A Designer’s View to Economics andFinance

Panos Y. Papalambros and Panayotis GeorgiopoulosOptimal Design Laboratory

University of MichiganAnn Arbor, Michigan 48109

January 7, 2005

Abstract

We review basic concepts from economics and finance relevant toproduct development. Case studies are used to demonstrate how theseconcepts relate to complex design decisions within the producing en-terprise, with a specific formulation for product portfolios. We con-clude with a proposed synthesis of engineering design, economics andfinance under a single decision-making model.

Nomenclature

C: cost

D: firm’s debt

E: firm’s equity

EP : price elasticity of product demand

f : analysis function

g: vector of inequality constraints

h: vector of equality constraints

H: horizon of product life-cycle

I: amount of a financial investment

K: capacity of production

M : firm’s market share

MR: marginal revenue

MC: marginal cost

P : product price

PV: present value of a future cash flow stream

Q: industry’s product demand

q: firm’s product demand

R: revenue

rf : risk-free interest rate

rm: financial market return

2

t: time

t0: commercialization time

U : capacity utilization

w: criterion preference weight

WACC: weighted average cost of capital

x: product decisions

xb: business decisions

xe: economic decisions

xf : financial decisions

z: Wiener process

α: observed product characteristics

β: sensitivity of firm’s stock to market return

δ: mean utility level from consuming a product

θ: intercept of the product demand curve

3

λ: slope of the product demand curve

µ: growth of product demand

ξ: unobserved product characteristics

π: profit

σ: volatility of product demand

τ : product

1 Introduction

In recent years the engineering design community has expanded its questto address analytically design intent, particularly as it is manifested withina producing enterprise, see, for example, Thurston and Locascio (1994);Thurston et al. (1994); Thurston (1999); McConville and Cook (1996); Cook(1997); Donndelinger and Cook (1997); Hazelrigg (1998, 2000); Marston andMistree (1998); Marston et al. (2000); Scott and Antonsson (1999); Whit-comb et al. (1999); Allen (2000, 2001); Chen et al. (2000); Gu et al. (2002);Li and Azarm (2000, 2002); Markish and Willcox (2002a,b); Wassenaar andChen (2003); Wassenaar et al. (2003, 2004); Georgiopoulos (2003); Cooperet al. (2003); Georgiopoulos et al. (2005); Michalek et al. (2004); Kim et al.(2004). These investigators, while following different approaches, have shownthat designers can and should not only balance technical tradeoffs but alsoaccount more directly for the needs of users and producers, positioning en-gineering tradeoffs in a societal context. In such a view, technical objectivesnot only compete amongst each other but also generate a long chain of ef-fects that influence purchasing behavior, firm growth, regional economics andenvironmental policies.

In this article we review some basic concepts that are the building blocksfor modeling economics and finance considerations in the context of product

4

development. Along the way we will use case studies to demonstrate howto incorporate these concepts in design decision making. We conclude withproposing a simple synthesis of engineering design, economics and finance ina single design optimization model.

2 Users and producers

The product design decision-making process involves matching customers’demand for differentiation with the firm’s capacity to supply differentiation(Grant, 1998, p.221). Differentiation translates to uniqueness and involvesall the activities the firm performs. The direct utility that consumers gainfrom a bundle of observable product attributes (i.e., tangible differentiation)in conjunction with socioeconomic, psychographic and demographic consid-erations, (i.e., intangible differentiation) are driving purchasing behavior.

The ability of the firm to satisfy demand for differentiation profitablyto a large extent depends on a set of product decisions x that encompassengineering design xd and business xb decisions. In turn business decisionsxb encompass economic decisions xe, such as product pricing and productionoutput decisions, and financial decisions xf such as product valuation andinvestment timing. The optimal combination of xd, xe and xf would lead toincreased profitability π:

π(x), x = [xd,xb] (1)

First the relevant mathematical models of economic decisions will be re-viewed. Financial decision models review will follow. A case study will helpto demonstrate the respective disciplinary models. This case study will serveas a platform to present a holistic product development decision model atthe end.

3 Economic decisions

The evaluation criterion of economic decisions is the summation of monetarycosts and benefits at one instant of time. In this section we will focus on twodecisions: product pricing and production output.

The demand curve is the main tool used in economics that links thequantity of a product the consumer is willing to buy to the price of that

5

product. One can do a quick experiment to understand the idea behindthe demand curve. Let us assume that a university professor is consideringbuying laptops for her research laboratory. At a price of $2000 she may buytwo laptops, one for research presentations and one for herself. At a price of$1000 she may buy two more for her most productive students. At a price of$400 she may end up buying laptops for all six students. From this examplewe conclude that the professor’s demand curve is downward sloping. Letus assume that we can do the same experiment for all the professors at theuniversity. Assuming that a single price can be charged, then we can sum allthe individual demand curves and obtain the university professors’ aggregatedemand curve. The latter will tell us how many laptops a manufacturercan sell at $2000, $1000 and $400 to the university. Assuming that thedemand curve is indeed downward sloping and linear, then the followingequation models the relationship between price P and quantity Q (Pindyckand Rubinfeld, 1997, p.21,31)

Q = θ − λP, (2)

where θ is the intercept and λ =∆Q

∆Pis the slope of the demand curve; λ is

the change in quantity associated with a change in price.But what does the slope of the demand curve represent? From calculus

we know that λ = ∆Q/∆P captures the change in quantity demanded froma change in price. Dividing by the original level of quantity and price wederive a normalized quantity called the price elasticity of demand

EP =

∆QQ

∆PP

=%∆Q

%∆P. (3)

Given that when we increase price usually the quantity falls, the price elas-ticity of demand is a negative number. This fall in demand is the consumer’sresponse to a pricing decision set by the decision-maker. Therefore, it modelsconsumer preferences towards a specific product attribute, which is price. Ifthe price elasticity is less than one in magnitude the consumers are inelasticand a change in price will not affect substantially the quantity demanded.The opposite applies when the price elasticity is greater than one and cus-tomers are elastic with respect to changes in price.

In the absence of cost we can formulate the following unconstrained op-

6

Price (dollars per unit)

Quantity

Figure 1: The solution of Eq. (4) is the price and quantity that maximizesthe shaded rectangle area under the demand curve.

timization problem

maximize {revenue}with respect to {quantity}, (4)

where revenue is the product of price and quantity. If we were to decide ona single price then the solution of Eq. (4) is the largest rectangle under thedemand curve (see Figure 1). In the presence of cost Eq. (4) becomes

maximize {revenue - cost}with respect to {quantity}. (5)

Design scenario: Demand curve formulation

We consider decisions to be made by an automotive manufacturing firm thatmarkets premium-compact (PC), among other, vehicles in the US. This mar-ket segmentation adopted in the study follows the J.D. Power classificationfor vehicles in the United States. The firm is assumed to operate in a matureindustry where complementary assets (Teece, 1986), such as access to distri-bution channels, service networks, etc., are given. Finally, the decision-makeris assumed to be playing a game against nature, namely, the firm’s strategyis affected by an exogenously-generated random state not by competitive in-teraction. The firm wishes to design new engines and transmissions for thePC segment. Representative of the PC product, the automatic transmission

7

versions of the Chevrolet Cavalier LS Sedan has been selected and simu-lated using the Advanced Vehicle Simulator (ADVISOR) program (Wipkeand Cuddy, 1996).

The monthly profit is defined as

π = Pq − C (6)

where P is the price, q the quantity, and C the average total cost of producinga vehicle.

We draw the relationship between the price P of each product and thequantity q demanded from the observed demand for final goods. Knowingtwo different points on the demand curve we can use the arc elasticity ofdemand, see Eq. (3), which is defined as:

EP =∆q

∆P

P

q. (7)

where P , q are the averages of the known prices and quantities, at the twopoints.

In years 2000 and 1999, General Motors PC vehicles (Chevrolet Cavalierand Prizm, Pontiac Sunfire, Saturn S-series) did not undergo a major designchange. Using two pairs of data points (P1/99, q1/99), (P1/00, q1/00) (see Table1) (WardAuto, 1998-00) we compute the price elasticity of the GM PC seg-ment as equal to −4.9. The price elasticity of all US automobiles has been

Table 1: Price and quantities for the PC segment

P1/99 $14,512q1/99 43,507P1/00 $ 15,015 (’99 adj.)q1/00 36,755

found to be between −1 and −1.5 (Irvine, 1983). For individual models priceelasticities have been found to be between −3 and −6.7 (Berry et al., 1995).For example in Berry et al. (1995) the price elasticity of the Chevrolet Cava-lier was found to be −6.4. It is reasonable to assume that the price elasticityof demand for all GM PC vehicles is less than the elasticity of an individualmodel in that segment.

8

Although there was no observed change in quality – which could be a rea-son for a shift in the demand curve – of vehicles in GM PC segment from 1999to 2000, other factors that affect demand may have taken place. To use theestimated elasticity we assumed that between the two years there was no ma-jor change in consumer’s income, product advertising, product informationavailable to consumers, price and quality of substitutes and complementarygoods, and population (Clyde, 2001).

