Journal of Contemporary Management
Submitted on 21/01/2016
Article ID: 1929-0128-2016-03-42-05 Shapoor Vali
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An Interesting Case of Quadratic Demand Function
Or “Wal-Mart/Apple” Effect
Dr. Shapoor Vali
Department of Economics, Fordham University
113 West 60th Street, New York, NY 10023, U.S.A.
Tel: +1-212- 636-6240 E-mail: [email protected]
Abstract: Microeconomics and Mathematical Economics textbooks seldom use quadratic, log
linear, or other nonlinear demand functions in microeconomics optimization models. The general
argument is that there is not much additional insight to gain by introducing non-linearity in the
models. Interestingly, as this paper demonstrates, the use of nonlinear demand functions may lead
to results not studied in the literature, results which could very well explain the behavior of price in
markets for certain product or some firm’s pricing policies when there is possibility of “bulk”
purchase or sale.
Keywords: Nonlinear demand function, Noncompetitive firm, Shutdown price, Bulk sales, Wal-
Mart effect
JEL Classifications: A22, A23, C02, D41, D42
In spite of the fact that in many cases a realistic economic model requires use of nonlinear
functions, the existing Microeconomics and Mathematical Economics textbooks seldom use
nonlinear demand functions -- like quadratic, semi-log, or log linear -- in optimization models. The
general argument in favor of linearity seems to be that introduction of non-linearity may not lead to
much additional insight and it only adds to computational complexity. Interestingly, as the
following problem will demonstrate, use of nonlinear demand functions in microeconomics
optimizations may lead to very useful results not studied in the literature.
Let’s consider a simple optimization problem. Assume that a single period 1 profit maximizing
noncompetitive (monopolist or monopolistically competitive) firm has the total cost function
and faces the following quadratic demand function
for its product in the market. For reason that becomes clear soon, the intercept in the demand
equation is specified as a parameter. Equation (1) will be a legitimate demand function over the
following range of values for Q and :
1 I carefully distinguish between a single period and a multi-period profit maximization strategy. A
noncompetitive firm, specially a monopoly with substantial market power, may follow a strategy of maximizing profit over several production cycles rather than a single cycle (dynamic optimization).
Journal of Contemporary Management, Vol. 5, No. 3
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To maximize profit, we form the profit function
The first order condition for a maximum is
The second order condition for a maximum requires that
Therefore in (2) the correct answer must be
It is clear from this equation that for the optimization to have a real solution which also
satisfies the second order condition, must be greater than 200,
By substitute for Q from (3) in (1) we can express price as
It is expected that as the demand for a good produced by a firm increases, the profit
optimization routine generate higher quantity and price. Since in our case increases in demand are
signified by higher values of , then in (5) we expect
> 0.
After differentiating (5) and simplifying it, we have
To satisfy
> 0, we must have
which leads to
Before discussing the implication of and we need to determine
another important value of , namely the value that simultaneously satisfies the following two
conditions
ISSNs:1929-0128(Print); 1929-0136(Online) ©Academic Research Centre of Canada
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This occurs when the point of intersection of the marginal revenue MR and the marginal cost
MC curves are aligned with the point of tangency between the demand curve and the average
variable cost AVC curve. At this point the total variable cost is equal to the total revenue and the
associated price is the shutdown price. For our firm (7) is
After eliminating we have
which leads to a nonzero solution = 3. We next determine the price as
It is easy to verify that at the price/quantity combination of (122, 3) total revenue and total
variable cost are indeed the same. Therefore, P = 122 is the shutdown price.2 Using the demand
equation, we have
and
Hence the demand equation associated with the price/quantity combination of (122, 3) is
The implication of (4), (6) and (8) is that if is less than 200 there is no real solution to the
profit maximization model. If is greater than 200 but less than 207.5, there is a real solution to
the model, but the combination of price and output is such that the total revenue would be less than
the total variable cost, consequently our firm has no incentive to operate. For values of in the
range of 207.5 to 222.97, the optimization results in higher outputs but lower prices, and only for
values of greater than 222.97 does the optimization leads to higher outputs and higher prices.
