What Is the Price Decision?
A pricing problem begins by specifying what can be changed and what is being held fixed. The decision might be the premium offered to one policyholder, a common premium for a customer segment, or a vector of premiums across an entire portfolio. Those are different decision structures, even when they use the same cost and demand information.
For a single policy offer, write \(p\) for the premium under consideration. If the offered coverage is fixed, then coverage is an input to the decision problem, not a decision variable. The same is true of an expected cost estimate, the number of eligible customers, and a model describing how purchase likelihood changes with price. These inputs may affect how a price is assessed, but they are not themselves the price being selected.
A variable is not a decision variable just because it can be written with a symbol. A model may contain many changing quantities, such as purchase probability or expected policy count. Those quantities can vary as the selected price changes, while still being model outputs rather than independent choices. The key question is whether the decision maker selects the quantity directly or whether the model determines it from the selected price and fixed inputs.
Define the Scope Before Writing the Model
The unit at which price is set determines the number and meaning of the decision variables. If each policy receives its own price, a policy-level model may have one price variable per policy. If all customers in a segment receive one common price, the segment has one price variable. If several segments can be priced separately, the decision is a vector, with one component for each segment.
| Pricing scope | Typical decision variable | What is held common? |
|---|---|---|
| One policy | \(p\), the price offered to that policy | The decision concerns one specified risk and coverage arrangement |
| One segment | \(p_s\), the price for segment \(s\) | All eligible customers in the segment face the same price |
| Multiple segments | \((p_1,\ldots,p_S)\), a price vector | Customers within each segment share that segment's price; segments may differ |
| Uniform portfolio price | \(p\), one common price | All included policies face the same price, even if their characteristics differ |
These formulations are not interchangeable. Replacing a vector of segment prices with one common price removes the ability to make segment-specific choices. Conversely, assigning a separate price to every policy creates more decision variables than a common segment price. The intended pricing authority and restrictions determine which formulation is appropriate.
Inputs, Outputs, and Constraints
A useful setup explicitly separates the decision from the information used to evaluate it. For a hypothetical segment \(s\), let \(p_s\) be the selected price, \(N_s\) the number of eligible customers, and \(q_s(p_s)\) the modeled probability that an eligible customer purchases at that price. Coverage terms and the expected cost under those terms can also be treated as fixed inputs. In this notation, the price is chosen; the purchase probability is supplied by the demand-response model once a price is specified.
If each eligible customer either purchases once or does not purchase, the expected number of purchases in the segment is \(N_s q_s(p_s)\), provided each customer has the stated purchase probability. This is an expected count, not a guaranteed count and not a second decision variable. No independence assumption is needed to calculate this expectation; independence would matter for some questions about the distribution of the total count, but not for its mean.
The function \(q_s(\cdot)\) is meaningful only under its stated assumptions and domain. It may be estimated for a particular product, customer population, and period; it should not be treated as a universal relationship. If a proposed price lies outside the range for which the function is specified, the model does not automatically provide a valid response there. Similarly, a probability model must produce values between zero and one wherever it is used.
Constraints define which prices are allowed. For example, an explicitly stated price floor and ceiling would restrict the decision to a closed interval. A rule requiring one common price across the segment would be a structural restriction on the model, not a statement about cost or demand. A regulatory requirement could also restrict the feasible prices. A constraint does not become a fixed input merely because it limits the decision: it is part of the decision problem's feasible set.
Worked Example: One Policy and a Fixed Coverage Choice
Worked Example: One Policy and a Fixed Coverage Choice
A hypothetical insurer is considering an offer to one small-business policyholder. Coverage is fixed at a stated package, and the expected cost under that package is given as $510. The demand model for this policyholder is \(q(p)=0.90-0.001(p-500)\), for prices from $500 to $800 inclusive. The allowed price range is $540 to $760. Identify the decision variable and fixed inputs, then evaluate the model at an offered price of $650.
Step 1: Identify what is selected. The decision variable is the offered premium \(p\). The feasible set is \(540\leq p\leq760\), in dollars. Coverage and the expected cost of $510 are held fixed; neither is being selected in this problem.
