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: vali@fordham.edu

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