Cyclical Effects of the Composition of Government Purchases1

Contributor Notes

Author’s E-Mail Address: jaziz@imf.org, lleruth@imf.org

This paper constructs a general equilibrium model with monopolistically competitive firms and endogenous markups where government spending consists of both consumption and investment goods. It is shown that when markups are countercyclical, increases in the share of investment goods in aggregate government expenditure entail a trade-off between greater long- run efficiency and higher short-run volatility. Estimates based on the model, calibrated to the postwar U.S. economy, show that the effects on output, employment, and welfare can be significant

Abstract

This paper constructs a general equilibrium model with monopolistically competitive firms and endogenous markups where government spending consists of both consumption and investment goods. It is shown that when markups are countercyclical, increases in the share of investment goods in aggregate government expenditure entail a trade-off between greater long- run efficiency and higher short-run volatility. Estimates based on the model, calibrated to the postwar U.S. economy, show that the effects on output, employment, and welfare can be significant

I. Introduction

This paper shows how changes in the composition of government expenditure between consumption and investment goods can affect both the long run efficiency as well as short run fluctuations of an economy even when public expenditure is assumed to be unproductive. Using a model with horizontally differentiated goods and monopolistic competition we find that when markups are countercyclical, an increase in the share of investment goods in total public expenditure, raises output, employment, and capital stock in the steady state. While this leads to increases in welfare and productivity in the long run, it also raises the short run cyclical variability of the economy. In particular, variance of output and employment arising from technological and aggregate demand shocks increase as the long run share of government investment goes up. Quantitative estimates using the model calibrated, to the postwar U.S. economy, shows that the effects of changing the composition of public spending can be quite significant. Although these effects are reversed when markups are procyclical, empirical evidence indicates that in the postwar U.S. economy markups are in fact countercyclical.

The link between the composition of public expenditure and growth has been the focus of endogenous growth models such as those of Barro (1990) and Devarajan et al. (1996), where a part or all of government expenditure is assumed to be directly productive. Some authors have also explored the impact of aggregate government purchases on short run output and employment when public spending is not directly productive. In perfectly competitive economies, an increase in government spending raises households’ willingness to supply more labor in response to the corresponding fall in wealth arising from financing additional government spending and through an increase in real interest rate induced by the higher government outlay (see, for example, Aiyagari et al 1989 and the references therein). In a model of oligopolistic price setting, Rotemberg and Woodford (1992) showed that higher government spending can result in increased output and employment without any shift in households’ consumption-leisure choices. However, these studies have concentrated on the effect of aggregate public spending rather than on its composition. In this paper, we concentrate our attention on the impact of the composition of public spending on the behavior of a monopolistically competitive economy.

The economy we consider is one where final goods have two different end uses — consumption and investment. Under the assumption that firms cannot discriminate between the two sets of buyers, the price elasticity of aggregate demand is given by the average of the elasticities of consumption and investment, weighted by the shares of the two end uses in total demand. This implies that the composition of aggregate demand will determine the final price elasticity faced by the producers of the commodity. In a world where firms have some market power, the markup charged by a firm over its marginal cost is then determined by this average elasticity. Consequently, whenever the composition of the demand changes between the two end uses so does the elasticity and in turn the markup. Since decisions regarding the use of different inputs by firms depend on the markup, i.e., in equilibrium MPKt = μtrt, where MPKt is the marginal product of capital, rt its rental price, and μt the markup, and MPLt = μtwt, where MPLt is the marginal product of labor and wt the real wage rate, changes in the composition of aggregate output affect both investment and employment in the economy.

The model is based on a framework originally developed by Gali (1994). Goods are horizontally differentiated with each good being produced by a single firm in a monopolistic market The price elasticity of demand for each good is different for households — who use it for consumption, and firms — who use it for investment. In this environment, we show that the government can affect the efficiency and cyclical behavior of the economy by changing composition of aggregate demand by altering the composition of its own purchases. In order to isolate the effect of compositional changes in public expenditure on the economy, we assume throughout the analysis that government purchases neither enter households’ utility nor the production process.

When markups are procyclical, which is an established feature of the U.S. economy (Woodford and Rotemberg 1990, Bils 1987), increasing the share of public investment while keeping government expenditure unchanged increases output, employment, capital stock, and welfare in the steady state. In this model, for markups to be countercyclical, the price elasticity of investment needs to be larger than that of consumption. When this happens, an increase in the share of investment increases the average elasticity lowers the markup charged by profit maximizing firms leading to higher output, employment, investment and finally welfare. Moreover, the fall in the markup pushes the economy closer to the perfectly competitive equilibrium such that there are efficiency gains in terms of both labor and total factor productivity. However, if the elasticity of investment is lower than that of consumption so that markups become procyclical, increasing the share of public investment has the opposite effect.

Although, when the price elasticity of investment is larger than that of consumption, increases in steady state public investment raises steady state output and welfare by decreasing the markup, the increase in elasticity which leads to the fall in the markup, also makes the aggregate demand less steep. Consequently, both exogenous demand and supply shocks lead to a larger adjustment in quantity rather than prices, so that output and employment variability also increases, such that as the economy is moved closer to the perfectly competitive allocation, the cyclical variability is also increased. With the increased cyclical variations, depending on the level of risk aversion, the loss in welfare can overcompensate the gains in welfare arising from the added efficiency so that increases in public investment can decrease overall welfare. Thus, changes in the long run composition of public expenditure entail a trade-off between increased long run efficiency and increased short run volatility.

The mechanism by which shocks affect markups in this model differs significantly from the processes relied upon in Basu (1995) and Rotemberg and Woodford (1992). In Basu, each producer faces a state dependent “menu cost” for changing prices. This generates price rigidity over some range of aggregate shocks such that markups become countercyclical. Rotemberg and Woodford, uses a structure based on Rotemberg and Saloner (1986) where firms within an oligopoly collude to keep prices above marginal cost with the collusion being supported by the threat to revert to the competitive price in future in response to any deviation. An exogenous increase in demand raises the gains from undercutting the industry set price. To prevent a breakdown of the collusion, the industry reverts to a lower price and hence to a lower markup. In contrast to these mechanisms, in our model, all prices are flexible and there is no collusion. The importance of markup variations in understanding business cycle fluctuations dates back to Kalecki (1939) and Keynes (1938). And in this paper, we add to this literature by showing how permanent changes in the composition of public expenditure can affect the cyclical response of the economy by changing the elasticity of aggregate demand. Although, Bils (1989) provides a setup where the price elasticity changes in response to demand shocks, the paper does not make a persuasive case as to why this can occur. In contrast, in our model this link is made explicit.

Among others, Hall (1988) and Evans (1992) have shown that empirically productivity shocks can be correlated with variables, such as changes to government expenditure and monetary impulses, although in principle these innovations should be orthogonal to the technology. We show that, government expenditure shocks can be positively correlated to productivity (as in Basu 1995), measured as the standard Solow residual, when markups are procyclical. As a result, the standard measures of the Solow residual will tend to overestimate the contribution of technological shocks. It is also shown that when markups are procyclical if the average elasticity is sufficiently large then increases in public investment crowds in out private investment. As public investment is not used in production, the crowding in of private investment occurs because of gains in efficiency, which increases the marginal revenue product of capital despite a fall in the markup.

