measuring the welfare gain from personal computers · greenwood & kopecky: welfare gain from...

MEASURING THE WELFARE GAIN FROM PERSONAL COMPUTERS

JEREMY GREENWOOD and KAREN A. KOPECKY∗

The welfare gain to consumers from the introduction of personal computers (PCs)is estimated. A simple model of consumer demand is formulated that uses a slightlymodified version of standard preferences. The modification permits marginal utility, andhence total utility, to be finite when the consumption of computers is zero. This impliesthat the good will not be consumed at a high enough price. It also bounds the consumersurplus derived from the product. The model is calibrated/estimated using standardnational income and product account data. The welfare gain from the introduction ofPCs is 2%–3% of consumption expenditure. (JEL E01, E21, O33, O47)

I. INTRODUCTION

What is the welfare gain to consumers fromthe development of and improvements in per-sonal computers (PCs)? This is the questionaddressed here. The answer offered is that wel-fare increased by somewhere between 2% and3%, measured in terms of total personal con-sumption expenditure, due to the introductionof the PC and its subsequent price decline.This finding is obtained by employing a modelof consumer behavior, based upon more-or-less standard preferences, which is fit to aggre-gate national income and product account datausing a direct and simple calibration/estimationstrategy.

To estimate the welfare gain from the intro-duction of a new product one must know whatutility is in the absence of the good. A con-ventional isoelastic utility function has twoproblems. First, at zero consumption the util-ity function returns a value of minus infinity,whenever the elasticity of substitution is lessthan one. In this case the welfare gain fromthe introduction of the new good is infinitelylarge. Second, marginal utility at zero consump-tion is infinite, so long as the elasticity ofsubstitution is finite. Therefore, consumers will

*The referee is thanked for a helpful suggestion and MatsBergman is thanked for some inspiring comments.Greenwood: Professor, Department of Economics, Uni-

versity of Pennsylvania, Philadelphia, PA 19104-6297.Phone 1-215-898-1505, Fax 1-215-746-2947.

Kopecky: Research Economist, Research Department, Fed-eral Reserve Bank of Atlanta, 1000 Peachtree St. NE,Atlanta, GA 30309-4470. Phone 1-404-498-8974, Fax1-404-498-8956, E-mail [email protected]

always purchase some of the good in question,no matter how high the price is, albeit perhapsin infinitesimal quantities. To avoid these prob-lems a form for preferences will be adopted thatgives a finite level for marginal utility, and henceone for total utility, at zero consumption. Withthis utility function, high prices may result in theconsumer optimally choosing to buy none of thenew good. In addition, the consumer’s surplusassociated with the introduction of a new goodis generally finite.

This paper contributes to the growing lit-erature on measuring the welfare gains fromnew goods. A classic example is the work byHausman (1999), who studies the introductionof cellular telephones. He finds that their tardyinclusion in the Consumer Price Index (CPI),some 15 years after their debut, results in a biasof up to 2% per year in the telecommunications-services price index. To obtain this estimate,Hausman (1999) effectively integrates back theestimated demand curve for cellular telephonesto recover the indirect utility function for con-sumers. This function can be inverted to obtainthe expenditure function, from which welfaremeasures can be calculated. The procedure wasdeveloped earlier in work by Hausman (1981).

ABBREVIATIONS

BEA: Bureau of Economics AnalysisBLS: Bureau of Labor StatisticsCPI: Consumer Price IndexGDP: Gross Domestic ProductPC: Personal ComputerRAM: Random-Access Memory

336

Economic Inquiry(ISSN 0095-2583)Vol. 51, No. 1, January 2013, 336–347

doi:10.1111/j.1465-7295.2011.00447.xOnline Early publication February 17, 2012© 2012 Western Economic Association International

GREENWOOD & KOPECKY: WELFARE GAIN FROM PERSONAL COMPUTERS 337

Analytical solutions for the expenditure func-tion can be obtained when the demand equationis (ln) linear. When the demand equation is notlinear the indirect utility function may have tobe recovered by numerically solving a differen-tial equation. This procedure is dual to the onepresented here, which focuses on the consumermaximization problem. Some utility functions,such as the one employed in the current work,may not lead to a linear demand equation of theform that is conventionally estimated.

Hausman (1999) also suggests an approx-imate measure of welfare based on a lineardemand curve. While he states explicitly thatthis measure of welfare is a lower bound, thiscaveat is often forgotten in applied work. It maywork well in the cell phone example that hestudies. As illustrated in the PC example studiedhere, however, this approximate demand curvemethod can lead to a serious underestimate ofthe welfare gains arising from the introductionof a new product.

Similarly, the use of conventional priceindices, such as the Fisher Ideal and Tornqvistprice indices, may lead to inaccurate estimatesof the consumer surplus that arises from theintroduction of a new good and/or the good’ssubsequent price decline. The accuracy of suchmethods will depend on how fast the marginalutility for a new good rises as the quantity con-sumed goes to zero. Figuring this out is partand parcel of the new goods problem. That is,what utility function or demand equation shouldbe used for estimating the welfare associatedwith the introduction of a new good? In fact,it is shown here that in some cases an equiva-lent variation fails to exist. In these cases, theassumptions required for the theory underlyingthe Fisher Ideal and Tornqvist price indices arenot satisfied. While this is not true for the caseof computers, it is found to be true in the caseof another important new good, electricity. Toillustrate this, the technique for measuring thewelfare gain from new goods is also applied toelectricity. The welfare gain from the inventionand price decline in electricity, as measured bythe compensating variation, is found to be 92%of personal consumption expenditure.

