Difficulty: Intermediate | Prerequisites: basic algebra, understanding of supply and demand curves, familiarity with utility functions.
This material sits at the heart of microeconomics: how consumers respond to price changes, and how to measure that response precisely. If you can read a linear demand function and know what "quantity demanded" means, you are ready. The notes cover evaluating demand at specific prices, computing price elasticity, interpreting what elasticity tells you about revenue, and applying the Cobb-Douglas optimal-choice formula. These tools recur in every market-analysis topic from here onward, so getting comfortable now pays off later.
A demand function tells you the quantity consumers want at each price. Price elasticity of demand measures how sensitive that quantity is to a price change, and its value determines whether raising the price increases or decreases total revenue. For Cobb-Douglas utility, optimal consumption of each good follows a tidy formula based on the exponent share of income divided by price.
Demand function, D(p)
A rule that returns the quantity demanded, q, for any given price p. In this course, demand functions are typically linear: D(p) = a - bp, where a is the quantity intercept and b is the slope coefficient.
In simple terms, it answers the question "how much will people buy at this price?"
Quantity demanded (q)
The number of units consumers wish to purchase at a specific price. Demand cannot be negative, so q = max(0, D(p)).
Think of it as the actual, usable output of the demand function once you have plugged in a price.
Price elasticity of demand (|Ep|)
A unit-free measure of how responsive quantity demanded is to a change in price. Formally: |Ep| = |(P / Q) x (dQ / dP)|. The absolute value is taken because demand slopes downward, so the raw number would be negative.
In simple terms, it tells you the percentage drop in quantity you get for each one-percent rise in price.
Unit elastic
When |Ep| = 1 exactly. A price increase and the resulting quantity decrease cancel out perfectly, so total revenue does not change.
Think of it as the tipping point: revenue neither rises nor falls.
Elastic demand
When |Ep| > 1. Quantity is highly responsive to price. A price increase causes revenue to fall because the lost sales outweigh the higher per-unit price.
Inelastic demand
When |Ep| < 1. Quantity barely budges when price changes. A price increase raises revenue because the small drop in sales is more than offset by the higher price.
Cobb-Douglas utility function
A utility function of the form u(x1, x2) = x1^c x2^d. It produces smooth, convex indifference curves and a particularly clean optimal-choice formula.
In simple terms, it is the workhorse utility function in introductory micro because the maths stays manageable.
Residual claimant (RC) budget approach
The method used here for Cobb-Douglas optimisation. Each consumer i has income Mi; total income M = sum of all Mi. The optimal quantity of good 1 is x1* = (c / (c + d)) x (M / P1).
Think of it as a ratio-of-exponents shortcut that skips the full Lagrangian.
A linear demand function takes the form D(p) = a - bp.
a is the maximum quantity demanded when price is zero (the vertical intercept on a quantity axis).
b is the rate at which quantity falls as price rises (the absolute slope).
To find quantity at a given price, substitute the price into the function.
Example (Quiz A): D(p) = 50 - 5p. At p = 5, q = 50 - 25 = 25. At p = 10, q = 50 - 50 = 0.
Example (Quiz B): D(p) = 100 - 10p. At p = 5, q = 50. At p = 15, the formula gives -50, but demand cannot be negative, so q = 0.
Demand is always non-negative: q = max(0, D(p)). If the formula returns a negative number, quantity demanded is zero.
The point-elasticity formula is |Ep| = |(P / Q) x (dQ / dP)|.
For a linear demand function D(p) = a - bp, the derivative dQ/dP = -b, so |Ep| = (bP) / Q.
Substitute the price and the corresponding quantity to get a number.
Example (Quiz A): D(p) = 50 - 5p. At p = 5, q = 25. Slope = -5. |Ep| = |5 x 5 / 25| = 25/25 = 1. Unit elastic.
Example (Quiz B): D(p) = 100 - 10p. At p = 5, q = 50. Slope = -10. |Ep| = |10 x 5 / 50| = 50/50 = 1. Also unit elastic.
Notice: the midpoint of a linear demand curve is always unit elastic. This is a general result, not a coincidence.
Revenue = P x Q. When price changes, revenue depends on whether the quantity effect or the price effect dominates.
At unit elasticity (|Ep| = 1), revenue does not change when price rises by a small amount. The gain from the higher price is exactly cancelled by the loss from reduced quantity.
Graphically, the "orange rectangle" (revenue gained from the price increase) equals the "blue rectangle" (revenue lost from lower quantity).
If |Ep| > 1 (elastic): a price increase lowers revenue.
If |Ep| < 1 (inelastic): a price increase raises revenue.
Given u(x1, x2) = x1^c x2^d, prices P1 and P2, n identical consumers each with income Mi.
Total income: M = sum of Mi (if n consumers each with Mi, then M = n x Mi).
Optimal quantity of good 1: x1* = (c / (c + d)) x (M / P1).
