The role of Advertising in the AggregateEconomy: the Working-Spending Cycle

Preliminary and Incomplete

Benedetto Molinari∗

Universitat Pompeu FabraFrancesco Turino†

Universitat Pompeu Fabra

May 31, 2006

Abstract

The present work is intended to investigate the effects of advertisingin the aggregate economy. We begin by building a database of aggregateadvertising expenditures at quarterly frequencies in US economy. Then,we document the four main empirical regularities we observed.

To consider explanation of these facts, we set up a dynamic stochasticgeneral equilibrium model. The model is novel in the literature, and it isbased on few assumptions of advertising at individual level.

We analyze the implications of the model economy: we find that adver-tising increases the time devoted to work both in the long and short-run.Thus, a work and spend cycle is apparent in our model, which comes athand to explain why historically the aggregate hours worked have notfallen despite a raising trend of the real wages. The model is also shownto have a stronger propagation mechanism with respect to the standardRBC model. Last finding is that the mark up here is time-varying

JEL Classification: D10, E32, J22, M37

Key words: Advertising, Business Cycle, Labor Supply, work andspend cycle.

∗Ph.D. Candidate, Departament of Economics and Business, Universitat Pompeu Fabra(Barcelona, Spain — EU). Contact Email: benedetto.molinari@upf.edu

†Ph.D. Candidate, Departament of Economics and Business, Universitat Pompeu Fabra(Barcelona, Spain — EU), and Graduate Department of Economics, University of Bologna.Contact email: francesco.turino@upf.edu

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1 IntroductionThe phenomenon of Advertising has been traditionally analyzed in microeco-nomics context. In IO literature many papers have been devoted to explain therole of advertising in the market competition, and its effect on social welfare.The tradition in macroeconomics, instead, has been rather limited. Aggregateadvertising expenditures account for no more than two point five per cent indeveloped countries. Perhaps with this fraction in mind, macroeconomists haveconcentrated more on the study of variables such consumption and investmentrather than advertising.There is, however, a relatively minor branch of the literature that tried

to explain the aggregate effects of adv; to say, its effects on macroeconomicvariables. By its own nature, adv is supposed to tilt the demand of goods,so to affect consumption. Through this channel, it can have an effect on theaggregate demand, and consequently on the output dynamics. In this spirit,adv belongs to that set of variables which are not important for their directeffects on the output, but for the indirect mechanism they trigger within theaggregate dynamics, as in the case of inventories or menu-costs.The present work wants to support this idea. To do it, the first step is

to prove that the relationship between adv and consumption, which has beenlargely supported at individual good level, it is true also at aggregate level.During the last century this idea has been object of various investigations, andwas appealing to the most distinguished economists, like Marshall (1890, 1918),Kaldor (1950), Galbraith (1957), Stigler (1961), Arrow (1962) and Solow (1958)among the others. Those prominent opinions used to identify the adv one of thecauses of aggregate demand fluctuations. A wide empirical literature tried tofind evidences to support those views, mostly using econometric tools like thecointegration analysis, and the Granger causality. Despite the large ammount ofeffort none of these work neither theoretical, nor empirical, has been conclusive.Since todays macroeconomics focuses on cycle fluctuations to investigate

the economic dynamics, our first concern has been to build a dataset to analyzethe aggregate advertising at business cycle fluctuations. Unfortunately, thequarterly data af total advertising expenditures are not available among thestandard business cycle statistics, and we had to go through various sources toput together such dataset. In section 2 we explain the empirical work we carriedon, and the characteristics of the dataset we achieved to collect. The followingstep has been to set up a dynamic stochastic general equilibrium model whereAdvertising Expenditures plays a role in the aggregate economy. The model isintended to match the set of empirical regularities that we observed about theaggregate advertising expenditures.There isn’t such model in the literature, so we had to fill the gap. The most

delicate task we faced was to model the adv into the demand of an individualgood; that’s because the literature does not agree on which should be the effectof adv on consumers’ behavior. The way we accomplish the issue goes beyondthe purposes of this paper, and it is left to a specific working paper. In brief, oncewe choose how the adv affects the consumers preferences relationship, we derive

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the key microeconomic relationship of the model: the dependence of consumer’sdemand for each good on the level of advertising. With this relationship athand, we set the profits maximization problem of the firms, which now fits inthe category of supermodular games. Firms, indeed, are called to decide bothsales and advertising intensity, which are complementary strategies. Second,we embed these relationships in an otherwise conventional RBC model. Ourobjective is to to show how advertising intensity has a positive effect on thehours worked. Moreover, the advertising is shown to amplify and propagateexogenous shocks to the system.The results we found are comforting: we manage to replicate the main styl-

ized facts we identify about aggregate adv in the cycle. More interestingly, advis show to trigger a so-called work and spend cycle: the agent work more inorder to afford an higher level of consumption. This is true both in the long runand in the short run. With respect to this last effect, the IRF of the labor w.r.t.an exogenous shock are amplified by the presence of adv. In general, in ourframework the advertising endogenously magnifies the volatility of consumptionand output with respect to the standard RBC model. This is a crucial featureof our model from a macroeconomics point of view. It supports the old opinionof Kaldor (1950), "... as matter of fact, the scale of expenditures on advertisingvaries positively with the general level of economics activity, so that, in so faras the effect of marginal expenditures itself tends to accentuate the amplitude ofeconomic fluctuations....".Another interesting feature of our model is the time variability of the mark

up. This result depends on the demand function generated by the adv hypothe-sis, which is derived by the intratemporal consumer’s expenditure minimizationproblem.

2 Stylized Facts about Aggregate AdvertisingIn this section we analyze the cyclical behavior of advertising expenditures com-pared with the GDP and its main components. We rely on two series of dataabout advertising. One with the total annual expenditures in advertising in allthe media from 1948 to 2005. In the second one, we collected the quarterlyfigures of money spent in advertising by US firms in American Newspaper andInternet.1 This is a partial series of total aggregate expenditures, since we gath-ered data only for two out of the seven main media that are usually referred toas the standard channels for advertisments.2 Yet, our series accounts for almost30% of the total. And was a valuable part of present work to create the onlyseries of quarterly figures of aggregate expenditures in advertising in US we areaware of. All the other available series of have figures only up to 1968, which isthe last year when the federal administration collected data on advertising.

1All the details about the data are reported in Appendix A.2These media are: television, radio, newspapers, internet, billboards, direct mails, and

outdoor advertising.

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Figure 1:

In panel 1 of Figure 1 we plot the annual ratio of total advertising expendi-tures to GDP in US economy.

