Difficulty: Intermediate | Prerequisites: the companion notes on Multiple Regression Analysis (APEC 3002 Lab 3), basic understanding of demand curves from introductory microeconomics.
Once you have estimated a regression model linking sales to advertising and price, the next step in managerial economics is to use that model to derive demand curves and compute elasticities. This is where the statistical output becomes a practical decision-making tool. The demand curve shows how quantity demanded responds to price at a given level of advertising, and elasticity puts a number on how sensitive that response is. You need to be comfortable with the regression equation from the companion notes before tackling this material.
By plugging different advertising levels into the regression equation, you can draw separate demand curves showing how sales respond to price at each ad spend. Price elasticity of demand computed from these curves is less than 1 in absolute value at every point tested, meaning the good is price-inelastic. Higher advertising makes the good even more inelastic by expanding the base of sales.
Demand curve
A graph (or equation) showing the relationship between the price of a good and the quantity demanded, holding other factors constant. In simple terms, it answers: "If we set the price at X, how much will people buy?"
Price elasticity of demand (Eₚ)
A measure of how responsive quantity demanded is to a change in price. Formally, the percentage change in quantity demanded divided by the percentage change in price. Think of it as "how much do sales flinch when we nudge the price?"
Inelastic demand
Demand is inelastic when |Eₚ| < 1. A 1% increase in price leads to less than a 1% decrease in quantity demanded. Revenue rises when price rises, because the drop in volume is small relative to the price increase.
Elastic demand
Demand is elastic when |Eₚ| > 1. A 1% increase in price leads to more than a 1% decrease in quantity demanded. Revenue falls when price rises.
Unit elastic demand
Demand is unit elastic when |Eₚ| = 1. A 1% price change produces exactly a 1% change in quantity demanded. Revenue is maximised at the unit-elastic point on a linear demand curve.
Shift in the demand curve
A change in a non-price factor (here, advertising) that moves the entire demand curve left or right. An increase in advertising shifts the demand curve to the right, meaning more is demanded at every price level.
Point elasticity
Elasticity calculated at a specific point on the demand curve, using the derivative (slope) of the demand function at that point, rather than an arc between two points.
Start with the estimated regression: S = 284.81 + 0.5497A - 1.2716P.
To draw a demand curve, fix A at a chosen level and vary P. This gives you a linear relationship between S and P.
At A = 600: S = 284.81 + 0.5497(600) - 1.2716P = 614.63 - 1.2716P
At A = 900: S = 284.81 + 0.5497(900) - 1.2716P = 779.54 - 1.2716P
At A = 1200: S = 284.81 + 0.5497(1200) - 1.2716P = 944.45 - 1.2716P
Each of these is a demand curve at a specific advertising level. The slope with respect to price is always -1.2716 (the coefficient on P does not change), but the intercept shifts rightward as advertising increases.
Increasing advertising from $600 to $900 to $1200 shifts the demand curve to the right. At any given price, a higher advertising level predicts higher sales.
The graph from the lab (Figure 2) shows three downward-sloping demand curves at these three advertising levels. The curve for A = 1200 sits furthest to the right, A = 600 furthest to the left.
The slope of each curve (the price sensitivity) is identical, because in a linear model the coefficient on P does not depend on the level of A. What changes is the starting point (the intercept).
Point price elasticity is calculated as: Eₚ = (dS/dP) × (P/S).
From the regression, dS/dP = -1.2716 (the coefficient on P).
You then plug in the specific P and S values for the scenario you are examining.
Results from the lab:
P = $90, A = $900, S = 665.08: Eₚ = -1.2716 × (90/665.08) = -0.172
P = $110, A = $900, S = 639.65: Eₚ = -1.2716 × (110/639.65) = -0.219
P = $130, A = $900, S = 614.21: Eₚ = -1.2716 × (130/614.21) = -0.269
P = $110, A = $600, S = 474.74: Eₚ = -1.2716 × (110/474.74) = -0.295
P = $110, A = $1200, S = 804.55: Eₚ = -1.2716 × (110/804.55) = -0.174
Every elasticity value is between 0 and -1 in absolute terms, confirming the good is price-inelastic at all tested points.
As price rises (holding A constant), demand becomes somewhat more elastic. At P = $90 the elasticity is -0.17; at P = $130 it is -0.27. This is normal for a linear demand curve: elasticity increases in absolute value as you move up the curve.
As advertising increases (holding P constant at $110), demand becomes more inelastic. At A = $600 the elasticity is -0.29; at A = $1200 it is -0.17. Higher advertising expands the sales base (S rises), so the same price change represents a smaller percentage of quantity demanded.
Because demand is inelastic, raising the price would increase total revenue (the percentage drop in quantity is smaller than the percentage increase in price).
Because advertising significantly drives sales, increasing the advertising budget is a viable way to boost volume without cutting price.
The combination of inelastic demand and a significant advertising effect suggests the firm has some pricing power, likely due to brand differentiation reinforced by advertising.
