Advertising vs Pricing Levers

A firm seeking to increase demand faces a fundamental allocation problem: should it lower prices or increase advertising? These two levers interact in subtle ways—advertising shifts the demand curve outward, while price moves along it. The classical results of Dorfman and Steiner (1954) and Nerlove and Arrow (1962) provide elegant conditions for the optimal mix, showing that the ratio of advertising spend to sales revenue depends on just two elasticities.

The Marketing Mix Problem

Consider a monopolist choosing both a price $p$ and an advertising expenditure $A$ to maximize profit. Demand depends on both instruments: higher advertising increases demand, while higher prices reduce it. We model demand using a constant-elasticity form:

D(p, A) = D_0 \cdot A^{\gamma} \cdot \left(\frac{\bar{p}}{p}\right)^{\varepsilon}

(1)

where $D_0$ is a base demand scalar, $\gamma \in (0, 1)$ is the advertising elasticity (the percentage increase in demand from a 1% increase in ad spend), $\varepsilon > 1$ is the absolute value of the price elasticity of demand, and $\bar{p}$ is a reference price.

Definition — Advertising Elasticity

The advertising elasticity $\gamma$ measures the responsiveness of demand to changes in advertising expenditure, holding price constant. Formally, $\gamma = \frac{\partial D}{\partial A} \cdot \frac{A}{D}$ . Empirical estimates typically fall in the range 0.01 to 0.20 for mature products and 0.20 to 0.50 for new products or categories where brand awareness is still being built.

The firm’s profit function is:

\pi(p, A) = (p - c) \cdot D(p, A) - A

(2)

where $c$ is the marginal cost of production. The firm must pay for advertising out of its operating margin, so there is a direct tradeoff: every dollar spent on advertising must be justified by the additional margin it generates through increased demand.

The Dorfman-Steiner Theorem

Dorfman and Steiner (1954) derived a remarkably simple condition for the profit-maximizing advertising intensity. Taking the first-order conditions of the profit function with respect to both $p$ and $A$ , and solving for the optimal advertising-to-sales ratio, yields:

Dorfman-Steiner Theorem

At the profit maximum, the optimal ratio of advertising expenditure to sales revenue satisfies:

\frac{A^*}{p^* \cdot D^*} = \frac{\gamma}{\varepsilon}

(3)

That is, the firm should set its advertising-to-sales ratio equal to the ratio of advertising elasticity to price elasticity.

The intuition is powerful. When advertising is highly effective (high $\gamma$ ), the firm should spend more on advertising relative to revenue. When demand is highly price-elastic (high $\varepsilon$ ), the firm should rely more on competitive pricing and less on advertising. The ratio $\gamma / \varepsilon$ captures exactly the relative effectiveness of the two levers.

Consumer Electronics

A consumer electronics brand estimates a price elasticity of $\varepsilon = 3.0$ and an advertising elasticity of $\gamma = 0.15$ . The Dorfman-Steiner condition gives an optimal A/S ratio of $0.15 / 3.0 = 0.05$ , or 5% of revenue. If the brand earns $10 million in revenue, it should spend approximately $500,000 on advertising. Spending significantly more would yield diminishing returns; spending significantly less would leave demand-shifting potential on the table.

The interactive chart below lets you explore the joint profit landscape over price and advertising spend. The heatmap shows profit levels, with the optimal point marked. Adjust the marginal cost and elasticity parameters to see how the optimal allocation shifts.

Notice how increasing the advertising elasticity $\gamma$ pulls the optimal point toward higher ad spend, while increasing the price elasticity $\varepsilon$ favors lower prices. The A/S ratio displayed in the metrics panel tracks the Dorfman-Steiner condition precisely.

Advertising Response

A central feature of the advertising response function is diminishing returns. Because the advertising elasticity $\gamma$ lies strictly between 0 and 1, the demand function $D(p, A) = D_0 A^{\gamma} ({\bar{p}}/{p})^{\varepsilon}$ is concave in $A$ . Each additional dollar of advertising generates less incremental demand than the previous one.

The marginal revenue of an additional dollar of advertising, holding price fixed, is:

\frac{\partial (p \cdot D)}{\partial A} = p \cdot D_0 \cdot \gamma \cdot A^{\gamma - 1} \cdot \left(\frac{\bar{p}}{p}\right)^{\varepsilon}

(4)

Since $\gamma - 1 < 0$ , this expression is decreasing in $A$ . The profit-maximizing ad spend occurs where the marginal revenue from advertising equals $1—the cost of the last dollar spent.