This ratio fluctuates around 2% throughtout all the sample, with peaks ofalmost 2.3%. In other words, it tells us that in US economy the advertisingabsorbed around 220 billions dollars in year 2000, and the message it gives us istwofold: first, advertising per se is a non-negligible industry for the US economy;second, this money was spent to tilt consumers’ demand, and are big enough tohave had some consequences on that demand.Panel 2 of Figure 1 has the per capita real advertising expenditures: adver-

tising displays a strong upward trend conditional on population growth. Thus,the phenomenon seems to enlarge in recent times. Switching to the quartely fig-ures, panel 3 of figure 1 plot the real advertising expenditures in Newspaper andInternet (join together) along with the cyclical component of real GDP, for the

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period 1970-2005. All the variables are taken in logs, and have been detrendedusing the band pass filter (Baxter and King, 1995).3 In the Advertising serieswe took away seasonal components. Panel 3 shows two facts:

• Advertising Expenditures are positively correlated with GDP.• The advertising has a bigger volatility, which has increased in the last partof the sample.

We check these two facts with the standard business cycle statistics.

Table1. Business cycle statistics (Quarterly data)

Xt σ(Xt)σ(Xt)σ(Gdpt)

ρ σ(Xt,Gdpt) σ(Xt,Gdpt+1) σ(Xt,Gdpt−1)

Gdp 1.54 1 0.93 1 0.93 0.93Adv 3.61 2.34 0.93 0.80 0.82 0.69AdvGdp 2.65 1.65 0.92 0.52 0.41 0.41News 3.05 1.99 0.92 0.83 0.85 0.71Note: All variables have been logged and detrended using Band Pass (6,32)ρ is the sample first-order serial correlation coefficient. Adv is the sumbetween newspaper and internet advertising. Data sample goes form 1971.q12005.q1.

Table 1 confirms that advertising has a positive correlation coefficient of 0.8with GDP, and it is two times more volatile. Plus, it appears very persistentover the cycle, with a the point estimate of the first order autocorrelation equalsto 0.9. That is true both for the aggregate series and for the newspaper alone.Regarding the single components of our advertising series, Internet data do notmake any sensible difference, except for bringing some extra volatility to theaggregate series. Finally, the positive correlation (0.52) of the ratio — advertisingexpenditures to GDP — with the GDP itself suggests that the advertising is nota constant proportion of GDP. In the next section, we set up a model where thefirms’ optimal policy rule for advertising is able to match these figures.As we said, we have only a partial series of advertising expenditures at

quarterly frequencies. It can be questioned whether our figures are really rep-resenatative of the total expenditures. To address an answer, we check therobustness of previous facts by computing the same statistics using data at an-nual frequencies, whose figures include the whole expenditures in all the media.The results are reported in Table 2.

3Since the band pass is an approximation of the optimal filter, to control for sputiousrelationships by calculating the statistics twice: with the band pass and with the Hodrick-Prescott filter. To save space HP statistics are not reported, but they are available uponrequest.

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Table 2. Business cycle statistics (Yearly data)

σ(Xt)σ(Xt)σ(Gdpt)

ρ σ(Xt,Gdpt)

Gdp 1.40 1 0.07 1Adv 2.40 1.70 0.16 0.69AdvGdp 1.77 1.23 0.14 0.10

News P. 2.90 2.02 0.16 0.63Magazines 3.60 2.53 0.19 0.76Radio 2.40 1.68 0.12 0.57

Television 7.70 5.40 -0.03 0.54Outdoor 3.80 2.65 0.01 0.51

Note: all the data have been logged and detrended with theBand Pass (2,8). Data sample goes from 1947 to 2000.

The annual figures confirm what we got using quarterly data. Total adver-tising expenditures are procyclical and more volatile than output (second rawof Table 2). In particular, the expenditures in all the media show positive corre-lation with GDP (the last column), but with very different degree of volatility.Most important, the comparison between the figures about the newspaper andthe total advertising series, e.g. standard deviation, autocorrelation, and corre-lation with GDP, indicate that the join series of advertising in newspaper andinternet can be considered a good proxy for total advertising.To return to the quarterly series, we now investigate the relationships of ad-

vertising with consumption and investment. The literature about the macroeco-nomics effects of advertising has always focused on the relationship of advertisingon consumption. Since the adv is supposed to tilt consumers choices, it is nat-ural to explain the effects of adverising on the aggregate economy through thischannel.Panel 4 in Figure 1 plots quarterly advertising along with consumption and

investment. We find that advertising is positive correlated with both consump-tion and investment. Plus, it appears to be more volatile than Consumptionbut less volatile than Investment. Table 3 provides the accompanying businesscycle statistics.

Table 3. Statistics: Adv, Consumption and InvestmentConsumption Invest.

Total Non Dur. Dur. Serv. Total

100*σ(Adv)σ(X) 2.87 3.30 0.88 4.72 0.49

σ(Advt;Xt) 0.83 0.80 0.83 0.75 0.83All the data have been logged and detrended with B.P(6,32). Datasample goes from 1971.q1 to 2005.q1

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The correlation coefficient between advertising and consumption is 0.83,which is slightly higher than the one it has with GDP. The relative standarddeviations is 2.87, i.e. advertising more than twice more volatile than consump-tion, but it has half the volatility of investment. In details, the advertising 4times more volatile than Services, and 3 times more volatile than non-durableconsumption, but less volatile than durable goods (relative standard deviationequal to 0.88).Interestingly, the advertising turns out to have a cyclical behavior similar to

the investment variables, i.e. it is strongly procyclical and highly volatile. Arrow(1962) made the same point. He argued that a good advertising campaign couldinfluence the demand for many periods of time. Advertising seems to add upto a stock rather than been a single-period-lasting flow. Accordingly with ourfindings, we support this view and we will model advertising to influence presentand future demand of goods, and so the present and the future revenues of thefirm that advertizes. This point will be more clear in next section where we setup the model. For the moment, we just anticipate that such temporal effectof advertising is captured by the concept of the goodwill,4 where advertising ismodeled like a flow that adds up into a stock that accumulates (and depreciates)as the time goes by, exactly as the stock of physical capital.The last issue of this section is about the dynamic cross correlations between

advertising and consumption and investment, which are provided in Table 4.The dynamic correlations show that advertising and consumption move con-temporaneously (the stronger correlation occurs at k=0), and that both tend tolead GDP over the cycle.5

Table 4. Dynamic cross correlationsσ(X(t);gdp(t+k))

K -4 -3 -2 -1 0 1 2 3 4Adv. 0.00 0.25 0.50 0.69 0.80 0.82 0.72 0.58 0.41Cons 0.04 0.29 0.54 0.76 0.90 0.93 0.86 0.70 0.47Inv 0.12 0.42 0.70 0.89 0.95 0.88 0.71 0.49 0.26

σ(X(t);adv(t+k))

Cons 0.25 0.47 0.67 0.79 0.83 0.78 0.65 0.48 0.30Inv 0.06 0.30 0.52 0.71 0.83 0.82 0.72 0.54 0.33

All the data have been logged and detrended with B.P(6,32). Data samplefrom 1971.q1 to 2005.q1.