Using the average values of two data points (P , q) we derive the demandcurve for the PC segment:

P = 17753− 0.075q. (8)

Assuming a linear relationship between cost and output,

C = c0q, (9)

the total profit for one time period is

π =θ

λP

q − 1

λP

q2 − c0q. (10)

Eq. (9) assumes that the marginal cost is constant, that is, for every unitincrease in output, the average total cost increases by c0 (Pindyck and Ru-binfeld, 1997, p. 236), which is set at $13,500 for the PC segment. Eq. (9)also assumes that the firm is operating at its minimum efficient scale.

The economic profit is

π = θλP

q − 1λP

q2 − c0q. (11)

and the optimum is

q∗t =θ

λP− c0

2λP

. (12)

3.1 The demand curve of the product developmentfirm

Let us review again Eq. (2). At each period of a multi-period decision timehorizon the decision-maker is facing pricing and output decisions. This ob-servation modifies Eq. (2) as follows:

qt = θ − λPt, (13)

9

Price (dollars per unit)

Quantity

Figure 2: A shift in the demand curve (Eq. 2) asks for new decisions

where t is the period we are at. At time period t the decision-maker willchoose the price of the product Pt and the quantity produced qt. This wouldessentially mean that economic decisions are moving along the demand curve,Fig. 1. But given that there is a unique rectangle with maximum area underthe demand curve, what is the motivation behind a time sequence of pricingand output decisions? The answer is that there are many factors that havechanged Eq. (2). To name a few, since the previous period (t−1) the incomeof the consumer may have contracted, the price and quality of substitutesmay have risen, and the population in the area under consideration mayhave decreased. All those factors would translate to a shift in the demandcurve. This shift asks for new decisions. The decision-maker is updating herdecisions to estimate the maximum area rectangle under the new demandcurve, see Fig. 2. The shifts in the demand curve are represented as follows(Pindyck (1988); Trigeorgis (1998, p.290); Bollen (1999)):

qt = θt − λPt, (14)

where θt is now a time-dependent intercept that moves away from or towardsthe origin. The assumption here is that the slope of the demand curve λ is notchanging as frequently as θ. Going back to our question, is it reasonable toassume that λ is not changing as frequent as θ? In the short-run the answeris positive. It takes time for people to change their consumer habits (Pindyckand Rubinfeld, 1997, p.35). However, one should pay special attention to theproduct under consideration and the elasticities of both supply and demand.Here it is assumed that consumer behavior will not change during the life-

10

cycle of the product.In the case where an evaluation of an investment opportunity is under

consideration the decision-maker is facing the ever-changing demand curve.How could he cope with this uncertainty? Fortunately, there is availabletheory that allows treatment of it. Using information available now, namely,information on past changes in θ and the deviation of those changes fromtheir long-run trend this source of uncertainty can be modeled. Still a criticalpiece will be missing, which is the future information that generates shockson the demand of the product. Therefore, the uncertainty model shouldcombine deterministic with stochastic information. The deterministic oneshould include past observations (old information) and the stochastic onewill involve a simple random number generation (or Monte Carlo Simulation)simulating new information that is unknown to the decision-maker today.

Two possible ways to model the future values of θ are the additive andmultiplicative models:

θ(i + 1) = aθ(i) + u(i)θ(i + 1) = u(i)θ(i).

(15)

where u(i), i = 0, 1, . . . , N−1 are random disturbances that cause the θ valueto fluctuate in the i-period (i = 0, 1, 2, . . . , H), and a is a constant. Once u(0)is given, based on θ(0), which is the intercept of the demand curve today,progressively we can find future values. Therefore, values of θ depend onlyon the value at the most recent previous time and the random disturbance.By taking the natural logarithm of both sides of the multiplicative model wehave

ln θ(i + 1) = ln θ(i) + ln u(i) ⇒∆ ln θ(t) = ln u(i).

(16)

We proceed by modeling the random variable ln u(i). We define the random-walk process z as

z(ti+1) = z(ti) + ε(ti)√

∆t,ti+1 = ti + ∆t,

(17)

ε(ti) is a normal random variable with mean 0 and variance 1. By taking thelimit of the random walk process Eq. (17) at ∆t → 0 the Wiener process (orBrownian motion) is obtained (Luenberger, 1998, p. 306):

∆z = ε(t)√

∆t. (18)

11

The generalized Wiener process is defined as

∆X(t) = a∆t + b∆z,

∆z = ε(t)√

∆t(19)

where z is a Wiener process, a, b are constants and X(t) is a random variable.The Ito process (Luenberger, 1998, p. 308)

∆X(t) = a(θ, t)∆t + b(θ, t)∆z,

∆z = ε(t)√

∆t.(20)

is a generalization of Eq. (19). We assume that the random variable ln u(i)has expected value E[ln u(i)] = ν and variance σ2. We can then describeln u(i) as a generalized Wiener process

ln u(i) = ν∆t + σ∆z, (21)

where z is a Wiener process. Then from Eqs. (16), (21)

∆ ln θ(t) = ν∆t + σ∆z ⇒∆θ(t)

θ(t)= µ∆t + σ∆z ⇒

∆θ(t) = (µ∆t + σ∆z)θ(t) ⇒∆θ(t) = µθ(t)∆t + σθ(t)∆z,

(22)

where µ = 12σ2 +ν is a correction term necessary when one changes variables

in Ito processes (Luenberger, 1998, p.309). Eq. (22) is termed the GeometricBrownian Motion. The full equation scheme is as follows:

∆θ(t) = µθ(t)∆t + σθ(t)∆z,

∆z = ε(t)√

∆t,ε ∼ N(0, 1).

(23)

Historical data on θ allows the estimation of µ and σ. By generating randomnumbers ε(i) and having the value of θ today then we can estimate ∆θ(t).To obtain θ(1) = θ(0) + ∆θ(t).

If one follows Eq. (14) as the representation of the demand curve, oneassumes that all factors other than the price of the product are random dis-turbances. While many factors are indeed random, others are not. As statedearlier, in corporate strategy literature the product design decision-making

12

process involves matching the customer’s demand for differentiation with thefirm’s capacity to supply differentiation. Therefore, the supply of differenti-ation is a choice made by the firm. If the level of product differentiation attime t is different from that at time (t−1) a shift in Eq. (14) will be realized.A simplified version of the Berry Levinsohn and Pakes (BLP) method (Berry,1994; Berry et al., 1995, 1998; Nevo, 2000) will help us model the consumerdemand for product differentiation.

The BLP method, as it is presented here, requires an a priori consumermarket segmentation. Segmentation is different from differentiation (Grant,1998, p.220). Segmentation is concerned with where the firm competes, e.g.,consumer groups, and geographic regions. Differentiation is concerned withhow the firm competes in the particular segment making its product differentfrom that of its competitors. An assumption being made in this chapter isthat market segmentations at times t and (t + 1) are the same. Althoughin the short-run this assumption is most likely to hold, in the long-run itdoes not. The value network of the consumer is dynamic. Static marketsegmentations, which are based on socioeconomic categories, have failed toaddress consumers’ preferences in the past, see, e.g., the General Motorscase in Grant (1998, p.227). A change in market segmentation depends bothon supply and demand. For example, the elasticities of supplying productreliability in the automotive industry were inelastic in the late 1980s. Thedemand for reliability asked for redesign of manufacturing processes not onlyby the automotive manufacturers but by their suppliers as well. As a resultJapanese automotive firms, which were relatively more elastic and thus hadgreater capacity to supply differentiation, met first the demand for differen-tiation both in the European and in the U.S. markets. To conclude, bothmarket segment definition and the frequency of reviewing it are left at thediscretion of the decision-maker.

Let us define a few terms. α are observed product characteristics, e.g.,horsepower, fuel efficiency, and price P ; ξ are unobserved product charac-teristics, e.g, style, prestige, reputation, and past experience; δ is the meanutility level (numerical measure of consumer preferences) obtained from con-suming a product τ in the segment under consideration

δτ =∑

κ

ατκφ

τκ + ξτ ⇒

δτ = −P τφτP +

K∑κ 6=P

ατκφ

τκ + ξτ τ ∈ {0, 1, . . . , L}, P, κ ∈ K

(24)

13

where φ is the market segment aggregate observed component of utility forthe product characteristic κ. Let us assume that market share depends onlyon mean utility levels:

sτ = sτ (δ), τ ∈ {1, . . . , L}. (25)

At the true values of utilities δ and market shares s, Eq. (25) must holdexactly. If Eq. (25) can be inverted to produce the vector δ = s−1(s), then theobserved market segment shares uniquely determine the means of consumerutility for each product τ . If M is the market segment size, that is, thenumber of consumers in the market, the output quantity sold is

qτ = Msτ (α, ξ, P ). (26)

The consumer may choose to purchase an outside product τ = 0 instead ofL products within the segment. The market share of product τ is then givenby the logit formula

sτ (δ) =eδτ

K∑k=0

eδτk

. (27)

with the mean utility of the outside product normalized to zero,

ln sτ − ln s0 = δj(s) ≡ −PτφP +K∑

κ 6=P

ατκφ

τ + ξτ (28)

so δτ is uniquely identified directly from a simple algebraic calculation involv-ing market shares. Thus the logit suggests a simple regression of differencesin log market shares on (ατ , P τ ). From past observations on the numberof firms within the market segment φ can then be estimated. When (28) is

solved the product attribute κ sensitivity of demand would be equal to∂Ms

∂φκ

.