The original Gali (1994) framework was constructed to show how in the absence of exogenous shocks, self-fulfilling revisions of expectations or sunspots can generate cyclical fluctuations that are similar to those in real economies. Although this property of the model is retained in this paper, we find that in light of the more recent evidence on econometric estimates of markups it is difficult to justify sunspot equilibria to be reasonable mechanisms by which cyclical fluctuations occur. In particular, when markups are procyclical, the existence of sunspots requires that, at the steady state, the markup has to be larger than 2. Recent studies such as those by Domowitz et al. (1988), Morrison (1990) and Oliviera Martins et al. (1996) typically estimate markups to be less than 1.7.

The rest of the paper is organized as follows. Section II presents the environment including the assumptions made with respect to the behavior of government expenditure. Section III characterizes the equilibrium while Section IV discusses the properties of the steady state. In Section V the long run effects of changing the composition of public spending is discussed. Issues involving the plausibility of sunspot equilibria are discussed in section VI. Section VII shows the effects of changing the composition of government expenditure on the economy when exogenous demand and technological shocks are present. Section VIII concludes the paper.

II. The Environment

A. Households

We assume that there is a large number of identical consumers denoted by i = 1, 2, …N whose preferences are described by

E0t=0βt[u(cti)υ(lti)]

where E0 is the expectation operator at time t = 0 and β ∈ [0,1] is the discount factor. The utility u(cti) derived by consumer i at time t is given by

u(cti)=M11σ[s=1M(csti)σ1σ]σσ1,σ>0

where csti is the quantity of good s consumed by i at t. This is a constant elasticity of substitution utility function used previously in the literature on monopolistic competition (Spence 1976, Dixit and Stiglitz 1977). The disutility from working lti hours is given by

υ(lti)=ζ1+ζ(lti)1+ζζ,ζ>0.

Normalizing each period wage to unity and denoting by qth the price of the shares in firm h, we can write the budget constraint of consumer i as a function of his income, the return on the shares that he owns and the taxes he pays to the Government. Thus, if shti is the number of shares he owns in firm h at time t and dth the dividend he receives from these shares, the budget constraint is given by

s=1Mptscstilti+h=1M(dth+qth)shtih=1Mqthsh,t+1iTti

where, Tti is the tax paid by the ith household.

If we define the income of consumer i net of taxes as ati, with

atilti+h=1M(dth+qth)shtih=1Mqthsh,t+1i

and assume that all shares are distributed equally between consumers at time t = 0, the consumer’s problem can be written as

maxE0t=0βt[u(cti)υ(lti)]s.t. s=1MptscstiatiTtish0i=1N(1)

Let Pt be the consumer price index corresponding to the composite consumption good purchased by the consumers and defined as

Pt=[1M(s=1M(pts)1σ)11σ]

The solution to the consumer’s problem can then be written as

csti=(ptsPt)σ(atiTtiPtM),s=1,2,M,i=1,2,Νlti=Ptζ,i=1,2,NqthPt=βEt1Pt+1(dt+1h+qt+1h),h=1,2,M,i=1,2,Nlim infTEβTh=1M(qThPT) shTi=0,h=1,2,M,i=1,2,N(2)

The impact of the taxes clearly appears in the first equation characterizing the demand for the consumption goods. The second equation is the labor supply as a function of the composite price index with an elasticity of ζ as defined above. The third is the Euler condition and corresponds to the familiar capital asset pricing equation. The fourth equation is a limit condition that guarantees that the value function is well defined.

B. Firms

There are M firms, each producing a horizontally differentiated commodity indexed by j = 1, 2, …M. Without loss of generality assume that N = M. The technology to produce these goods is identical across firms and requires both labor and capital with a Cobb-Douglas production function given by

ytj=zt(ktj)α(ltj)1α,0<α<1(3)

where ytj,ktj, and ltj, denote respectively the jth firm’s output, capital stock and labor input. zt denotes the productivity shock and follows

zt+1=ϕzzt+εt+1z

where ϕz < 1, measures the persistence in the shock and εtz is an i.i.d. disturbance with mean ε¯ and a finite positive support [ε,ε+]. Capital accumulation requires the use of all M commodities and the law of motion of capital is given by the following equation

kt+1j=(1δ)ktj+f(itj),(4)

where

f(itj)=(1M)1η1[h=1M(ihtj)η1η]ηη1,η>0(5)

and where δ ∈ [0,1] represents the rate of depreciation. Thus the capital stock of each firm is increased by purchasing goods from the other firms that are mixed together (through the same function for all firms) to produce a composite investment good f(itj). Basu (1995) uses a technology that is similar in some respects. There output is produced using a constant returns to scale technology in labor and intermediate goods, Qtj=(Ltj)α(Itj)1α, where Itj=(01Iki1ϕdk)11ϕ. However, ϕ which measures the elasticity of substitution among different goods in production is also the same among the goods in consumption so that the price elasticities of firms and households are identical. Consequently, in Basu the transmission mechanism by which shocks filter through to firms is different from the process relied upon in this model.

If we denote, as above, by pth, h = 1,2, …M, the price of good h, the cost of the composite investment good purchased at each period (i.e. the total investment expenditure of the firm) is given by vtj=h=1Mpthihtj. Implicitly, this framework requires that firms are not able to discriminate between types of buyers. When a buyer purchases one unit of the good sold by a firm, that buyer can be a consumer purchasing a part of his consumption composite, a firm purchasing a part of its investment composite, or the government buying either. We assume here that in none of these cases can the firm discriminate and extract a different margin.

In each period, the firm chooses ihtj to maximize f(itj) given vtj. Consequently,

ihtj=(pthΠt)η(vtjΠtM),h=1,2,Mf(itj)=vtjΠt(6)

where πt given by,

Πt=[(1M)h=1M(pth)1η]11η(7)

is the price index of the composite investment good.

Equation (6), provides the solution to the static allocation problem faced by the firm choosing the optimal mix among M commodities to accumulate capital. Using this solution, one can then write the intertemporal problem of the jth firm as

maxE0t=0βtdtjPts.t.dtj=ptjytjltjνtjytj=zt(ktj)α(ltj)1αkt+1j=(1δ)ktj+νtjΠt(8)

where β is the discount factor, dtj is the dividend paid by firm j at time t, Pt the price index for the consumption goods defined similarly to ∏t, and E0 is the expectation operator at time t =0. Since the intertemporal problem is also identical across firms, we can drop the subscript j.