Another example in the literature is theGoolsbee and Klenow (2006) study of the ben-efit to consumers of the internet. They esti-mate the demand for the internet by relatingthe time spent using the product to the oppor-tunity cost of time, an approach which, theyargue, makes sense since internet access is a

good whose marginal cost consists almost solelyof the leisure time spent by the consumer. Theyfind very large welfare gains when taking aliteral interpretation of the model’s structure.This is due to the fact that in their specifica-tion the marginal utility of internet consump-tion approaches infinity as consumption goesto zero. To mitigate the impact of the zero-consumption region of the utility function, theyemphasize an alternative measure based upon alinearized leisure demand curve. Additionally,their setup requires the elasticity of substitutionto be greater than one. This is satisfied in thedata for internet consumption. But, it is not truefor all products. A case in point is cellular tele-phones, which Hausman estimates to have anelasticity less than one.

Finally, Petrin (2002) considers, as an ex-ample, the introduction of the minivan todemonstrate a technique for estimating welfaregains in the absence of consumer-level data.He shows how information describing the pur-chasing habits of different demographic groups,in conjunction with market-level data, can beused as a suitable substitute for consumer-leveldata. In his discrete-choice analysis minivanconsumption is a lumpy good, so the specifi-cation of the utility function is not central.1

II. COMPUTERS

Computers first became available in theUnited States in the 1950s but at prices so highand sizes so large that, for the most part, no indi-vidual would want to buy one. It was not untilthe early 1970s, with the invention of the micro-processor, that the first generation of microcom-puters—computers that were small enough to fiton a desk and inexpensive enough to be ownedby individuals—was born. Lasting from 1971 to1976, this period was characterized by a grow-ing interest in computers among engineers andhobbyists. The year 1977 marks the birth ofthe PC. The key difference between PCs andtheir microcomputer predecessors is the limitedamount of expertise that the use of the formerrequires. In other words, PCs are computers that

1. Greenwood and Uysal (2005) model a world withlumpiness in consumption and many consumer goods. Anincrease in income or a fall in price leads to a shift inconsumption along the extensive margin, as the set of goodsconsumed changes. The implications of indivisibilities inconsumption have not been fully explored in the literatureyet.

338 ECONOMIC INQUIRY

are both small in size and user friendly to indi-viduals with no technical training.

The first PC to be successfully mass producedwas the Apple II. Released in June of 1977it consisted of a microprocessor running at 1MHz, 4 kb of random-access memory (RAM),no hard disk, and an audio cassette interface forprogram loading and data storage. The computerretailed at approximately $1,200. Systems withlarger amounts of RAM were also availableup to a maximum of 48 kb and at a priceof approximately $2,600. Rapid technologicalprogress led to consistent improvement in theApple II following its release. For example, in1978 the floppy disk drive peripheral becameavailable. Far superior to cassettes, the additionof floppy disks greatly improved the quality ofApple II computing.

Since the introduction of the Apple II, rapidtechnological progress in computer developmenthas fueled continual quality improvements anddeclining costs of production. Compared to theApple II, today’s computers are often equippedwith multi-core processors running at over 3,000MHz, gigabytes of RAM, and hard drives capa-ble of storing hundreds of gigabytes of data.

Quality improvements in computers and com-puter production have resulted in an enormousfall in quality-adjusted PC prices. In fact, priceshave dropped at an astounding rate of 25% peryear. Thus, a PC today is more than 700 timescheaper than one in 1977. Starting from virtu-ally zero demand for computers in and preceding1977, the fall in prices throughout the last 30years has been synonymous with a rapid risein demand. Figure 1 shows price and quantityindices for computers and computer peripheralsfor the period 1977–2004—see the Appendixfor information on the data. As the demand forPCs rose, so did their share of total expendi-ture. Figure 2 shows computer’s share of totalexpenditure since 1977. It rose from 0 in 1977to more than 0.6% in 2004.

III. MODEL

Consider an individual with income, y, thatcan be used to purchase general consumptiongoods, c, and computers, n. Computers aremeasured in some sort of standardized, quality-adjusted units and sell at a price of p interms of consumption. Let the person’s tastesbe described by

θU(c) + (1 − θ)V (n), with 0 < θ < 1.(1)

FIGURE 1Price and Quantity Indices for Computers,Peripherals, and Software for the Years

1977–2004

1975 1980 1985 1990 1995 2000 2005

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

YearQ

uant

ity In

dex,

Pric

e In

dex

Price

Quantity

FIGURE 2Computers, Peripherals, and Software’s Share

of Total Personal Consumption Expenditure forthe Years 1977–2004

1975 1980 1985 1990 1995 2000 2005

0

0.1

0.2

0.3

0.4

0.5

0.6

Year

Sha

re o

f Tot

al E

xpen

ditu

re (

%)

Take the utility function for the consumptionof general goods to be of the standard constant-relative-risk-aversion variety, so that U(c) canbe written as

U(c) = c1−ρ/(1 − ρ), with ρ ≥ 0.(2)

Notice that U(c) satisfies the standard proper-ties U1(c) > 0, U11(c) < 0, limc→∞ U1(c) = 0,and limc→0 U1(c) = ∞. Represent the utilityfunction for PCs by

V (n) = (n + ν)1−ρ/(1 − ρ), with 0 < ν < ∞.

(3)


FIGURE 3Tastes for Computers When ρ ≥ 1—Model

and Conventional Formulation (dashed line)

The function V (n) is completely standard exceptthat

V (0) = ν1−ρ/(1 − ρ) > −∞ and V1(0) = ν−ρ.