Having estimated the sensitivity we can then estimate the product attributeκ elasticity of demand.

One needs to make a few back-of-the-envelope calculations before exer-cising Eq. (28). First a definition of market segment size M is needed. Oftenpublicly-held firms are listing their sales in dollar values per segment. In thiscase M is equal to the sum of all sales in the market. If this is not the case,one can acquire volume data from market research firms. The estimation of

14

the market size of the outside product requires some judgement. In the casewhere one wants to exercise Eq. (28) for compact cars, then it is reasonableto assume that the outside product consists of all other market segmentsand the market of used cars. It cannot be overstated, that the calculation ofEq. (28) is very sensitive to the definition of the outside product.

In the case where one needs to take into account the price and attributelevels of a product’s substitutes, Eq. (28) will give unreasonable cross-priceelasticities. The cross-price elasticities represent percentage change in quan-tity demanded by one percent change in the price of competing products(Pindyck and Rubinfeld, 1997, p.32). For the decision-maker interested onsubstitution patterns, the full version (Berry et al., 1995) of the BLP methodaddresses this problem.

We conclude this section by laying out the multidimensional demandcurve of the firm for a given market segment, namely:

qt = θt − λP Pt + λTαα. (29)

Here Eq. (14) is augmented to include the impact of product characteristicson demand. Now we can assemble the entire demand model by including thecalculation of θ from Eq. (23):

qt = θt − λP Pt + λTαα

∆θ(t) = µθ(t)∆t + σθ(t)∆z,

∆z = ε(t)√

∆t,ε ∼ N(0, 1).

(30)

Here are the necessary steps to construct the demand curve of the productdevelopment firm Eq. (30):

• The product market segment needs to be defined first. One can usepublicly-available definitions from market research firms.

• Observations of market shares of competing products and their respec-tive levels of product attributes need to be collected. The time bracketsof these observations ask for special attention. The decision-maker maybe tempted to collect a large sample to improve the fidelity of descrip-tive statistics. However, technological changes may have improved thequality of the product, regardless of consumer preferences.

15

• Estimation of the market size and the size of the outside product comesnext. Based on Eq. (28) product attributes sensitivities of demand λα

are then estimated.

• Having completed the study of the market segment we shift our focusto the firm itself. A time interval where the product attributes arefixed needs to be identified. For the defined time interval observationsof price and demand need to be collected.

• Based on these observations the drift µ and volatility σ need to beestimated. Given that θ follows a stochastic process, calculation ofthe (deterministic) µ and volatility σ need to be treated appropriately.Having estimated µ, σ, λα Eq. (30) can be now constructed. This isthe product demand curve for a specific market segment and will beused for product decisions for the estimated life-cycle.

The estimation of Eq. (30) is decomposable. It captures sensitivities of theconsumer towards product attributes and product demand uncertainty. Thelatter allows us to generate random paths of the future, which allows thedecision-maker to project future cash-flows necessary for investment valua-tion.

Design scenario: Single vehicle decision model

The design of new engines and transmissions allows the firm to market aproduct with improved performance. From Berry et al. (1995) we know themiles per dollar and horsepower to vehicle weight elasticities of demand forChevrolet Cavalier are 0.52 and 0.42, respectively. That is a 10% increase inmiles per dollar and horsepower to vehicle weight ratio will boost demand by5.2% and 4.2% respectively. We will use these elasticities as representativefor the PC segment.

From Eq. (29) we define the demand curve as follows,

q = θ − λP P + λHPw

HPw

+ λM$

M$. (31)

Using the average values of two data points (P , q) we derive the demandcurve for the PC segment:

P = 14943− 0.075q + 2401HPw

+ 805M$. (32)

16

where HP is horsepower, w weight in tens of pounds, and M/$ the numberof ten miles increment one could travel for one dollar.

The fuel economy ratings for a manufacturer’s entire line of passenger carsmust average at least 27.5 miles per gallon (mpg). Failure to comply with thecorporate average fuel economy (CAFE) limit, Li, results in a civil penalty of$5 for each 0.1 mpg the manufacturer’s fleet falls below the standard, multi-plied by the number of vehicles it produces. For example, if a manufacturerproduces 2 million cars in a particular model year, and its CAFE falls 0.5mpg below the standard, it would be liable for a civil penalty of $50 million.Specifically, for each vehicle τ , the penalty (or credit) cCAFE due to CAFEis

CostCAFE = cCAFEq,

cCAFE =

[5× L− fe

0.1

].

(33)

Fuel economy fe is measured in miles per gallon M/G and it is an engineeringattribute computed in terms of the design decisions by the ADVISOR model(Wipke and Cuddy, 1996).

The CAFE regulation can only hurt the firm’s profits, not contribute tothem. The PC segment allows the firm to get credit for less fuel efficientvehicles that yield higher profits. Therefore each PC vehicle sold generatesthe opportunity for the firm to reap those profits. In a following section wewill study the impact of CAFE in a portfolio of two vehicles, one fuel efficientand one fuel inefficient.

The economic profit is

π = θλP

q − 1λP

q2 + 1λP

(λHPw

HPw

+ λM$

M$

)q − c0q − cCAFEq. (34)

where CAFE is equal either to CAFE penalty or to contribution of credit.The optimum is

q∗t =

θλP

+ 1λP

(λHPw

HPw

+ λM$

M$

)− c0 − cCAFE

2λP

. (35)

We portray Eq. (35) in Fig. 3 where monthly profits are depicted onthe acceptable values of engine size in kW and final drive ratio. Profits havebeen calculated based on Eq. (34). We can observe that the lower and upperengine size bounds yield the maximum profit for the firm. The lower enginesize bound corresponds to a fuel efficient vehicle, while the upper boundcorresponds to a fuel inefficient one.

17

50

100

150

2.5

3

3.5

4

4.560

62

64

66

68

70

Engine Size (kW)Final Drive Ratio

Pro

duct

Val

ue (

in m

illio

ns o

f $)

Figure 3: Optimal profitability for different design decisions. Optimal designslie on the points where technical constraints are active.

Given that we have not yet accounted for technical requirements someof the designs may be infeasible. Therefore, Eq. 35 was derived from anincomplete decision model.

We formulate a design problem by taking into account the CAFE reg-ulation of Eq. (33). Then we will solve the problem without Eq. (33) tounderstand the impact of government intervention in the supply of the finalproduct.

18

The decision model is:

maximize πwith respect to: x = {engine size, final drive ratio}

subject to: fuel economy ≥ 27.3mpgt0−60 ≤ 12.5st0−85 ≤ 26.3st40−60 ≤ 5.9s(max acceleration) ≥ 13.0ft/s2

(max speed) ≥ 97.3mph(max grade at 55mph) ≥ 18.1%(5-sec distance) ≥ 123.5ft

(36)

This is the first step towards the synthesis model that will emerge in theconclusion of this chapter.

We will solve the design model Eq. (36) twice. First by taking into accountthe cost of CAFE in computing profits π as stated in Eq. (34). The impactof regulation on the incentives of the firm to supply differentiation will bequantified by estimating profits without taking into account the CAFE cost(or credit) component (Eq. 33).

The optimization algorithm employed to solve Eq. (36) is DIRECT (Jones,2001). In Table 2 the solution in each of the two cases is presented.

The two designs deviate substantially. In the case where CAFE is takeninto account the firm has the incentive to design a fuel efficient vehicle (leftpoint on Fig. 3). The time 0 to 60 and 0 to 80 constraints are active. In theabsence of regulations the fuel economy constraint becomes active. Underthis scenario the vehicle has 80% increased horsepower (right point on Fig.3). The firm realizes a 6% increase in profits. As the case study continue toevolve we will see that this increase is insignificant compared to the profitsrealized by supplying differentiation to the higher segment.

4 Investment decisions

The evaluation criterion of investment decisions is the summation of mon-etary costs and benefits across time. The decision of interest is whether ornot one should sacrifice current consumption for future consumption.