One can thus summarize the problem in (8) by

maxE0t=0βt1Pt[ptytzt11αyt11αkkα1αΠt(kt+1(1δ)kt)](9)

where zt11αyt11αktα1α is the variable cost at time t. Solving (9) by choosing {yt, kt+1} one gets

ptyt+pt=11αzt11α(ytkt)α1α

where pt represents the derivative of pt with respect to yt. The above equation can be rearranged as

pt(ptytpt+1)=11αzt11α(ytkt)α1α

or,

pt(11ξt)=11αzt11α(ytkt)α1α

where ξt, is the price elasticity of the commodity and 11αzt11α(ytkt)α1α is the marginal cost. This gives us the standard price setting rule in the case of monopolistic competition namely, that the price is set equal to a mark-up over the marginal cost, i.e.,

pt=μtωt(10)

where μt=ξtξt1 is the mark-up and

ωt=11αzt11α(ytkt)α1α

is the marginal cost. After rearranging, the optimal capital accumulation policy function, obtained by taking the first order condition in kt+1 is given by

ΠtPt=βEt1Pt+1[α1αzt+111α(yt+1kt+1)11α+(1δ)Πt+1](11)

where α1αzt+111α(yt+1kt+1)11α corresponds to the decline in the variable or labor cost due to a unit increase in the capital stock. Equations (6), (10), and (11) characterize the optimal policy functions of the firms in terms of output at time t and capital stock at time t + 1, thus fully characterizing the solution to the firm’s problem.

C. The Government

As discussed previously, we assume that the government collects lump sum taxes from consumers for a total of Tt, with Tt=i=1NTti. The government spends the entire amount on purchasing consumption and investment goods i.e., Gt = Tt. Once Tt is determined, the budget allocation is carried out in a two step process. First, the overall sectoral expenditure limits are determined. We denote them by θtGt for public investment and (1 – θt)Gt for public consumption, with θt ∈ [0,1]. Alternatively, the budget constraint of the government can be written as

ptscgts(1θt)GtpthigthθtGt(12)

Two assumptions regarding the behavior of government purchases are made next. First, θt = θ for all t, and second we assume that for any fixed amount earmarked for public consumption (public investment), the government allocates the money among different goods exactly in the same way as it would be done by the private sector. This would in effect occur if the aggregator function for the government is a linear combination of u(cti) and f(itj) i.e., the government’s allocation problem is

maxωM11σ[s=1M(csti)σ1σ]σσ1+(1ω)(1M)1η1[h=1M(ihtj)η1η]ηη1

subject to

ptscgts(1θt)GtpthigthθtGt

where θt and Gt is given and the weight ω ∈ [0, l]1. The following two equations are obtained as the solution to the spending patterns of the government2.

cgts=(ptsPt)σ((1θ)GtPt),s=1,2,Migth=(pthΠt)η(θGtΠt),h=1,2,M(13)

Finally, assume that government expenditure is stochastic in the sense that gtGtYt=(1ϕg)g¯+ϕggt1+εtg, where εtg has a finite support [g ,g+] containing g¯ and has an i.i.d. distribution. This constitutes the demand shock to the system as opposed to zt which is the supply shock.

III. Symmetric Market Equilibrium

Equations (2), (6), (10), (11), and (13) together with the overall feasibility conditions and the price indices describe the market equilibrium. However, before proceeding further with the characterization of the equilibrium it is useful to discuss the aggregate demand faced by firms.

Note that the demand faced by the firm j comprising of the demand by consumers, other firms and the government is given by

ytj=(ptsPt)σ(atGtPt)+(pthΠt)η(vtΠt)+(ptsPt)σ((1θ)ϒtPt)+(pthΠt)η(θϒtΠt)ytj=(ptsPt)σ(atθGtPt)+(pthΠt)η(vt+θGtΠt)(14)

As is evident from (14), if θ = 0, then aggregate demand is not affected at all by government expenditure.

Now let λtj denote the share of investment demand. Then the price elasticity of the jth good is given by

ξtj=(1λtj)σ+λtjη(15)

The price elasticity ξtj is a weighted sum of the consumers and investors elasticities. And therefore, the markup will depend on the weight, λtj. Furthermore, note that if the two elasticities are the same then μt=σ, a constant.

Focusing our attention to only symmetric equilibria where all firms produce the same quantity and use the same amount of inputs. Moreover, all consumers consume and save the same amount. Furthermore, since firms cannot discriminate between consumers and investors price charged for any particular commodity will be the same, regardless of its end use. As a result Pt = ∏t. With the assumption of symmetry, all firm and household specific indices are dropped.

The demand for labor by firms can be written as lt=(xtzt)11σkt where, xtytkt, the output-capital ratio. Using this and equation (11) one derives the Pt as

Pt=(xtzt)1ζ(1α)kt1ζ(16)

The real wage in the economy wt is then given by

wt=1Pt

Replacing (16) in (8) provides a closed form expression for the output-capital ratio

xt=Axzt1+ζ1+αζμtζ(1α)1+αζkt(1α)1+αζ,Ax=(1α)ζ(1α)1+αζ(17)

Using (17) one can subsequently derive

yt=Axzx1+ζ1+αζμtζ(1α)1+αζktα(1+ζ)1+αζ(18)
lt=Alztζ1+αζμtζ1+αζktαζ1+αζ,Al=(1α)ζ1+αζ(19)

Note that both output and labor are functions of capital stock kt and λt. One can use (18) to find solutions for consumption, ct, and investment, it. However, λt is an endogenous variable and needs to be solved. We next turn out attention to solving for λt. To do so, consider equation (11). Under the assumption that Pt = ∏t, (11) becomes

ρ+δ=Et1Pt[α1αzt+111α(yt+1kt+1)11α]ρ=1β1(20)

Using the fact that at period t, Kt+1 is known the above can be written as

kt+1=Ak(Et[(zt+1μt+1)1+ζ1+αζ])1+αζ1αAk=(αp+δ)1+αζ1α(1α)ζ(21)

Note that μt+1 is a function of λt+1. Therefore, if λt+1 is perceived to increase and μ(⋅) ≠ 0, then private investors will increase or decrease kt+1. If, for example, μ(⋅) < 0, then kt+1 will increase. However, this will imply that λt will also rise. The law of motion of capital is given by (4)

kt+1=[(1δ)+(λtθgt)xt]kt(22)

Rearranging the above yields,

(λtθgt)xt=kt+1kt(1δ)

equivalently,

(λtθgt)xt=Akkt(Et[(zt+1μt+1)1+ζ1+αζ])1+αζ1+α(1δ)(23)

Using the derived first order conditions a symmetric market equilibrium for this economy can be defined as:

Given k0 and z0, a sequence of {zt, gt} and government policy described by {θ, τt}, a symmetric equilibrium for this economy is a sequence {yt, ct, lt, kt+1t, μt, Pt} such that

  1. xt=Axzt1+ζ1+αζμtζ(1α)1+αζkt(1α)1+αζ

  2. yt=Axzt1+ζ1+αζμt−ζ(1−α)1+αζktα(1+ζ)1+αζ

  3. lt=Alztζ1+αζμtζ1+αζktαζ1+αζ

  4. Pt=(xtzt)1ζ(1ζ)kt1ζ

  5. ωt=1Pt

  6. μt=ξtξt1

  7. ξt=(1λt)σ+λtη

  8. τt=ϒtyt

  9. (λtθgt)xt=kt+1kt(1δ)

  10. kt+1=Ak(Et[(zt+1μt+1)1+ζ1+αζ])1+αζ1α

  11. gt=τt

IV. The Steady State

Conditions (1)(11) completely characterize the competitive equilibrium. The only intertemporal equations are (9)-(11). Assuming that τt = τ and z¯ and g¯ are the mean values of zt and gt and letting ψθg¯, in a deterministic steady state i.e., zt=z¯ and gt=g¯, the steady state values of capital, share of investment, and mark-up are given by

(λ¯ψ)μ¯=αδρ+δ(24)
k¯=Akz¯1+ζ1α(1μ¯)1+ζ1α(25)
μ¯=((ησ)λ¯+σ(ησ)λ¯(1σ))(26)

Note that from (26) it follows that when μλ>0 as long as σ > p. When λ =ψ, the RHS of (24) is 0 and monotonically increasing if σ > p. Therefore, as long as αp+<(1ψ)(ηη1) there will exist some 0 ≤ λ < 1for which (24) is satisfied and a steady state exists. In particular, given the monotonicity of (λ¯ψ)μ¯, the steady state will also be unique. However, if σ < p then μλ<0 and in which case multiple steady states can exist (Gali, 1994).

