Customer-Level And Uniform Pricing Decisions

Two Different Pricing Decisions

A pricing model can allow a distinct price for each customer, or require the same price for everyone in a portfolio. These are different decision problems, even when they use the same costs and purchase-response assumptions. Separate prices can respond to differences in a customer’s expected cost and demand response. A uniform price must balance those differences across the portfolio.

The comparison below concerns expected contribution: expected premium less expected cost, weighted by the probability that a customer buys. It does not decide whether customer-specific pricing is permissible or practical. It isolates the mathematical consequence of allowing one price per customer rather than one price for the group.

Definition: For customer $i$, let $p_i$ be the quoted annual premium, $c_i$ the expected cost per policy if purchased, and $q_i(p_i)$ the probability of purchase at that price. The expected contribution from that customer is $(p_i-c_i)q_i(p_i)$. For a group of $n_i$ comparable customers, the expected contribution is $n_i(p_i-c_i)q_i(p_i)$. A customer-level decision chooses prices separately; a uniform-price decision chooses one $p$ for all customers in the portfolio.

Here, $c_i$ can include expected claim cost and expenses that are treated as fixed per policy for this calculation. The model assumes that a customer who buys generates expected cost $c_i$, while a customer who does not buy generates neither premium nor policy cost. It also assumes that the stated purchase probabilities describe the relevant policy term and that costs and purchase probabilities are supplied inputs.

For a uniform price, the portfolio objective is the sum of the customer-group contributions, not the contribution at an average cost or an average demand response:

\Pi(p)=\sum_{i=1}^{k} n_i(p-c_i)q_i(p).

The price variable is common across groups in this expression. The group sizes, costs, and demand responses can still differ. For separate prices, each term can instead be optimized independently, provided there are no interactions between customers and no shared restrictions on the prices.

Linear Purchase Probability and the Interior Optimum

A useful supplied demand model is a linear purchase probability, $q_i(p)=a_i-b_i p$, where $a_i$ and $b_i$ are given and $b_i>0$. The model is used only over a price range where the resulting probability is between zero and one. Assume for now that the optimum is interior to that range.

Result: Under $q_i(p)=a_i-b_i p$, the expected contribution per customer is $(p-c_i)(a_i-b_i p)$. Its unique interior maximum is at $p_i^*=(a_i+b_i c_i)/(2b_i)$, if this price lies in the valid probability range.

To justify the result, expand the contribution and differentiate with respect to price:

(p-c_i)(a_i-b_i p) =a_i p-a_i c_i-b_i p^2+b_i c_i p, $$ $$ \frac{d}{dp}\big[(p-c_i)(a_i-b_i p)\big] =a_i+b_i c_i-2b_i p.

At an interior maximum the derivative is zero, so $2b_i p=a_i+b_i c_i$, giving the stated price. The second derivative is $-2b_i<0$, so the contribution is strictly concave and this stationary point is the unique maximum within the model’s valid range. Multiplying by a positive group size $n_i$ does not change the price that maximizes that group’s contribution.

For one uniform price, differentiate the summed objective instead. Under the same linear model for each group, the first-order condition is:

\Pi'(p)=\sum_{i=1}^{k} n_i(a_i+b_i c_i-2b_i p)=0, $$ $$ p_{\mathrm{uniform}}^* =\frac{\sum_{i=1}^{k}n_i(a_i+b_i c_i)} {2\sum_{i=1}^{k}n_i b_i}.

This formula also requires an interior optimum and valid purchase probabilities for all included groups at the selected price. The second derivative is $-2\sum_i n_i b_i<0$, so the portfolio objective is strictly concave when group sizes and slopes are positive. If the stationary price gives an invalid probability for any group, the linear model cannot justify using that result as a feasible price; the appropriate endpoint or a different demand specification must be considered.

Separate Prices Versus One Portfolio Price

Worked Example: A Uniform Price for Two Customer Groups

A hypothetical insurer has 80 customers in group A and 20 in group B. For a one-year policy, expected costs are $c_A=\$500$ and $c_B=\$900$. Their supplied purchase probabilities are $q_A(p)=0.9-0.0005p$ and $q_B(p)=1.3-0.0008p$, with $p$ in dollars. Compare the separately optimized prices with the best uniform price and calculate expected contribution under each approach.

Method. Optimize the group contributions separately first, then sum their derivatives to optimize the portfolio under one common price. The calculation assumes no cross-customer interactions and uses the linear probability models only where their probabilities are valid.

Separate prices. For group A, $a_A+b_Ac_A=0.9+0.0005(500)=1.15$, and $2b_A=0.001$. Thus $p_A^*=1.15/0.001=\$1{,}150$. Its purchase probability is $0.9-0.0005(1{,}150)=0.325$. Expected contribution is $(1{,}150-500)(0.325)(80)=650(0.325)(80)=\$16{,}900.00$.