Observe that since ρ ≥ 0, the magnitudeof the elasticity of demand for computers isunrestricted. The implications of these assump-tions on the utility function for PCs are por-trayed in Figure 3, for the case where ρ ≥1. The conventional formulation is illustratedby the dashed line. Note that one could setV (0) = 0 by redefining the utility function tobe [(n + ν)1−ρ − ν1−ρ]/(1 − ρ); such a nor-malization has no implication for the anal-ysis. Also, observe that the utility functionfor computers is nonhomothetic due to thepresence of ν.

The individual’s static maximization problemwill read

w ≡ W(y, p) = maxc,n

[θU(c) + (1 − θ)V (n)],(4)

subject to his budget constraint

c + pn = y,(5)

and the non-negativity conditions

c, n ≥ 0.

Note that W(y, p) represents the person’sindirect utility function, which gives his max-imal level of welfare at the income level ywhen he faces the price for computers p.The non-negativity constraint on c will neverbind and can be safely disregarded, becauselimc→0 U1(c) = ∞. Since the marginal utilityof zero computers is finite, the solution to theindividual’s maximization problem could be ata corner where n = 0.

The solution to the above problem can beobtained by using the budget constraint (5) tosubstitute out for c in the objective function (4)and then maximizing with respect to n. Thisleads to the Kuhn–Tucker conditions

θ(y − pn)−ρp − (1 − θ) (n + ν)−ρ ≥ 0,

(6)

n ≥ 0,[θ(y − pn)−ρp − (1 − θ) (n + ν)−ρ

]n = 0.

Equation (6), in conjunction with the budgetconstraint (5), determines the demand functionsfor c and n:(7)

c = C(y, p)

=

⎧⎪⎪⎨⎪⎪⎩y,

if p ≥ P̂ (y) ≡ (1 − θ)/θν−ρyρ;(y + pν)/

{1 + [(1 − θ)/θ]1/ρp(ρ−1)/ρ

},

if p < P̂ (y),

and(8)

n = N(y, p)

=

⎧⎪⎪⎨⎪⎪⎩0,

if p ≥ P̂ (y);(y + pν)/

{p + [(1 − θ)/θ]−1/ρp1/ρ

} − ν,

if p < P̂ (y).

Observe from Equation (8) that for any givenincome level, y, there exists a threshold price,P̂ (y), such that the optimal expenditure oncomputers will be zero whenever p ≥ P̂ (y).The price P̂ (y) coincides with Hick’s (1940)virtual price. It is the minimum price that setsthe demand for n equal to 0, or is the price atwhich the demand curve for n will touch thevertical axis. This threshold price is increasingin income. This implies that along a falling pricepath, the rich will buy the good before the poordo. Finally, note that the demand curve givenby Equation (8) is not of a form that an appliedeconomist would typically estimate.


Last, suppose that computers can be producedfrom final output according to the productionfunction

n = zo,

where o is the use of output in computer pro-duction and z is the level productivity in thecomputer sector. Under this assumption the priceof computers is simply given by p = 1/z. Thus,the decline in the price of computers over timecan be identified with exogenous technologicalprogress in the production of computers. It ishard to comprehend how the observed astonish-ing 25% annual price decline can be anythingelse.

A. Welfare Gain

What is the welfare gain to consumers in2004 from the invention of PCs and the fall intheir relative price since 1977? The welfare gainwill be measured in terms of both the equivalentand compensating variations.

First, suppose it is the year 2004 and com-puters have never been invented. As is readilyapparent from Equation (8), this is the same asassuming that computers exist but sell at someprohibitively expensive price, say p = ∞. Howmuch more income would you have to give tothe consumer so that his welfare level withoutcomputers is equivalent to the one he obtainedwith them? This is the equivalent variation. LetλEV be the additional income required, mea-sured as a percentage of actual 2004 income,y2004. When computers do not exist the personwill spend his entire income on the aggregatemarket good. His maximal utility will be

W((1 + λEV )y2004,∞)(9)

= θ((1 + λEV )y2004)1−ρ

/(1 − ρ) + (1 − θ) ν1−ρ/(1 − ρ).

In the year 2004 the consumer actually didpurchase computers at the price p2004. Assumingthat he undertook his purchases optimally, hisindirect utility function specifies a welfare levelof W(y2004, p2004).

The equivalent variation is determined, whenit exists, by solving the following equation forλEV :

W((1 + λEV )y2004,∞) = W(y2004, p2004)

(10)

≡ w2004.

That is, the equivalent variation, λEV , rendersthe individual indifferent between consuming(1 + λ)y2004 of the market good, c, and zerocomputers, on the one hand, and consuming hisactual 2004 consumption bundle when comput-ers exist and are available at 2004 prices, on theother. Equations (9) and (10) yield

λEV = [(1 − ρ)W(y2004, p2004)

− (1 − θ)ν1−ρ]1/(1−ρ)/(θ1/(1−ρ)y2004)− 1.

Notice that the equivalent variation can becomputed given data on total expenditures andprices, and estimates of the three preferenceparameters, ν, θ, and ρ.

The second measure of welfare that will beconsidered is the compensating variation. Thecompensating variation is similar to the equiv-alent variation. In fact, for quasi-linear prefer-ences the two are equivalent. The compensatingvariation is the amount of income that wouldhave to be taken from the consumer in 2004 togive him the level of welfare that he would haverealized if computers had never been invented.Denote the compensating variation, measuredas a percentage of the agent’s 2004 incomelevel, by λCV . The compensating variation, λCV ,solves the equation

W((1 − λCV )y2004, p2004) = W(y2004,∞).(11)

Although λCV cannot be written explicitly, it isuniquely defined by Equation (11) and can becomputed numerically, given estimates of thepreference parameters in conjunction with thedata on prices and expenditures.2

B. The Measurement of Welfare Gains UsingPrice Indices

Approximations of the compensating andequivalent variations can be computed froma Tornqvist price index as follows. DefineE(w, p) to be the expenditure function associ-ated with the welfare level w and the price p.

