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Simple Dynamic Pricing Setting Price and Gaining Sales The corporation is a large fast-food chain that makes pizza. For pizza prices between $10 and $15, we believe that sales per year are equal to ten million times the difference between “product desirability” and price. Product desirability can then be inferred from actual sales and the set price. It represents many factors including competitors’ prices and market conditions. Product Desirability is currently 18. If Product Desirability drops below the price, then sales are zero. We believe that for the next ten years (at which point the model will end and the company’s average value will be its “future value”, see below), Product Desirability has a growth rate of -20%, a volatility of 30%, a correlation of 60% with the Market, and a correlation of 10% with variable costs. At any point in time, our decision is what pizza price to set. As product desirability changes, we must change the price to maximize True Shareholder Value(e.g., if desirability drops, we may need to drop prices, depending on other factors). It costs nothing to raise prices; however, to gain the benefit of lower prices we must expend $10 million in advertising. Costs, Salvage Value, and Future Value Variable costs are currently $9 per pizza. Our information leads us to say that variable costs have a –5% expected growth rate, 15% volatility, and a correlation of 20% with the Market. Other costs total $50 million per year. It is costless to raise prices. Corporate taxes are 39% and apply to all costs and revenue. The corporation can receive $20 million in salvage value at any time. If operations are continuing in ten years, the company’s expected value at that time is equal to onetenth of current free cash flow (except allowing for negative sales and negative profit margins) minus $203.05 million plus $10 million times the square of the difference between desirability and variable cost (only for desirability greater than variable cost). Model Properties The following figure shows a summary of the model properties. True Shareholder Value and Strategy The True Shareholder Value of the corporation in this model is $237.09M and the optimal current price is $13.50. The graph layout below shows up to the second pricing decision. Queries and Sensitivity Analysis The derivative analysis of the percentage model properties is shown below. Model Property Desire Market Correlation Cost Growth Cost Market Correlation Cost Desire Correlation Risk Free Rate Corporate Tax Rate Desire Volatility Cost Volatility Desire Growth d(NPV) / d(%) ($1,501,46 3) ($4,521,15 5) $292,953 ($607,547) ($3,149,29 8) ($3,603,80 6) $8,385,828 ($291,505) $11,586,08 5 It is no surprise that a higher growth rate of desirability and a lower growth rate for pizza costs have a large, positive impact on value. Higher volatility of desirability also has a large, positive impact on value. There must be a lot of option value embedded in this company’s value. On the other hand, higher volatility of pizza costs actually has a negative impact on value. Therefore, the extra Market risk involved with the variability of the pizza costs must outweigh any extra option value. This could be because the pizza costs have no direct interaction with pizza sales like desirability. The following plot shows how the root NPV is affected by different expected growth rates of desirability. There is no surprise here either. As the expected growth rate increases, the value of the pizza company increases dramatically. However, this analysis provides us with some quantitative insight as to the value of increasing the desire growth, e.g., through advertising. Thus, if we could change from -20% to -10% growth, we generate an additional ~$200M in True Shareholder Value. From this information, we might augment the model by adding decisions on when/how much to advertise, along with uncertainties reflecting advertising’s effect on the growth of desire for our pizzas. To get an idea of how the optimal prices are being set, below is a plot of the optimal pricing map for various values of cost and desire at year 8 when the previous price was $12. The trend isn’t surprising: if desirability is higher, we charge a higher price, and if costs are higher we also charge a higher price. If desirability is too low or cost to high, we cease sale of pizzas. Notice that there are no small drops in price, e.g. to $11.50. This could be due to the advertising costs. It simply isn’t worth advertising a small change. So, we see a region (small orange circles) in which we should drop the price to the minimum.
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