Let us assume that the university professor will discussed earlier is nowconsidering to invest $100,000 in a real estate property that will yield a

19

Table 2: Impact of government regulations on design decisions and profitabil-ity

Solution with Solution withVariable CAFE no CAFE

quantity 29,121 29,994price $15,670 $ 15,735

engine size 97 (HP) 174 (HP)final drive 3.52 2.92

(fuel economy) 37.78 (mpg) 27.30 (mpg)(acceleration 0 to 60) 12.42 (s) 8.02 (s)(acceleration 0 to 80) 26.27 (s) 15.91 (s)

(acceleration 40 to 60) 5.66 (s) 3.82 (s)(5-sec distance) 130 (ft) 182.55 (ft)

(max acceleration) 16 (ft/s2) 16.24 (ft/s2)(max speed) 110.73 (mph) 137.93 (mph)

(max grade at 55mph) 18.41 (%) 25.77 (%)CAFE ($514) n/a

Profit (single period) $63M ($78M including CAFE) $67M

return of $110,000 a year from now. However, she is also considering a tripto Jamaica after correcting the final exam two weeks from now. Assumingthat the return of the real estate investment is riskless, there should be afinancial institution willing to lend her money today. Assuming that theinterest rate at the capital markets is 5 percent, the university professorcould borrow $110,000/1.05, which is equal to $104,762. That is

present value =future value

1 + interest rate. (37)

Therefore, the net present value of the real estate investment is $104,762 -$100,000 = $4,762. That is

net present value =future value

1 + interest rate− investment. (38)

A year from now she will be able to pay off the loan from the return that theinvestment will yield.

20

Now, let us assume that a colleague of the university professor has accessto this investment opportunity but has different preferences towards con-sumption. Unlike her, he prefers to travel to Michigan’s Upper Peninsulaafter the exams and consume the accumulated wealth a year from now. Thepresent value of the investment is $110,000/1.05 = $104,762. Although theyhave different preferences towards consumption they both agree that the netpresent value of the investment is $110,000/1.05 - $100,000 = $4,762.

The case where the two university professors are shareholders of the samefirm is now considered. Let us assume that the required real estate investmentis equal to $10M. Regardless of their preferences, they would both authorizethe management of the firm to invest in this opportunity. The net presentvalue would now be equal to $0.47M. If the two colleagues own 100 shareseach, out of the total 10M outstanding shares, then from this investmentopportunity they would increase their wealth by 100 × $0.47M/10M = $4.7.

In the previous example we did valuation of the real investment oppor-tunity under certainty. The risk of the firm not getting $11M return waszero. Unfortunately, this is rarely the case. Investments in assets other thanTreasury bills encompass many uncertainty factors that translate to risk. Re-laxing the certainty assumption we demonstrate valuation under uncertainty.

Let us assume that the probability the publicly-held firm will gain $11Mfrom the real estate investment is 80%. Many economics analysts have raisedconcerns regarding a possible burst in property prices. In that case the realestate is expected to have a value of $9M. Let us assume that the probabilityof such an event to occur is 20%. The net present value of this investmentis $11M×0.8+$9M×0.2

1.05− $10M = $95K. Therefore, it is advisable that the firm

should undergo this investment.So far, we made the assumption that the firm needs to make the in-

vestment decision today. Let us assume that the real estate company hasenough opportunities available such that the firm can invest a year fromnow. Essentially, this relaxes the assumption of absence of flexibility in thedecision-making process. The firm can adopt a “wait and see” approach.A year from now the decision-maker would be able to see if the economistswere right in their projections regarding the assumed property-price bub-ble. The valuation question now turns to the following: How much is theoption to invest a year from now worth to the firm? In the case where theanalysts are right, then the net present value is $9M−$10M

1.052 = −$900K. Inthe case where, say, the chairman of the Federal Reserve Board is right andproperty prices will indeed rise, then the net present value of the investment

21

is $11M−$10M1.052 = $900K. In both cases we have a three-period investment

(today, period one and period two) and therefore we need to discount twicethe expected profit to obtain the net present value. Taking into accountthe probability of each likely event the value of this option is equal to 0.8× $900K + .2 × (-$900K) = $540K. Given that the firm could exercise theoption today and get $95K, the price of the option is equal to $540K - $95K= $445K. If the real estate company asked for a fee of more than $445K toreserve the right for the publicly-held firm to decide a year later whetherto exercise the option to buy, then the firm should forego the investmentopportunity.

Let us now assume that the university professor was hired by the realestate firm as a consultant. Her project is to find the net present value of aninvestment decision the firm is facing. In her personal investment decisionwe assumed that the discount factor or opportunity cost of capital is equal to5% (equal to the rate of a Treasury bill). What is the appropriate discountrate for the firm’s investment decision?

Let us assume that the stock of the firm is listed in the stock exchange.By using historical data of the past performance of the stock one can exercisethe following regression:

r = a + βrm + ε ⇒E(r) = a + βE(rm)

(39)

where r is the return of the firm’s stock, rm is the return of the capitalmarkets as proxied by the S&P500 index, ε is the error of the regression,a and β are constants and E denotes expectation. The slope β models thesensitivity of the firm’s stock return for 1% variation in the market return.This can be represented as follows:

β =Cov(r, rm)

Var(rm), (40)

where Cov starts for covariance and Var for variance.Based on certain assumptions that concern the behavior of the investors

and the conditions for perfect and competitive capital markets (Trigeorgis,1998, p.44) it is assumed that any investor can invest a portion (β%) of herwealth at the S&P500 with E(rm), so that its covariance with the capitalmarkets is unity (β = 1). She can then borrow (1−β%) at a “risk-free” raterf so that its covariance with the capital markets is 0 (β = 0). Her return

22

would beE(r) = βE(rm) + (1− β)rf orE(r) = rf + β[E(rm)− rf ].

(41)

The intercept of Eq. (41) is the risk-free interest rate rf and its slope themarket risk premium [E(rm) − rf ]. Therefore an investor should be willingto hold a stock with a particular β only if is compensated by a return equalto Eq. (41).

In our example, the expected risk premium on the stock of the firm shouldbe equal to

E(r)− rf = β[E(rm)− rf ] or

β =E(r)−rf

E(rm)−rf

(42)

which is called the Capital Asset Pricing Model (CAPM). The CAPM isessentially modeling how much more (or less) risky is the stock of a firmrelative to the market. If the β of the firm’s stock is two that means that itis twice as risky as the expected market risk premium [E(rm) − rf ]. The βof the stock is calculated based on Eq. (39).

If the firm does not have any debt then the appropriate discount factorfor the investment that the firm is considering should be calculated basedon Eq. (42). In the case were the firm does have debt then this makes theinvestment more risky. Assuming the firm has an amount of debt equal to D,and market capitalization (number of shares multiplied by the stock price)equal to E, where E stands for equity, then the total value of the firm isequal to (D + E). Then the Weighted Average Cost of Capital (WACC) isequal to

WACC =D

Vrd +

E

Vr, (43)

where r is the cost of equity calculated by Eq. (42) and rd is the current bor-rowing rate of the firm. If the riskiness of the investment under considerationis that of the firm then the university professor, and now consultant, shoulduse Eq. (43) as the discount factor to evaluate this investment. SubstitutingEq. (42) in Eq. (43)

WACC =D

Vrd +

E

V(rf + β[E(rm)− rf ]). (44)

This would be the situation we assume in the studies described in the fol-lowing chapters.

23

The investment decision that the firm is facing can be summarized in thefollowing unconstrained optimization model

maximize {net present value(future cash flows,investment cost, risk)}

with respect to {invest now or,invest never or,invest later}.

(45)

Design scenario: Portfolio decision model

We now expand the design scenario to consider an automotive manufacturingfirm that markets premium-compact (PC) and full-size sport utility (SUV)vehicles. This market segmentation follows the J.D. Power classification forvehicles in the United States.

The firm wishes to design new engines and transmissions for both PCand SUV segments. The PC and SUV segments are low and high profitmargin segments, respectively. The Energy Policy and Conservation Act of1975 required passenger car and light truck manufacturers to meet corporateaverage fuel economy (CAFE) standards applied on a fleet-wide basis. Thereare K units of monthly capacity currently in place for both segments, and soK is fixed, representing a capacity constraint. It is assumed that this capacityis not expandable. The decision-maker faces the following decisions: Howshould the units of capacity be allocated between the two segments in orderto maximize the firm’s value? What should the performance specificationsfor engines and transmissions be and how do these specifications affect theresource allocation decision? How much is this investment worth to the firm’sstock owners?

We formulate the following decision model as

maximize NPVwith respect to {xτ ,qτ}, τ = 1, 2

subject to2∑

τ=1

qτ = K

2∑τ=1

CostτCAFE ≤ 0

gτ (xτ ) ≤ 0, τ = 1, 2,

(46)

24

where NPV is the net present value, q is the vector of supply decisions(monthly production quantities q1, q2), xτ is the vector of engineering designdecisions (engine sizes and final drive ratios) for each vehicle. The equalityis a production constraint that fixes the total available production, the firstinequality is an enterprise constraint that will not allow Corporate AverageFuel Economy (CAFE) penalties to be paid for the selected product mix, andthe two vector inequalities are engineering constraints. Eq. (46) is a specialcase of the design portfolio problem (Eq. 83) where government intervenesin the decision-making process of the firm. We will use the index τ = 1, 2 forthe two products, but we will also use the subscripts PC and SUV insteadof τ = 1 and τ = 2, respectively when convenient.