2. In particular, note that

W((1 − λCV )y2004, p2004)

= [C((1 − λCV )y2004, p2004)]1−ρ/(1 − ρ)

+ [N((1 − λCV )y2004, p2004) + ν]1−ρ/(1 − ρ),

where the functions C(·) and N(·) are specified by (7) and(8). Also, recall that W(y2004, ∞) is given by (9), when λEV

is set to zero.


Thus

E(w, p) = minc,n

[c + pn],

subject to

θU(c) + (1 − θ)V (n) = w.

Suppose that the expenditure function can beapproximated by a translog function. Then, onecan write

E(w, p) � exp(α + β ln p + κ ln p2(12)

+ ψ ln w + σ ln w2 + μ ln w ln p).

Under this assumption, approximations ofthe equivalent and compensating variations ofa price decline from p1977 to p2004 can beconstructed from the Tornqvist price index, P T,using the formulae

λTEV = 1/P T − 1 and λT

CV = 1 − P T,(13)

where

P T ≡ (p2004/p1977)(s1977+s2004)/2,

and

st ≡ ptnt/(ct + ptnt ), for t = 1977, 2004.

(With some abuse of notation, in the discussionhere think about p1977 as representing the priceof the new good in some initial period andp2004 as giving the price in some final period,etc.) If p1977 is Hick’s (1940) virtual price atwhich demand is zero then λT

EV and λTCV are

approximations of the welfare gain from theintroduction of the good and its subsequent pricedecline to p2004. A magnificent discussion ofthe theory underlying this approach is given inDiewert (2008). Measures of the equivalent andcompensating variation can also be constructedfrom the Fisher Ideal price index, at least whenthe expenditure function can be approximatedby a quadratic function—again, see Diewert(2008). This works best when the utility functionis homothetic, which it is not here.

Interestingly, there may not exist an equiv-alent variation large enough to compensate aperson living today to forgo a new good. Thatis, there is no value for λEV that will solve (10).

LEMMA 1. (Nonexistence of an equivalentvariation) Let tastes be given by Equations(2) and (3) and assume that ρ ≥ 1. There arep2004, y2004, and ν combinations such that theredoes not exist a λEV that can satisfy (10); i.e.,such that an equivalent variation does not exist.

Proof. See Appendix. �If the situation described in the Lemma hap-

pens, the assumptions underlying the theoryjustifying the use of price indices to measurewelfare changes fail to hold. This occurs because,when the equivalent variation does not exist,the expenditure function cannot be well approx-imated by either quadratic or translog functions.

COROLLARY 1. (Nonexistence of a translogapproximation) There are p2004, y2004, and νcombinations such that the expenditure function,E(w, p), cannot be adequately approximated byeither translog or quadratic functions.

Proof. Again, see Appendix. �

REMARK 1. The argument can be extended tothe case of a low initial demand at some highprice. Set p1977 at Hick’s (1940) virtual price,P̂ (y2004), so that p1977 = P̂ (y2004). This is theminimum price that sets the demand for the newgood equal to zero as defined in Equation (7). Bycontinuity, when the equivalent variation doesnot exist, the expenditure function cannot be wellapproximated for any price p < p1977 that isclose enough in value to p1977. But, for thesep’s, the demand for the new good, N(y2004, p),will be small but positive.

Even though the use of the Tornqvist priceindex to calculate measures of the welfare gainsfrom new goods may work well in many sit-uations, the above discussion is not merely atheoretical curiosity. As will be seen below,while the Tornqvist price index approach pro-vides a reasonable approximation of the welfaregain from the introduction of computers, it doesnot work well for electricity.

IV. QUANTITATIVE EXPERIMENT

The task is to compute the welfare gainto consumers in 2004 due to the invention ofthe PC in 1977 and the subsequent decline inits price. In order to compute this, informa-tion about appropriate values for the preferenceparameters must be obtained. There are threepreference parameters to pin down: the coeffi-cient of relative risk aversion, ρ, the weight onutility from aggregate market consumption netof computers, θ, and the parameter ν, which isimportant for specifying the marginal utility ofzero computer consumption. These parameters


are determined using the following calibra-tion/estimation procedure.

To begin with, for each year t between 1977and 2004 let pt represent the quality-adjustedprice of computers relative to aggregate marketconsumption net of computers, and yt denotetotal expenditure at date t in the data. Simi-larly, let nt be the aggregate quantity of quality-adjusted computers purchased. The price andexpenditure data are taken from the Bureau ofEconomic Analysis (BEA)—see the Appendixfor more detail. The BEA adjusts these series,using hedonic price methods and other tech-niques, for the quality improvement in comput-ers, peripherals, and software that occurs overtime. Out of the 16.5% annual decline in thequality-adjusted prices for PCs that occurredbetween 2001 and 2005, Wasshausen and Mou-ton (2006) report that 11.5% was due to qualityadjustment and 4.9% was attributable to changesin unit value. Given the quality adjustment thatthe BEA does to the data, think about nt asrepresenting the quantity of computers denomi-nated in some sort of standardized unit. Figure 1shows the PC price and quantity indices.3

Next, note that given values for the pref-erence parameters, the model’s prediction forthe quantity of computers consumed at somedate t , n̂t , can be computed by plugging thecorresponding price and income levels, pt andyt , into the demand functions specified by (8)to obtain n̂t = N (yt , pt ). These demand func-tions also depend on the model’s underlyingparameters, ρ, θ, and ν. Denote this mappingfrom the preference parameters to the predic-tion for the quantity of computers consumed byn̂t = N(ρ, θ, ν; yt , pt ).

