Demand Elasticities and Diversion Ratios

Why Derivatives Reveal Substitution

When a product’s price rises, its demand generally falls. The more informative question is where the lost demand goes: to a close competing product, to several alternatives, or out of the market. Price elasticities describe the percentage response of demand to a percentage price change. Diversion ratios describe how the marginal loss in demand for one product is distributed across alternatives. Both are calculated from derivatives of market shares with respect to prices.

The distinction matters in differentiated-product demand. Two products can have similar market shares but very different substitution patterns. A logit model imposes a particular pattern; random-coefficient models allow consumers’ price sensitivities and product tastes to vary, changing the derivatives and the implied diversion. The calculations below treat a price change as a local change, holding the market’s other prices and product characteristics fixed.

Model and Definitions

Consider one market with $J$ inside products and an outside option indexed by $0$. Consumer $i$’s indirect utility from product $j$ is $V_{ij}+\varepsilon_{ij}$, where $V_{ij}$ is its deterministic utility and $\varepsilon_{ij}$ is a type-I extreme-value choice shock. The outside option has deterministic utility $V_{i0}$, often normalized to zero. Conditional on consumer characteristics, the logit choice probability is

P_{ij}=\frac{\exp(V_{ij})}{\exp(V_{i0})+\sum_{\ell=1}^{J}\exp(V_{i\ell})}.

Suppose the price coefficient for consumer $i$ is $\alpha_i<0$, and utility depends on product $j$’s own price $p_j$ through a term with derivative $\partial V_{ij}/\partial p_k=\alpha_i\mathbf{1}\{j=k\}$. Thus, raising product $k$’s price directly changes the utility of product $k$, not the utilities of other alternatives. The market share is $s_j=\int P_{ij}\,dF(i)$, where $F$ is the distribution of consumer characteristics and tastes. In a homogeneous logit model, all consumers have the same price coefficient and probabilities, so $s_j=P_{ij}$.

Definition: The price derivative $\partial s_j/\partial p_k$ measures the local change in product $j$’s market share when product $k$’s price changes, holding other prices fixed. The price elasticity of product $j$’s share with respect to price $p_k$ is $\varepsilon_{jk}=(p_k/s_j)(\partial s_j/\partial p_k)$. The diversion ratio from product $j$ to a different product $k$ is $D_{j\to k}=(\partial s_k/\partial p_j)/(-\partial s_j/\partial p_j)$, when the own-price derivative is negative.

An elasticity is dimensionless: it is the approximate percentage change in product $j$’s share for a one-percent change in price $p_k$. An own-price elasticity has $j=k$ and is generally negative. A cross-price elasticity has $j\ne k$ and, in the model above, is nonnegative because the products are substitutes. A diversion ratio is also dimensionless, but it is a ratio of share derivatives: it measures the fraction of product $j$’s marginal lost share that appears as a gain for product $k$.

Deriving Logit Share Derivatives

Differentiate the conditional choice probability with respect to a product’s price. The derivative of a logit probability with respect to its deterministic utility is $P_{ij}(\mathbf{1}\{j=\ell\}-P_{i\ell})$. Applying the chain rule to the price coefficient gives

\frac{\partial P_{ij}}{\partial p_k} =\alpha_i P_{ij}\bigl(\mathbf{1}\{j=k\}-P_{ik}\bigr).

For the own derivative, set $j=k$; for the cross derivative, set $j\ne k$. Integrating these probability derivatives over consumers gives the market-share derivatives. The integration and differentiation can be interchanged here because the probability derivatives are bounded in absolute value by $|\alpha_i|$, which is integrable when the price coefficient has a finite absolute mean.

\frac{\partial s_j}{\partial p_j} =\int \alpha_i P_{ij}(1-P_{ij})\,dF(i), \qquad \frac{\partial s_j}{\partial p_k} =\int -\alpha_i P_{ij}P_{ik}\,dF(i),\quad j\ne k.

Since $\alpha_i<0$, the own derivative is negative and each cross derivative is positive. The elasticity formula then scales each derivative by the price of the product whose price changes and divides by the share whose response is being measured. In particular, the price in $\varepsilon_{jk}$ is $p_k$, not $p_j$.

For homogeneous logit, $\alpha_i=\alpha$ and $P_{ij}=s_j$ for every consumer. The formulas simplify to $\partial s_j/\partial p_j=\alpha s_j(1-s_j)$ and $\partial s_j/\partial p_k=-\alpha s_js_k$ for $j\ne k$. Therefore, the own-price elasticity is $\alpha p_j(1-s_j)$, while the cross-price elasticity is $-\alpha p_k s_k$.

Worked Example: Homogeneous-Logit Elasticity and Diversion

Suppose a hypothetical market has shares $s_A=0.20$, $s_B=0.30$, and outside share $s_0=0.50$. Product A costs $p_A=\$10$, and the common price coefficient is $\alpha=-0.05$ per dollar. Calculate A’s own-price elasticity, the elasticity of B’s share with respect to A’s price, and diversion from A to B.