Net present value is the aggregation of future monthly cash flows or profitsπτ minus the investment cost I over the life H of product τ with a weightedaverage cost of capital:

NPV = −I +

∫ H

0

πτe−(WACC)tdt. (47)

The monthly profit is defined as

πτ = P τqτ − Cτ (48)

where P τ is the price, qτ the quantity, and Cτ the average total cost ofproducing vehicle τ .

We will first derive the model for the problem where instead of the NPVwe maximize simply the monthly profits

∑πτ .

In Section 3 using price and quantity data points from years 2000 and1999 we estimated the demand curve for the PC segment. For the GM SUVsegment (Chevrolet Tahoe and Suburban, GMC Suburban/Yukon and YukonXL) we used data points from the years 1999 and 1998 (P1/98, q1/98), (P1/99, q1/99)(WardAuto, 1998-00) where no major design change took place as well, find-ing the elasticity to be equal to −2.3.

We assume that for each segment between the two years there was no ma-jor change in consumer’s income, product advertising, product informationavailable to consumers, price and quality of substitutes and complementarygoods, and population (Clyde, 2001). We further assume that the two goodsare independent, namely, a change in the price of the compact car has noeffect on the quantity demanded for the sport utility vehicle.

For demonstration purposes we assume that the horsepower to weightelasticity of demand of the traditional luxury segment is close to the SUV

25

Table 3: Price and quantities for the PC and SUV segments

P1/99 q1/99 P1/00 q1/00

PC $14,512 43,507 $ 15,015 (’99 adj.) 36,755P1/98 q1/98 P1/99 q1/99

SUV $28,628 24,658 $29,596 (’98 adj.) 22,813

one. In Berry et al. (1995) the Cadillac Seville horsepower to weight elasticityof demand is found to be 0.09. In this segment the miles per dollar elasticity ofdemand is found to be close to 0. This essentially means that the customer ofthat segment is satisfied with the current level of fuel economy performance.

The demand curves for both segments are

P PC = 14943− 0.075qPC + 2401HPw

+ 805M$

P SUV = 40440− 0.525qSUV + 2071HPw

(49)

where HP is horsepower, w weight in tens of pounds, and M/$ the numberof ten miles increment one could travel for one dollar.

Assuming a linear relationship between cost and output,

Cτ = c0τqτ , (50)

Eq. (73) becomes

πτ =

(θτ

λτP

− 1

λPτqτ +

1

λPτ(λτ

α)T ατ

)qτ − c0q

τ . (51)

Eq. (50) assumes that the marginal cost is constant, that is, for every unitincrease in output, the variable cost increases by c0, which is set at $13,500and $18,5001 for the PC and SUV segments, respectively (Pindyck and Ru-binfeld, 1997, p. 236). We have assumed that the firm is operating at itsminimum efficient scale (Besanko et al., 2000, p.73).

For the period 1992 to 2001 the CAFE penalty was non-positive andapproximately zero for DaimlerChrysler, Ford, and General Motors (WSJ,1/28/2002). This is represented as follows:

2∑τ=1

CostτCAFE ≤ 0. (52)

1These are estimates from private discussions with automotive industry professionals.

26

Note that the CAFE regulations can only hurt the firm’s profits, not con-tribute to them. They function as a set of internal taxes (on fuel inefficient ve-hicles) and subsidies (on fuel efficient vehicles) within each firm (Koujianou-Goldberg, 1998).

Hence, the cost of each product, Eq. (50), is modified to be

Cτ = c0τqτ + Costτ

CAFE

= (c0τ + cτ

CAFE)qτ

= ξτqτ

(53)

where ξτ , c0τ + cτ

CAFE, and cτCAFE is equal either to CAFE penalty or

to contribution of credit to the portfolio. For example, if the PC vehiclegenerates $3M of credit but the penalty incurred to the SUV is $2.5M thenCostPC

CAFE would be equal to $2.5M. For a portfolio of n products the valueof Costτ

CAFE is then redefined as:

cτCAFEqτ =

cτCAFEqτ ,

∑n1 cτ

CAFE ≥ 0

cτCAFEqτ +

U(cτCAFE)cτ

CAFEqτ∑n1 U(cτ

CAFE)|cτCAFE|qτ

n∑1

cτCAFEqτ

∑n1 cτ

CAFEqτ < 0.

(54)where U(cτ

CAFE) is equal to 1 when cτCAFE is negative and 0 when non-

negative.As mentioned early in this section, the microeconomic model suggests

that in each monthly period the firm should produce the quantity that max-imizes total profit during that period:

maximize2∑

τ=1

πτ =2∑

τ=1

(θτ

λτP

− 1

λτP

qτ +1

λτP

(λτα)T ατ

)qτ − ξτqτ

with respect to {qτ}, τ = 1, 2

subject to2∑

τ=1

qτ = K

2∑τ=1

CostτCAFE = c1

CAFEq1 + c2CAFEq2 ≤ 0.

(55)For positive quantities of production qτ , Eq. (55) can be solved analyti-

27

cally and the global optimum is

qPC∗= max(

( θPC

λPCP

− θSUV

λSUVP

)+( 1

λPCP

(λPCαPC )T αPC− 1

λSUVP

(λSUVαSUV )T αSUV )+2 1

λSUVP

K−(ξPC−ξSUV )

2( 1

λPCP

+ 1

λSUVP

),

KcSUVCAFE

cSUVCAFE − cPC

CAFE

),

qSUV ∗= K − qPC∗

,(56)

where( θPC

λPCP

− θSUV

λSUVP

)+( 1

λPCP

(λPCαPC )T αPC− 1

λSUVP

(λSUVαSUV )T αSUV )+2 1

λSUVP

K−(ξPC−ξSUV )

2( 1

λPCP

+ 1

λSUVP

)is the

interior optimum, andKcSUV

CAFE

cSUVCAFE − cPC

CAFE

is the boundary optimum.

From the derivation above we see that the enterprise-wide problem Eq. (46)can be partitioned to production and design problems (see also section Sec-tion 5). The production problem is Eq. (55) above, which can be solvedseparately for q∗. With this partial optimization (Papalambros and Wilde,2000, p. 93) πτ in Eq. (47) can be replaced by πτ ∗ computed from Eq. (56),and the problem of Eq. (46) is reduced to Eq. (57) for fixed investment costs.Of course, q∗ depends on the design variables via the quantities (λτ

α)T ατ andcτCAFE.

maximize NPV (x) = −I +

∫ H

0

2∑τ=1

πτ ∗e−(WACC)tdt

with respect to xsubject to gτ (x)τ ≤ 0, τ = 1, 2.

(57)

The firm’s product demand (qτt )D for product τ at time t is expressed as

a product of the two sources of uncertainty defined above, namely, marketproduct demand Q and market share M :

(qτt )D =

{0 0 ≤ month ≤ 24Qτ

t Mτt 25 ≤ month ≤ 84.

(58)

(qτt )D is a 1× 84 vector and represents a random walk in the future. During

the first 24 months of product development and production start-up time wehave null sales.

If the actual demand exceeds the optimal capacity of the plant the firmwill sell only at capacity. Thus, if the actual demand (qτ

t )D is less than the

28

optimal capacity qτ ∗ of the plant then the firm will supply only as much asthe demand permits. That is, the actual supply (qτ

t )S is given by

(qτt )S =

{(qτ

t )D if (qτt )D < qτ ∗,

qτ ∗ otherwise.(59)

The assumption is that the firm does not possess flexibility in adjustingcapacity. If that were not the case then an appropriate theory of capitalbudgeting would have been needed to model the resource allocation decision.

We assume that future cash flows generated by a product’s commercial-ization are only imperfectly predictable from the current observation. Theprobability distribution is determined by the present, but the actual path re-mains uncertain (Dixit, 1992). We consider product demand and the firm’smarket share as the two main sources of uncertainty.

To describe future product demand we assume that the automotive prod-uct demand Q follows Geometric Brownian motion. Seasonality (Tseng andBarz, 1999) and life-cycle considerations (Bollen, 1999) can be also taken intoaccount.

∆Qt = µQt∆t + σQt∆z

∆z = ε√

∆tε ∼ N(0, 1)

(60)

Here µ∆t and σ√

∆t are the expected value and the standard deviation,respectively, of ∆Qt

Qtin ∆t. To simulate the path followed by Q we divide

the life of the source of uncertainty into 73 monthly intervals from January1995 until January 2001. The value of Q at time ∆t (i.e., February 2001)is calculated from the initial value of Q (i.e., January 2001), the value attime 2∆t is calculated from the value at time ∆t and so on (Hull 2000).One simulation trial involves constructing a complete path for Q using 73random samples from a normal distribution. Data provided by J.D. Power& Associates for the period between January 1995 and January 2001 (seeTable 4) have been employed for the estimation of the expected growth rateµ and volatility σ.

The Ornstein-Uhlenbeck or mean-reverting process is used to model mar-

29

Month Units Market ShareJan-95 135,928 26.25%Feb-95 154,139 26.19%Mar-95 176,348 30.82%

......