Finally, the preference parameters are deter-mined by minimizing the sum of the squaresof the difference between the logarithm of thequantity of computers purchased as observed inthe data and the logarithm of the quantity con-sumed as predicted by the model over the period1977–2004. This estimation is undertaken sub-ject to a constraint that is discussed now. TheBEA reports that a zero quantity of computers(probably some trivial amount) was consumed

3. Berndt and Rappaport (2001) report quality-adjustedprice declines for desktop and mobile PCs. For the period1976–1999, they found that their prices fell somewherebetween 25% and 27% a year. This is similar to the25% calculated here using the BEA data for the period1977–2004. Pakes (2003) finds that his hedonic index yieldsan average price decline of 16% for the 1995–1999 sub-period. This compares with 16% obtained using the BLS’shybrid method for the CPI and 22% for the PPI.

in 1977 while they were available at a positiveprice. Very small amounts are reported in theyears immediately after 1977. In order to trackaccurately the small amounts of computer con-sumption in the early years a restriction will beimposed on the estimation requiring that the pre-dicted purchases of computer in 1977 must bezero. In other words, the parameters are chosenby solving

minρ,θ,ν

2004∑t=1977

[ln nt − ln N(ρ, θ, ν; yt , pt )]2,

subject to

N(ρ, θ, ν; y1977, p1977) = 0.

Essentially, one can think of the constraint asidentifying ν.

The calibration/estimation procedure resultsin a value for ρ of 0.326, a value for θ of0.994, and one for ν of 4 × 10−5. The valuesare reasonable. The value for ρ suggests thatcomputers and general consumption goods aresubstitutes. This makes sense given that comput-ers and their accessories have substituted for avariety of other goods such as calculators, type-writers, stationery, travel agents, photo albums,etc. At high levels of expenditure, the coefficient1 − θ is approximately computer’s share of totalexpenditure. In 2004 this was exactly 0.006, sothat the share of general goods in spending was0.994. Last, ν is only 0.005% of the quantityof computer purchases in 2004. The model’sprediction for the logarithm of the quantity ofcomputers demanded along with the logarithmof the quantity consumed from the data are givenin Figure 4. As can be seen, the model fits thedata remarkably well. The R2 is 0.9813.

Plugging the parameter values and the 2004data into the formulas for the equivalent andcompensating variations yields an equivalentvariation of 2.14% and a compensating varia-tion of 2.19%. This compares with Hausman’s(1996) estimate of 0.002% due to the introduc-tion of Apple-Cinnamon Cheerios, which is aminor product innovation. Petrin (2002) reportsa welfare gain of 0.029% associated with theadvent of the minivan, a more substantive prod-uct. A much smaller fraction of the populationowns minivans as compared with computers,though. Furthermore, this product has not seenthe remarkable price decline that computershave. In a similar vein, Goolsbee and Petrin(2004) find a 0.035% gain tied to the genesis ofsatellite TV. The current estimate is far below


FIGURE 4Log (ln) Quantity Indexes for Computers,Peripherals, and Software for the Years

1977–2004

1975 1980 1985 1990 1995 2000 2005−12

−10

−8

−6

−4

−2

0

2

Year

Log

Qua

ntity

Inde

x

Model

Data

Goolsbee and Klenow’s (2006) one of 26.8%resulting from the internet, at least when themodel is interpreted literally. They note that withthe isoelastic utility function they choose “theutility from the first units of consumption is sohigh” that the welfare gains will be large.4 Aswas mentioned, Goolsbee and Klenow (2006)present an alternative, smaller, estimate (theirpreferred one) based upon a linearized leisuredemand curve to reduce the sensitivity of thewelfare gain estimate to this region of the utilityfunction.

Hausman’s (1999) approximate demand mea-sure of the welfare gain from the introductionof a new good is 0.5×(share of new good inexpenditure)/(price elasticity of demand). Thetime series data for computer consumption sug-gest a price elasticity of 1.83. This implies that

4. Modeling the consumption of internet services istrickier than other goods. Most consumers purchase internetservices at a fixed monthly price. Therefore, they can use asmuch of the services in a month as they desire. The limitingfactor is the amount of time that an individual wants tospend on the internet. This is why Goolsbee and Klenow(2006) estimate the demand for the internet by relating thetime spent using the product to the opportunity cost oftime. Modifying the utility function to bound the marginalutility of internet services at zero consumption would lowerthe welfare estimate. It should also help explain Goolsbeeand Klenow’s (2006) fact that 37% of people were not on-line. Putting more concavity in the utility function at highlevels of consumption would help lower the welfare estimateas well. Last, as a technical aside, Goolsbee and Klenow(2006) mix internet consumption and time spent surfing in aCobb–Douglas fashion. It would be an interesting exerciseto remodel things with a Leontief structure, because thiswould explicitly bind internet consumption by the timedevoted to it.

a 10% increase in the quality-adjusted price ofcomputers will result in an 18% decrease indemand. Taking their 2004 expenditure shareof 0.006 (one of the larger values recorded ascan be seen from Figure 2) suggests a welfaregain of only 0.16%. As was mentioned, Haus-man (1999) states that his approximate demandmeasure is best interpreted as a lower boundon the welfare gain. It is a far cry from thetrue value that obtains if tastes take the formspecified in (1) with the estimated parametervalues. The small number obtained from Haus-man’s approximation procedure results from thefact that computers constitute a small share ofexpenditure. Yet, they still are important in gen-erating utility. His linear demand approximationmethod is likely to perform better for the intro-duction of more minor products, such as Apple-Cinnamon Cheerios, which can be viewed as asmall change from the status quo. In fact, Haus-man’s measure performs worse than the wel-fare measures obtained using the Tornqvist priceindex, as given by Equation (13). The indexperforms well yielding estimates of 2.07% and2.03% for the equivalent and compensating vari-ations respectively. The Fisher Ideal price indexdoes not work well in this situation, yieldingestimates of 137% and 58%.5

