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?
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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.
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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.
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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.
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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.
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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.
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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.
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Next slide

Line 7 numbers from slide #8

Y=377717x^-2.154
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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.
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Using the Solver

The Solver dialog box is shown here.
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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.
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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.
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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.
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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.
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