...Jan-01 163,253 25.04%

Table 4: Market share data of a major US automotive manufacturer for thepremium-compact segment

ket share uncertainty M .

∆Mt = α(Mt −Mt)∆t + σ′∆z

∆z = ε√

∆tε ∼ N(0, 1)

(61)

Here α is the speed of reversion, M is the “normal” level of M , i.e., the levelto which M tends to revert. By running the nonlinear regression (Dixit andPindyck 1994)

∆M = p + bMt−1 + εt, (62)

from data provided by J.D. Power & Associates (see Table 4) for the periodbetween January 1995 and January 2001 we estimate

M = p̂/b̂,

α̂ = −log(1 + b̂),

σ̂′ = σ̂′ε

√log(1+b̂)

(1+b̂)2−1,

(63)

where σ̂ε is the standard error of the regression.As we noted in in the presentation of Eq. (23) in the mathematical finance

and economics literature (Bollen, 1999; Pindyck, 1988) product demand un-certainty is also described by the intercept θ of Eq. (32). That essentiallydescribes random shifts of the demand curve – with the same slope λ. Thisapproach would require in addition to historical demand data, pricing datathat are unavailable. In the present treatment we decided not to use hypo-thetical data, thus maintaining the credibility of actual results in the contextof the assumptions made.

30

Collecting Eqs. (51), (53), (54), (56) and (59) for the PC segment we getthe complete calculation of the monthly profit. A similar set of equationsholds for the SUV segment.

πPCt

∗= P PC(qPC

t )S − ξ(qPCt )S

(qPCt )S = min{(qPC

t )D, qPC∗},

qPCt

∗= max(

( θPC

λPC − θSUV

λSUV )+2 1

λSUV K+( 1

λPC λPCα

TαPC+ 1

λSUV λSUVα

TαSUV )−(ξPC−ξSUV )

2( 1

λPC + 1

λSUV ),

KcSUVCAFE(xSUV )

cSUVCAFE−cPC

CAFE),

ξPC = c0PC + cPC

CAFE,

cτCAFEqτ =

cτCAFEqτ ,

∑m1 cτ

CAFEqτ ≥ 0

cτCAFEqτ +

U(cτCAFE)cτ

CAFEqτ∑m1 U(cτ

CAFE)|cτCAFE|qτ

m∑1

cτCAFEqτ

∑m1 cτ

CAFEqτ < 0.

(64)From Eq. (51) after substituting qτ for (qτ

t )S using Eq. (59) and Cτ usingEq. (53) we get the monthly profit πt

τ ∗ over the eighty four month salesperiod for product τ .

πτt∗ = P τ (qτ

t )S − (ξτ )(qτt )S. (65)

During the first 24 months we have null profits.Recall that we assume the decision to develop the new engines and trans-

missions has already been made, and so the decision facing the firm now isone of resource allocation. Upon determining the optimal production ratio,this decision will be implemented immediately. Hence, the decision containsno embedded real option, which simplifies the net present value calculation.The time period T for both products is estimated to be seven years or eightyfour months, and includes the product development, production start-up,and sales periods. The present value, PV , of discounted future payoffs πτ

t∗

is formally represented as an integral over the space of sample paths of theunderlying stochastic processes X and Y

PV ≈

n∑1

[∫ H

0

(2∑

τ=1

πτtn∗)e−(WACC)t dt

]n

(66)

where n = 100, 000. The exponent r, the weighted average cost of capital, isestimated as in Eq. (43)

WACC = rd(1− tc)D

E + D+ re

E

E + D(67)

31

where rd is firm’s cost of debt, tc tax rate, D market value of debt, E equity,and re is the cost of equity estimated by the Capital Asset Pricing Model(CAPM) (Brealey and Myers (2000) and Section 4):

re = rf + β(rm − rf ), (68)

where rf is the risk-free rate, (rm − rf ) the market risk premium, β firmstock’s sensitivity to fluctuation of the market as a whole. Using the valuesin Table 5 we estimate the weighted average cost of capital of a publiclytraded US automotive manufacturer to be 9.4%. By using a single r for all

Table 5: Parameters values in Eqs. (67, 68)

tc 36%D $81.3Brf 6.7%β 1.1

Long term interest $2.8BLong term Debt $31.3B

Number of Shares 756MStock price $58

Market risk premium 8.4%

84 months we are making the assumptions that the firm’s capital structure(the debt to equity ratio D/E) and the risk of the firm to its shareholderswould remain the same for all 84 months.

To estimate the PV we generate a 100,000 random walks resulting in a100, 000×84 matrix. Discounting all future payoffs πτ

tn∗ across the probability

space we get a 100, 000× 1 matrix. The present value is the average of those100,000 numbers (see also Fig. 4). Note that Eq. (66) does not take intoaccount working capital.

Subtracting the fixed capital investment needed (I = $3B) we calculatethe net present value

NPV = PV − I

≈ −I +

n∑1

[∫ H

0

(2∑

τ=1

πτtn∗)e−WACCt dt

]n

.

(69)

32

Future Future Cash Cash FlowsFlows

TimeTime

Present Value =Present Value =

Future Future Cash Cash FlowsFlows

TimeTime

Present Value =Present Value =

Future Future Cash Cash FlowsFlows

TimeTime

Present Value =Present Value =

…Iteration 2Iteration 2

Iteration 1Iteration 1

Iteration 10000Iteration 10000

EXPECTED EXPECTED PRESENTPRESENT

VALUEVALUE==

Today Month 1 … …… Month 8425 26 27

AVERAGEAVERAGEToday Month 1 … …… Month 8425 26 27

Today Month 1 … …… Month 8425 26 27

Figure 4: Expected present value

Other investment costs are ignored; for example, the cost of building theproduction facility plant is considered a sunk cost because we assume theplant has already been built.

The NPV expression in Eq. (69) is the stochastic calculation of the ob-jective in the model of Eq. (57). This NPV criterion is based on the optimaleconomic conditions (Bollen (1999); Trigeorgis (1998, p.291)) computed inEq. (56), the uncertainty of future cash flows Eqs. (60) and (61), and theengineering performance Eq. (33).

Bounds on vehicle performance attributes define the constraints for eachproduct and its corresponding market segment. These “engineering” con-straints are expressed in terms of the design variables using the ADVISOR

33

program, cited earlier.The model in Eq. (70) involves four variables and sixteen constraints

(eight each for the PC and SUV segments of the engineering design model).The complete model of Eq. (70) is now assembled using the expressions de-rived in the preceding sections.

maximize −$3B +

n∑1

[∫ T

0

(2∑

τ=1

πτtn∗)e−(WACC)t dt

]n

with respect to {x}subject to (final drive)PC ≥ 2.5

(final drive)PC ≤ 4.5(final drive)SUV ≥ 2.5(final drive)SUV ≤ 4.5(engine size)PC ≥ 50(kW)(engine size)PC ≤ 150(kW)(engine size)SUV ≥ 150(kW)(engine size)SUV ≤ 250(kW)(fuel economy)PC ≥ 27.3(mpg)(acceleration 0 to 60)PC ≤ 12.5(s)(acceleration 0 to 80)PC ≤ 26.3(s)(acceleration 40 to 60)PC ≤ 5.9(s)(5-sec distance)PC ≥ 123.5(ft)(max acceleration)PC ≥ 13(ft/s2)(max speed)PC ≥ 97.3(mph)(max grade at 55mph)PC ≥ 18.1(%)(fuel economy)SUV ≥ 12.8(mpg)(acceleration 0 to 60)SUV ≤ 9.8(s)(acceleration 0 to 80)SUV ≤ 22.8(s)(acceleration 40 to 60)SUV ≤ 5.0(s)(5-sec distance)SUV ≥ 154.5(ft)(max acceleration)SUV ≥ 15.4(ft/s2)(max speed)SUV ≥ 100.4(mph)(max grade at 55mph)SUV ≥ 18.6(%)

(70)

We now present the results obtained by solving the valuation problemposed in Eq. (70). The divided rectangles (DIRECT) optimization algo-rithm (Jones, 2001) was used. DIRECT can locate global minima efficiently

34

without derivative information, when the number of variables is small, as inthis case. It is often inefficient at refining local minima, and so a sequen-tial quadratic programming algorithm was combined with DIRECT to findsolutions in a more efficient manner.

Optimization problems were solved for two different production capaci-ties: 50,000 and 20,000. The optimal solutions found are shown in Table 6.

Prior to interpreting the results recall that Eq. (70) is a portfolio designproblem with scarce resources. That is, the sum of the optimal quantities ofeach segment is greater or equal to the manufacturing resources of the firm.One can interpret the results by paying attention to production quantityq, the regulatory penalty (or credit) per vehicle, and the vector of designvariables (i.e., engine size, final drive). Specifically:

• At the small capacity level of 20,000, the small quantity of producedcompact vehicles calls for high fuel efficiency of the compact car andaverage performance of the sport utility vehicle.