A. Some Sensitivity Analysis

Given the outstanding fit of the model to thedata, room for an improved estimation wouldappear to be small. One could try generalizingthe utility function over general goods consump-tion and computers. Specifically, assume that theutility functions for general goods consumptionand computers have different degrees of curva-ture:

U(c, n) = θc1−ρ/(1 − ρ) + (1 − θ)

× (n + ν)1−γ/(1 − γ),

with 0 ≤ θ ≤ 1 and ρ, γ ≥ 0. Calibrating thismodified version of the model using the proce-dure described above yields θ = 0.97, ρ = 1.8,γ = 0.76, and ν = 4 × 10−5. The equivalent and

5. This price index is the geometric mean of theLaspeyres and Paasche prices indices. The Laspeyresprice index is given by (c1977 + p2004n1977)/(c1977 +p2004n1977) = 1, because the initial consumption of comput-ers is zero so that n1977 = 0. The Paasche index is defined as(c2004 + p2004n2004)/(c2004 + p1997n2004), which is a smallnumber given the rapid decline in the price of computers;i.e., p1997 is very high relative to p2004. Thus, the FisherIdeal price index shows an extremely steep decline.


compensating variations under this version ofthe model turn out to be 3.3% and 3.2% of 2004consumption, respectively. The upshot of theabove analysis is that it may be best to viewthe welfare gain from PCs as lying in the rangeof 2.14%–3.3%.

One can also compute the welfare gain thatcomputers generated at any year from the timeof their introduction in 1977 to 2004, usingthe estimated parameter values and the priceand quantity data. The average welfare gainsfor each year, computed as the average of theequivalent and compensating variations gener-ated from both versions of the model, are plot-ted in Figure 5. It is possible that there issome error in the measurement of the quality-adjusted prices for PCs. Thus, it is interestingto assess the sensitivity of the welfare esti-mates to variations in the observed rate of pricedecline. Using the price data to compute a95% confidence interval around the mean annualrate of price decline generates a lower boundfor the price decline of 23% and an upperbound of 26%. Now, construct two rescaledprice series that have average rates of declinethat are consistent with these bounds; i.e., justmultiply the period-t price for computers by(1.23/1.25)t−1977 and (1.26/1.25)t−1977. Corre-sponding lower and upper bounds on the welfareestimates can then be computed, using these tworescaled price series. These bounds on the wel-fare estimates are also presented in Figure 5.Notice how the welfare benefit grows over timeas the price for computers declines. Brynjolfsson(1996) estimates that the welfare gain from com-puters was somewhere between 0.2% and 0.3%

FIGURE 5Year-by-Year Welfare Gains for Personal

Computers

1975 1980 1985 1990 1995 2000 20050

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Year

Fra

ctio

n of

200

4 C

onsu

mpt

ion

Exp

endi

ture

lower bound

upper bound

of gross domestic product (GDP) in 1987. Thiscompares with the 0.2% found here.6

The analysis above shows that a simplemodel of a representative consumer with a slightmodification to the standard isoelastic utilityfunction can lead, using a straightforward cali-bration/estimation procedure and aggregate data,to a reasonable measure of the welfare gainrealized from the introduction of PCs. A fewwords of caution are in order, however. First,since the parameter values are not determinedthrough a statistical estimation, the analysis issilent on standard errors and other tests of themodel. Second, the welfare measures are con-ditional on the parametric form chosen for util-ity. This is a problem that many econometricapproaches suffer from as well. On the latterpoint, there may be nonseparabilities in utilitybetween computers and other specific goods. Forexample, Gloosbee and Klenow (2006) assumethat internet services and leisure are Edgeworth-Pareto complements in the utility function. Thiscould be true of computers more generally, ofcourse. Think about playing computer games.Internet services could also be an Edgeworth-Pareto substitute with housework, if they can beused to reduce time spent on chores such as pay-ing bills, shopping, etc. Gloosbee and Klenow(2006) also mention that there may be spillovereffects across consumers that are important forhousehold computer adoption. Entering a net-work externality into tastes may provide anotherroute for modeling the low initial demand forcomputers. More sophisticated specifications oftastes would probably require more data in orderto estimate the structure well, such as the time-use data used by Gloosbee and Klenow (2006).

V. ELECTRICITY, ANOTHER NEW GOOD

The method outlined above is a reason-able and simple way to estimate the welfaregain from the invention of, and advancementsin, major new goods whose price decreasesover time are primarily driven by technologicalprogress. One of the most important innovationsin modern history is electricity. Following theapproach that is used for computers, the welfaregain is computed from the invention of electric-ity and the subsequent technological improve-ments that led to a decrease in its price. The

6. Brynjolfsson (1996) discusses several measures ofconsumer welfare, such as the Hicksian and Marshalliannotions of consumer surplus. They all give more or lessthe same answer.


model with electricity is calibrated using priceand expenditure data for electricity taken fromthe BEA—details are provided in the Appendix.Prices are available for each year from 1929to 2006. At the first available price observa-tion the quantity of electricity consumed wasfar from zero, unlike in the case for computers.In fact, electricity was already approximately0.8% of total consumption expenditure in 1929,more than half of its 2006 level of 1.6%. Con-sequently, the price that sets demand to zero,which was used earlier to identify the parameterν, is not the 1929 price nor any of the otherprices observed over the 1929–2006 period.Thus, the calibration is carried out exactly asin the computer case, but without any constrainton the demand function.

