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Example 7.3. Pricing Models. Background Information. Suits cost the Men’s Warehouse $320. The current price of suits to customers is $350, which leads to annual sales of 300 suits. The elasticity of the demand for men’s suits is estimated to be –2.5.
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Example 7.3 Pricing Models
Background Information • Suits cost the Men’s Warehouse $320. • The current price of suits to customers is $350, which leads to annual sales of 300 suits. • The elasticity of the demand for men’s suits is estimated to be –2.5. • Each purchase of a suit leads to an average of 2 shirts and 1.5 ties being sold. • Each shirt contributes $25 to profit and each tie contributes $15 to profit. • Determine a profit-maximizing price for suits.
Solution • As in the previous two examples, we first need to determine the demand function for suits. • Although this could be a linear function or some other form, we will again assume a constant elasticity function of the form D = aPb, where the exponent b is the elasticity. • Then we can find the constant a from the current demand and price for suits 300=a350-2.5, so a=300/350-2.5.
Solution – continued • The solution from this point is practically the same as the solution to Example 7.1 except for the profit function. • Each suit sold also generated demand for 2 shirts and 1.5 ties, which contributes 2(25) +1.5(15) extra dollars in profit. • Therefore, it makes sense that the profit-maximizing price for suits will be lower than in the absence of shirts and ties – the company wants to generate more demand for suits so that it can reap the benefits from shirts and ties.
PRICING3.XLS • The model appears on the next slide. • This file can be used to build the model. • We originally set up two models, one where shirts and ties are ignored and one where they are not, to see how the profit-maximizing price changes. • However, a clever use of SolverTable allows us to treat both cases in one model.
Developing the Model • The details are as follows. • Inputs. Enter all inputs in the shaded regions. • Constant for demand function. Calculate the constant for the demand function in cell B9 with the formula =CurrDemSuits/CurrPriceSuits^Elast. • Sensitivity factor. We will treat both cases, when shirts and ties are ignored and when they are not, by using SolverTable with a sensitivity factor as the input cell. When this factor is 0, the complementary products are ignored; when it is 1, they are taken into consideration. Enter 1 in the SensFactor cell for now. In general, this factor determines the average number of shirts and ties purchases with the purchase of a suit – we multiply this factor by the values in the UnitsPerSuit range. When this factor is 1, we get the values in the statement of the problem.
Developing the Model – continued • Price, demand. Enter any price in the PriceSuits cell, and calculate the corresponding demand for suits in the DemandSuits cell with the formula =Const*PriceSuits^Elast. • Profit. The total profit is the profit from suits alone, plus the extra profit from shirts and ties that are purchased along with suits. To calculate profit, enter the formula =(PriceSuits-UnitCostSuits+SUMPRODUCT(SensFactor*UnitsPerSuit,UnitProfit))*DemandSuits in the Profit cell. • Solver setup. The Solver setup is the same as in Example 7.1. We maximize profit, with the price of suits as the only changing cell, and we constrain this price to be at least as large as the unit cost of suits.
Developing the Model – continued • SolverTable. Run SolverTable with the SensFactor cell as the single input cell, varied, say, from 0 to 2 in increments of 0.5, and keep track of price, demand, and profit.
Solution – continued • The SolverTable results show that when the company ignores shirts and ties, the optimal price is set high at $533.33. • However, as more ties and shirts are purchased by purchasers of suits, the optimal price of suits decreases fairly dramatically. • As we would imagine, as more shirts and ties are purchased with the suits, the company makes more profit – if it properly takes shirts and ties into account.
Solution – continued • For the situation in the problem statement, where the sensitivity factor should be set at 1, the optimal price is $412.50. • How much profit does the company lose in this case if it ignores shirts and ties? • You can answer this by entering $533.33 in the PriceSuits cell, keeping the sensitivity factor equal to 1. • If you do so, you will find that profit decreases from $32,826 to $29,916, a drop of about 9%.
