Elasticities And Diversion In Nested Logit

Why Elasticities and Diversion Matter

A price change can alter the probability that a consumer chooses the priced alternative and the probabilities of choosing every other option. An elasticity summarizes the size of one such response in percentage terms. A diversion ratio instead asks where the probability lost by one alternative goes, locally, when its price rises. These summaries are useful for describing substitution, but their values depend on the choice probabilities and on the nested-logit structure.

The key distinction is whether a destination alternative is in the same nest as the price-changing alternative. Nested logit allows a different substitution pattern within a nest than across nests. The formulas below make that distinction explicit. They are local responses to an infinitesimal price change, not predictions for a large price change without recalculating the probabilities.

Model and Notation

Consider a finite choice set partitioned into nests. Alternative $j$ belongs to nest $m$. Its systematic utility is $V_j$, and its choice probability conditional on choosing nest $m$ is $q_{j\mid m}$. The probability of choosing nest $m$ is $P_m$, so the unconditional probability of choosing alternative $j$ is $P_j=P_m q_{j\mid m}$. For compactness, write $q_j=q_{j\mid m}$ when the relevant nest is clear.

For a standard nested-logit probability specification, with nest scale parameter $\lambda_m>0$, define $D_m=\sum_{r\in m}\exp(V_r/\lambda_m)$ and inclusive value $I_m=\lambda_m\log D_m$. The conditional and nest probabilities are $q_{j\mid m}=\exp(V_j/\lambda_m)/D_m$ and $P_m=\exp(I_m)/\sum_l\exp(I_l)$. A price-invariant outside option, if present, is included in this choice set and assigned to a nest. The standard random-utility-consistent restriction is $0<\lambda_m\leq 1$; the derivative calculations require positive $\lambda_m$.

Definition: Suppose a price change for alternative $k$ affects its utility through $V_k=\widetilde V_k-\beta_k p_k$, where $p_k>0$, $\beta_k>0$, and all other systematic utilities are held fixed. The price elasticity of $P_j$ with respect to $p_k$ is $\varepsilon_{jk}=(p_k/P_j)(\partial P_j/\partial p_k)$. It is an own-price elasticity when $j=k$, and a cross-price elasticity when $j\ne k$. A local diversion ratio from $k$ to $j$, for $j\ne k$, is $D_{k\to j}=(\partial P_j/\partial p_k)/(-\partial P_k/\partial p_k)$, provided the denominator is positive.

An elasticity is dimensionless: it gives the approximate percentage change in $P_j$ for a one-percent increase in $p_k$, holding other prices and model components fixed. A diversion ratio is also dimensionless. It describes the fraction of the marginal probability loss for $k$ that appears as a marginal probability gain for $j$. Both interpretations are local.

Deriving the Probability Responses

Let alternative $k$ belong to nest $n$. A price increase lowers $V_k$ at rate $\partial V_k/\partial p_k=-\beta_k$. First, within nest $n$, differentiating the conditional probability gives $\partial q_{j\mid n}/\partial p_k=-(\beta_k/\lambda_n)q_{j\mid n}(\mathbf{1}\{j=k\}-q_k)$. For $j$ outside nest $n$, its conditional probability in its own nest has no direct response to $p_k$.

The inclusive value of nest $n$ changes at rate $\partial I_n/\partial p_k=-\beta_k q_k$. This follows by differentiating $\lambda_n\log D_n$: the derivative of $D_n$ is $(1/\lambda_n)\exp(V_k/\lambda_n)(-\beta_k)$, and $\exp(V_k/\lambda_n)/D_n=q_k$. Differentiating the upper-level nest probability therefore gives $\partial P_m/\partial p_k=-\beta_k P_m q_k(\mathbf{1}\{m=n\}-P_n)$. Combining the conditional and nest responses yields the following derivatives.

\frac{\partial P_j}{\partial p_k}= \begin{cases} -\beta_k P_k\left[\dfrac{1-q_k}{\lambda_n}+q_k(1-P_n)\right], & j=k,\\[6pt] \beta_k P_j q_k\left[\dfrac{1}{\lambda_n}-1+P_n\right], & j\ne k,\ j\in n,\\[6pt] \beta_k P_j q_k P_n, & j\notin n. \end{cases}

For example, when $j\in n$, the product rule gives $\partial(P_n q_j)/\partial p_k=q_j(\partial P_n/\partial p_k)+P_n(\partial q_j/\partial p_k)$. For $j\ne k$, the first term is $\beta_k P_n q_j q_k(1-P_n)$, and the second is $\beta_k P_n q_j q_k/\lambda_n$. Since $P_j=P_nq_j$, their sum is the same-nest expression above. When $j\notin n$, only the nest-probability response applies, giving $\beta_kP_jq_kP_n$.