Step 2: Check the proposed price. The proposed \(p=\$650\) satisfies the restriction because \(540\leq650\leq760\). It is therefore feasible.
Step 3: Calculate purchase probability. Substitute the proposed price into the stated demand function: \(q(650)=0.90-0.001(650-500)=0.90-0.001(150)=0.90-0.15=0.75\).
The model assigns a purchase probability of 0.7500 at $650. The model output is a probability; the insurer's decision variable remains the price \(p\). This calculation does not select a best price or change coverage. It simply evaluates the stated response model at one feasible decision.
Method and trap avoided: Define the decision from the controllable quantity, then substitute it into the response model. A common mistake is to call the $510 expected cost the decision variable because it is an important pricing input. It is fixed by the problem statement, while the premium is the quantity being chosen.
Worked Example: One Common Price for a Segment
Worked Example: One Common Price for a Segment
A hypothetical insurer has 500 eligible customers in a defined equipment-insurance segment. All receive the same coverage package and the same offered price \(p\). For this exercise, the expected cost per purchased policy is fixed at $430, and the specified purchase probability is \(q(p)=0.88-0.001(p-400)\) for prices between $400 and $700. The feasible price interval is $450 to $650. At \(p=\$520\), identify the decision and inputs, and calculate the purchase probability and expected number of purchases.
Step 1: State the decision structure. There is one segment-level decision variable, \(p\), because all 500 eligible customers face a common price. The feasible set is \(450\leq p\leq650\). The customer count \(N=500\), coverage, expected cost of $430, and function \(q(\cdot)\) are fixed inputs.
Step 2: Verify feasibility. Since \(450\leq520\leq650\), the proposed price is allowed.
Step 3: Evaluate demand response. \(q(520)=0.88-0.001(520-400)=0.88-0.001(120)=0.88-0.12=0.76\). Thus the probability for an eligible customer is 0.7600.
Step 4: Calculate expected purchases. Multiplying the 500 eligible customers by the common purchase probability gives \(500(0.76)=380\). The expected number of purchases is 380 policies.
The result is an expectation, not a promise that exactly 380 customers will purchase. The expected cost is not used to calculate this number because the stated demand model and customer count are sufficient. The example describes the consequences of one candidate price; it does not determine which price should be selected.
Method and trap avoided: A common segment price produces one decision variable, even though it affects many customers. Do not describe the 500 individual purchase outcomes as 500 separately selected prices when the problem imposes one shared price.
Why Expected Counts Follow From Purchase Probabilities
The expected-count calculation is justified by indicator variables. For each eligible customer \(j\), define \(I_j=1\) if that customer purchases and \(I_j=0\) otherwise. If each customer has purchase probability \(q(p)\), then \(E[I_j]=q(p)\). The total number of purchases is \(K=\sum_{j=1}^{N} I_j\). By linearity of expectation,
The equality does not require the \(I_j\) values to be independent. It does require the stated probability to apply to each of the \(N\) customers. If purchase probabilities differ by customer, then the corresponding expression is \(E[K]=\sum_{j=1}^{N}q_j(p)\), where \(q_j(p)\) is the specified probability for customer \(j\). In either case, the expected count is a consequence of the price decision and demand inputs, not a separately chosen price.
Worked Example: Separate Prices Across Two Segments
Worked Example: Separate Prices Across Two Segments
A hypothetical insurer considers two customer segments with different stated demand responses. Segment A has 300 eligible customers and \(q_A(p_A)=0.90-0.001(p_A-500)\), with an allowed price range of $500 to $700. Segment B has 400 eligible customers and \(q_B(p_B)=0.75-0.0005(p_B-700)\), with an allowed range of $700 to $900. Coverage and expected costs are fixed within each segment. The candidate prices are \(p_A=\$560\) and \(p_B=\$760\). Specify the decision vector, check feasibility, and calculate the expected purchase count in each segment and in the combined portfolio.
Step 1: Specify the decision vector. Since the segments may receive different prices, the decision is \((p_A,p_B)\), not one common portfolio price. The fixed exposure counts are \(N_A=300\) and \(N_B=400\). The coverage, costs, and demand functions are inputs.
