Real-Time Price Optimization

Abstract

Demand falls as price rises, so profit is a curve with a peak rather than a line to climb. This project finds that peak under a constant-elasticity demand model — 18.00 for a unit costing 8.00 — then asks the harder question: what does it cost when the elasticity is estimated from noisy sales rather than known? Twelve observations recover 1.829 against a true 1.80, and the resulting price gives up 0.03% of profit.

Prerequisites

• Python 3.8 or above
• A code editor or IDE
• Basic understanding of ML and analytics
• Required libraries: pandas, scikit-learn, matplotlib

Before you Start

Install Python and the required libraries:

Install dependencies

pip install pandas scikit-learn matplotlib

Getting Started

Create a Project

1. Create a folder named real-time-price-optimization.
2. Open the folder in your code editor or IDE.
3. Create a file named real_time_price_optimization.py.
4. Copy the code below into your file.

Write the Code

Real-Time Price Optimization

pch.viewSource

Real-Time Price Optimization

"""Real-time price optimization.

Demand falls as price rises, so profit is a curve with a peak rather than a
line to climb. The optimiser finds that peak, and the interesting part is what
happens when the elasticity it assumes is wrong -- which it always is, because
elasticity has to be estimated from noisy sales.
"""

import matplotlib.pyplot as plt
import numpy as np
from scipy.optimize import minimize_scalar


class PriceOptimizer:
    """Constant-elasticity demand: quantity = base * (price / anchor) ** -e."""

    def __init__(self, cost, base=1000.0, anchor=20.0, elasticity=1.8):
        self.cost = cost
        self.base = base
        self.anchor = anchor
        self.elasticity = elasticity

    def demand(self, price):
        return self.base * (price / self.anchor) ** -self.elasticity

    def profit(self, price):
        return (price - self.cost) * self.demand(price)

    def best_price(self):
        result = minimize_scalar(lambda p: -self.profit(p),
                                 bounds=(self.cost + 0.01, self.anchor * 5),
                                 method="bounded")
        return float(result.x)

    def estimate_elasticity(self, prices, quantities):
        """Fit the elasticity from observed sales, in log space."""
        slope, _ = np.polyfit(np.log(prices), np.log(quantities), 1)
        return float(-slope)


def observed_sales(truth, seed=0, points=12, noise=0.08):
    rng = np.random.default_rng(seed)
    prices = np.linspace(truth.cost * 1.2, truth.anchor * 2.2, points)
    quantities = truth.demand(prices) * rng.lognormal(0, noise, points)
    return prices, quantities


def main():
    truth = PriceOptimizer(cost=8.0, elasticity=1.8)
    optimal = truth.best_price()
    print("Real-Time Price Optimization")
    print(f"  unit cost               : {truth.cost:.2f}")
    print(f"  true elasticity         : {truth.elasticity:.2f}")
    print(f"  profit-maximising price : {optimal:.2f}")
    print(f"  profit at that price    : {truth.profit(optimal):,.0f}")
    print(f"  profit at cost + 50%    : {truth.profit(truth.cost * 1.5):,.0f}")

    print(f"\n  {'assumed elasticity':>19} {'chosen price':>13} "
          f"{'true profit':>13} {'lost vs best':>13}")
    assumed_rows = []
    for elasticity in (1.2, 1.5, 1.8, 2.2, 3.0):
        assumed = PriceOptimizer(cost=truth.cost, elasticity=elasticity)
        price = assumed.best_price()
        realised = truth.profit(price)
        assumed_rows.append((elasticity, price, realised))
        print(f"  {elasticity:>19.2f} {price:>13.2f} {realised:>13,.0f} "
              f"{truth.profit(optimal) - realised:>13,.0f}")

    prices, quantities = observed_sales(truth)
    estimated = truth.estimate_elasticity(prices, quantities)
    fitted = PriceOptimizer(cost=truth.cost, elasticity=estimated)
    fitted_price = fitted.best_price()
    print(f"\n  elasticity estimated from 12 noisy observations: {estimated:.3f}"
          f" (true {truth.elasticity:.2f})")
    print(f"  price it recommends: {fitted_price:.2f} against the true "
          f"optimum {optimal:.2f}")
    gap = truth.profit(optimal) - truth.profit(fitted_price)
    print(f"  profit given up by that error: {gap:,.0f} "
          f"({gap / truth.profit(optimal):.2%})")

    grid = np.linspace(truth.cost + 0.5, truth.anchor * 3, 300)
    figure, axes = plt.subplots(1, 2, figsize=(9.5, 3.6))
    axes[0].plot(grid, truth.profit(grid), linewidth=1.6)
    axes[0].axvline(optimal, linestyle="--", linewidth=1.2,
                    label=f"optimum {optimal:.2f}")
    axes[0].axvline(fitted_price, linestyle=":", linewidth=1.2,
                    label=f"from estimate {fitted_price:.2f}")
    axes[0].set_xlabel("price")
    axes[0].set_ylabel("profit")
    axes[0].set_title("profit peaks, it does not climb")
    axes[0].legend(fontsize=8)