From (25) it follows that when μ¯ is high k¯ is low. Since steady state output is given by

y¯=Axz¯1+ζ1+αζμ¯ζ(1α)1+αζk¯α(1+ζ)1+αζ

a fall in the steady state markup unambiguously increases steady state output. Similarly, when the markup falls both employment,

l¯=Alz¯ζ1+αζμ¯ζ1+αζk¯α1+αζ

and labor productivity,

y¯l¯=const×z¯11αμ¯α1α

also increase.

Recall that yt=ztktαlt1α or equivalently, ln yt = In zt+α In kt + (1–α) In lt. One method of computing total factor productivity, TFP, is to define TFPt = In yt – α In kt (1 –α) In lt. In particular, this formulation has been used to derive the contribution of productivity shock zt in growth and cyclical variability. In computing the total factor productivity, following Solow (1957), researchers have used the share of labor income in national income to approximate (1 – α). In the presence of perfect competition the share of labor income approximates (1 – α) quite well. However, under imperfect competition i.e., μ > 1, the share of labor income is 1αμ and that of capital αμ. If these shares are used to compute productivity then

TFPμ=Inyμ(αμ)Inkμ(1αμ)Inlμ

It turns out that

TFPμ=(ζ(1α)2+α(1+ζ)2)(11μ)(27)

For μ>1,TFPμ<0 rather than 0 if α, the true labor elasticity was used. Consequently, if the steady state markup increases, not only does output increase as shown before but also total factor productivity. This, however, should not be surprising. The presence of the markup causes the equilibrium to be suboptimal in the first place so that when the markup falls there are efficiency gains reflected in the increase in total factor productivity. But, more importantly, it also implies that factors that can change the markup, such as changes in government expenditure (of a permanent kind for the steady state analysis) will be correlated with the Solow residual, i.e., the measured factor productivity. In particular, since μ=(11ξ)1, where ξ = (η – σ)λ + σ, if λ increases and η – σ > 0, then μ falls as a higher elasticity implies a lower markup as predicted by standard microeconomic theory. If η – σ < 0 the opposite holds true. We summarize these findings in the following proposition.

A decrease in the steady state markup is associated with higher steady state levels of capital stock, employment and output and also higher labor and total factor productivity. Moreover if (η – σ) is positive (negative) then the share of investment in output is positively(negatively) correlated with total factor productivity in the long run.

V. Public Expenditure Policy — The Long Run Effects

Proposition 1 makes it clear that changes in the equilibrium markup can have important consequences for the economy in general. In this section we look at a particular way of changing the markup, namely by altering the share of public investment, ψ. To see how government policies reflected in the composition of public expenditure can affect the markup of producers consider the expression for the weighted elasticity ξt = (1 –λt)σ + λtη. Denote by λtp, the share of investment in total private spending, then λt=(1gt)λtp+θtgt and

ξt=(ησ)[(1gt)λtp+θtgt)]+σ

Government expenditure, gt, by itself can only have a second-order effect on the elasticity since if θt = 0 then ξt=(ησ)(1gt)λtp+σ and gt can have an effect only if λtp changes, i.e., if changes in government spending affects the way private agents allocate their income between consumption and investment. In this model since investment is determined by firms, who care only for the overall breakdown between investment and consumption, and not who the end-users are (by assumption all end-users for the same functional purposes have the same elasticity) changes in gt will be completely offset by changes in λtp such that (1gt)λtp remains constant. If in contrast, households decided on the level of investment then λtp would probably not move enough to completely offset the change in gt such that (1gt)λtp would change leading to a change in the elasticity. In particular it would depend on how government expenditure changes both disposable income of households and the interest rate in the economy and the reaction of private investment to such changes. However, if θt ≠ 0 then changes in both θt and gt matter in equilibrium. When gt or θt is altered, the overall composition of output changes causing the elasticity to change which in turn leads to a change in the markup.

In the steady state, (λ¯ψ)μ¯=αδp+δ and μ¯=((ησ)λ¯+σ(ησ)λ¯(1σ)). Substituting the second equation in the first we have (λψ)((ησ)λ+σ(ησ)λ(1σ))αδp+δ=0. Since (ησ)λ¯+σ(ησ)λ¯(1σ)0 for 0 ≤ λ ≤ 1 and η ≥ 0, the steady state relationship between λ and ψ can be expressed as λ=f (ψ).

If η < σ then 0<dfdψ<1. Otherwise, dfdψ>1 if ξ(ξ1)>(λψ)(ησ) and dfdψ<0 if ξ(ξ1)<(λψ)(ησ).

Follows directly from differentiating (f(ψ)ψ)((ησ)f(ψ)+σ(ησ)f(ψ)(1σ))αδp+δ=0.

From the above lemma it follows that when η < σ, if public investment increases, although share of total investment goes up — dλdψ>0 — the increase is less than the amount by which public investment is raised — dλdψ<1. This implies that the share of private investment must fall. Since λ increases when ψ is raised and η – σ < 0, it follows from ξ=(η – σ)λ + σ and μ=(11ξ)1 that the markup also increases. With μ being raised steady state output falls (Proposition 1) so that not only does the share of private investment decrease but also its level. Permanent increases in public investment, in this case, crowds out private investment. However, if η – σ > 0 and the price elasticity of aggregate demand is sufficiently high, i.e., ξ(ξ – 1) > (λ – ψ)(η – σ), not only does the share of total investment increase but also induces an increase in the share of private investment — dλdψ>1. Given that η – σ > 0, the steady state markup falls and output also increases (again due to Proposition 1) so that the level of private investment also increases. In contrast to the previous case, public investment crowds in private investment. Since increasing θ or g while keeping the other unchanged in equivalent to increasing ψ we have the following proposition.

If θ is increased keeping g constant or g is increased keeping θ constant and if η – σ > 0 and ξ(ξ – 1) > (λ – ψ) (η – σ) then the steady state levels of output, private investment, and employment are increased along with labor productivity and total factor productivity. The opposite occurs if η – σ < 0.

Follows from proposition 1 and lemma 1.

The long run effects of changing government expenditure and its composition depend on whether investor elasticity is larger than consumer elasticity and the weighted average of the two elasticities — the elasticity of aggregate demand. As we argue later in the paper, it is more likely that η – σ > 0 for the post war U.S. economy. In which case, increasing the share of investment in government expenditure does in fact increase output, employment and productivity. The same effects are also obtained if the share of government spending is increased keeping its composition unchanged. To ascertain the short run effects we need to compute the temporal equilibria of the economy. However, given the nature of the environment closed form solutions cannot be obtained. To get around the problem, we construct a linear approximation to the equilibria and study the short run impact of government spending using the approximate solution.