Step 2: Check each price. For A, \(500\leq560\leq700\). For B, \(700\leq760\leq900\). Both candidate prices are feasible.
Step 3: Calculate segment A's purchase probability and expected count. \(q_A(560)=0.90-0.001(560-500)=0.90-0.06=0.84\). Therefore \(E[K_A]=300(0.84)=252\) expected purchases.
Step 4: Calculate segment B's purchase probability and expected count. \(q_B(760)=0.75-0.0005(760-700)=0.75-0.0005(60)=0.75-0.03=0.72\). Therefore \(E[K_B]=400(0.72)=288\) expected purchases.
Step 5: Combine the expected counts. By adding the two segment expectations, the combined expected number is \(252+288=540\) policies. This addition does not assert that the realized portfolio count will equal 540.
The price decision has two components because segment-specific prices are permitted. If the problem instead required one uniform price, the decision would have one component and the stated candidate pair would violate that structural requirement. The given costs remain fixed inputs in either formulation.
Method and trap avoided: Match the dimension of the decision vector to the pricing authority. A common error is to report one price for the portfolio even though the problem allows separate segment prices, or to combine the segment counts before evaluating their distinct demand functions.
Common Setup Errors
- Treating every model quantity as a choice. Purchase probability and expected purchase count can change with price, but in a specified demand model they are outputs, not independently selected decisions.
- Changing a fixed product term without being asked. If coverage is fixed, the decision is the price for that coverage. Changing the coverage would define a different problem.
- Ignoring the pricing unit. One common segment price is one variable; separate prices for \(S\) segments form an \(S\)-component vector.
- Using a price outside the model's domain. A stated demand function or feasible-price interval may not support extrapolation beyond its defined range.
- Confusing expected and actual counts. \(Nq(p)\) is the expected count under the stated model, not a guaranteed number of purchases.
- Assuming that an important input is a decision. A cost estimate can matter greatly while remaining fixed. The decision-variable test is whether the problem permits the decision maker to select that quantity.
Check Your Understanding
For each problem, identify the decision structure and calculate the requested model output. Treat all stated coverage and cost information as fixed unless the question explicitly says otherwise.
- A hypothetical individual policy has a specified purchase probability \(q(p)=0.82-0.0004(p-600)\), valid from $600 to $900. The feasible price range is $650 to $850. At a proposed price of $700, identify the decision variable and calculate the purchase probability. (Answer: Decision variable \(p\); feasible; \(q(700)=0.7800\).)
- A segment has 250 eligible customers, one common price \(p\), and \(q(p)=0.95-0.001(p-300)\). At \(p=\$420\), calculate the expected number of purchases. (Answer: \(q(420)=0.8300\); expected purchases \(250(0.83)=207.5\).)
- Two segments have separate prices. Segment X has 200 eligible customers and \(q_X(p_X)=0.80-0.001(p_X-500)\); segment Y has 150 eligible customers and \(q_Y(p_Y)=0.90-0.002(p_Y-400)\). At \(p_X=\$550\) and \(p_Y=\$450\), calculate each expected count and their total. (Answer: X: \(q_X=0.7500\), 150 expected; Y: \(q_Y=0.8000\), 120 expected; total 270 expected purchases.)
- A portfolio has 100 eligible customers and requires one uniform price. The stated demand function is \(q(p)=0.90-0.001(p-500)\), and the feasible interval is $500 to $700. A proposed plan assigns $600 to one segment and $650 to another. Is it feasible under the stated pricing rule? If the required uniform price is instead set at $600, calculate the expected portfolio count. (Answer: The two-price plan is infeasible; at the uniform $600 price, \(q=0.8000\) and expected purchases are 80.)
- A segment has 80 eligible customers with customer-specific purchase probabilities of 0.70 for 30 customers and 0.50 for the other 50 customers at a specified candidate price. Calculate the expected total purchases and state whether independence is needed for this expectation. (Answer: \(30(0.70)+50(0.50)=21+25=46\) expected purchases; independence is not needed.)