For group B, $a_B+b_Bc_B=1.3+0.0008(900)=2.02$, and $2b_B=0.0016$. Thus $p_B^*=2.02/0.0016=\$1{,}262.50$. Its purchase probability is $1.3-0.0008(1{,}262.50)=0.29$. Expected contribution is $(1{,}262.50-900)(0.29)(20)=362.50(0.29)(20)=\$2{,}102.50$. Total expected contribution with separate prices is $\$16{,}900.00+\$2{,}102.50=\$19{,}002.50$.

Uniform price. The derivative of the portfolio objective is $80(1.15-0.001p)+20(2.02-0.0016p)$. Expanding gives $132.4-0.112p$. Setting this derivative to zero gives $p=132.4/0.112=\$1{,}182.142857\ldots$. The second derivative is $-0.112<0$, confirming a maximum.

At this price, $q_A=0.9-0.0005(1{,}182.142857)=0.3089286$, and $q_B=1.3-0.0008(1{,}182.142857)=0.3542857$. Group A contributes $80(1{,}182.142857-500)(0.3089286)=\$16{,}858.67$. Group B contributes $20(1{,}182.142857-900)(0.3542857)=\$1{,}999.18$. The uniform-price total is $\$18{,}857.86$, rounding money to the nearest cent at the final step.

Both probabilities are between zero and one at the selected prices. Separate pricing yields $\$19{,}002.50-\$18{,}857.86=\$144.64$ more expected contribution in this unconstrained mathematical comparison. The uniform price is not the simple customer-count-weighted average of the two separate optima: that average is $0.8(1{,}150)+0.2(1{,}262.50)=\$1{,}172.50$. Different slopes affect how strongly each group’s objective changes with price.

How Group Size and Demand Response Shape the Uniform Price

The uniform-price formula can be read as a weighted average of the group-specific interior optima, but the weights are proportional to $n_i b_i$, not just to group size. In fact, since $a_i+b_ic_i=2b_i p_i^*$, the portfolio formula can be written as:

p_{\mathrm{uniform}}^* =\frac{\sum_i n_i b_i p_i^*}{\sum_i n_i b_i}.

This identity holds for the same interior linear-demand setup. A larger group receives more weight, and, all else equal, a group with a steeper purchase-probability slope also receives more weight in the uniform-price first-order condition. This is a property of the specified objective and linear demand model; it is not a universal rule for all demand curves.

Worked Example: Unequal Group Sizes with the Same Demand Slope

A hypothetical portfolio has 60 customers with expected cost $\$400$ and purchase probability $q_A(p)=0.8-0.0005p$. It has another 40 customers with expected cost $\$700$ and purchase probability $q_B(p)=1.1-0.0005p$. Find the separate optimum for each group and the portfolio’s uniform optimum, then compare expected contribution.

Separate-price calculation. Group A’s optimum is $(0.8+0.0005(400))/(0.001)=1.0/0.001=\$1{,}000$. Its probability is $0.8-0.0005(1{,}000)=0.30$, so its expected contribution is $60(1{,}000-400)(0.30)=\$10{,}800.00$.

Group B’s optimum is $(1.1+0.0005(700))/(0.001)=1.45/0.001=\$1{,}450$. Its probability is $1.1-0.0005(1{,}450)=0.375$, so its contribution is $40(1{,}450-700)(0.375)=\$11{,}250.00$. Total expected contribution with separate prices is $\$22{,}050.00$.

Uniform-price calculation. The derivative is $60(1.0-0.001p)+40(1.45-0.001p)=118-0.1p$. Its zero is $p=\$1{,}180$, and the second derivative, $-0.1$, is negative. The probabilities are $q_A=0.8-0.59=0.21$ and $q_B=1.1-0.59=0.51$.

Group A contributes $60(1{,}180-400)(0.21)=60(780)(0.21)=\$9{,}828.00$. Group B contributes $40(1{,}180-700)(0.51)=40(480)(0.51)=\$9{,}792.00$. The uniform total is $\$19{,}620.00$, which is $\$2{,}430.00$ below the separate-price total. The uniform price is also the group-size-weighted average of the separate optima here: $0.6(1{,}000)+0.4(1{,}450)=\$1{,}180$, because both groups have the same demand slope.

Worked Example: Demand Slopes Change the Weighting

A hypothetical insurer has 20 customers in group A, each with expected cost $\$400$ and purchase probability $q_A(p)=0.8-0.0004p$. Group B has 80 customers, expected cost $\$600$, and purchase probability $q_B(p)=1.0-0.0008p$. Calculate the separately optimized contribution and the contribution at the uniform optimum.