The procedure generates a value for ρ of 9.26,a value for θ of 1 − 10−8, and a value for ν of0.062. The high value of ρ means that generalconsumption goods and electricity are strongcomplements in utility. This makes sense giventhat, for the most part, people do not deriveutility from electricity alone but by combiningelectricity with other consumption goods. Aswith computers, the model is able to do agood job of matching the demand for electricityobserved in the data. The R2 is 0.9776.

The compensating variation computed fromthe model for electricity is 92.0% of total 2006consumption expenditure. How does this com-pare to Hausman’s (1999) approximate demandmeasure? Reiss and White (2005) estimate theprice elasticity of demand for electricity in Cal-ifornia to be 0.39. Plugging this plus electric-ity’s share of expenditure in 2006, 1.6%, intoHausman’s formula results in a welfare gainof only 2.0%. According to Reiss and White(2005), a lower bound on the price elasticitybased on other studies is 0.15. Electricity’s shareof expenditure over the 1920–2006 period wasthe highest in 1984 when it reached 2.4%. Usingthese extreme values, Hausman’s measure gen-erates a welfare gain of 8.0%.

Measures of the welfare gain arising from theintroduction of electricity can also be computedusing the Tornqvist price index. To do this,use the model to compute Hicks (1940) virtualprice; that is, the minimum price that sets thedemand for electricity equal to zero. Supposethat this was the price of electricity in 1881,the year when the Niagara Falls HydraulicPower & Manufacturing Company opened asmall power station that generated a smallamount of electricity to light the village of

Niagara Falls. This price can then be pluggedinto the Tornqvist price index. The estimatesof the compensating and equivalent variationsderived from the Tornqvist price index are22.8% and 29.6%, respectively.7 Even thoughthe Tornqvist price index performs much betterthan Hausman’s measure, the resulting estimatefor the compensating variation is still a long wayfrom the 92.0% welfare gain generated by themodel.

In line with Lemma 1, there does not exist anequivalent variation large enough to compensatea person living in 2006 to forgo electricity. Thatis, there is no value for λEV that will solve(10). This illustrates that it may not be possibleto give people enough income in 1881 (theequivalent variation) to make them as well offliving then without electricity (the new good) asthey do today with it. That is, individuals cannotobtain enough utility from the “old goods” tocompensate them for the loss of the “new good.”This makes intuitive sense. On the other hand,it is always possible to take enough incomeaway from someone today (the compensatingvariation) to make him/her as bad off as anindividual living in 1881. For example, thinkabout taking away all of his income. He wouldthen be worse off than someone living in 1881.

Last, Figure 6 shows the plot of the equiv-alent variation, based on the estimated valuesfor θ, ρ, and ν, as the price for electricity risesfrom some conjectured initial starting value,denoted by p, toward the virtual price P̂ (y2006).The final price is the 2006 price for compu-ters. Observe that the equivalent variation growswithout bound. The figure also shows the plotof the Tornqvist approximation to the equiva-lent variation. As the initial price for electricitybecomes large the gap between the two mea-sures widens.

An interesting application of the techniquedeveloped here is undertaken in Hersh and Voth(2009). They estimate the welfare gains from theintroduction of new products in the 17th century,namely coffee, sugar, tea, and tobacco. Whiletoday’s consumer takes these simple goods forgranted, life was Spartan at that time. The con-sumption basket was weighted heavily toward

7. Specifically, the welfare gains are given by (13) with

P T = [p2006/P̂ (y2006)]s2006/2,

where p2006, y2006, and s2006 are electricity’s price, income,and expenditure share in the 2006 U.S. data and where thevirtual price P̂ (y2006) is defined in (7).


FIGURE 6The Equivalent Variation and Its Tornqvist

Approximation

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

50

100

150

200

250

Equ

ival

ent V

aria

tion

Actual

Tornqvist

Initial Price, ptowards virtual price,P (y2006)

food. The European diet was essentially com-posed of beer, grains, meat, milk, and water.Hersh and Voth (2009) discuss how these newgoods dramatically transformed European eat-ing habits. After their introduction, the price ofgoods such as sugar and tea declined rapidly[see Hersh and Voth (2009, figures 2–4)]. Theyestimate that by 1850 the introduction of tea andsugar alone was worth about 15% of consump-tion to the average consumer.

VI. CONCLUSION

What is your PC worth to you? About2%–3% of total consumption expenditure is thefinding. This answer is predicated upon a simplemodel of consumer demand. A slight modifica-tion of the standard isoelastic variety of pref-erences results in a well-behaved demand forcomputers: demand drops to zero as prices riseto some well-defined level, and the consumer’ssurplus associated with a new good is gener-ally bounded. The model of consumer demandis fit to national income and product data, usinga straightforward calibration/estimation proce-dure, to uncover the taste parameters neededfor the welfare analysis. The parameter valuesobtained are reasonable and the framework fitsthe aggregate data well. In addition, the resultsare robust to a modification of preferences.Finally, the method for computing the welfaregain for computers is found to work well forestimating the welfare gain from electricity. In

this case the theoretical assumptions required forprice indices to approximate the gain in wel-fare are violated. Here, a compensating varia-tion of 92% is found. No equivalent variationexists, implying that it would be impossible tocompensate a person living today to go withoutelectricity.