VI. Sunspot Equilibria

In a recent paper, Gali (1994) used this environment to show how equilibria that are driven by self-fulfilling revisions of expectations can generate time series with properties that are similar to those observed in U.S. postwar business cycles. The belief that sunspot equilibria can generate empirically reasonable time series properties have been shared by others such as Farmer and Guo (1994) and Woodford (1991) among others. However, unlike in Gali, the other models rely on some form of increasing returns or externalities to deliver the sunspot equilibria. Although Gali’s approach is appealing since it eliminates the need to rely on exogenous shocks to generate cycles, we find that for the U.S. postwar economy it is not a plausible mechanism to use in studying cyclical fluctuations.

A. Existence of sunspots

To see how sunspot equilibria may be generated in this economy and why they are not plausible we turn our attention to studying the model’s cyclical behavior. First, equilibrium conditions (1), (6), (9), and (10) are linearized around the steady state, implicitly given by (24) and (25) to obtain approximate closed form solutions. Let x¯={λ¯,k¯,z¯} denote the steady state of the system. The linearized equations are

x^t=a1z^ta2μ^ta3k^t(28)
μ^t=b1λ^t(29)
λ^t=1c1k^t+11c1k^tδc1x^t+c2g^t(30)
k^t+1=d1Et[z^t+1μ^t+1](31)

where the coefficients are the partial derivatives evaluated at their steady state values (see the appendix for details). By using (28), (29), (30), (31) becomes

k^t+1=Et[e1z^t+1+e2k^t+2+e3g^t+1](32)

Now suppose | e2 |> 1. Consider first the perfect foresight equilibrium i.e., z^t+1=0 and ĝt = 0 ∀t = 0,…∞. The first equation in this system is k^1=e2k^2. Consequently, there is no restriction on k1 imposed by the initial capital stock k0 i.e., the system of equations, k^t+1=e2k^t+2t=0,. are unrestricted by the initial stock of capital. There is an unique steady state given by k^t+1=0t=0,. Moreover, the perfect foresight equilibrium is stable. Any sequence of {k^t+1} starting from an arbritrary k1 will converge to the steady state k^t+1= 0. Since, k0 does not restrict the choice of k1, there is an indeterminate number of perfect foresight equilibria. However, of these equilibria there is one that is stationary namely, {k^t+1}=0t=0, which is the steady state equilibrium. But this is not the only stationary equilibrium. Around a sufficiently small neighborhood of the steady state there may be more than one such equilibria (Woodford 1991, Gali 1994). To see why the steady state is not the only stationary equilibrium note that Et[k^t+1e2k^t+2]=0 is the version of (32) when there is no intrinsic uncertainty (εtz=0 and εtg=0) which is the optimal capital accumulation rule followed by investors. Lagged one period, the rule is given by Et1(k^t+11e2k^t)=0. Now suppose investors’ believe that investment follows k^t+1=1e2k^t+st, where the random variable st is a realization of an i.i.d. probability distribution with Et1(k^t+11e2k^t)=0 and described over a closed support [a, b] containing the steady state. Although this rule still implies that Et1(k^t+11e2k^t)=0 which is consistent with the optimal policy (32), the extrinsic uncertainty st can generate investment paths that depend solely on it. If | e2 |< 1 then the solution to k^t+1=1e2k^t+st is given by k^t+1=Etj=0(e2)jst+jt=1,. As Etst+j=0,k^t+1=0t=1, is the unique stationary solution. But if | e2 | > 1, then the solution is

k^t+1=j=0(1e2)jstjt=0,(33)

with k^1=e2s0 and the equilibrium path of capital stock is determined entirely by current and past realizations of the sunspot variable st3. However, if agents do not believe that the sunspot matters for the capital stock i.e., k^t+1=1e2k^t+st but k^t+1=1e2k^t, then the only stationary equilibrium possible is the one where {k^t+1}=0t=0,.

Since the existence of sunspots depend on the parameter e2 we next provide its characterization.

|e2|<1 if and only if1δ(1α)1+ζ[1+(λ¯λ¯ψ¯)1ε¯μ]>14.

See appendix.

While the theoretical possibility of sunspots existing have been established it still remains to be shown whether empirically such equilibria should be considered seriously. To this end, in the next section we use a suitably parameterized economy to provide the answer.

B. The Likelihood of Sunspots

In this section, parameters of the model are calibrated to be compatible with the post-war U.S. economy using long-run and micro-level data and econometric studies. Following, Kydland and Prescott (1990) set λ¯=0.21, the share of investment in aggregate output. Depending on whether investment in defense equipment is included ψ¯=0.023 or 0.03. The depreciation rate is set at δ = 0.016 per quarter such that the steady state capital-output ratio is around 10. This depreciation rate is different from 0.025 which is normally used approximating an annual rate of 10 percent. Using the average risk-free interest rate per quarter in the postwar U.S. economy ρ = 0.01 or β = 0.99. Typically, equilibrium business cycle models assume extremely large labor supply elasticities. However, estimated elasticities have been in general very low (close to zero for adult white males) and have rarely been over 2 (Killingsworth and Heckman 1986). In this model ζ = 1. This leaves behind unresolved the values for μ, α, η and σ. Note that the steady state condition (λ¯ψ)μ=αδρ+δ implies that for any value of μ there is an unique α consistent with it.

Using industry data, Hall (1987, 1989) estimated μ of over 1.8 for all the one-digit industries. Subsequent work by Domowitz, Hubbard and Petersen (1988) and Morrison (1990) have shown that estimates of markups that range from 1.2 to 1.7 are more reasonable. This range of values for μ seem more reasonable also on the ground that for the postwar U.S. economy profit margins have been rather small. More recently, Martins, Scarpetta and Pilat (1996) estimated the average markup for the U.S. economy to be around 1.15. On sectoral basis, they found that markups for most industries were between 1.05 and 1.54. However, Gali (1994) assumes a steady state value of μ [2.0,3.03] using evidence from Hall (1988). These two ranges i.e., [2.0, 3.03] and [1.2,1.7] have significant behavioral differences. To see these differences in a more transparent manner consider the data in Table 1. It describes the values of {η, σ} consistent with a given μ Table 1 also lists the corresponding sign of e2 and whether it is greater or less than 1. Recall that if e2 is greater than 1, then sunspots are possible.

Table 1.

Sunspot Zones and Markups

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For every value of μ considered, for some region of {η,σ}, | e2 |> 1 such that it is possible that sunspot equilibria may exist for a wide range of markups. Moreover, as can be seen in Table 1, the intervals of {η,σ} for which sunspots exist are also quite large. Consequently, the existence of such equilibria in the environment discussed in this paper does not depend on very specific values of parameters. Note, that for values of {η,σ} close to one another | e2 |< 1 and for sufficiently apart values | e2 |> 1. This implies, as described by Gali, for sunspots to exist the two elasticities need to be sufficiently far apart. If η and σ are close to one another ε¯μ is extremely small such that 1ε¯μ is very large which in turn implies that |e2|={1δ(1α)1+ζ[1+(λ¯λ¯ψ)1ε¯μ]}1 will be small and sunspot equilibria cannot exist.