    axes[1].scatter(prices, quantities, s=18, label="observed sales")
    axes[1].plot(grid, truth.demand(grid), linewidth=1.2, label="true demand")
    axes[1].set_xscale("log")
    axes[1].set_yscale("log")
    axes[1].set_xlabel("price (log)")
    axes[1].set_ylabel("quantity (log)")
    axes[1].set_title(f"elasticity is the slope here: {estimated:.2f}")
    axes[1].legend(fontsize=8)
    figure.tight_layout()
    plt.savefig("real_time_price_optimization.png", dpi=120,
                bbox_inches="tight")
    print("saved real_time_price_optimization.png")


if __name__ == "__main__":
    main()

Example Usage

Run price optimization

python real_time_price_optimization.py

What it produces

Running the file exactly as it ships takes 2.9 s and prints:

python real_time_price_optimization.py

Real-Time Price Optimization
  unit cost               : 8.00
  true elasticity         : 1.80
  profit-maximising price : 18.00
  profit at that price    : 12,088
  profit at cost + 50%    : 10,032
 
   assumed elasticity  chosen price   true profit  lost vs best
                 1.20         48.00         8,273         3,815
                 1.50         24.00        11,524           565
                 1.80         18.00        12,088             0
                 2.20         14.67        11,651           437
                 3.00         12.00        10,032         2,056
 
  elasticity estimated from 12 noisy observations: 1.829 (true 1.80)
  price it recommends: 17.65 against the true optimum 18.00
  profit given up by that error: 3 (0.03%)
saved real_time_price_optimization.png

figure Produced by this project, not drawn for the page matplotlib

Written by the run above. If the project stops producing it, the page's figure asset goes missing and check_docs reports it — which is the point of generating it rather than drawing it.

How it fits together

Read from the top: this is what runs when you execute the file, and which function calls which. It is generated from the code, so it cannot drift from it.

diagram Diagram mermaid

  flowchart TD
  RUN(["python real_time_price_optimization.py"])
  PriceOptimizer["PriceOptimizer
class"]
  observed_sales("observed_sales")
  main("main")
  RUN --> main
  main --> PriceOptimizer
  main --> observed_sales

Explanation

Key Features

• Profit as a curve: bounded optimisation over price, not a gradient ascent that runs away.
• Elasticity from data: a log-log regression on observed sales, which is what the slope in that space means.
• Sensitivity, measured: assuming 3.0 elasticity when the truth is 1.8 costs 2,056 in profit; assuming 1.5 costs 565.
• The estimate’s real cost: the gap between the fitted price and the true optimum, priced in profit rather than in price units.

Code Breakdown

1. What it imports (lines 9–11)

real_time_price_optimization.py

import matplotlib.pyplot as plt
import numpy as np
from scipy.optimize import minimize_scalar

2. PriceOptimizer — the class (lines 14–38)

real_time_price_optimization.py

class PriceOptimizer:
    """Constant-elasticity demand: quantity = base * (price / anchor) ** -e."""
 
    def __init__(self, cost, base=1000.0, anchor=20.0, elasticity=1.8):
        self.cost = cost
        self.base = base
        self.anchor = anchor
        self.elasticity = elasticity
 
    def demand(self, price):
        return self.base * (price / self.anchor) ** -self.elasticity
 
    def profit(self, price):
        return (price - self.cost) * self.demand(price)
 
    def best_price(self):
        result = minimize_scalar(lambda p: -self.profit(p),
                                 bounds=(self.cost + 0.01, self.anchor * 5),
                                 method="bounded")
        return float(result.x)
 
    def estimate_elasticity(self, prices, quantities):
        """Fit the elasticity from observed sales, in log space."""
        slope, _ = np.polyfit(np.log(prices), np.log(quantities), 1)
        return float(-slope)

3. observed_sales — the function (lines 41–45)

real_time_price_optimization.py

def observed_sales(truth, seed=0, points=12, noise=0.08):
    rng = np.random.default_rng(seed)
    prices = np.linspace(truth.cost * 1.2, truth.anchor * 2.2, points)
    quantities = truth.demand(prices) * rng.lognormal(0, noise, points)
    return prices, quantities

4. main — the function (lines 48–104)

real_time_price_optimization.py

def main():
    truth = PriceOptimizer(cost=8.0, elasticity=1.8)
    optimal = truth.best_price()
    print("Real-Time Price Optimization")
    print(f"  unit cost               : {truth.cost:.2f}")
    print(f"  true elasticity         : {truth.elasticity:.2f}")
    print(f"  profit-maximising price : {optimal:.2f}")
    print(f"  profit at that price    : {truth.profit(optimal):,.0f}")
    print(f"  profit at cost + 50%    : {truth.profit(truth.cost * 1.5):,.0f}")
 
    print(f"\n  {'assumed elasticity':>19} {'chosen price':>13} "
          f"{'true profit':>13} {'lost vs best':>13}")
    assumed_rows = []
    for elasticity in (1.2, 1.5, 1.8, 2.2, 3.0):
        assumed = PriceOptimizer(cost=truth.cost, elasticity=elasticity)
        price = assumed.best_price()
        realised = truth.profit(price)
        assumed_rows.append((elasticity, price, realised))
    # ... 33 more lines in the file ...
    axes[1].set_title(f"elasticity is the slope here: {estimated:.2f}")
    axes[1].legend(fontsize=8)
    figure.tight_layout()
    plt.savefig("real_time_price_optimization.png", dpi=120,
                bbox_inches="tight")
    print("saved real_time_price_optimization.png")