4This concept was introduced by Arrow (1962).5The fact that consumption leads GDP is a well known characteristic of the US economy.

See Benhabib and Wey (2002) for a theoretical explanation.

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Such time path of advertising contrasts with the one found in Blank (1962).He reported evidences that advertising tends to lag the output, and similarresults are found in Yang (1962). The difference may be due to the different datasamples, or to the different detrending filters used in their papers.6 Anyway, theevidences we found here seem to dismiss the conventional idea that advertisingcan be a leading indicator of the cycle.To summarize, the main features we found in the data are the following:

• The annual ratio between advertising and GDP shows that the amountof resources invested in advertising are consistent. The ratio fluctuatesbetween 1.7 and 2.3 per cent. The per capita real advertising shows astrong upward trend.

• Advertising is strongly procyclical and high volatile. This is true at bothquarterly, and yearly frequencies. In quarterly data, advertising is quitepersistent over the cycle.

• Advertising expenditures are positive correlated with both consumption,and investment. However they are more volatile than consumption (andits non durable component), but less than investment, and durable con-sumption. The evidence is compatible with the hypothesis that advertisingacts in the cycle as an investment.

• From the dynamic cross-correlations it emerges that advertising expendi-tures and total consumption tend both to lead GDP over the cycle, whilethe biggest cross-correlation between consumption and advertising is thecontemporaneous one. Such evidence is compatible with a model in whichfluctuation in consumption generates fluctuation in advertising, and bothgenerate fluctuations in output. Potentially, advertising seen as a sellingcost, could amplify the effect of any shocks that lead to fluctuations inconsumption.

3 The model

3.1 Overview and basic assumptions

The model we present here is a two sector neocalssical growth model. The twosectors are consumption and investment. The consumption sector is modeledas a monopolistic competition market, where there is a continuum of imper-fect substitute goods. This assumption is not essential. Any framework with adownward slope demand function is suitable for our purposes. The adv affectsthe demand of each good, by triggering some urge to consume into the agent.The investment is a perfect competition sector, the adv does not affect its de-mand, and the agent uses it to bring purchasing power to the future. The firms

6Both the papers used first differenced data, which has been argued to be not a validmethod to isolate the business cycle component in the time series.

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in the consumption sector advertize their good, and the advertising is a costthat absorbs resources in term of the good they produced.From this framework we derived a nonconventional demand schedule, and we

plug it in an otherwise conventional Multisectorial Real Business Cycle model.

3.2 The demand of goods and the role of advertising

Key ingredient of this model, we now study the effect of advertising on con-sumer’s demand. The immediate intuition is that advertising acts as a positivetaste shock on the consumption of individual goods. In other words, the morethe firm spends to advertize a good, the higher it fosters the sales, but with de-creasing returns. At firm level, this is one of the few empirical evidences aboutadv that are not source of controversy in the literature, and is well documentedin large number of empirical studies.7

The positive relationship between advertising intensity and sales in the de-mand of individual goods, is consistent with a preferences system á la Geary-Stone, where the utility of a good is measured by the distance from the actualconsumption and the desired one. The standard interpretation of "desired level"is the level of substistence, which sounds funny concept in contemporaneouseconomies. More reasonably, the desired level of consumption of an individualgood is function of different variables, like how vivid is the good in the memoryof the consumer when he shops; and/or how many usages are possible of thatgood with respect to other similar goods; and/or whether there is a social statusassociate with the consumption of that specific good. So the adv may trigger theconsumer’s need either because it hits the consumer with new information aboutthe good, or because it shows some added value in purchasing the advertizedgood. Either way, we may assume, then, that current and past advertisementsadd up to create the reputation of a good, the so called the producer’s "good-will". The evidence that the effect of advertising lasts in time is another reliableempirical regularity that we borrowed from microeconomic literature to buildup the model.8 In particular, the utility of a good perceived by the consumer islikely to be affected not only by the current advertising expenditures, but alsoby the past advertisements, with an intensity that fades out as the time goesby. Thus, we define the intangible asset goodwill as the stock of advertising thataffects the consumer’s utility at time t. Such stock is supposed to summarizethe effects of current and past advertising outlays on the demand.For the good i at time t, we postulate the law of motion of the goodwill Gi,t

as:Gi,t = (1− δg)Gi,t−1 + Zi,t (1)

where Zi,t is the current advertising outlays, and δg is the depreciation rate ofthe goodwill.As we said, in our model economy there is a monopolistic competition mar-

ket with a continuum of imperfect substitute goods indexed i in the interval

7See Bagwell (2005) Section 3.2 for a survey of the empirical studies.8 See Clark (1976) for a survey of empirical results.

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(0, 1). The consumer chooses the optimal basket of intratemporal consumptionamong all the goods, by solving the standard expenditure minimization prob-lem. Accordingly with the Geary-Stone preference system, the Dixit-Stiglitzaggregator of consumption goods in this problem is the following:

eCt =

⎛⎝ 1Z0

¡Ci,t −Gθ

i,t

¢ ε−1ε di

⎞⎠ε

ε−1

(2)

where ε > 1 is the usual price elasticity of the demand, and θ ≤ 1 is a structuralparameter that controls for the effects of the advertising on the utility.For the consumer Gi,t is given, since the level of goodwill is decided by the

firms. So, he chooses the best combination of goods Ci,t to minimize the totalexpenditures for any given level of utility eCt and goodwill Gi,t. The result is thesystem of demand equations:

Ci,t =

µPi,tPt

¶−ε eCt +Gθi,t Xi ∈ (0, 1) (3)

where Pi,t is the price of good i at time t, and Pt is the price aggregator. Thislast is shown to be the Lagrange multiplier of the minimization problem, or:9

Pt =

∙Zp1−εi,t di

¸ 11−ε

(4)

The advertising modeling we have chosen has the important property that theprice index is not affected by adv. It seems to us neither reasonable, nor possi-bly supported by empirical studies, to imply an aggregate effect of adv on theconsumer price index, as in the model by Benhabib and Bisin (2000).Equation (3) is a key relationship in the advertising model; first, it shows

that the goodwill raises the level of the demand, with a concave effect whichdepends on the parameter θ. As a consequence, the firm has now two comple-mentary policies to push the demand, and we will see in the section on thegeneral equilibrium how this modifies the resource constraint of the economy.Second, the advertising affects the price elasticity of the demand, and makesit time dependent. The demand schedule (3) is composed by two terms: thefirst one, (Pi,t/Pt)

−ε eCt, which has elasticity ε, and the second one, Gθi,t, which

in inelastic. Thus, the price elasticity of the demand of good i is a weightedaverage of the two. In details, the price elasticity we obtain from (3) is:

|εc,p| = ε

Ã1− Gθ

i,t

Ci,t

!(5)

9As in the standard expenditure minimization problem (Dixit-Stiglitz, 1977), the Lagrangemultiplier is the rise in consumer’s expenditures for a marginal increase in the utility. Thismultiplier is a prices index omogeneous of degree one, and fulfils the requirements to be theconsumption price index of this economy.