However, when μ is small e.g., 1.2 or 1.5, sunspots exist only in the region η < σ. For larger values of the markup e.g., for values considered by Gali i.e., μ ∈ [2, 3.03], | e2 |> 1 in regions where η < σ as well as η < σ. The difference in the behavior of the economy for larger values of the steady state μ turns out to be quite significant. In the region η < σ, the elasticity of the markup to λ,ε¯μ=(σρ)λ¯ξ¯(ξ¯1) is negative. At the empirical level, the U.S. economy displays two strong regularities

(a) a strongly procyclical investment share,

(b) a strongly countercyclical markup5.

If these two features are to hold simultaneously then when output increases, λ should increase and μ should fall. This implies that εμ < 0. Therefore, if the model is to be consistent with the regularities displayed by the U.S. economy it follows that η < σ since ε¯μ=(σn)λ¯ξ¯(ξ¯1). While for parameter values consistent with a steady state markup μ > 2.0, sunspots can exist (| e2 |> 1) when η > σ , in the range μ ∈ [1,2) this is not true. In fact for μ ≤ 2, sunspots fail to exist for η > σ.

If η > 0, σ > 0, and η > σ, then at the steady state μ > 2 for sunspot equilibria to exist.

See appendix.

Note that in the proof of Proposition 3, we did not use any specific parameter values except for restricting the parameters to be positive and the markup to be greater than 1. Consequently, the proposition holds quite generally and for this class of economies if the share of investment is to be procyclical and the optimal markup countercyclical, at the steady state the markup has to be over 100 percent. Given the more recent studies discussed previously, it is unlikely that the long-run markup for the U.S. economy can be that large. In fact, the Martins, Scarpetta and Pilat (1996) study shows that for most OECD countries markups are substantially lower than 2.

Simplifying the parameters e1 and e3, (32) becomes

k^t+1=Et[d1(1e2)z^t+1+e2k^t+2+(ρ+δα)ψ2e2g^t+1]

This implies that if e2 > 1 then k^t+1 is negatively correlated with technological shocks as d1(l — e2) < 0. In turn, e2 > 1 also implies a negative correlation for investment which we find to be counter intuitive. Furthermore, if μ ∈ [2, 3.03], the values used by Gali, α ∈ [0.53, 0.8] and the share of wages in national income lies between [0.24, 0.07]. This value is significantly different from the observed share of around 0.64 (Prescott, 1986). Consequently, given the preponderance of both empirical evidence and the implied inconsistencies with empirical regularities it seems unlikely that the version of this model with sunspots will be a reasonable approximation of the real economy.

VII. Equilibrium with Exogenous Fluctuations and Public Expenditure Policy

In keeping with the restrictions derived in the previous section we follow studies that have used a similar framework such as Rotemberg and Woodford (1992) and Basu (1995) to restrict steady state markups to the interval [1.2,1.7] that matches the more recent econometric estimates. As shown in proposition 3, this restriction forces |e2| < 1, so that sunspot equilibria are ruled out. Moreover, in the rest of this section unless otherwise noted it will be assumed that η > σ so that εμ < 0. Therefore, in order to generate cyclical fluctuations the model will depend exclusivelyon the two exogenous shocks: (i) the technological shock z^t and (ii) the demand shock g^t. Consider first the perfect foresight equilibrium i.e., z^t=0 and g^t=0,t=0,. The first equation in this system is k^1=e2k^2. Consequently, there is no restriction on k1 imposed by the initial capital stock k0. The unique steady state of the system is k^=0 and since | e2 | < 1 it is unstable, such that the only rational expectations equilibrium is the one where {k^t+1}=0t=0,. Therefore, given any initial capital stock k0, there is an unique perfect foresight equilibrium which is also stationary. The instability of the steady state allows the stochastic version of the difference equation (32) to be forward stable such that there is an unique rational expectations equilibrium given by

k^t+1=e1Et[j=0e2jz^t+j+1]+e3Et[j=0e2jg^t+j+1](34)

Since z^t+1=ϕzz^t+ε^t+1z and Et[ε^t+1]=0 and g^t+1=ϕgg^t+ε^t+1g and Et[ε^t+1g]=0 we have,

k^t+1=ϕze1Et[j=0(ϕze2)jz^t]+ϕge3Et[j=0(ϕge2)jg^t](35)

which can be expressed as

k^t+1=ϕze11ϕze2z^t+ϕge31ϕge2g^t(36)

This is the capital accumulation law of motion generated by the equilibrium conditions of the economy and using the linearized system of equations (28)(4) all other variables can be expressed in terms of z^t and g^t. In particular6,

λ^t=λzz^tλkk^t+λgg^t(37)
μ^t=b1λzz^tb1λkk^t+b1λgg^t(38)
y^t=yzz^t+ykk^t+ygg^t(39)
l^t=lzz^t+lgg^t+lkk^t(40)

We investigate the effects of changing the long run share of investment goods in overall government purchases, θ, on the cyclical behavior of aggregate output yt, investment it, consumption ct and labor lt keeping the long run level of overall government expenditure g¯ constant. Using the expression for equilibrium output (39) derived in the appendix and ignoring the variations in the stock of capital which are generally small,

υar(y^)=(y1a)σz2+(y3a)2σg2

since Coυ(z^,k^)=0,Coυ(z^,g^)=0,Coυ(g^,k^)=0. Consequently, the way output responds to the exogenous shocks, z^t and g^t,, depends on the parameters yz and yg These two parameters are affected by θ. As a result, when the long-run composition of government outlay changes, it affects the way the economy reacts to both supply and demand shocks. The cyclical variability of output can therefore, be accentuated or reduced by the choice of θ.

As noted in the discussion on the long run effects of changes in the share of public investment, when θ changes the price elasticity of the commodity also changes at the steady state also changes since

ξ¯=(ησ)[(1g¯)λ¯p+θg¯]+σ

In particular, when (ησ) > 0, then an increase in θ will increase the elasticity as long as ξ¯(ξ¯1)>(λ¯θg¯)(ησ) (see lemma 1). This will lead to a reduction in the markup. Apart from increasing welfare by reducing the markup and increasing productivity (proposition 1) an increase in θ also impinges on the way the variability of output and variables such as employment, consumption and investment. This occurs since an increase in the elasticity results in a “flattening” of the aggregate demand function. With a flatter demand function, exogenous shocks have a larger effect on quantity than prices. Consequently, the variability in aggregate output for the same exogenous shocks will be larger resulting in a larger short-run volatility leading to the following situation. On one hand, increased government investment enhances productivity and welfare while on the other it also raises the volatility in income. Depending on the risk aversion of households, the loss in welfare due to increased volatility could, in principle, overshadow the welfare gains due to increased productivity leading a to fall in the total welfare.

A. Cyclical Effects of Public Investment

Although it is possible to derive closed form solutions expressing all the variables in terms of the exogenous shocks (see the appendix) evaluating the effects of increased θ analytically is still too cumbersome. Instead, we use the parametric restrictions derived in the previous section to simulate the effects. In these experiments we keep g constant at the long run average for government purchases which is around 0.21. The steady state value of θ is around 0.15 which includes investment goods for defense. This implies that government investment is about 3.1 percent of GDP.

