 − 0.25) = 2 / 0.75 = $2.67
The markup over cost is (P − MC) / MC = 0.67 / 2 = 0.333, and the margin is (P − MC) / P = 0.67 / 2.67 = 0.25.
On the inelastic part of a linear demand curve, MR is negative. Since MC is always non-negative, MR = MC cannot hold on the inelastic portion. A firm there could raise its price, which would increase revenue and decrease costs (fewer units produced). So it is always profitable to move toward the elastic region.
Two conditions for effective price discrimination:
The firm must have some market power (i.e., it faces a downward-sloping demand curve).
The firm must be able to prevent resale (arbitrage) between consumer groups.
Three types:
First-degree (perfect) price discrimination
Each consumer is charged their exact willingness to pay. The firm captures all consumer surplus. Quantity sold equals the competitive quantity (where demand meets MC).
Second-degree price discrimination
Different prices for different quantities or versions of the product (e.g., bulk discounts, versioning of information goods). Consumers self-select into price tiers.
Third-degree price discrimination
Different prices for different identifiable groups of consumers (e.g., student discounts, geographic pricing). The firm sets MR equal across segments and equal to MC.
A practical method for implementing price discrimination without directly identifying each consumer's willingness to pay.
The firm offers a discount that requires some effort (clipping a coupon, waiting for a sale, buying online vs in-store).
Price-sensitive consumers jump the hurdle; less-sensitive consumers pay the full price.
Effectively segments the market by self-selection.
Simple: find the profit-maximising price and quantity for each segment separately (set MR = MC in each), given that resale is impossible.
With a quantity constraint: when the firm has limited output, allocate units so that MR is equalised across segments. If total quantity is fixed, the last unit sold should generate the same MR in every segment.
Price elasticity of demand (point method): Ep = (dQ/dP) × (P/Q)
Direct elasticity: Ep = (%ΔQ) / (%ΔP)
Revenue-elasticity link: MR = P × (1 + 1/Ep)
Margin-elasticity relationship: (P − MC) / P = −1/Ep
Rule of thumb price: P = MC / (1 + 1/Ep)
Income elasticity: E_I = (%ΔQ) / (%ΔI)
Cross-price elasticity: E_xy = (%ΔQ_x) / (%ΔP_y)
⚠️ Elasticity changes along a linear demand curve. The slope is constant, but elasticity is not. This is one of the most commonly tested conceptual points.
⚠️ Never price on the inelastic portion. MR is negative there, and MC cannot be negative, so MR = MC is impossible on that stretch.
⚠️ Know the course definitions: margin = (P − MC) / P, markup = (P − MC) / MC. These differ from the textbook.
⚠️ When using the point-slope elasticity formula, use the slope of the demand function (dQ/dP), not the inverse demand function (dP/dQ). A common error is to grab the slope from P = a − bQ and forget to invert it.
⚠️ At P = a (the vertical intercept of a linear demand curve, where Q = 0), demand is perfectly elastic (|Ep| = ∞). At Q = a/b (the horizontal intercept, where P = 0), demand is perfectly inelastic (|Ep| = 0).
⚠️ For price discrimination problems with a quantity constraint, equalise MR across segments.
Q: For Skateboards, Inc., the inverse demand curve is P = 10 − 2Q. At P = 10, what is the absolute value of the elasticity of demand?
A: At P = 10, Q = 0. The demand function is Q = 5 − 0.5P, so dQ/dP = −0.5. Ep = (−0.5)(10/0). Division by zero means |Ep| = ∞, i.e. infinitely elastic. Answer: (d).
Q: The price of good X rises from $5 to $8 and quantity demanded falls from 100 to 80. Is demand elastic, inelastic, or unit elastic?
A: %ΔP = 3/5 = 60%. %ΔQ = 20/100 = 20%. Ep = 20/60 = 0.33. Since |Ep| < 1, demand is inelastic. Answer: (b).
Q: A firm's MR is $20 and |Ep| = 5. What is the profit-maximising price?
A: MR = P(1 + 1/Ep). So 20 = P(1 + 1/(−5)) = P(4/5). P = 20 × 5/4 = $25. Answer: (b).
Q: Find the rule-of-thumb price when MC = $2 and |Ep| = 4.
A: P = MC / (1 + 1/Ep) = 2 / (1 − 1/4) = 2 / 0.75 = $2.67.
Q: From the same problem, find the margin and the markup.
A: Margin = (2.67 − 2) / 2.67 = 0.25 (or 25%). Markup = (2.67 − 2) / 2 = 0.333 (or 33.3%).
Q: Find the marginal cost if P = $5 and |Ep| = 3.
A: (P − MC) / P = 1/|Ep| = 1/3. So P − MC = 5/3, and MC = 5 − 5/3 = 10/3 ≈ $3.33.
Q: A hot dog vendor's demand is Q = 180 − 15P and he sells 30 hot dogs a day. How much revenue does he collect?
A: At Q = 30: 30 = 180 − 15P, so P = 10. Revenue = 30 × 10 = $300.
Q: What is the price elasticity at that point?
A: dQ/dP = −15. Ep = (−15)(10/30) = −5. |Ep| = 5.
Q: Should the vendor raise or lower price to increase revenue?
A: Demand is elastic (|Ep| = 5 > 1), so a price decrease would increase total revenue. The vendor should lower his price.
Q: What are the two conditions for effective price discrimination?
A: The firm must have market power (downward-sloping demand), and it must be able to prevent resale between consumer groups.
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