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For sake of comparison, the price elasticity of demand in the standard modelof RBC with monopolistic competition is ε.10 The finding of a steeper demandschedule for an advertized good is a well know effect in the literature, namedthe fidelization of the consumer.More interestingly from the point of view of this paper, the demand elasticity

(5) implies that the mark up is time dependent in the model economy. Suchimplication matches the last empirical findings about the time variation of themark up in US economy, and address a solution to the recent critics to theconstant mark up in the standard RBC model. We will investigate more thisissue in the section of the paper about the results.

3.3 The Firms’ Policies: Price and Advertising

Our model economy has two sectors: Investment and Consumption, but theadv enters only in the consumption sector. This is the simplest framework toassure that the model fulfils a set of reasonable conditions about the adv. Inparticular,

1. The consumption in equilibrium cannot coincide with the aggregate de-mand. Against this option, Young and Seldom (1994) argued that a posi-tive effect of advertising on consumption is not enough to draw conclusionsabout the macroeconomic effects of adv, because this last is likely to crowdout the Investments. Thus a one sector model with bonds, rather than atwo sector model with investment, would not have been the appropriatechoice here because the net supply of bonds is always zero in equilibrium,and the aggregate demand coincides with aggregate consumption. Hence,we introduced the Investment sector.

2. While it is reasonable to suppose that advertising affects the Consumption,it is highly improbable that advertising affects also the Investment. In thismodel, the decisions about how much to invest are taken by the consumers,in order to bring purhasing power to the future. Now, one implication ofadv in the Investment sector would be a direct negative effect of adv onthe interest rate, which of course is a silly hypothesis. Therefore, adv mustaffect only the Consumption sector. Any effect of adv on Investment mustbe indirect, and we analyze the solution of the model to see whether thisis true or not.

3. As a matter of fact, in present model adv behaves as an endogenous tasteshock. It is per se interesting to compare the general equilibrium re-sponses to an endogenous advertising shock and to an exogenous tasteshock. Eventually, any qualitative difference can be used in the estima-tion of the DSGE in order to disentangle the endogenous component of ademand shock from the exogenous (pure) taste shock.

10Our model nests the standard one as a particular case. In facts, when the goodwill asno effect on the demand (θ = 0), the mark up is constant and equal to ε/ (ε− 1) as in thestandard case.

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Beyond these conditions, the framework we use is quite standard: the twosectors are in competition on common factor markets, i.e. Labor and Capital,where there is perfect competition. Thus, there are two equilibrium prices: onein the market for labor (the wage Wt), and one in the market for capital (theinterest rate Rt), both endogenously determined.To keep the model as simple as possible, we model the Investment sector as

perfect competition market. It produces an omogeneous good It in each period,which adds up to the capital stock Kt with the standard Law of Motion. Thissector, of course, does not generate any profit, and it has no impact on therepresentative Consumer’s budget constraint.On the contrary, the Consumption sector is a monopolistic competition mar-

ket with a continuum of firm indexed i ∈ (0, 1) , each of them producing animperfect substitute good. The demand for each good is derived above in (3).As usual, the firm minimizes the cost function to choose the optimal factorscombination. The cost function for the i− th firm is

C¡Y ci,t

¢= Bc

R1−αct Wαct

AtUct

¡Y ci,t + Zi,t

¢(6)

where Bc =³1−αcαc

´αc ³1

1−αc

´is a constant term, At is the technology shock

common to the two sectors, and Uct is an idiosyncratic shock of the consumption

sector. From (6) is apparent that the adv does not enter into the marginal cost ofthe firm. This is an interesting feature of the model, because it is in accordancewith the marketing rules where adv is a financial cost, not a production cost.Moreover, it stresses the difference of adv with other non-pricing policies of thefirm, like the R&D. This characteristic is due to the production function weused:

Y ci,t = U c

tAtNαci,tK

1−αci,t − Zi,t (7)

In (7) adv is measured in units of the consumption good produced by the firm.This last is produced using a standard cobb-douglas production function, andpart of the production is used to pay the cost of advertizing.Also, this production function (7) is crucial to satisfy the assumption that

adv does not get into Investment sector. It can be shown, indeed, that anymodel where Zt has its own production technology that uses labor, or capital,or investment, or all of them, in turns implies that adv has a direct effect on thewage, or the interest rate, or on both of them. Again, this assumption soundsweird to us, and we carefully avoided it.Althought expenditures in adv absorbe resources, they don’t change the cost

structure. Now, if the cost minimization problem is not affected by the levelof adv chosen by the firm, and we can solve the firm optimization with a twostep procedure: first, the firm decides the optimal combination of factors in theproduction. Then, it decides how much to produce and how much to spend inadvertising.The cost minimization problem is standard, and we briefly give the main

conditions in the next section. The profit maximization, instead, is a noncon-

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ventional problem and we solve it here. Differently from the standard monopo-listic competition case, the firms here solve an intertemporal problem in order tomaximize the profits under the demand contraint given by (3), and the Law ofMotion of the goodwill (1). In this case, one can show that the optimal pricingrule is given by:

p∗i,t, =ε

µ1− Gθ

i,t

Ci,t

¶ε³1− Gθ

i,t

Ci,t

´− 1| z

≡MKi,t

MCi,t (8)

As in the standard case, the firm optimal strategy is to set the price equalto the marginal cost times a mark up coefficient. The difference is that themark up is time varying because now the price elasticity of the demand is notconstant anymore, as we pointed out in previous section. Hence, we claim thatan important corollary of the adv model is the microfoundation of the variabilityof the mark up in the cycle. This implication is specially interesting in relationwith the recent literature about the time variability of the mark up and the so-called deep habits of the consumers11 and, therefore, it will be object of furtherinvestigations.From the profits maximization we obtained also the other policy of the firm,

which gives the optimal advertising level:

(MKi,t − 1)MCi,t · ∂Ci,t

∂Gi,t+ β (1− δg)E (MCi,t+1)| z

marginal benefit

= MCi,t| z marginal cost

(9)

Equation (9) is a special case of the dynamic Dorfman-Steiner condition pro-vided in Arrow and Nerlove (1962). It states that the firm invests in adv untillthe marginal benefit of an extra unit of adv equals the marginal costs of produc-ing it. Because of the Law of Motion of the goodwill, the marginal benefit hastwo components: one is the increase in the revenues associated with a marginalincrease in adv; the other one is the discounted opportunity cost for producingthe (depreciated) goodwill that survives tomorrow.