We take as the benchmark the steady state implied by the parameter values enumerated in Table 2. The unresolved parameters are η and σ. However, given the procyclicality of markups as discussed in the previous section we know that η > σ. In this region we consider two sets of values for these variables {η = 2.75, σ = 2.65} and {η = 3.5, σ = 2.46}. In the first set the difference between the two elasticities is small and in the second relatively large. For each set of parameters we vary the θ keeping g constant at 0.21. A five percentage point increase (decrease) in θ implies a one percentage point increase (decrease) in share of government investment in GDP. The steady state share of public investment, ψ, is around 0.03. Tables 3a and 3b summarizes the effects of changing this share by one percentage point or equivalently changing θ from its steady state value of 0.15 by five percentage points.

Table 2.

Parameters used in Simulations

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When ψ is changed by 1 percentage point, the share of total investment λ changes by more than 1 percentage for both cases i.e., η and σ are close to one another (Table 3a, column A) and η and σ relatively far apart (Table 3a, column B). However, the multiplier effect is somewhat larger when the two elasticities are further apart — the multiplier increases from 1.01054 to 1.01096. On the other hand, when η and σ are close to one another, the effect on the steady state markup is very small. The markup, μ remains almost unchanged. This occurs because the average elasticity, which determines the level of the markup, given by (ησ)λ + σ is approximately constant at σ when η – σ is small. When the difference between η and σ is large the change in the markup is by almost half a percentage point. In a similar vein, capital stock, labor, output and private consumption rise in both cases although the increase is sharper when the two elasticities are further apart.

Table 3a

Steady State Effects of a One Percentage Change in Public Investment*

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Column A corresponds to {η = 2.75, σ = 2.65}and column B to {η =3.5, σ= 2.46}

When η = 3.5 and σ = 2.46, the effect of percentage increase in public investment leads to almost half a percent increase in employment. If θ was held constant at 0.15, to achieve the same effect government expenditure as a percentage of output would have had to increase by 5 percentage points, i.e., from 0.21 to about 0.26. Thus keeping the composition of public expenditure constant, permanent increases in government spending permanently increases employment and output. In models with constant markup, although such effects are present the causality works through the labor supply decisions of households and depends almost entirely on the income effect on leisure. An increase in government expenditure leads to a decline in private wealth and raises real interest rates. Both these raise the marginal utility of wealth and causes labor supply to rise (Barro 1981). However, in this environment the causality works from the demand side. With an increase in the share of investment, the price elasticity of a firm’s output decreases causing the markup to fall and consequently leading to an increase in labor demand (for a related analysis see Rotemberg and Woodford 1992).

The short run effects of permanently changing θ are quite significant even when the difference between ρ and σ is small (column A, Table 3b). While the percent increase in the variance of output from the technology shock is by almost 1 percent, for labor it is by over 2 percent. The effect is more dramatic for the variance caused by demand shocks. For both employment and output the increase is by over 85 percent which increases to over 100 percent when the elasticities are further apart. This increase in the variance of employment and output leads to a fall in households’ welfare. The loss in welfare is around 0.03 percent. When the difference between the two elasticities is larger, the loss of welfare increases as do the variances in output and employment (column B, Table 3b). As discussed previously, an increase in the public investment increases the elasticity and decreases the markup. While this leads to an increase in welfare due to efficiency gains, the increase in elasticity also makes aggregate demand “flatter”. With a more flat demand, adjustments in quantity (output and therefore employment) are larger than in prices. This increase in volatility tends to decrease the level of welfare.

Table 3b.

Cyclical Effects of a One Percentage Change in Public Investment*

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Column A corresponds to {η = 2.75, σ = 2.65} and column B to {η = 3.5, σ =2.46}

B. Correlation of Total Factor Productivity with Demand Shocks

Assume in this section that z^t=0, so that the only exogenous shock to the system is g^t.

Letting x^txtx¯x¯In(xtx¯), total factor productivity is given by

q^t=yk(11μ¯)k^t+yg(11μ¯)g^t

where yg=ς(1α)ξμλg1+αζ. As long as the elasticity of λ with respect to markup is negative, a condition satisfied if η > σ — which is the same requirement to ensure that the markup is procyclical — the Solow residual, q^t, will be positively correlated to g^t, namely the aggregate demand shock. Consequently, in this economy if Solow residuals are measured in the standard way i.e., the difference between actual output and the share weighted contribution of capital and labor, it will be biased upwards as shocks such as g^t that are exogenous to the true technological shock, will be picked up. While, Hall (1988) and Evans (1992) have empirically shown that productivity is correlated with variables such as government expenditure shocks, Basu (1995) provides an argument why this may take place in a model where final good production requires intermediate goods. In this environment however, the driving force behind the result is the wedge between the elasticities of firms and households for the same goods.

VIII. Concluding Remarks

That aggregate demand affects the cyclical behavior of an economy by changing the markup is well established in the literature with markups behaving in a countercyclical manner as increases in demand raise the elasticity of demand faced by monopolistically competitive firms such that profit maximization leads to a reduction in their markups. Building on the framework developed by Gali (1994), who showed that the composition of aggregate demand can have similar impact, we found that changes in government purchases can have efficiency effects as well as affect the short run volatility of macroeconomic variables like output and employment. In particular, when the price elasticities of investment is larger than that of consumption, markups behave countercyclically. In such a situation if the share of investment in total public expenditure is increased aggregate demand becomes more elastic and consequently, optimal markup is reduced. With a fall in the markup the economy is pushed towards the competitive equilibrium with the resulting increase in productivity raising the marginal revenue products of both capital and labor, such that in equilibrium, firms hire more workers and accumulate more capital, which in turn increases the steady output, employment and capital stock. However, the increase in the elasticity of aggregate demand implies that in response to exogenous shifts in demand or supply quantity adjustment will be more than price adjustment thereby increasing the cyclical variability of output and employment. This trade-off between welfare gains due to increased efficiency and welfare losses due to greater variability is the critical decision that policy makers need to make when changing the composition of public expenditure.

For the postwar U.S. economy, a one percentage increase in the share of public investment, keeping the share of total government spending constant, raises steady state output by almost 0.07 percent and welfare by 0.8 even when the elasticities of consumption and investment are close to one another. For larger differences in the two elasticities while the increase in output is larger the welfare gains are less since there is an offsetting increase in disutility from increased hours of work. However, this also increases the short run variability of the economy. The variance of output from the same technology shock increases by almost one percent while the variance of employment is raised by about 2 percent. The resulting loss in welfare from this increased uncertainty is about 0.03 percent.

We conclude the paper by discussing briefly a particular shortcoming of the model. Typically government expenditure is made of consumption and investment goods and labor, i.e., public sector employees. If a government changes the number of its employees, the economy can also be affected. However, this effect will be through the supply side. By changing its level of employment, the government can affect the amount of labor input available to the private sector, i.e., the effective labor supply faced by firms in the private sector. The effect of such changes could be significant both quantitatively as well as in terms of its direction. One cannot draw a strict analogy between changes in the composition of government purchases and public sector employment since the latter will affect the economy primarily by changing the composition of available inputs and their relative prices. In this paper, however, we did not consider the effects of changes in the public sector employment.