Demand curve at a fixed advertising level:
S = (\beta_0 + \beta_1 A) - \beta_2 PPlug in the advertising level to get the intercept; the slope with respect to P is always β₂ = -1.2716.
Point price elasticity of demand:
E_p = \frac{dS}{dP} \times \frac{P}{S} = \beta_2 \times \frac{P}{S}Where β₂ = -1.2716, P is the price at the point of interest, and S is the predicted sales at that point.
Example calculation (P = $110, A = $900):
S = 284.81 + 0.5497(900) - 1.2716(110) = 639.65E_p = -1.2716 \times \frac{110}{639.65} = -0.2187Pricing teams at consumer goods companies use elasticity estimates exactly like these to set prices. If demand is inelastic, the firm can raise prices without losing much volume, a finding that directly informs revenue management. The advertising-shift analysis is standard in media planning: it quantifies how much additional ad spend is needed to expand the market at a given price point. Streaming services, airlines, and retailers all run this kind of analysis when deciding whether to invest in advertising or adjust prices.
Students often think elasticity is the same as slope. It is not. The slope of the demand curve (dS/dP = -1.2716) is constant along a linear demand curve, but elasticity changes at every point because it also depends on the ratio P/S.
Students sometimes believe that "inelastic" means people do not respond to price at all. Inelastic means the response is proportionally small, not that it is zero. Sales still fall when price rises; they just fall by a smaller percentage than the price rose.
A common error is forgetting that a rightward shift in the demand curve (from more advertising) changes the elasticity calculation because S increases in the denominator, even though the slope stays the same.
Students occasionally confuse a movement along the demand curve (caused by a price change) with a shift of the demand curve (caused by a change in advertising or another non-price factor). These are different things and must be described separately.
⚠️ Be able to derive a demand curve from a multiple regression equation by fixing one variable and varying price. This is a standard exam question format.
⚠️ Know the point elasticity formula: Eₚ = (dS/dP) × (P/S). Expect to calculate it at a given price and advertising level.
⚠️ Understand why elasticity varies along a linear demand curve even though the slope is constant.
⚠️ Be able to explain the managerial implication: if demand is inelastic, a price increase raises total revenue.
⚠️ Know the distinction between a movement along the demand curve (price changes) and a shift of the demand curve (advertising changes). Exams often test whether you can identify which is which.
True or False: Elasticity is the same as the slope of the demand curve. (False. Elasticity = slope × P/S, so it changes along the curve even when slope is constant.)
Fill in the blank: When |Eₚ| < 1, demand is said to be ____. (Inelastic.)
True or False: Increasing advertising shifts the demand curve to the right. (True.)
Fill in the blank: If Eₚ = -0.22, a 10% price increase leads to approximately a ____% decrease in quantity demanded. (2.2%.)
True or False: When demand is inelastic, raising the price will decrease total revenue. (False. Revenue increases because the percentage drop in quantity is smaller than the percentage increase in price.)
Q: Given S = 284.81 + 0.5497A - 1.2716P, derive the demand curve when A = 900.
A: Substitute A = 900: S = 284.81 + 0.5497(900) - 1.2716P = 284.81 + 494.73 - 1.2716P = 779.54 - 1.2716P. This is a linear demand curve with intercept 779.54 and slope -1.2716.
Q: Calculate the point price elasticity of demand when P = $110 and A = $600.
A: First find S: S = 284.81 + 0.5497(600) - 1.2716(110) = 284.81 + 329.82 - 139.88 = 474.75. Then Eₚ = -1.2716 × (110/474.75) = -0.295. Demand is inelastic at this point.
Q: Explain why increasing advertising from $600 to $1200, holding price at $110, changes the price elasticity from -0.295 to -0.174.
A: When advertising increases, predicted sales rise (the denominator S gets larger). The slope dS/dP remains -1.2716 and P stays at $110, so the numerator of the elasticity formula is unchanged. Because S is larger, the ratio P/S shrinks, and the absolute elasticity falls. The good becomes more inelastic because the same absolute price effect represents a smaller share of a larger quantity.
Q: If demand is inelastic, should the firm raise or lower its price to increase revenue? Explain.
A: The firm should raise its price. With inelastic demand, the percentage decline in quantity demanded is less than the percentage increase in price, so total revenue (P × S) increases. This continues to hold as long as |Eₚ| remains below 1.
Q: What is the difference between a movement along the demand curve and a shift of the demand curve in this model?
A: A change in price (P) causes a movement along the demand curve: you slide to a different point on the same line. A change in advertising (A) shifts the entire demand curve left or right: at every price, the predicted quantity changes. In the regression, price is the variable on the axis of the demand curve; advertising is a shifter.
The elasticity calculations here build directly on the regression output covered in the companion notes on Multiple Regression Analysis. Understanding elasticity is also essential for the pricing and revenue optimisation topics that follow later in APEC 3002, where you will use the condition MR = MC to find the profit-maximising price. The demand-shift analysis connects to broader marketing economics: how firms allocate budgets between price promotions and brand advertising.
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