The homogeneous-logit own derivative for A is $\partial s_A/\partial p_A=\alpha s_A(1-s_A)$. Substituting the values gives $(-0.05)(0.20)(0.80)=-0.00800$ share units per dollar. Thus, a one-dollar increase in A’s price locally reduces A’s share by 0.00800, or 0.8 percentage points.

A’s own-price elasticity is $\varepsilon_{AA}=(p_A/s_A)(\partial s_A/\partial p_A)$. The calculation is $(10/0.20)(-0.00800)=-0.400$. A one-percent increase in A’s price is associated, locally, with a 0.400% decrease in A’s share.

The cross derivative of B’s share with respect to A’s price is $\partial s_B/\partial p_A=-\alpha s_Bs_A$. Its value is $(0.05)(0.30)(0.20)=0.00300$ share units per dollar. The cross-price elasticity is $\varepsilon_{BA}=(p_A/s_B)(\partial s_B/\partial p_A) =(10/0.30)(0.00300)=0.100$. So a one-percent increase in A’s price locally raises B’s share by 0.100%.

Finally, diversion from A to B is $D_{A\to B}=0.00300/0.00800=0.375$. In this model, 37.5% of the marginal share lost by A goes to B. This is a local derivative ratio, not a claim that a finite price increase transfers exactly 37.5% of A’s lost share to B.

Diversion in Random-Coefficient Models

With random coefficients, consumers can differ in price sensitivity and in their tastes for products. Their probabilities must be differentiated first, then averaged over the consumer distribution. For $j\ne k$, the cross derivative is an average of $-\alpha_iP_{ij}P_{ik}$; the own derivative is an average of $\alpha_iP_{ij}(1-P_{ij})$. These averages are generally not obtained by substituting market-average shares and a mean price coefficient into the homogeneous-logit formulas. Consumers’ probabilities and price coefficients can vary together.

When all consumers have $\alpha_i<0$, diversion from $j$ to $k$ is

D_{j\to k} = \frac{\int (-\alpha_i)P_{ij}P_{ik}\,dF(i)} {\int (-\alpha_i)P_{ij}(1-P_{ij})\,dF(i)}, \qquad j\ne k.

This expression weights consumers by their contribution to the marginal loss from product $j$. A consumer with a large absolute price coefficient and a high probability of choosing $j$ can contribute substantially to the derivative. Diversion therefore depends on both which consumers are at the margin and which alternative they are likely to choose.

Worked Example: Integrating Derivatives Across Consumer Types

Consider a hypothetical two-type market, with each type representing half of consumers. Type 1 has $\alpha_1=-0.04$, probabilities $(P_A,P_B,P_0)=(0.20,0.30,0.50)$, and type 2 has $\alpha_2=-0.10$, probabilities $(0.40,0.20,0.40)$. Product A costs $\$8$. Calculate the market shares, A’s own-price elasticity, the cross-price elasticity of B with respect to A’s price, and diversion from A to B.

The market shares are probability averages: $s_A=(0.20+0.40)/2=0.300$ and $s_B=(0.30+0.20)/2=0.250$. The outside share is $(0.50+0.40)/2=0.450$; the three shares sum to 1.

The own derivative is the average of the type-specific derivatives: \[ \frac{\partial s_A}{\partial p_A} =\frac{(-0.04)(0.20)(0.80)+(-0.10)(0.40)(0.60)}{2} =\frac{-0.0064-0.0240}{2} =-0.0152. \] Thus, a one-dollar increase in A’s price locally reduces its market share by 0.0152, or 1.52 percentage points.

A’s own-price elasticity is $\varepsilon_{AA}=(8/0.300)(-0.0152)=-0.4053$. For B, the cross derivative is \[ \frac{\partial s_B}{\partial p_A} =\frac{-(-0.04)(0.30)(0.20)-(-0.10)(0.20)(0.40)}{2} =\frac{0.0024+0.0080}{2} =0.0052. \] Therefore, $\varepsilon_{BA}=(8/0.250)(0.0052)=0.1664$: locally, a one-percent increase in A’s price raises B’s share by 0.1664%.

The diversion ratio is $D_{A\to B}=0.0052/0.0152=0.3421$. About 34.2% of A’s marginal lost share goes to B under these model probabilities. The type-specific price sensitivities matter: the type with the larger absolute price coefficient has different probabilities across A, B, and the outside option, so replacing the types with a single average consumer would generally change this derivative ratio.

Comparing Substitution Across Product Pairs

For a fixed product $j$, compute each diversion ratio using the same own-derivative denominator $-\partial s_j/\partial p_j$. The ratios to all other inside products and to the outside option add to one in this model. To see why, choice probabilities sum to one for each consumer. If $p_j$ rises, the derivatives of every other inside probability and the outside probability sum to the negative of the own derivative. Thus the gains across all destinations exactly offset the marginal loss from product $j$.