Appendix

The linearization of the following equations
xtAxzt1+ζ1+αζμtζ(1α)1+αζkt(1+α)1+αζ=0(1)
μt(ησ)λt+σ(ησ)λt(1σ)=0(2)
(λtθgt)xtkt+1kt+(1δ)=0(3)
kt+1Ak{Et[(zt+1μt+1)1+ζ1+αζ]}1+αζ1α=0(4)
resulted in
x^t=a1z^ta2μ^ta3k^t(5)
μ^t=b1λ^t(6)
λ^t=1c1k^t+11c1k^tδc1x^t+c2g^t(7)
k^t+1=d1Et[z^t+1μ^t+1](8)
where, a1=(1+ζ1+αζ),a2=ζ(1α)1+αζ,a3=1α1+αζ,b1=(αδρ+δ)((σp)((ησ)λ¯+σ)2)λ¯(λ¯ψ),c1=δλ¯(λ¯ψ),c2=(ψλ),d1=(1+ζ1α), and ω^t=ωtω¯ω¯ for ω=x,λ,k,z,v,g. Note that at the steady state (λ¯ψ)((ησ)λ¯+σ(ησ)λ¯(1σ))=αδρ+δ. Consequently,
(αδρ+δ)((ση)((ησ)λ¯+σ)2)=(λ¯ψ)((ση)((ησ)λ¯(1σ))((ησ)λ¯+σ))
Since ξ¯=(ησ)λ¯+σ, the b1 becomes λ¯((σρ)ξ¯(ξ¯1)). From the definition of the elasticity of the markup with respect to λ, we know that ε¯μ=(ση)λ¯ξ¯(ξ¯1). Consequently, b1=ε¯μ. Replacing (5) and (6) in (7) we have under the assumption that c1−δa2b1 ≠ 0
λ^t=1c1δa2b1k^t+1(1δa3)c1δa2b1k^tδa1c1δa2b1z^t+c1c2c1δa2b1g^t(9)
Using the above and leading (6) by one period
μ^t+1=b1(1c1δa2b1k^t+2(1δa3)c1δa2b1k^t+1δa1c1δa2b1z^t+1+c1c2c1δa2b1g^t+1)
Note that c1c2=δψλψ. Replacing the above in (8) we have, provided c1b1(d1−δ) ≠ 0,
k^t+1=Et[e1z^t+1+e2k^t+2+e3g^t+1](10)
where,
e1=d1[c1+b1δ(a1a2)]c1b1(d1δ)e2=d1b1c1b1(d1δ)=d1ε¯μc1ε¯μ(d1δ)e3=d1c1c2b1c1b1(d1δ)=d1c1c2ε¯μc1ε¯μ(d1δ)
Next, we list below some of the other coefficients of the linearized system of equations:
kz=ϕze11ϕze2,kg=ϕge31ϕge2,λz=kzδa1c1δa2b1,λk=(1δa3)c1δa2b1,λg=kg+c1c2c1δa2b1yz=((1+ζ)ζ(1α)b1λz1+αζ),yk=(α(1+ζ)+ζ(1α)b1λk1+αζ),yg=ζ(1α)b1λg1+αζlz=ζ1+αζ(1b1λz),lg=ζb1λg1+αζ,lk=ζ1+αζ(α+b1λk)
Using the expression for d1, c1 and b1 one gets
e2={(αδρ+δ)((ση)((ησ)λ¯+σ)2)δ(1α1+ζ)[1(1+ζδ(1α)1)(αδρ+δ)((ση)((ησ)λ¯+σ)2)]}1e2={δ(1α1+ζ)[1((1+ζ)δ(1α)1)(αδρ+δ)((ση)((ησ)λ¯+σ)2)](1+ζ1α)(αδρ+δ)((σρ)((ησ)λ¯+σ)2)}=δ((1α)1+ζ)[1(1+ζ1α)(αδρ+δ)((ση)((ησ)λ¯+σ)2)+1(1+ζδ(1α))]={δ(1α)1+ζ[1+1(αδρ+δ)((ση)((ησ)λ¯+σ)2)]1}
Using that fact that the steady state (αδρ+δ)((ση)((ησ)λ¯+σ)2)=(λ¯ψλ¯)ε¯μ, we have
1e2={δ(1α)1+ζ[1+(λ¯λ¯ψ)1ε¯μ]1}

Taking modulus, 1|e2|=1δ(1α)1+ζ[1+(λ¯λ¯ψ)1ε¯μ].If|e2|<1,then|1e2|>1i.e.,1δα1+(1ζ)[1+(λ¯λ¯ψ)1ε¯μ]>1.

Consider the expression for e2={1δ(1α)1+ζ[1+(λ¯λ¯ψ)1ε¯μ]}1. Since ε¯μ=(ση)λ¯ξ¯(ξ¯1) and ξ¯=(ησ)λ¯+σ, using the steady state condition (λ¯ψ)μ=αδρ+δ, we get e2=C2[C12ησ+C3]1 where C1=(ησ)λ+ σ>0,C2=α(1+ζ)(ρ+δ)(1α)>0 and C3=C2αδ(ρ+δ)>0. Now if | e2 |> 1 when η – σ > 0 then C2>[C12ησ+C3]. This implies that C2C3>C12ησ,i.e.αδ(ρ+δ)>C12ησ which in turn results in ησ>C12(αδρ+δ)1. Since, ησ=C1λσ, the inequality can be rearranged as C1λσ>C12(αδp+δ)1. Again using the steady state condition (λ¯ψ)μ=αδρ+δ and rearranging the expression for optimal markup such that C1=(11μ)1, the inequality becomes μC1C12(λλψ)>μσ. Now if σ > 0 then μC1(λλψ)>0i.e.,μ(μμ1)(λλψ)>0. This implies that μ>1+λλψ. Since λλψ>1 it implies that.μ > 2.

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1

We are grateful to Ben Klotz, George Kopits, and Eswar Prasad for useful comments and suggestions. The usual disclaimer applies.

1

In particular, note that if an overall budget constraint is only imposed, i.e., ptscgts+pthigthGt, then θ t =ω.

2

Note that the elasticities of government’s consumption goods and investment goods expenditures are the same as for individual agents i.e., the elasticities are σ and n.

3

The reason why such sunspot equilibria can exist is identical to that in Woodford (1991). Without loss of generality assume that in period t the realization of the sunspot is a positive st. Furthermore, suppose all investors believe that the choice of k^t+2 will follow the rule k^t+2=j=0(1e2)jstj+1. Then the positive realization of st increases the expected value of Etk^t+2. However, this also implies that investment in t + 1 will need to be higher such that Ety^t+1 will be increased. In order to support a higher equilibrium output in t + 1 investors will then, as an optimal response, increase their desired capital stock in period t i.e., k^t+1. Consequently, the seemingly inconsistent behavior of k^t+1=1e2k^t+st instead of k^t+1=1e2k^t as determined by the fundamentals of the environment – is justified.

4

This lemma is identical to the result in Gali (1994) except that it is modified to include ψ.

6

The appendix lists the complete expressions for the different coefficients.