APPENDIX

Theory

Proof of Lemma. First, suppose that

w2004 ≡ W(y2004, p2004) > (1 − θ)ν1−ρ/(1 − ρ).(A1)

When this transpires an equivalent variation cannot existbecause the right-hand side of Equation (9) approaches(1 − θ)ν1−ρ/(1 − ρ) from below as λ → ∞. But W(y2004,p2004) > (1 − θ)ν1−ρ/(1 − ρ), by assumption. Second, thereexists a ν such that (A1) holds. To see this, note that(1 − θ)ν1−ρ/(1 − ρ) → −∞ as ν → 0. Now, with a littleeffort, it can be shown that when both goods are consumed

W(y2004, p2004)

= (y2004 + p2004ν)1−ρ/(1 − ρ)

×[

1/[1 +(

1 − θ

θ

)1/ρ

(p2004)(ρ−1)/ρ]

]1−ρ

×[θ + (1 − θ)

[((1 − θ)/θ)(1/p2004)

](1−ρ)/ρ].

By picking p2004 and y2004 appropriately this expressioncan be clearly made bigger than (1 − θ)ν1−ρ/(1 − ρ). Thereare many combinations that can do this. For example,W(y2004, p2004) → θ(y2004)

1−ρ/(1 − ρ) as p2004 → 0 So,pick y2004 > [(1 − θ)/θ]1/(1−ρ)ν for some small p2004. �

Proof of Corollary. In line with the lemma, pick combina-tions for p2004, y2004, and ν such that an equivalent variationdoes not exist. Now, suppose that the expenditure functioncan be approximated by a translog function in line with (12).Under this assumption, the equivalent variation of a pricedecline from Hick’s (1940) virtual price p∗ ≡ P̂ (y2004) top2004 is given by

λEV = E(w2004, p∗)/y2004 − 1

� [α + β ln p∗ + κ ln(p∗)2 + ψ ln w2004

+ σ ln w22004 + μ ln w2004 ln p∗]/y2004 − 1.

Clearly, λEV given by the formula above is finite, acontradiction. The argument for the quadratic function isthe same. �

Data

All the computer data derives from the U.S. NationalIncome and Product Accounts, Tables 2.4.4 and 2.4.5, andspans the period 1977–2004. The electricity data is fromthe same source; however, it spans the period 1929–2006.These tables are available on the website for the BEA. Whenmapping the model into the computer data, the variable n

is taken to be real personal consumption expenditure on


computers, peripherals, and software. This series is con-structed by deflating nominal personal consumption expen-diture on computers, peripherals, and software by the priceindex for this particular series. Note that BEA adjusts theseseries, using hedonic price methods and other techniques,for the quality improvement in computers, peripherals, andsoftware that occurs over time. The variable c represents realpersonal consumption expenditure on all other goods. Thisis obtained by subtracting nominal personal consumptionexpenditure on computers, peripherals, and software fromtotal nominal personal consumption expenditure, and thendeflating by the price series for personal consumption expen-diture. The relative price p is simply taken to be the ratioof the price index for personal consumption expenditure oncomputers, peripherals, and software to the price index forpersonal consumption expenditure. Last, real income, y, issimply defined by y = c + pn, which is total personal con-sumption expenditure.

The mapping from the model to the data for the case ofelectricity is done in exactly the same ways as for computers.The only difference is that instead of expenditure and pricesfor computers, peripherals, and software, the variables n andp are taken to be real personal consumption expenditure andprices for electricity services, respectively.

REFERENCES

Berndt, E. R., and N. J. Rappaport. “Price and Quality ofDesktop and Mobile Personal Computers: A Quarter-Century Historical Overview.” American EconomicReview, 91(2), 2001, 268–73.

Brynjolfsson, E. “The Contribution of Information Tech-nology to Consumer Welfare.” Information SystemsResearch, 7(3), 1996, 281–300.

Diewert, E. W. “Consumer Surplus and the Measurementof Ex Post Welfare Change.” Economics 581: LectureNotes, University of British Columbia, 2008.

Goolsbee, A., and P. J. Klenow. “Valuing Consumer Prod-ucts by the Time Spent Using Them: An Applicationto the Internet.” American Economic Review, 96(2),2006, 108–13.

Goolsbee, A., and A. Petrin. “The Consumer Gains fromDirect Broadcast Satellites and the Competition withCable TV.” Econometrica, 72(2), 2004, 351–81.

Greenwood, J., and G. Uysal. “New Goods and the Transi-tion to a New Economy.” Journal of Economic Growth,10(2), 2005, 99–134.

Hausman, J. “Exact Consumer’s Surplus and DeadweightLoss.” American Economic Review, 71(4), 1981,662–76.

. “Valuation of New Goods under Perfect and Imper-fect Competition,” in The Economics of New Products,edited by T. Bresnahan and R. Gordon. Chicago: Uni-versity of Chicago Press, 1996, 209–37.

. “Cellular Telephone, New Products, and the CPI.”Journal of Business and Economic Statistics, 17(2),1999, 188–94.

Hersh, J., and H. J. Voth. “Sweet Diversity: Colonial Goodsand the Rise of European Living Standards after1492.” Mimeo, Department of Economics, UniversitatPompeu Fabra, 2009.

Hicks, J. R. “The Valuation of the Social Income.” Econom-ica, 7(2), 1940, 105–24.

Pakes, A. “A Reconsideration of Hedonic Price Indexes withan Application to PCs.” American Economic Review,93(3), 2003, 1578–96.

Petrin, A. “Quantifying the Benefits of New Products: TheCase of the Minivan.” Journal of Political Economy,110(4), 2002, 705–29.

Reiss, P. C., and M. W. White. “Household ElectricityDemand, Revisited.” Review of Economic Studies,72(3), 2005, 853–83.

Wasshausen, D., and B. R. Mouton, “The Role of Hedo-nic Methods in Measuring Real GDP in the UnitedStates.” Mimeo, Bureau of Economic Analysis, U.S.Department of Commerce, 2006.

measuring the welfare gain from personal computers · greenwood & kopecky: welfare gain from...

Documents