3.4 The General Equilibrium

For the complete design of the general equilibrium model we need few morerelationships. First we characterize the consumer’s intertemporal allocation ofconsumption and saving. From this exercise we obtain the aggregate labor sup-ply and the aggregate demand schedule (the euler equation). Next, we placethese relationships, toghether with the pricing and non-pricing optimal policiesof the firm within a standard multisector framework. Finally we solve the modelto find the non-cooperative symmetric equilibrium. The equilibrium relation-ships make the influence of advertising on the model economy trasparent. For

11See Ravn et al. (2005), forthcoming RES.

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the interested reader, a more detailed derivation of all the equations is providedin Appendix B, together with the log-linearized form of the model.

3.5 The Aggregate demand

Once the optimal composition of the goods aggregator is chosen, the consumermaximizes the intertemporal utility function. We follow the traditional for-mulation of the intertemporal problem in the RBC, where the representativeconsumer is a infinitely-lived agent who consume, work, and save. The utility isa CRRA separable function of two arguments: the consumption aggregator eCt,and the labor Nt.12 We already discussed about the consumption aggregator inpresence of adv, which is given in (2). The consumer is also endowed with one(normalized) unit of time, which can be devoted to work or leisure. As usual,the utility is time separable. The budget constraint is slightly different in ourmodel, because now the goods demand is affected by the advertising, which onone side enters in the aggregator of consumption, and on the other side drawsresources from the system. The nominal budget constraint can be written as:

Pt eCt +

1Z0

Pi,tGθi,tdi+ P I

t It ≤WtNt +RtKt +Πt (10)

We already defined all the variables that appear in (10), except for Πt whichare the profits that the consumption sector redistributes to the consumer.The intertemporal optimization programme is solved maximizing the utility

function under the infinite sequence of budget constraints (10) for t = 0, ...,∞,and under the standard Law of Motion for the capital. From the FOCs weidentify the Lagrangian multiplier,

λt = eC−σt (11)

which turns out to be the marginal utility of consumption, which now depends onboth consumption goods and the goodwill levels. Plus, we derived the aggregatesupply of labor,

γNϕt =Wtλt (12)

and the euler equation,

1 = βE

∙PtPt+1

λt+1λt

1

P It

¡P It+1 (1− δ) +Rt+1

¢¸(13)

The relationships (12) and (13) are the same optimal conditions than inthe standard model, unless that now the marginal utility depends on the levelsof goodwill. Through its effect on the marginal utility, advertising modifiesboth the labor supply and the intertemporal condition, which determines theproportion of todays consumption versus the consumption of tomorrow.

12See the Appendix B.

14

3.5.1 Partial Equilibrium analysis

When the firms invest an extra unit of advertising, the higher level of goodwill in-creases the marginal utility of consumption. In equation (12) the RHS becomesbigger than LHS, and the agent will raise the supply of labor to recover theequality. In the euler equation (13), an increase in Zi,t will make the marginalutility higher both today and tomorrow, since the goodwill is an autoregressiveprocess. The consumption will raise in t, and then follows a declining monotonepath back to the steady state. This partial analysis mechanism casts light onthe the results we will find in the aggregate dynamics.Since the advertising increases both the labor supply and the consumption,

it triggers a "work and spend cycle" — which owes its name to J. Schor (1992).We obtained such result using just few standard assumptions about adv atindividual good level rather than building up an ad-hoc formulation of adv inthe aggregate. In our opinion, this is one relevant contribution of this paper.

3.6 The complete model

The relative price of Investment is derived from the cost function in the Invest-ment sector, which is:

C¡Y It

¢= BI

R1−αIt WαIt

AtU It

Y It (14)

where BI =³1−αIαI

´αI ³1

1−αI

´, At is the (common) technology shock, and U I

t

is an idiosincratic shock on the Investment production funtion. For the assump-tions we made, the derivative of (14) w.r.t Y I

t is the price of the Investmentgood P I

t . The price for the consumption aggregator is given in (4), and in thesolution of the model we normalize it to one.To close the model we impose the market clearing conditions for the fac-

tor markets (capital and labor), and for the goods markets (investment andconsumption goods). Plus, the two resource constraints about the productionfactors must be satisfied:

Nt =

1Z0

Nci,tdi+NI

t

Kt =

1Z0

Kct di+KI

t

Finally, the exogenous shocks are assumed to satisfy:

At = Aρa

t−1eεat

Uct =

¡Uct−1¢ρc

eεct

UIt =

¡Uct−1¢ρI

eεIt

15

where ρa, ρc, ρI ∈ (0, 1) and⎛⎝ εat

εctεIt

⎞⎠ ∼ N

⎡⎣⎛⎝ 000

⎞⎠ ;⎛⎝ σ2a 0 0

0 σ2c 00 0 σ2I

⎞⎠⎤⎦ .As next step we solve the model to find the non cooperative equilibrium,

which is defined as follows:

Definition 1 Given the sequence of stochastic process©εat , ε

ct , ε

It

ª∞t=0

and a vec-tor of initial conditions, the general equilibrium is the sequence of prices andquantities

©Pt, P

It ,Wt, Rt,Πt, Ci,t, It,K

ct ,K

It , N

ct , N

It ,Kt+1, Nt, Gi,t, Zi,t

ª∞t=0

s.t.(i) the consumer maximes the intertemporal utility funtion under the budgetcontraint and the law of motion of the capital; (ii) the firms maxime the profitsunder the technology contraint and the law of motion of the goodwill; (iii) allthe markets are in equilibrium.

It can be shown that the equilibrium is symmetric for all the firm, since theyall face the same marginal cost. This implies that all the prices are equals, which,in turns, lead to the same goodwill and consumption level for all the firms. Aformal proof of the solution is given in the Appendix B. In what follows we wantto stress the main mechanism through advertising in the aggregate dynamics.