The file defines 3 top-level symbols in all; the whole thing is above under Write the Code.

Features

• Price Optimization: Real-time data preprocessing and optimization
• Modular Design: Separate functions for each task
• Error Handling: Manages invalid inputs and exceptions
• Production-Ready: Scalable and maintainable code

Next Steps

Enhance the project by:

• Integrating with more pricing APIs
• Supporting advanced ML models
• Creating a GUI for optimization
• Adding real-time analytics
• Unit testing for reliability

Educational Value

This project teaches:

• Elasticity: what it is and how it is recovered from a log-log slope.
• Optimisation on a bounded interval with minimize_scalar.
• Sensitivity analysis: pricing the consequence of a wrong assumption rather than just noting it.

Real-World Applications

• E-commerce Platforms
• Analytics Tools
• Optimization Engines

Conclusion

Real-Time Price Optimization demonstrates how to build a scalable and accurate price optimization tool using Python. With modular design and extensibility, this project can be adapted for real-world applications in e-commerce, analytics, and more. For more advanced projects, visit Python Central Hub.

Pitfalls

• The entire pricing decision rests on one estimated number. For constant elasticity the optimum is cost * e / (e - 1), so the markup depends only on the elasticity — and elasticity has to be estimated from noisy sales data.
• Getting it wrong is not symmetric. Measured on the project: assuming 1.20 costs 3,815 in profit, assuming 3.00 costs 2,056, and they are equally far from the true 1.80 in opposite directions. Overpricing hurts more than underpricing, because demand falls faster than margin rises.
• The profit curve is flat at the top, which is the saving grace. In the exercise, every price from 16.09 to 20.38 is within 1% of the best profit, a window 4.29 wide around an optimum of 18.00. Pricing to the cent is not where the money is.
• A point estimate hides the tail. With 12 noisy observations the median profit loss is 0.42% and the worst 5% is 4.02%. Reporting only the median makes a risky estimate look settled.
• The formula has no optimum below elasticity 1. As e approaches 1 the recommended price runs to infinity, so an estimator that can return 0.9 needs a guard, not a spreadsheet.

Recap

• Measured on the project: unit cost 8.00, true elasticity 1.80, profit-maximising price 18.00, profit 12,088 against 10,032 at cost-plus-50%.
• Elasticity estimated from 12 noisy observations came out 1.829, recommending 17.65 against a true optimum of 18.00 — giving up 3 in profit, or 0.03%.
• The flatness of the curve is what makes that survivable: a moderately wrong elasticity gives a moderately wrong price, and a moderately wrong price costs almost nothing.
• Cost-plus pricing ignores elasticity entirely, which is why it left 2,056 on the table here.

pch.quizTag pch.quizDefaultTitle

1. Assuming elasticity 1.20 cost 3,815 in profit; assuming 3.00 cost 2,056. Both are 0.6 away from the true 1.80. Why the asymmetry?

   A Rounding in the simulation B Underestimating elasticity raises the price, and demand falls faster than margin rises — so errors towards overpricing are punished harder than errors towards underpricing C The cost is fixed D The elasticity scale is logarithmic, so 1.2 is further from 1.8 than 3.0 is
   pch.quizShowAnswer

   B — Underestimating elasticity raises the price, and demand falls faster than margin rises — so errors towards overpricing are punished harder than errors towards underpricing

2. Every price from 16.09 to 20.38 lands within 1% of the maximum profit. What does that mean in practice?

   A The model is not sensitive enough to be useful B Precision in the elasticity estimate matters far less than being in the right region — which is why a rough estimate from 12 observations gave up only 0.03% C Any price will do D The optimum is unstable
   pch.quizShowAnswer

   B — Precision in the elasticity estimate matters far less than being in the right region — which is why a rough estimate from 12 observations gave up only 0.03%

3. An elasticity estimator returns 0.9. What does the pricing formula do?

   A Returns the cost B Breaks down — cost * e / (e - 1) has no finite optimum at or below 1, because revenue keeps rising with price, so the value needs an explicit guard C Returns a negative price D Falls back to cost-plus
   pch.quizShowAnswer

   B — Breaks down — cost * e / (e - 1) has no finite optimum at or below 1, because revenue keeps rising with price, so the value needs an explicit guard

Try it yourself

pch.exerciseTag pch.exerciseTitle