Dynamic Pricing with Risk Analysis and Target Revenues Baichun Xiao Long Island University

Outline  Risk-neutral- a basic assumption of most RM models; its inability to deal with short-term behavior;  Literature review;  What affects short-term risk?  A RM model with a target revenue and penalty function;  Concluding remarks.

Risk Neutral Assumption of RM Models  Decision makers are risk neutral; i.e., all models attempt to maximize the expected revenues at the end of the disposal period;  Optimal in the long run (the law of large numbers): no single realization has the potential for severe revenue impacts on the company;  Not necessarily the best option in the short run.

Risk-Neutral May Not Apply to Short- Term Compelling reasons for concerning short- term revenues:  Financial constraints;  Uncompromised revenue goals or minimum probability of achieving these goals;  Shareholders’ requirements;  All of the above are escalated by the perishability of products.

Risk with Short-term Revenues  Short-term revenues can swing drastically from their long-term estimates because: Uncertainty of demand; Forecast errors; Speculations; Unexpected capacity changes.  A single poor performance can be very damaging and may not be compromised by the long-term average.

Example One

Practitioners’ Solutions Sacrifice expected revenues in return for higher probability of achieving a revenue goal.  Liquidations;  Clearances;  Negotiated discounts;  Favorable price for large-volume demand.

Example Two

Theoretical Framework? Current RM models are  unable to explain why a dynamic control policy leads to a steep dive of prices during liquidation periods;  unable to explain discount policies for large- volume demand.

Related Research  McGill and van Ryzin (1999): addressed importance and challenge of pricing group demand.  Bitran and Caldentey (2002): “essentially all the models that we have discussed assume that the seller is risk neutral.”  Feng and Xiao (1999): a risk-sensitive pricing model to maximize sales revenue of perishable products.

Related Research  Kleywegt and Papastavrou (1997), Slyke and Young (2000), Brumell and Walczak (2003): pricing group demand, but not from the perspective of risk.  Lim and Shanthikumar (2004): (i) a model built upon erroneous forecast parameters may perform badly and present a risk; (ii) robust dynamic pricing; (iii) equivalent to single product dynamic RM with exponential utility function without parameter uncertainty;

Related Research  Levin, McGill, and Nediak (2005): (i) motivated by inventory clearance of high-value items (automobiles, electronic equipment, appliances, etc.); (ii) permit control of the probability that total revenues fall below a minimum acceptable level; (iii) augment the expected revenue objective with a penalty term for the probability that revenues drop below a desirable level.

Related Research  Feng and Xiao (2005): (i) Maximize the risk- averse utility function instead of risk-neutral revenue; (ii) The risk-averse utility model retains monotone properties of the optimal policy; (iii) Risk-neutral models are special cases of risk-averse models; (iv) The risk- averse model explains behaviors that cannot be rationalized by the risk-neutral assumption; e.g., group discount policy.

How is Risk Measured?  Risk is normally measured by variance and standard deviation;  Variance and standard deviation are policy dependent;  Variance and standard deviation are affected by the remaining inventory and time-to-go;  Penalty function using the standard deviation is not a proper choice when the time-to-go is diminishing.

Risk is Affected by Policy

Variance vs. Remaining Inventory Variance increases with the remaining Inventory.

Variance vs. Remaining Inventory  A single-policy model ( ):

Variance vs. Remaining Inventory

General cases (inventory control)

Variance and Time Remaining

Coefficient of Variation

Risk of Selling Perishable Products  For a given policy, risk increases if the remaining inventory increases;  For a given policy, risk increases if the time- to-go diminishes;  Risk can be controlled by policy.

Alternatives for Reducing Risk  Expected revenue with a penalty function when the target revenue is not met;  Expected revenue with a penalty function when the probability of not achieving the target revenue is above a threshold;  Risk-averse expected utility function;

Distribution of Revenues

A Continuous-Time Pricing Model with Target Revenue Assumptions:  Management has a target revenue in the short run;  If the target is not met, a penalty is incurred;  The penalty is proportional to the deficit of revenue;  Management makes price decisions to maximize the expected revenue with a penalty of not meeting the target.

Notations

Objective Function  optimal expected revenue at T given the remaining inventory and realized revenue at t be n(t) and r(t), respectively;  The revenue function has three parameters t, n, and r.

Boundary Conditions when t = T Note: (i) Penalty is incurred if r < r 0 ; (ii) The penalty function is piecewise linear.

Boundary Conditions when t = T Implication: sell remaining inventory in the neighborhood of T for any price.

Boundary Conditions when n = 0 Note: V(t, 0, r) is independent of t.

Boundary Conditions when n = 0 For r 1 > r 2,

Optimality Condition  Only one price is accepted at any given time;  Optimal price is chosen from the price set P ( may not be the highest price).

Optimality Condition  If p i is the optimal solution at t with realized revenue r, then

Optimality Condition  Result: If then the optimal price in the neighborhood of T is the lowest price ;  The lowest price has the highest revenue rate.

Solve V(t, 1, r)  In the neighborhood of T, leads to

Solve V(t, 1, r)

 Let  In the left neighborhood of, the optimal price becomes  Other thresholds are similarly defined.

Solve V(t, n, r)  Assume has been obtained;  In the neighborhood of T, is the optimal price, V(t, n, r) is given by

Solve V(t, n, r)  Let  In the left neighborhood of, the optimal price becomes  Other thresholds are similarly defined.

Numerical Experiment Data:

Expected revenue with the target r 0 r0r0 V(0, M, 0)

Expected Revenue with the Target r 0 Expected Revenue r0 V(0,M,0) V(t,n,r) is a decreasing function of r 0.

Expected revenue as Function of c cV(0, M, 0) Note: r 0 =700

Expected revenue as Function of c V(t,n,r) is a decreasing and linear function of c.

Expected revenue as Function of n V(t,n,r) is increasing and concave in n for fixed t and r. (r 0 =700)

Expected Revenue as Function of r V(t, n, r) is an increasing function of r for fixed t and n (r 0 =700).

Expected Revenue as Function of t V(t,n,r) is a decreasing and concave function of t for given n and r; but V(t,n,r)-V(t,n-1,r) may not decrease in t (r 0 =700).

Property of V(t,n,r+s) -V(t,n,r)

Concluding Remarks  Decision makers are risk-averse in financial market and many other areas, revenue management should not be an exception;  The proposed pricing model handles risk with a target revenue and a penalty function;  Many properties of risk-neutral models seem to hold except the marginal expected revenue;

Concluding Remarks  More structural properties of the value function need to be uncovered;  Whether pricing policy for group demand can be dealt with by the proposed model need to be explored.