4 Results

4.1 Long run effects: The Steady-State

The non-stochastic steady state of the model is determinated setting the ex-ogenous variables to their mean value. In what follows we focus on the longrun effects of advertising expenditures. Under the assumption of Stone Gearypreferences advertising expenditures affect in the long run both prices and laborsupply with important consequences on all the others variables of the model.The Euler equation can be used to determine the steady-state ratio of the

investment sector capital to the total capital of the economy

KI

K=

βδk (1− αI)

1− β (1− δk)(15)

Using the capital resourse constraint, and the optimal demand of capital andlabor in both sectors we get the followings

Kc

K= 1− KI

K(16)

KI

NI=

µδkK

KI

¶− 1αI

(17)

Kc

Nc=

µ1− αcαc

¶µαI

1− αI

¶KI

NI(18)

16

Hence, advertising does not modifies the steady-state ratios of all the pro-ductive factors. Indeed, such values are exactly the same as in the standard twosectors model.Calculating the steady-state mark up allows as to draw another trasparent

comparition with the standard two sectors RBC setup. In particular, indicatingwith µs the mark up in the standard two sector model, it is trivial to verify thatthe following inequality holds

µ =ε³1− Gθ

C

´ε³1− Gθ

C

´− 1

>ε

ε− 1 = µs ∀Gθ

C> 0

that is, Advertising increases the steady-state value of the mark-up. Chang-ing the persived differentiation of the goods, advertising increases the degree ofmonopoly power of the firms, and, consequently, the equilibrium mark up in thelong run.Advertising affects through the mark up all the others prices of the econ-

omy. In fact, from equilibrium condictions in the factor markets turns out thatthe steady-state values of the wage, the interest rate, and the relative price ofinvestment are (decrising) functions of the mark up. The steady state values ofsuch variables are given by the following equations

W = αcµ−1µKc

Nc

¶1−αc(19)

R = (1− αc)µ−1µKc

Nc

¶−αc(20)

pI = µ−1µ1− αc1− αI

¶µKc

Nc

¶−αc µKI

NI

¶αI(21)

It is worth to note that advertising generates a competitive effect betweenconsumption and investment. The relative price of investment decreases withadvertising, and, consequently, in the long run investment becomes more conve-nient than consumption. In this perspective, the aggregate advertising expen-ditures negativelly affects consumption.The consumer intratemporal condiction can be used to analyze the impact

of advertising on the equilibrium labor supply. From equation, after a bit of al-gebra, it turns out that the long run labor supply is determined by the followingequation

γNϕ =

µ1− gθ

C

¶−σC−σW (22)

Equation (22) is exactly the same as in the standardard two sector model

except for the term³1− gθ

C

´−σ, which positively affects the marginal utility of

consumption, and so the labor suply. Nevertheless, advertising also decreases

17

the steady-state values of the wage and, consequently, modifies the long runlabor supply in two opposite directions. Hence, the total impact depend uponthe relative force of these two factors. Moreover, the actual impact of advertisingon both the wage and marginal utility can be determined analytically only oncewe solve for the steady-state value of advertising. Such a value depends uponthe optimal advertising policy, which is given in steady state by the following

θ(µ− 1)gθ−1 = 1− β(1− δg) (23)

Unfortunately, equation (23) cannot be esplicitelly solved for advertising.Nevertheless, it is still possible to characterize a sufficient condiction underwhich advertising unequivocally increases the long run labor supply.

Proposition 2 In simmetric equilibrium a sufficient condiction to advertisingunequivocally increases the long run labor supply is σ > 1

Proof. See apendix C.Since advertising does not modify the ratio of capital to total labor (as well

as the ratios of the productive factors to total labor) then the above conditionalso implies that all the others productive factors increase with advertising.The consumption market clearing condiction can be used to find an expresion

for the steady-state value of consumption

C = N

µNc

N

¶µKc

Nc

¶1−αc− Z (24)

The effect of advertising on consumption in the long run is not clear. Indeed,advertising positively affects the labor supply but, at the same time, tends tocrowd out the consumption. As before, the actual impact depends upon thesteady state value of advertising. In order to qualitatively explore the relation-ship between advertising and consumption, in Figure 1.C (see appendix C) wegraphed the numerical derivative of consumption with respect to advertising.From the picture turns out that, in our model, advertising tends to push

up the consumption in the long run. The derivative is, in fact, always positive.Consequently, for a wide range of parameters the positive effect fo advertis-ing on the labor supply tends to over compensate its crowding out effect onconsumtpion .The ratio of consumption to GDP can be recovered from equation (mettere

il numero)

C

Y= 1− pII (25)

Again, the actual impact of advertising on this ratio is not clear. However,from figure 2.C turns out that also such ratio increases with advertising for awide range of parameters. Consequently, the model predict that with advertisingthe economy tends to be more consumption based.In the previous analysis we have showed that esplicitly considering advertis-

ing in an otherwise standard two sector model changes the long run equilibrium

18

in many directions. In particular, we have proved that advertising affects bothnominal and real variables. Particularly interesting is the effect of advertisingon the real variables.

4.2 Short run effects: the Aggregate Dynamics

We now characterize quantitatively the response of the adv model to a variety ofexogenous shocks. We consider three source of aggregate fluctuations: a technol-ogy shock, a taste preferences (i.e. demand) shock, and finally an idiosyncraticshock to consumption sector. We choose fairly standard values for the tasteand technology parameters. The value of the depreciation of the goodwill ischoosen to match the average duration of the effects of an advertising campainon consumers’ minds, which are been shown to last for around nine months.13

The following tables summarizes the whole calibration set of parameters.

[INSERT CALIBRATION TABLE HERE]

The following two Figures give the overall picture of the aggregate dynamicsin our model with respect to the standard two sector RBC model. The blucontinuous line is the adv model, the red dashed line is the standard RBCmodel. Recall that with the standard model the mark up is constant

13See Clarks (1976).

19

20

The propagation mechanism of the adv is apparent in the three cases. Inparticular, adv is procyclical as in the stylized facts (always positive IRF). Theconsumption has always an higher reaction to the shock. And finally, the Im-pulse of the hours worked is bigger than in the case of the standard model.Moreover, in the model economy the mark up is time-varying, which is a fea-ture that deserves further attention.Our mechanism is that the agent wants to consume more when there is adv.

To achieve ah higher consumption he works more. We provide some sensitiveanalysis to show the mechanism. In particular we show the effect on IRF ofconsumption (Figure 4) and output (Figure 5) when the parameter that controlsfor Fisher elastyicity of labor varies. φ = 0 is the classical linear case of Hansenand Prescott (1982).