The chart below illustrates two perspectives. The first view shows how the demand curve shifts outward with increasing advertising spend—each curve represents a different level of $A$ . The second view shows the total revenue curve and the marginal revenue of advertising, making the diminishing returns visible.

In the Demand Curves view, observe how the vertical gap between successive curves narrows as $A$ increases. This is the visual signature of concavity. In the Marginal ROI view, the dashed curve declining toward the breakeven line shows exactly where additional advertising ceases to pay for itself.

Goodwill Dynamics

The Dorfman-Steiner theorem treats advertising as a static decision: spend today, generate demand today. In practice, advertising has persistent effects. Nerlove and Arrow (1962) introduced the concept of a goodwill stock to capture these dynamics.

Definition — Goodwill Stock

The goodwill stock $G_t$ represents the accumulated effect of past advertising on current demand. It evolves according to the law of motion:

G_t = (1 - \delta) \cdot G_{t-1} + A_t

(5)

where $\delta \in (0, 1)$ is the depreciation rate (the fraction of goodwill that decays each period) and $A_t$ is advertising expenditure in period $t$ . Demand then depends on $G_t$ rather than on $A_t$ directly.

Under constant spending $A_t = A$ for all $t$ , the goodwill stock converges to a steady state:

G^* = \frac{A}{\delta}

(6)

This expression reveals a key insight: the long-run stock of goodwill is inversely proportional to the depreciation rate. In industries where consumers forget quickly (high $\delta$ )—such as fast food or daily deals—maintaining goodwill requires continuous spending. In industries with durable brand associations (low $\delta$ )—such as luxury goods—the same budget builds a much larger goodwill stock.

The half-life of goodwill is $t_{1/2} = \ln 2 / \delta$ . If $\delta = 0.10$ , goodwill has a half-life of about 7 periods; if $\delta = 0.30$ , it has a half-life of only 2.3 periods.

The chart below compares three spending strategies—constant, pulsing, and front-loaded—all with the same total budget. Adjust the depreciation rate and budget to observe how the goodwill stock evolves under each approach.

Pulsing vs Constant Advertising

Marketing practitioners sometimes advocate “pulsing” strategies, where spending is concentrated in periodic bursts rather than spread evenly. The Nerlove-Arrow model shows that pulsing produces a sawtooth pattern in goodwill: sharp spikes during bursts followed by exponential decay. Whether this outperforms constant spending depends on whether the demand function is convex or concave in goodwill. Under the standard concave assumption, constant spending is weakly preferred because it avoids the costly rebuilding after each decay trough.

Implications for Practice

The Dorfman-Steiner and Nerlove-Arrow frameworks yield several actionable principles for pricing practitioners.

When to invest in brand over price cuts. The Dorfman-Steiner ratio $\gamma / \varepsilon$ provides a direct test. When the advertising elasticity is large relative to the price elasticity—as is typical for differentiated products, luxury goods, and new categories—the firm should allocate more budget toward advertising. Conversely, in commodity markets where price elasticity dominates, price reductions are the more effective lever.

Advertising and price are complements. The model reveals that advertising and pricing are not independent decisions. Advertising shifts the demand curve, which changes the optimal price. In the constant-elasticity model, a higher goodwill stock increases demand at every price point, but the optimal price itself depends only on the markup rule $p^* = c \cdot \varepsilon / (\varepsilon - 1)$ . This means that advertising increases volume (and hence profit) without necessarily changing the optimal price level—it operates through the demand intercept rather than the price sensitivity.

Dynamic considerations favor consistency. The Nerlove-Arrow model suggests that, under concave demand, steady advertising is preferred to erratic bursts. Pulsing can work when there are threshold effects in awareness or when the demand function is S-shaped (convex at low goodwill, concave at high goodwill), but the burden of proof falls on the practitioner to demonstrate that such nonlinearities exist.

Industry depreciation rates matter enormously. A firm in an industry with a high goodwill depreciation rate (say, $\delta = 0.3$ ) must spend three times as much per period as a firm with $\delta = 0.1$ to achieve the same steady-state goodwill. This explains why consumer packaged goods firms allocate a much larger share of revenue to advertising than durable goods manufacturers, even when their advertising elasticities are similar.

References

Dorfman, R. & Steiner, P. O. (1954). “Optimal Advertising and Optimal Quality.” American Economic Review, 44(5), 826–836.
Nerlove, M. & Arrow, K. J. (1962). “Optimal Advertising Policy under Dynamic Conditions.” Economica, 29(114), 129–142.