Multiplying each derivative by $p_k/P_j$ gives the elasticities:

\varepsilon_{kk}=-\beta_kp_k\left[\frac{1-q_k}{\lambda_n}+q_k(1-P_n)\right], \qquad \varepsilon_{jk}= \begin{cases} \beta_kp_kq_k\left[\dfrac{1}{\lambda_n}-1+P_n\right], & j\ne k,\ j\in n,\\[6pt] \beta_kp_kq_kP_n, & j\notin n. \end{cases}

With $0<\lambda_n\leq1$, positive prices, and $\beta_k>0$, the own-price elasticity is nonpositive. It is strictly negative if $q_k<1$ or $P_n<1$; if the choice structure has no competing probability mass, the bracket can be zero. The cross-price elasticities are nonnegative under these assumptions. They are positive when their relevant probability terms are positive. These signs describe the model’s local response, not evidence that a price change causes an observed market outcome.

Worked Example: Elasticities Within and Across Nests

Suppose alternative 1 belongs to nest A, where $\lambda_A=0.5$, $P_A=0.60$, and $q_1=0.40$. Its price is $p_1=\$2.00$, and its utility price coefficient is $\beta_1=0.80$ per dollar. Alternative 2 is also in nest A, with $q_2=0.60$. Alternative 3 is in another nest with probability $P_3=0.25$; a price-invariant outside option has probability $0.15$. These probabilities are exhaustive: $P_1=0.60(0.40)=0.24$, $P_2=0.60(0.60)=0.36$, and $0.24+0.36+0.25+0.15=1.00$.

The common price multiplier is $\beta_1p_1=0.80(2.00)=1.60$. The own-price elasticity is $\varepsilon_{11}=-1.60[(1-0.40)/0.5+0.40(1-0.60)]=-1.60[1.20+0.16]=-2.176$. Locally, a one-percent increase in alternative 1’s price corresponds to an approximately $2.176\%$ decrease in its choice probability.

For alternative 2, the same-nest cross-price elasticity is $\varepsilon_{21}=1.60(0.40)[1/0.5-1+0.60]=0.64(1.60)=1.024$. For alternative 3, which is in a different nest, it is $\varepsilon_{31}=1.60(0.40)(0.60)=0.384$. Thus both other alternatives gain probability locally when alternative 1’s price rises, and the model gives a larger percentage response for alternative 2 in this example.

The corresponding derivatives, in probability units per dollar, are $\partial P_1/\partial p_1=(0.24/2.00)(-2.176)=-0.26112$, $\partial P_2/\partial p_1=(0.36/2.00)(1.024)=0.18432$, and $\partial P_3/\partial p_1=(0.25/2.00)(0.384)=0.04800$. For the outside option, its cross-price elasticity is $1.60(0.40)(0.60)=0.384$, so its derivative is $(0.15/2.00)(0.384)=0.02880$. The changes sum to zero: $-0.26112+0.18432+0.04800+0.02880=0$, as required because the probabilities sum to one.

From Derivatives to Diversion Ratios

For the price increase considered here, the own-probability loss per dollar is $\beta_kP_k A_k$, where $A_k=(1-q_k)/\lambda_n+q_k(1-P_n)$. If $A_k>0$, the local diversion ratios follow by dividing each cross-probability derivative by this loss. For an alternative $j\ne k$ in the same nest and an alternative $j$ in a different nest, respectively, the results simplify to

D_{k\to j}= \begin{cases} \dfrac{q_j\left(1/\lambda_n-1+P_n\right)}{A_k}, & j\ne k,\ j\in n,\\[8pt] \dfrac{P_j}{A_k}, & j\notin n. \end{cases}

The simplification uses $P_k=P_nq_k$ and, for an alternative in nest $n$, $P_j=P_nq_j$. The coefficient $\beta_k$ cancels: it scales the probability loss and gains together, while diversion describes their relative allocation. This does not mean that $\beta_k$ is irrelevant to the elasticities or to the magnitude of the derivatives.