21

22

5 ConclusionsIn this paper, we assessed the ability of aggregate advertising expenditures toaffect the aggregate economy. We analysis this aspect explicity modelling adver-tising in a DSGE model. A central finding of our investigation is that advertisingexpenditure can potentially affects the aggregate economy in both long run andshort run.According with a Stone Geary preferences system, the firm’s advertising

positively affects the household demand of good. In aggregate terms, it impliesthat advertising positively affects the total consumption of the economy as if itwere an exogenous demand shock. According with Galbraith (1967), our modelpredicts that advertising affects not only the composition but also the totalammount of consumption expenditures.In the short run advertising modifies the marginal utility of consumption

inducing a urge in consumption. At the same time, advertising modifies themarginal rate of substitution between consumption and leisure inducing a strongersubstitution effect. Consequently, the household is willing to work more to con-sume more. Such an effect allowed us to identify the meccanism through whichadvertising hits the economy. If fact, advertising increases the fluctuations ofboth consumption and laisure, and so amplifies the fluctuations of the economy.According with Kaldor (1950), advertising in our model is able to amplifies anykind of shocks that hit the economy.In the long run, under a wide range of parameters empirically plausible, ad-

vertising implies an increasing in both consumption and labor supplies. More-over, the ratio of consumption to Gdp increases with advertising. Consequently,like in Galbraith (1967), our model predicts that advertising tends to generatean economy more consumption based.

23

References[1] Ashley, R. and Granger, C.W.J. and Schmalensee R. (1980): "Advertising

and Aggregate Consumption: An Analysis of Causality"; Econometrica vol.48 no5 pp.1149-1168.

[2] Ashley, R. (1998): "A New Technique for Post Sample Model Selection andValidation"; Journal of Economic Dynamics and Control vol. 22 pp.647-665.

[3] Bagwell, Kyle (2003): "The Economics of Advertising"; Elgar ReferenceCollection. International Library of Critical Writings in Economics, vol.136. Cheltenham, U.K. and Northampton

[4] Baxter, M. and King, R. G. (1995): "Measuring business cycles. Approx-imate band-pass filters for economic time series"; NBER Working PaperSeries no5022.

[5] Benhabib, Jess and Bisin, Alberto (2000): "Advertising, Mass Consump-tion and Capitalism"; Working Paper Department of Economics, NYU .

[6] Benhabib, Jess and Wen, Yi (2004): "Indeterminacy, Aggregate Demand,and the Real Business Cycle"; Journal of Monetary Economics vol. 51 no3,pp. 503-530.

[7] Blank, David (1961): "Cyclical Behavior of National Advertising"; Journalof Business vol. 35 pp. 14-27.

[8] Brack, J and Cowling, K. (1983): "Advertising and Labor Supply: Woek-week and Workyear in U.S. Manufacturing Industries, 1919-76"; KyklosVol. 36:2 pp.285-303

[9] Clarke, D. G. (1976): "Econometric Measurament of the Duration of Adver-tising Effect on Sales"; Journal of Marketing Research vol 13:4 pp 345-357.

[10] Cogley, Timothy and Nason, James M. (1995): "Output Dynamics in Real-Business-Cycle Models"; American Economic Review Vol. 85 no3 pp. 492-511.

[11] Datamonitor (2004): "Advertising in The United States";www.datamonitor.com

[12] Dixit, Avinash K. and Stiglitz, Joseph (1977): "Monopolistic Competitionand Optimal Product Diversity", American Economic Reviw vol.67 no3 pp.267-308

[13] Dixit, Avinash K. and Norman, Victor (1978): "Advertising and Welfare";Bell Journal of Economics vol.9 no1 pp.1-17.

[14] Fraser, J. and Paton, D. (2003): "Does advertising increases the laborsupply? Time series evidences from the UK"; Applied Economics vol. 35,pp.1357-1368.

24

[15] Galbright, J.K. (1967): “The New Industrial State” Boston: HoughtonMifflin

[16] Kaldor, N.V. (1950): "The Economic Aspects of Advertising"; Reviw ofEconomic Studies vol.18 pp.1-27.

[17] Marshall, A. (1890): "Principles of Economics"; MacMillan and Co. (Lon-don).

[18] Marshall, A. (1918): "Industry and Trade: A Study of Industrial Techniqueand Business Organization; and of Their Influences on the Conditions ofVarious Classes and Nations"; MacMillan and Co. (London).

[19] Nerlove, M. and Arrow, Kennet J. (1962): "Optimal Advertising Policyunder Dynamic Conditions"; Economica vol. 29 pp. 129-142.

[20] Robinson, J. (1933): "Economics of Imperfect Competition"; MacMillanand Co. (London).

[21] Stigler, G. J. (1961): “The Economics of Information,” Journal of PoliticalEconomy vol. 69 pp. 213-25.

[22] Stock, J. and Watson, M. (1999): "Business Cycle Fluctuations in USMacroeconomic Time Series"; in Taylor, J. and Woodford, M. (eds.) Hand-book of Macroeconomics vol. 1 pp. 3-64.

[23] Yang, Y. C. (1964): "Variations in the cyclical behaviour of advertising",Journal of Marketing 28:2 pp. 25-30.

25

A Source of Data

A.1 Quarterly Data

• Advertising expenditures in news paper advertising: Newspaperassociation of America. Avaible data sample from 1971q1 to 2005q4. Thedata have been seasonal adjusted using X11. Web site :http://www.naa.org/

• Advertising expenditures in internet advertising: Interactive Ad-vertising Bureau. Available data sample from 1996.q1 to 2005.q1 Website: http://www.iab.net/resources/ad_revenue.asp.

• All the Macrodata: Federal Reserve Bank of St. Luise. Web site:http://research.stlouisfed.org/fred2

In details:

• Gdp: Real Gross Domestic Product (GDPC96)• Consumption and components: Real Personal Consumption Expen-ditures and components (PCEC96)

• Investment: Real Private Fixed Investment (FPICA)• Defletor: Gdp Implicit Price Deflator (GDPDEF)

A.2 Yearly Data

Total Advertising expenditures and components: Universal McCann.Coen’s Annual Report.