For example, the outside-option probability satisfies $\partial P_{i0}/\partial p_j=-\alpha_iP_{ij}P_{i0}$. Its derivative is nonnegative when $\alpha_i<0$, and its diversion ratio is $(\partial s_0/\partial p_j)/(-\partial s_j/\partial p_j)$. Including this destination distinguishes a gain by competing products from a consumer leaving the inside market altogether.

Worked Example: Comparing Diversion From A to B and C

Suppose a hypothetical random-coefficient market has two equally weighted consumer types. Their probabilities and price coefficients are shown below. Calculate the derivative-based diversion from A to B, from A to C, and from A to the outside option, then compare the destinations.

Type	$\alpha_i$ per dollar	$P_A$	$P_B$	$P_C$	$P_0$
1	-0.05	0.30	0.20	0.10	0.40
2	-0.10	0.20	0.10	0.40	0.30

First calculate the magnitude of A’s own derivative: \[ -\frac{\partial s_A}{\partial p_A} =\frac{(0.05)(0.30)(0.70)+(0.10)(0.20)(0.80)}{2} =\frac{0.0105+0.0160}{2} =0.01325. \] The denominator is the same for each destination. The derivative to B is $\partial s_B/\partial p_A=[(0.05)(0.30)(0.20)+(0.10)(0.20)(0.10)]/2 =(0.0030+0.0020)/2=0.00250$. Hence $D_{A\to B}=0.00250/0.01325=0.1887$.

The derivative to C is $\partial s_C/\partial p_A=[(0.05)(0.30)(0.10)+(0.10)(0.20)(0.40)]/2 =(0.0015+0.0080)/2=0.00475$. Thus, $D_{A\to C}=0.00475/0.01325=0.3585$. The outside derivative is $\partial s_0/\partial p_A=[(0.05)(0.30)(0.40)+(0.10)(0.20)(0.30)]/2 =(0.0060+0.0060)/2=0.00600$, so $D_{A\to 0}=0.00600/0.01325=0.4528$.

The ratios sum to $0.1887+0.3585+0.4528=1.0000$, up to rounding. In this example, the outside option receives the largest share of A’s marginal lost demand, while C receives more diversion than B. This comparison follows from the type-specific probability patterns and price sensitivities, not from the market shares alone.

Practical Interpretation and Common Pitfalls

Elasticities and diversion ratios answer related but distinct questions. An elasticity measures the response to a percentage price change, while a diversion ratio compares marginal share changes following a particular product’s price increase. A large cross-price elasticity does not by itself mean that the destination receives a large fraction of the source product’s lost demand: the cross elasticity also depends on the source price and destination share.

Using the wrong price in an elasticity. For $\varepsilon_{jk}$, the price is $p_k$, the price being changed. The response share in the denominator is $s_j$.
Confusing a derivative with a finite change. Derivatives and diversion ratios are local. For a substantial price change, probabilities and derivatives can change along the way.
Assuming diversion equals a share ratio in random-coefficient logit. The homogeneous-logit result $D_{j\to k}=s_k/(1-s_j)$ follows from its particular derivative formulas. It does not generally hold when probabilities and price sensitivities vary across consumers.
Ignoring the outside option. Diversion to inside products need not sum to one. When the outside option is included, all destination ratios sum to one under the stated model and a negative price coefficient for each consumer.
Interpreting positive cross derivatives as causal evidence without qualification. The derivatives describe the fitted demand model’s response to changing a price while holding other inputs fixed. A causal interpretation depends on the demand model and the identification of its price effects.

In empirical work, the probabilities and derivatives in a random-coefficient model are evaluated at estimated parameters and integrated over the consumer distribution. If integration uses simulation or numerical quadrature, the computed elasticities and diversion ratios are approximations; their accuracy depends on the quality of that numerical integration. The formulas themselves clarify what is being approximated and how each measure should be interpreted.

Key takeaway: Derive each price derivative from individual choice probabilities, average across consumers, and only then scale the derivatives into elasticities or diversion ratios. This preserves the distinction between percentage responsiveness and the allocation of marginal lost demand.

Check Your Understanding

Use the definitions and derivative formulas to answer the following questions.

In the elasticity $\varepsilon_{jk}$, which product’s price appears in the numerator, and which product’s share appears in the denominator?
For a homogeneous logit model with $\alpha<0$, explain why the cross-price derivative $\partial s_j/\partial p_k$ is positive when $j\ne k$.
Why can random-coefficient diversion from product $j$ to product $k$ differ from $s_k/(1-s_j)$?
If the diversion ratios from a product to its other inside alternatives sum to less than one, what destination should be checked?
Explain why a derivative-based diversion ratio should not automatically be interpreted as the exact share of lost demand transferred after a large price increase.