Worked Example: Interpreting Local Diversion

Use the probabilities and parameters from the preceding example. The denominator is $A_1=(1-0.40)/0.5+0.40(1-0.60)=1.36$. Diversion from alternative 1 to alternative 2 is $D_{1\to2}=0.60(1/0.5-1+0.60)/1.36=0.96/1.36\approx0.7059$. Equivalently, using derivatives, $0.18432/0.26112\approx0.7059$.

Diversion to alternative 3 is $D_{1\to3}=0.25/1.36\approx0.1838$, and diversion to the outside option is $D_{1\to\mathrm{out}}=0.15/1.36\approx0.1103$. The three ratios sum to approximately $1.000$. In this local calculation, about $70.59\%$ of alternative 1’s marginal probability loss goes to alternative 2, $18.38\%$ goes to alternative 3, and $11.03\%$ goes to the outside option.

Comparing Substitution Across Nest Structures

The within-nest formula includes the factor $1/\lambda_n-1+P_n$; the across-nest formula depends on the destination probability $P_j$. A smaller $\lambda_n$, when it satisfies the model’s restrictions, can raise the within-nest response relative to what the same-nest formula would imply at a larger scale parameter. But it does not guarantee that every same-nest alternative receives more diversion than every alternative in another nest. The destination’s probability and the full denominator $A_k$ also matter.

Worked Example: Comparing Diversion Under Different Nest Probabilities

Suppose an alternative $k$ has $q_k=0.30$ in a nest with $P_n=0.50$ and $\lambda_n=0.60$. There is one same-nest competitor $j$, with $q_j=0.70$, and a different-nest destination $r$, with $P_r=0.20$. Other alternatives account for the remaining probability. First calculate $A_k=(1-0.30)/0.60+0.30(1-0.50)=1.1667+0.1500=1.3167$.

The same-nest diversion ratio is $D_{k\to j}=0.70(1/0.60-1+0.50)/1.3167 =0.70(1.1667)/1.3167\approx0.6203$. The different-nest diversion ratio is $D_{k\to r}=0.20/1.3167\approx0.1519$. Thus this particular same-nest competitor receives a larger share of the local probability loss. The remaining probability loss is diverted to other alternatives in the choice set, including any outside option; the two calculated ratios alone need not sum to one.

Checks, Limits, and a Common Mistake

When all alternatives, including an outside option if present, are included and probabilities sum to one, differentiating that identity gives $\sum_j\partial P_j/\partial p_k=0$. If the own derivative is negative, the positive derivatives for the other alternatives therefore sum to its absolute value, and their diversion ratios sum to one. A failure of this check can indicate that an option was omitted, a derivative was calculated incorrectly, or the assumed model treats some probability mass outside the listed choice set.

A common mistake is to call $(p_k/P_j)(\partial P_j/\partial p_k)$ an own-price elasticity for every $j$. It is an own-price elasticity only for $j=k$; for $j\ne k$, it is a cross-price elasticity. Another mistake is to interpret a diversion ratio as the probability that a particular consumer switches to $j$, or as the response to a large price increase. It is a ratio of aggregate marginal probability changes in the model, evaluated at specified prices and probabilities.

Key takeaway: Nested-logit price responses are determined by both conditional choice within a nest and the probability of choosing that nest. Calculate the own response first, then compare each destination’s cross response with the resulting probability loss to obtain local diversion.

Check Your Understanding

Use the definitions and formulas above to answer these questions.

In the elasticity notation $\varepsilon_{jk}$, when is the elasticity own-price, and when is it cross-price?
For a price increase in alternative $k$, what model probabilities and parameters enter the own-price elasticity?
Why can the own-price elasticity be zero in a degenerate choice set even when $p_k>0$ and $\beta_k>0$?
What is the distinction between a same-nest and a different-nest cross-price elasticity in the nested-logit formulas?
Under what conditions do the diversion ratios from $k$ to all other options sum to one?

Elasticities And Diversion In Nested Logit

What You'll Learn

Why Elasticities and Diversion Matter

Model and Notation

Deriving the Probability Responses

Worked Example: Elasticities Within and Across Nests

Worked Example: Elasticities Within and Across Nests

From Derivatives to Diversion Ratios

Worked Example: Interpreting Local Diversion

Comparing Substitution Across Nest Structures

Worked Example: Comparing Diversion Under Different Nest Probabilities

Checks, Limits, and a Common Mistake

Check Your Understanding

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