B The Model

B.1 The households’ problem

In the first state the consumer solves the following constrained minimitationproblem

Mincsi,t

1Z0

pi,tci,tdi

s.t

⎛⎝ 1Z0

¡ci,t − gθi,t

¢)ε−1ε di

⎞⎠ε

εe−1

≥ Ct

From the solution of this problem we find the condictional demand sistem(3), and the consumer price index (4). In the optimum, using (3) the totalexpenditures can be rewrited in the following way

26

1Z0

pi,tci,tdi =

1Z0

pi,t

õpi,tPt

¶−εCt + gθi,t

!di = PtCt +

1Z0

pi,tgθi,tdi (26)

Using (26), the (real) intertemporal budget constraint that consumer facescan be rewrite in the following way

Ct +

1Z0

µpi,tPt

¶gθi,tdi+

pI,tPt

It ≤ Wt

PtNt +

Rt

PtKt +

πtPt

The intertemporal household problem can be now stated as consisting inchosing processes Ct, and Nt as to maximize (mettere l’utilitá) under the cos-traint (mettere il numero della law of motion). Formally

MaxCs

t+i∞i=0Nst+i∞i=0

P∞i=0 β

iEt

∙Cs(1−σ)t+i −11−σ − γ

Ns(1+ϕ)t+i

1+ϕ

¸

s.t.Ct +

1Z0

µpi,tPt

¶gθi,tdi+

pI,tPt

It ≤ Wt

PtNt +

Rt

PtKt +

πtPt

It=Kt+1 − (1-δk)Kt

The first-order conditions assosiated with the problem are

λst = C−σt

γNϕ = λtWt

Pt

λst = βEt

∙PtPt+1

λst+1pI,t

(pI,t+1 (1− δk) +Rt+1)

¸

B.2 The firms’ behaviour

B.2.1 Consumption sector

The Lagrangian of firm i ’s first stage problem can be written as

MinNCi,t,K

Ci,t

WtNci,t +Ri,tK

ci,t + λt

¡AtU

ctN

αci,tK

1−αci,t − zi,t − yci,t

¢The first-order conditions associated with this problem are the following

Wt = αcλtAtUctN

αc−1i,t K1−αc

i,t (27)

Rt = (1− αc)µtAtUctN

αci,tK

−αci,t (28)

27

Combining the above conditions we get the cost function (6).The optimal prices and advertising policy are derived by maximizing the

profit function. The dynamical nature of the goodwill makes the problem in-tertemporale. Formaly

Maxpi,t+j∞j=0gi,t+j∞i=0

P∞j=0 β

jEt

£pi,t+jci,t+j − CTc

i,t+j(YCi,t+j)

¤s.t.ci,t =

1Z0

µpi,tPt

¶−εCst ds+ gθi,t

zi,t = gi,t − (1− δg)gi,t

CTci,t (Y

Ci,t) = B

R1−αct Wαc

AtUct

(yci,t + zi,t) =Mci,t(YCi,t)(ci,t + zi,t)

or, equivalently, in term of Lagrangian function

MaxL =pi,t,gi,tci,t

P∞j=0 β

jEt [pi,t+jci,t+j −Mci,t+j(ci,t+j + gi,t+j − (1− δg)gi,t+j)]+

+P∞

j=0 βjEt

∙λt+j

µ³pi,t+jPt+j

´−εCt+j + gθi,t+j − ci,t+j

¶¸(29)

The first order condition associated with the problem are

pi,t −Mci,t = λt (30)

pi,tci,t = ελt

µpi,tPt

¶−εCt (31)

θλtgθ−1i,t + β(1− δg)Et [Mci,t+1] =Mci,t (32)

Combining the above equations, we finally get the optimal prices (8) andadvertising policy (9).

B.3 Symmetric Equilibrium

To prove the existence of the symmetrical equilibrium, it is enough to showthat both prices and goodwill are common among the firms. First note thatthe assumption on production function implies that the all firms face the samemarginale cost (i.e. Mct,i = Mct ∀ i ∈ [0, 1]). Second, an explicit value of theoptimal goodwill stock can be determinated once we substitute equation (30)in equation (32)

gi,t =

∙Mct − β(1− δg)Et [Mct+1]

θ (pi,t −Mct)

¸ 1θ−1

= f(pi,t;Mct;Et [Mct+1]) (33)

Now, substituing the above equation and the condictional demand in equation(3), we find that

28

pi,t =ε (pi,t −Mct)

³pi,tPt

´−εCt³

pi,tPt

´−εCt + gi,t

(34)

which implicitely shows that pi,t = g(Ct;Mct;Et [Mct+1]). Consequently, theoptimal price does not depend on index i (i.e. it is common among the firms).Substituing the implicit optimal prince into equation (33) we finally get thatalso the goodwill stock is common among the firms.

C Steady State

Proof of proposition 1. First note that the first term in the RHS of equation(22) can be rewrited in the following wayµ

1− gθ

C

¶−σ=

∙1

ε

µµ

µ− 1¶¸−σ

(35)

Second, observe that the ratio of consumption to labor supply is a decresingfunction of advertising

C

N=

µNc

N

¶µKc

Nc

¶1−αc− Z

N(36)

Substituing the above equations and the steady state value of wages (19) inequation (22), it is possible to find an expresion for the long run labor supply

N =

"1

γ

∙1

ε

µµ

µ− 1¶¸−σ µ

C

N

¶−σαcµ

−1µKc

Nc

¶1−αc# 1(ϕ+σ)

(37)

∝"µ

C

N

¶−σΩ(µ)

# 1(ϕ+σ)

where Ω(µ) = µ−(σ+1)

(µ−1)−σ . Now, differentiating (37) with respect to advertising,you get the following

∂N

∂Z∝"∂¡CN

¢−σ∂Z

Ω(µ) +∂Ω(µ)

∂Z

µC

N

¶−σ#(38)

Since CN is a decreasing function of advertising, then the first term in the

RHS of equation (38) is positive for every σ bigger than zero. Consequently,it turns out that whenever ∂Ω(µ)

∂Z = 0 implies ∂N∂Z > 0 . Hence, the inequality

∂Ω(µ)∂Z = 0 rappresents a sufficient condiction under which advertising increases

29

the long run labor supply. Now since ∂Ω(µ)∂Z = ∂Ω(µ)

∂µ∂µ∂Z and ∂µ

∂Z it is alwayspositive, it is trivial to verify that the following inequality holds

∂Ω(µ)

∂Z= 0 ⇐⇒ ∂Ω(µ)

∂µ= 0

Differenting Ω(µ) with respect to µ, you get

∂Ω(µ)

∂µ=

Ω(µ)h(µ− 1)−σ

i2 ∙− (σ + 1) 1µ + σ1

µ− 1¸

(39)

which implies that

∂Ω(µ)

∂Z= 0 ⇐⇒

∙− (σ + 1) 1

µ+ σ

1

µ− 1¸= 0

or

∂Ω(µ)

∂Z= 0 ⇐⇒ 1

µ− 1 =1

σ(40)

since 0 < µ− 1 < 1, then we conclude that∂Ω(µ)

∂Z= 0 ∀ σ = 1

This proves the statement.

30

Figure 2:

31

Figure 3:

32

Figure 4:

33