• At the high capacity level of 50,000, the quantity of compact vehicleshas enough scale to mitigate the regulatory penalty, while improvingthe performance of the sport utility vehicle by a 28% increase in enginesize.

From a public policy point of view, the CAFE regulation can be in-terpreted as an active constraint to consumer preferences. As long as theconsumer asks for more horsepower, and the producer can materialize thispreference in terms of profit, the fuel efficiency regulatory constraint wouldbe active. A question of interest is whether or not new technologies can makethe CAFE constraint inactive and at what cost.

We proceed in the next section with the synthesis of the economic, in-vestment and engineering decision models.

35

5 The design decision model of the product

development firm

For a single segment and a single product affecting demand, at time instancet the inverse demand curve is from Eq. (30)

Pt =θt

λP

− 1

λP

qt +1

λP

λTαα. (71)

Revenue Rt is equal to price times quantity

Rt =θt

λP

qt −1

λP

q2t +

1

λP

λTααqt. (72)

and profit πt is equal to revenue minus cost

πt =θt

λP

qt −1

λP

q2t +

1

λP

λTααqt − Ct(qt)qt, (73)

where the cost is time dependent due to learning curves and economies ofscale and is also a function of the production quantity. Let us assume thatthe cost function is a quadratic function of the quantity produced (Pindyck(1988); Pindyck and Rubinfeld (1997, p.239); Trigeorgis (1998, p.290); Bollen(1999))

C(qt) = c0 + c1qt + c2q2t (74)

Product observed attributes α are either functions of design decisions α(xd)or design decision themselves α(xd) = xd. Eq. (73) then becomes

πt =θt

λP

qt −1

λP

q2t +

1

λP

λTααqt − c0qt − c1q

2t − c2q

3t . (75)

The profit maximization problem is as follows

maxqt

πt =θt

λP

qt −1

λP

q2t +

1

λP

λTααqt − c0qt − c1q

2t − c2q

3t . (76)

Setting the first derivative with respect to quantity equal to zero, the opti-mum is found to be

q∗t =2(c1 + 1

λP) +

√4(c1 + 1

λP)2 + 12c2(

θt

λP+ 1

λPλT

αα− c0)

−6c2

. (77)

36

The second derivative of Eq. (76) is negative and Eq. (77) maximizes profits.Let us analyze further Eq. (77). Based on the random values of θt the

decision-maker will respond with q∗t . One of the assumptions here is thatproduct design decisions are irreversible. That means product observableattributes α cannot be substantially altered once design decisions x are made.However as one can see from Eq. (77) x have partially determined the optimalproduction decisions q∗t . Under what condition the design decisions will notaffect q∗t ? One can easily infer that from Eq. (77). Whenever 1

λPλT

αα equalsto c0 then q∗t is not affected by x.

The latter can be explored further. Let us assume a scalar α. Also,

1λP

λα = 1∆Q∆P

∆Q∆α

, or1

λPλα = ∆P

∆Q∆Q∆α

.(78)

∆P∆Q

∆Q∆α

denotes how much a change in the product attribute will affect product

demand that in turn affects price. Therefore, the product (∆P∆Q

∆Q∆α

)α measuresthe demand side for product differentiation. The cost coefficient c0 is the onlycoefficient in Eq. (74) that is independent of the unit amount of production q.It simply denotes how much the first unit of production costs. It is reasonableto assume that is affected from design decisions x. c0(x) measures the supplyside of differentiation. Therefore, when the capacity to supply differentiationequals to the demand for differentiation engineering decisions do not affectbusiness ones:

∆P

∆Q

∆Q

∆αα(x) = c0(x). (79)

Under Eq. (76) and in the case where the market conditions are such that thefirm can pass all the cost of differentiation to a price increase then businessand engineering decision-making are independent of each other.

Let us assume that the time period as of when the design decisions arerealized is known. The present value of the future cash flows stream πt

realized from the commercialization time t0 = 0 until the end of the life-cycleH is

PV = E

(H∑0

θt

λPq∗t − 1

λP(q∗t )

2 + ( 1λP

)λTαα(x)q∗t − c0(x)q∗t − c1q

∗t2 − c2q

∗t3

(1 + WACC)t

),

(80)where t ∈ {0 . . . H}. The product life-cycle period is (H−0). The net presentvalue is the difference between the present value future cash flows and the

37

required investment cost.

NPV = −I+E

(H∑0

θt

λPq∗t − 1

λP(q∗t )

2 + 1λP

λTαα(x)q∗t − c0(x)q∗t − c1q

∗t2 − c2q

∗t3

(1 + WACC)t

).

(81)At the product commercialization time t optimal design decisions need to bemade

maximize −I + E

(H∑0

θt

λPq∗t − 1

λP(q∗t )

2 + 1λP

λTαα(x)q∗t − c0(x)q∗t − c1q

∗t2 − c2q

∗t3

(1 + WACC)t

)with respect to x

subject to g(x) ≤ 0(82)

where g(x) are functional engineering constraints, which are part of a stan-dard engineering decision model.

For initial resources R the firm will form a portfolio of n products τ ∈{1, 2, . . . , n}. We let the portfolio be defined by w = w1, w2, . . . , wn, whichdenotes the resources allocated to each product, and corresponding designsx = x1,x2, . . . ,xn. The portfolio problem of the firm is

maximize wTnNPVn

with respect to w,xsubject to

∑wn = 1

wTn In ≤ R

g(x) ≤ 0.

(83)

6 Conclusion

Product design involves different disciplines that are linked with each other.The designer needs to acknowledge this linking for optimal decision-making.Engineering decisions and investment decisions are linked with respect totime. Both are long-term irreversible decisions (with respect to economicdecisions) that will affect the firm during the product life-cycle. Engineeringdecisions and economic decisions are linked with respect to customers’ pur-chasing choice. Price, technical characteristics, macroeconomic conditionsand demographics will play a role in shaping consumer’s behavior. The keyconcept that emerges is that product technical characteristics are outcomes

38

of engineering decisions and need to be treated simultaneously with all otherenterprise factors affecting final choice. Technical characteristics will remainlargely unchanged during the life cycle, while it is most likely that economicdecisions and macroeconomic conditions will change. Therefore, engineeringdecisions will affect economic decisions across the life-cycle.

One can argue that the uncertainty present in economics and finance pre-vents us from relying seriously on results of models such as those presentedabove. However, as economist Arnold Harberger puts it: “...I think we musttake it for granted that our estimates of future costs and benefits (particu-larly the latter) are inevitably subject to a fairly wide margin of error, in theface of which it makes little sense to focus on subtleties aimed at discrim-inating accurately between investments that might have an expected yieldof 101

2percent and those that would yield only 10 percent per annum. As

the first order of business we want to be able to distinguish the 10 percentinvestments from those yielding 5 or 15 percent...(Harberger, 1972)”. Thistype of thinking is consistent with that of engineers who build models ofphysical artifacts to predict their behavior under uncertainty.

We conclude by recalling the hypothesis stated in a seminal paper of Sta-bell and Fjeldstad (1998). According to their proposition firms that are fo-cusing on problem-solving activities can be modeled as “value shops”: Theirflow of activities is not linear, but iterative between activities and cyclicalacross the activity set. In this chapter we validated their hypothesis. Opti-mal decisions in one discipline depend on the optimal decisions of the other.Therefore product design is not a linear forward-looking process but ratheran iterative process among disciplines.

Acknowledgments

Acknowledgments The work described here was partially supported by theAntilium Project of the H.H. Rackham Graduate School, the AutomativeResearch Center and the General Motors Collaborative Research Laboratory,all at the University of Michigan. This support is gratefully acknowledged.The opinions expressed are only those of the authors.

39

References

B. Allen. A toolkit for decision-based design theory. Journal of EngineeringValuation and Cost Analysis, 3(1):85–105, 2000.

B. Allen. On the aggregation of preferences in engineering design. In ASMEDETC, Pittsburgh, PA, USA, 2001. DETC2001/DAC-21015.

S. Berry. Estimating discrete choice models of product differentiation. RANDJournal of Economics, 23(2):242–262, 1994.

S. Berry, J. Levinsohn, and A. Pakes. Automobile prices in market equilib-rium. Econometrica, 63(4):841–890, 1995.

S. Berry, J. Levinsohn, and A. Pakes. Differentiated products demand sys-tems from a combination of micro and macro data: The new car market.Technical Report 6481, National Bureau of Economic Research, 1998.

D. Besanko, D. Dranove, and M. Shanley. The economics of strategy. JohnWiley and Sons, New York, NJ, USA, 2 edition, 2000.

N.P.B. Bollen. Real options and product life cycles. Management Science,45(5):670–684, May 1999.

R.A. Brealey and S.C. Myers. Principles of Corporate Finance. McGraw-Hill,New York, NY, USA, 2000.

W. Chen, K. Lewis, , and L. Schmidt. Decision-based design: An emergingdesign perspective. Journal of Engineering Valuation and Cost Analysis,3(1):57–66, 2000.