The counter intuitive feature of our quadratic demand case is that as the demand condition
improves and increases from 207.5, the level of output increases but the price declines. This
process continues until reaches 222.97 and price reaches is minimum of 117.5. As increases
beyond the value of 222.97, both output and price increase. The trace of optimal price, which
displays the profit maximizing prices for different values of is depicted in Figure 1.
2 There are cases that the price determined by (7) may not be the shutdown price. For detail see Vali
(2014).
Journal of Contemporary Management, Vol. 5, No. 3
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Figure 1. Trace of optimal prices for demand equation (1)
Table 1. Optimal output, price, and profit for selected values of β
β
Output
Price
Profit
207.5
3.000
122.0000
-5.0000
208.5
3.065
121.2584
-1.9674
209.0
3.095
120.9275
-0.4273
209.5
3.125
120.6328
1.1279
210.0
3.155
120.3270
2.6980
212.0
3.265
119.3801
9.1193
215.0
3.414
118.4077
19.142
217.0
3.506
117.9660
26.063
220.0
3.633
117.6093
36.773
222.0
3.712
117.5295
44.120
222.9
3.747
117.5100
47.477
225.0
3.825
117.5653
55.429
230.0
4.000
118.0000
75.000
240.0
4.309
120.0137
116.584
250.0 4.582 123.0376 161.066
267.5 5.000 130.0000 245.000
275.0 5.162 133.4562 283.114
300.0 5.651 146.4251 418.432
The optimal values of output, price, and profit for selected values of β are given in Table 1.
As the table indicates, increase in demand, signified by increase in β, leads to systematic increase
in output and profit, but price declines till β moves beyond 222.9.
117
117.5
118
118.5
119
119.5
120
210 215 220 225 230 235 240
P
r
i
c
e
Beta
ISSNs:1929-0128(Print); 1929-0136(Online) ©Academic Research Centre of Canada
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Should we treat the demand equation (1) as a miss-specification? The answer is no. The
quadratic equations similar to (1) are very helpful in capturing the dynamics of the market for
certain industries, like retailing and hand-held electronic devices. The law of demand works both
ways. As due to a combination of income and socio-cultural factors, demand for smart phone
dramatically increased, price of smart phone appreciably declined. Part of this decline is
attributable to more efficient production technology. But also a big increase in the market size must
be considered as another contributing factor. Firms are willing to charge lower unit prices for bulk
sales. In today’s global market, bulk sales and purchases propagate through the entire supply chain
and ultimately lead to lower prices for consumers. This phenomenon could be termed “Wal-Mart”
or “Apple” effect. The sheer size of bulk purchase and sales of variety of personal and household
items by Wal-Mart makes it possible for the company to offer these items at cheaper prices. The
same is true for Apple iPod, iPhone and iPad.
Conclusion
An article by Kevin Roose (2012) in the Business section of the New York Times in July 11,
2012 explains how large firms, including private equity firms, has learned to save millions of
dollars by buying in bulk. The article cites example of a group of companies that “collectively
bought 16 million reams of copy paper, 35 million FedEx shipments and 900,000 days’ worth of
rental cars from National and Avis” and shaved millions of dollars off their costs.
This paper is an attempt to show that a quadratic demand function can be used to model the
bulk purchase strategy which is now widely used by large firms. Success of giants’ mega-cap
companies like Wal-Mart, Apple, and Amazon is partially rooted to their ability to buy raw
materials and intermediate or finished goods in bulk and realize substantial bulk discount.
Determining the relationship between the coefficients of linear and quadratic terms in the
quadratic demand equation, which leads to the situation similar to our case, must be subject of
further investigation.
References
[1] Roose, Kevin (2012). “Private Equity Giants Use Size to Lean on Suppliers”, New York Times
(July 11, 2012).
[2] Vali, Shapoor (2014). Principles of Mathematical Economics. Paris: Atlantis Press. ISBN: 978-
94-6239-035-5