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Example 7.1 Pricing Models Background Information The Madison Company manufactures and retails a certain product. The company wants to determine the price that maximizes profit from this product. The unit cost of producing and marketing the product is $50. Madison will certainly charge at least $50 for the product to ensure that it makes some profit. However there is a very competitive market for this product, so that Madison’s demand will fall sharply as it increases its price. How should the company proceed? 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 Solution If Madison charges P dollars per unit, then its profit will be (P –50)D, where D is the number of units demanded. The problem, however, is that D depends on P. As the price P increases, the demand D decreases. Therefore the first step is to find how D varies with P – the demand function. In fact, this is the first step in almost any pricing problem. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 Solution – continued We will try two possibilities – A linear demand function of the form D = a – bP – A constant elasticity demand function of the form D = aPb. You might recall from microeconomics that the elasticity of demand is the percentage change in demand caused by a 1% increase in price. The larger the (magnitude of) elasticity is, the more demand reacts to price. The advantage of the constant elasticity demand function is that the elasticity remains constant over all points on the demand curve. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 Solution – continued For example, the elasticity of demand is the same when price is $60 as when price is $70. Actually, the exponent b is approximately equal to this constant elasticity. For example, if b= -2.5, then demand will decrease by about 2.5% if price increases by 1%. In contrast, the elasticity changes for different price levels if the demand function is linear. Nevertheless, both forms of demand functions are commonly used in economic models. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 Solution – continued Regardless of the form of the demand function, the parameters of the function (a and b) need to estimated before any price optimization can be performed. This can be done with Excel trend curves. Suppose that Madison can estimate two points on the demand curve. Specifically, suppose the company estimates demand to be 400 units when price equals $70 and 300 units when price equals $80. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 Solution – continued Then we create two X-Y charts of demand versus price from these two point and use Chart/Add Trendline menu item with the option to list the equation of the trendline on the chart. For a linear demand curve, we select the Linear trendline, and for the constant elasticity demand curve, we select the Power trendline. The results appear on the next slide. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 PRICING1.XLS Once Madison has determined the demand function, the pricing decision is straightforward as shown on the next slide for the constant elasticity model. This file contains the spreadsheet model. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 Developing the Model To develop this model, proceed as follows. – Inputs. The inputs for this model are the unit cost and the parameters of the demand function found earlier. Enter them as shown. – Price. Enter any trial value for price. It will be the single changing cell. – Demand. Calculate the corresponding demand from the demand function by entering the formula =Const*CEPrice^Elast in the Demand cell. – Profit. Calculate the profit as net price times demand with the formula =(CEPrice-UnitCost)*CEDemand in the Profit cell. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 Using the Solver The Solver dialog box is shown here. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 Using the Solver – continued We maximize profit subject to the constraint that price must be at least as large as unit cost, and price is the only decision variable. However, do not check the Assume Linear Model box under Solver options. This model is nonlinear for two reasons. First, the demand function is nonlinear because Price is raised to a power. But even if the demand function were linear, profit would still be nonlinear. The reason is that it involves the product of price and demand, and demand is a function of price. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 Using the Solver – continued This nonlinearity can be seen easily with the data table and corresponding chart shown earlier. These show how profit varies with price – the relationship is clearly nonlinear. Profit increases to a maximum, then declines slowly. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 Sensitivity Analysis From an economic point of view, it should be interesting to see how the profit-maximizing price varies with the elasticity of the demand function. To do this, we use SolverTable with the elasticity in cell C7 as the single input cell, allowing it to vary from - 2.4 to 1.8 in increments of 0.1. The results appear in the figure on the next slide. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11 Sensitivity Analysis – continued When the demand is most elastic, increases in price have a greater effect on demand. Therefore, the company cannot afford to set the price as high in this case. 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | 7.8 | 7.9 | 7.10 | 7.11