P. Clyde. Business Economics 501. University of Michigan Business School,Ann Arbor, MI, USA, 2001.

H.E. Cook. Product Management: Value, Quality, Cost, Price, Profits, andOrganization. Chapman & Hall, Hingham, MA, USA, 1997.

A.B. Cooper, P. Georgiopoulos, H.M. Kim, and P. Y. Papalambros. Ana-lytical target setting: An enterprise context in optimal product design. InASME DETC, Chicago, IL, USA, 2003. DETC2003/DAC-48734.

40

A. Dixit. Investment and hysteresis. The Journal of Economic Perspectives,6(1):107–132, 1992.

J. Donndelinger and H.E. Cook. Methods for analyzing the value of auto-mobiles. In SAE International Congress, Detroit, MI, 1997. SAE Paper970762.

P. Georgiopoulos. Enterprise-wide Product Design: Linking Optimal DesignDecisions to the Theory of the Firm. PhD thesis, University of Michigan,Ann Arbor, MI, USA, 2003.

P. Georgiopoulos, Jonsson M., and P.Y. Papalambros. Linking optimal designdecision to the theory of the firm: The case of resource allocation. Journalof Mechanical Design, forthcoming 2005.

Robert M. Grant. Contemporary Strategy Analysis. Blackwell, 1998.

X. Gu, J.E. Renaud, L.M. Ashe, S.M. Batill, Budhiraja A.S., and L.J. Kra-jewski. Decision-based collaborative optimization. Journal of MechanicalDesign, 124(1):1–13, March 2002.

A.C. Harberger. Project Evaluation. MacMillan, London, GB, 1972.

G.A. Hazelrigg. “A framework for decision based engineering design”. Jour-nal of Mechanical Design, 120:653–658, 1998.

G.A. Hazelrigg. Special edition on decision-based design. Journal of Engi-neering Valuation and Cost Analysis, 2000.

F.O. Irvine. Demand equations for individual new car models estimatedusing transaction prices with implications for regulatory issues. Southern-Economic-Journal, 49(3):764–782, 1983.

D.R. Jones. The DIRECT Global Optimization Algorithm, volume 1, pages431–440. Kluwer Academic Publishes, Dordrecht, The Netherlands, 2001.

H.M. Kim, D.K.D. Kumar, W. Chen, and P. Y. Papalambros. Target feasibil-ity achievement in enterprise-driven hierarchical multidisciplinary design.In 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Con-ference, Albany, NY, USA, 2004. AIAA-2004-4546.

41

P. Koujianou-Goldberg. The effects of the corporate average fuel economystandards in the automobile industry. Journal of Industrial Economics,pages 1–33, March 1998.

H. Li and S. Azarm. Product design selection under uncertainty and withcompetitive advantage. Journal of Mechanical Design, 122(4):411–418,December 2000.

H. Li and S. Azarm. An approach for product line design selection underuncertainty and competition. Journal of Mechanical Design, 124(3):385–392, September 2002.

D. G. Luenberger. Investment Science. Oxford University Press, 1998.

J. Markish and K. Willcox. Multidisciplinary techniques for commercial air-craft system design. In 9th AIAA/ISSMO Symposium on MultidisciplinaryAnalysis and Optimization, volume 2, Atlanta, GA, USA, 2002a. AIAA-2002-5612.

J. Markish and K. Willcox. A value-based approach for commercial aircraftconceptual design. In 23rd ICAS Congress, Toronto, Canada, 2002b.

M. Marston, J. K. Allen, and F. Mistree. The decision support problemtechnique: Integrating descriptive and normative approaches in decision-based design. Journal of Engineering Valuation and Cost Analysis, 3(1):107–129, 2000.

M. Marston and F. Mistree. An implementation of expected utility theoryin decision based design. In ASME DETC, Atlanda, GA, USA, 1998.DETC1998/DTM-5670.

G. P. McConville and H. E. Cook. Estimating the value trade-off betweenautomobile performance and fuel economy. In SAE International Congress,Detroit, MI, 1996. SAE Paper 960004.

J. Michalek, F. Feinberg, and P. Y. Papalambros. Linking marketing and en-gineering product design decisions via analytical target cascading. Journalof Product Innovation Management: Special Issue on Design and Market-ing in New Product Development, forthcoming 2004.

42

A. Nevo. A practitioners guide to estimation of random coefficients logitmodels of demand. Journal of Economics & Management Strategy, 9(4):513–548, 2000.

P.Y. Papalambros and D. J. Wilde. Principles of Optimal Design. CambridgeUniversity Press, New York, NY, USA, 2 edition, 2000.

R. S. Pindyck. Irreversible investment, capacity choice, and the value of thefirm. The American Economic Review, 78(5):969–985, 1988.

R.S. Pindyck and D.L. Rubinfeld. Microeconomics. Prentice-Hall, New York,NY, USA, 4th edition, 1997.

M.J. Scott and E.K. Antonsson. Arrow’s theorem and engineering designdecision making. Research in Engineering Design, 11:218–228, 1999.

C.B. Stabell and Ø. D. Fjeldstad. Configuring value for competitive advan-tage: On chains, shops, and networks. Strategic Management Journal, 19:413–437, 1998.

D. Teece. Profiting from technological innovation: Implications for inte-gration, collaboration, licensing and public policy. Research Policy, 15:285–305, 1986.

D.L. Thurston. Real and perceived limitations to decision-based design. InASME DETC, Las Vegas, NE, USA, 1999. DETC1999/DAC-8750.

D.L. Thurston, J.V. Carnahan, and T. Liu. “Optimization of design utility”.Journal of Mechanical Design, 116:801–808, 1994.

D.L. Thurston and A. Locascio. “design theory for design economics”. En-gineering Economist, 40(1):41–72, 1994.

L. Trigeorgis. Real Options, Managerial Flexibility and Strategy in ResourceAllocation. The MIT Press, Cambridge, MA, USA, 1998.

C.L. Tseng and G. Barz. Short-term generation asset valuation. 32nd HawaiiInternational Conference, 1999.

WardAuto. Automotive Yearbook. Ward’s Communication, Southfield, MI,USA, 1998-00.

43

H. J. Wassenaar and W. Chen. An approach to decision based design withdiscrete choice analysis for demand modeling. Journal of Mechanical De-sign, 125:490–497, September 2003.

H. J. Wassenaar, W. Chen, J. Cheng, and A. Sudjianto. Enhancing dis-crete choice demand modeling for decision-based design. In ASME DETC,Chicago, IL, USA, 2003. DETC-DTM48683.

H. J. Wassenaar, W. Chen, J. Cheng, and A. Sudjianto. An integrated latentvariable choice modeling approach to enhancing product demand modeling.In ASME DETC, Salt Lake City, UT, USA, 2004. DETC2004-57487.

C.A. Whitcomb, N. Palli, and S. Azarm. “A prescriptive production-distribution approach for decision making in new product design”. IEEETrans. on Systems, Man and Cybernetics, Part C, 29(3):336–348, 1999.

K. Wipke and M. Cuddy. Using an advanced vehicle simulator (advisor)to guide hybrid vehicle propulsion system development. Technical ReportS/EV96, NREL, 1996.

WSJ. Detroit again attempts to dodge pressures for a ’greener’ fleet. WallStreet Journal, 1/28/2002.

44

Table 6: Solutions of the enterprise model for different capacities

Solution for Solution forVariable K = 20, 000 K = 50, 000

QuantityPC 5286 28675wPC 26 (%) 58 (%)

CAFEPC -$508 -$510enginePC 72.85 (kW) 72.50 (kW)

(final drive)PC 3.49 3.53(fuel economy)PC 37.67 (mpg) 37.70 (mpg)

(acceleration 0 to 60)PC 12.43 (s) 12.4 (s)(acceleration 0 to 80)PC 26.17 (s) 26.23 (s)

(acceleration 40 to 60)PC 5.7 (s) 5.62 (s)(5-sec distance)PC 130.46 (ft) 130.62 (ft)

(max acceleration)PC 16.05 (ft/s2) 16.05 (ft/s2)(max speed)PC 110.7 (mph) 111.09 (mph)

(max grade at 55mph)PC 18.37 (%) 18.52 (%)QuantitySUV 14714 21325

wSUV 74 (%) 42 (%)CAFESUV $183 $314engineSUV 194 (kW) 250 (kW)

(final drive)SUV 3.97 2.58(fuel economy)SUV 16.65 (mpg) 14 (mpg)

(acceleration 0 to 60)SUV 8.24 (s) 7.54 (s)(acceleration 0 to 80)SUV 19.09 (s) 18.3 (s)

(acceleration 40 to 60)SUV 4.06 (s) 3.55 (s)(5-sec distance)SUV 193.29 (ft) 196.7 (ft)

(max acceleration)SUV 19.24 (ft/s2) 19.24 (ft/s2)(max speed)SUV 132.07 (mph) 119.18 (mph)

(max grade at 55mph)SUV 26.57 (%) 36.93 (%)Fleet CAFE 0 - $7.9M

NPV(7 year period) $7.3B $9.36B

45