Separate prices. Group A’s optimum is $(0.8+0.0004(400))/(0.0008)=0.96/0.0008=\$1{,}200$. Its probability is $0.8-0.0004(1{,}200)=0.32$, giving $20(1{,}200-400)(0.32)=\$5{,}120.00$.

Group B’s optimum is $(1.0+0.0008(600))/(0.0016)=1.48/0.0016=\$925$. Its probability is $1.0-0.0008(925)=0.26$, giving $80(925-600)(0.26)=\$6{,}760.00$. The total with separate prices is $\$11{,}880.00$.

Uniform price. The derivative is $20(0.96-0.0008p)+80(1.48-0.0016p)$, which expands to $137.6-0.144p$. Therefore the uniform optimum is $137.6/0.144=\$955.555556\ldots$. At that price, $q_A=0.8-0.0004(955.555556)=0.4177778$, and $q_B=1.0-0.0008(955.555556)=0.2355556$.

Group A’s expected contribution is $20(955.555556-400)(0.4177778)=\$4{,}641.98$. Group B’s is $80(955.555556-600)(0.2355556)=\$6{,}700.25$. The uniform total is $\$11{,}342.22$, rounding to cents only at the end. Both probabilities are valid, and the uniform-price objective is strictly concave because its second derivative is $-0.144$. Although group B is larger, its slope also differs; the uniform optimum is therefore not obtained by weighting the separate prices by group size alone.

Interpretation and a Common Exam Trap

If every customer can be priced separately, and each customer’s contribution is independent of the prices offered to others, the portfolio objective is a sum of independently maximized terms. Choosing each group’s own maximizing price therefore cannot produce less expected contribution than forcing all groups to use a single price. A uniform price is a restricted decision: every price available to the group must be the same. The difference in expected contribution measures the cost of that restriction under the supplied model; it is not, by itself, a recommendation about whether different prices should be offered.

A frequent error is to calculate separate optimum prices and then take their arithmetic or group-size-weighted average as the uniform optimum. That generally fails when slopes differ. In the third example, the incorrect group-size average would be $0.2(\$1{,}200)+0.8(\$925)=\$980$. The correct uniform price is $\$955.56$, obtained by aggregating the derivatives, or by weighting the separate optima by $n_i b_i$.

The correct procedure is to build the portfolio objective with the common price, include every group’s size, cost, and demand response, and then optimize that combined objective. Do not average costs first or maximize a representative customer’s contribution unless the problem’s data establish that this aggregation is equivalent.

State the decision.
Decide whether prices are independent by customer group or whether all groups share one price.

Write the relevant objective.
For separate prices, optimize each $n_i(p_i-c_i)q_i(p_i)$. For a uniform price, sum all group contributions using the same $p$.

Check the solution.
For an interior linear-demand optimum, confirm the second derivative is negative and every purchase probability at the resulting price is between zero and one.

Compare like with like.
Evaluate expected portfolio contribution under each pricing decision using the same customer counts, costs, and demand assumptions.

Key takeaway: Separate prices optimize customer groups individually; a uniform price optimizes the sum of their contributions at one shared price. For linear purchase probabilities, the uniform optimum weights group-specific optima by group size times demand slope, when the interior solution is valid.

Check Your Understanding

For each problem, use annual premiums and expected costs in dollars. Assume the supplied linear purchase probabilities are valid at the calculated optima.

A single customer has expected cost $\$500$ and purchase probability $q(p)=0.95-0.0005p$. Find the contribution-maximizing price, its purchase probability, and expected contribution. (Answer: price $\$1{,}200$; probability $0.3500$; expected contribution $\$245.00$.)
A group of 50 customers has expected cost $\$400$ per policy and purchase probability $q(p)=0.8-0.0004p$. If each customer can be offered the group’s individually optimal price, find the price and total expected contribution. (Answer: price $\$1{,}200$; total expected contribution $\$12{,}800.00$.)
A portfolio has 30 customers in group A, with cost $\$500$ and $q_A(p)=0.9-0.0004p$, and 70 in group B, with cost $\$700$ and $q_B(p)=1.2-0.0008p$. Find the uniform-price optimum. (Answer: $\$1{,}148.53$, rounded to cents.)
A portfolio has 25 customers with cost $\$300$ and $q_A(p)=0.75-0.0005p$, and 75 customers with cost $\$500$ and $q_B(p)=1.05-0.0005p$. Find the separate-price total expected contribution and the uniform-price optimum and its total expected contribution. (Answer: separate-price total $\$28{,}500.00$; uniform price $\$1{,}200$; uniform-price total $\$27{,}000.00$.)

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