This is the "RC" (residual claimant) budget approach: spend the fraction c/(c+d) of total income on good 1.
Example (Quiz A): u = x1^2 x2, so c = 2, d = 1. P1 = 2, P2 = 1, n = 4, Mi = 10. M = 40. x1* = (2/3)(40/2) = 40/3 = 13.33.
Example (Quiz B): u = x1^2 x2, c = 2, d = 1. P1 = 1, P2 = 2, n = 4, Mi = 8. M = 32. x1* = (2/3)(32/1) = 64/3 = 21.33.
Linear demand function
D(p) = a - bp, with q = max(0, D(p))
Point price elasticity of demand
|Ep| = |(P / Q) x (dQ / dP)| = bP / Q (for linear demand)
Cobb-Douglas optimal quantity (good 1)
x1* = (c / (c + d)) x (M / P1)
where M = total income across all consumers, c and d are the exponents of goods 1 and 2 in the utility function, and P1 is the price of good 1.
Price elasticity is the reason airlines charge different fares for the same seat: business travellers have inelastic demand (they will pay more because they have to fly), while leisure travellers are elastic (they will switch to a different date or destination if the price rises). Revenue management systems lean on elasticity estimates to set prices that maximise total revenue.
The Cobb-Douglas expenditure-share result shows up in consumer spending data: households tend to spend roughly fixed fractions of income on broad categories like food, housing, and transport, which is consistent with Cobb-Douglas preferences.
Students often confuse the slope of the demand curve with elasticity. They are related but not the same. Elasticity depends on both the slope and the price-quantity ratio at the point you are evaluating, so it changes along a linear demand curve even though the slope is constant.
Students sometimes forget to take the absolute value when computing |Ep|. The raw elasticity of a downward-sloping demand curve is negative. The convention in this course is to report the absolute value.
A common error is plugging in a price that drives D(p) below zero and treating the negative number as a real quantity. Demand is always max(0, D(p)).
Students sometimes assume that because the Cobb-Douglas formula uses total income M, they should plug in one consumer's income. M is the sum across all n consumers: M = n x Mi.
⚠️ Expect a question that gives you a linear demand function and asks for quantity at two prices, then asks for elasticity at one of them. This appeared in both Quiz A and Quiz B.
⚠️ The follow-up is almost always: "Based on your elasticity result, what happens to revenue if price rises?" Know the three cases (elastic, unit elastic, inelastic) cold.
⚠️ The Cobb-Douglas optimisation problem is a reliable exam question. You will be given the utility function, prices, the number of consumers, and individual income, and asked to find x1*. The formula is quick but watch the arithmetic, especially the exponent-share fraction c/(c+d).
⚠️ The midpoint of a linear demand curve is always unit elastic. If the numbers work out to |Ep| = 1, check whether you are at the midpoint price, which is p = a/(2b).
True or false: A steeper demand curve always means more inelastic demand. (False: elasticity also depends on where on the curve you are.)
If D(p) = 80 - 4p, what is q when p = 25? (q = max(0, 80 - 100) = 0.)
Fill in the blank: When |Ep| = 1, a small price increase causes total revenue to ______. (Stay the same.)
True or false: In the Cobb-Douglas formula x1* = (c/(c+d))(M/P1), M is a single consumer's income. (False: M is total income across all consumers.)
If u = x1^3 x2^2 and P1 = 5, what fraction of total income is spent on good 1? (3/(3+2) = 3/5 = 60%.)
Q: Given D(p) = 50 - 5p, find the quantity demanded at p = 3 and compute |Ep| at that point.
A: q = 50 - 15 = 35. |Ep| = (5 x 3) / 35 = 15/35 = 3/7 ≈ 0.43. Demand is inelastic at p = 3.
Q: If demand is inelastic at the current price, should a firm raise or lower price to increase revenue?
A: Raise it. With inelastic demand, the percentage drop in quantity is smaller than the percentage rise in price, so revenue increases.
Q: For u(x1, x2) = x1^2 x2, P1 = 4, P2 = 2, and 5 consumers each with income 20, find x1.*
A: c = 2, d = 1. M = 5 x 20 = 100. x1* = (2/3)(100/4) = (2/3)(25) = 50/3 ≈ 16.67.
Q: At what price is a linear demand function D(p) = a - bp unit elastic?
A: At p = a/(2b). This is the midpoint price of the demand curve.
Elasticity connects directly to the market equilibrium material (Chapter 16): once you know how to find equilibrium price, you can evaluate elasticity at that price to predict revenue effects of shocks. It also feeds into the taxation analysis, where the relative elasticities of supply and demand determine who bears the burden of a per-unit tax.
The Cobb-Douglas optimisation here is the demand side of general equilibrium. When you later study welfare and efficiency, the same utility framework reappears in consumer and producer surplus calculations.
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