Model

We incorporate unemployment and non-wasteful government spending into the standard medium-scale DSGE model (e.g. SW 2007). This section focuses on explaining how to introduce unemploy-ment and non-wasteful governunemploy-ment spending and on illustrating how non-wasteful fiscal expansions may affect unemployment. The entire model is described in Appendix.

5.2.1 Unemployment

Following the GSW framework, we consider a large household with a continuum of members repre-sented by the unit square and indexed by a pair (j, h)∈[0,1]×[0,1]. The first dimension indexed by j∈[0,1] represents a differentiated skill in which a given household member is specialized. The second dimension indexed byh∈[0,1] indicates member’s labor disutility. We can think intuitively of the first dimension as labor unions and the second dimension as members within each union.

Unions have market power due to their differentiated skills indexed by j ∈ [0,1], but they are assumed to face nominal wage rigidities a la Calvo in line with Erceg et al. (2000). Therefore, unions set their nominal wages, taking the nominal stickiness into consideration. It should be noted that setting wages simultaneously determines employment from labor demand for union j.

Members within each union have different labor disutilities with uniformly distributed as h ∈ [0,1]. Given the nominal wage determined by each union, members decide to work or not taking their labor disutilities into consideration. In addition, we assume the full risk sharing of consumption across members: Members can enjoy consuming with the same level.

Then, the preference of a member h (who has a disutility h) in any unionj at period tcan be written by

ζ_t^cln

C˜j,t−θC˜t−1

−1t(j, h)ζ_t^hχ^h_tAHh^σh (5.1)

C˜j,t stands for consumption of member h in union j and ˜Ct ≡ R1

0 C˜j,tdj stands for aggregate consumption. The term θC˜t−1 indicates (external) habits on consumption and the parameter θ∈

5.2. MODEL 191 (0,1) depicts the importance of the habit formation. 1_t(j, h) is the indicator function, which takes a value equal to one if the memberhis employed at periodt, and zero otherwise. It is worth noting that the indicator function means members decide to work with fixed hours (normalized as unity) or not. χ^h_t stands the endogenous preference shifter defined as the following equation:

χ^h_t ≡ Zχ,t

C˜_t−θC˜t−1

, (5.2)

Zχ,t=Z_χ,t−1^1−v

C˜t−θC˜t−1

v

, (5.3)

This preference specification leads marginal labor disutility decreases during (aggregate) consump-tion booms. Two structural shocks are embedded: ζ_t^c is the preference shock and ζ_t^h is the labor supply shock. AH is the scale parameter and σh is the inverse Frisch elasticity.

LetH_j,t be defined as employment of unionj. Then, aggregating the member’s utility regarding h, we derive utility of unionj at periodtas follows:

ζ_t^cln

C˜j,t−θC˜t−1

−ζ_t^hχ^h_tAH

Z Hj,t

0

h^σhdh

= ζ_t^cln

C˜_j,t−θC˜t−1

−ζ_t^hχ^h_tA_HH_j,t^1+σh 1 +σh

(5.4) Thus, the preference of unionj falls into the standard functional form.

Now, we explain how to introduce unemployment into the medium-scale DSGE model. Because of full risk sharing of consumption, the marginal utility of consumption becomes common across members. Let the marginal utility of consumption denoted byϕ^c_t. Given the (real) wage wj,t, the memberhis willing to work as long as the real wage is greater than the marginal rate of substitution (MRS) between labor supply and consumption:

(1−τ_t^h)w_j,t≥ ζ_t^hχ^h_tA_Hh^σH

ϕ^c_t (5.5)

τ_t^hstands for labor income tax rate. The left-hand side (LHS) indicates the marginal benefit of labor supply after tax adjustment. The right-hand side (RHS) is MRS which corresponds to reservation wage of member h. Letting the marginal supplier of unionj’s member be denoted byLj,t, we have:

(1−τ_t^h)wj,t= ζ_t^hχ^h_tA_HL^σ_j,t^H

ϕ^c_t (5.6)

As mentioned before, unionj decides nominal wageWj,t which simultaneously determines employ-mentH_j,tfrom the labor demand for unionj. Let aggregate employment denoted byH_tdetermined by unions and let aggregate labor supply denoted byLtdetermined by members. Then, unemploy-ment rateUt is defined as the following equation.

Ut≡ Lt−Ht

Lt

(5.7) Figure 5.2 (a) illustrates the occurrence of unemployment in our model where three curves are depicted: Labor demand curve, marginal revenue (MR) curve and MRS curve (labor supply curve). Consider three members {a, b, c} with different reservation wages (equivalently, different labor disutilities). Memberahas the lowest reservation wage (the lowest labor disutility), member bhas the medium one and member c has the highest one.

In a steady state, unions should set wage at A so as to match MR curve with MRS curve and aggregate employment H is simultaneously determined. Given the wage at A, members a and b are willing to work since the wage determined by the union is higher than their reservation wages.

Meanwhile, member c enjoys leisure since the wage determined by the union is lower than her reservation wage. In other words, membercisvoluntarily unemployed. Thus, given the wage at A, aggregate labor supply is determined atL.

The difference between L(aggregate labor supply) andH (aggregate employment) corresponds toinvoluntary unemployment. In Figure 5.2 (a), for instance, the reservation wage of member bis lower than the wage at Abut the memberb is not employed.

5.2.2 Non-wasteful Government Spending Edgeworth complimentarities

Government consumption is assumed to directly affect household’s utility as the following way:

C˜j,t=Cj,t+νgG^c_t (5.8)

C˜_j,t consists of private consumption C_j,t and government consumption G^c_t. The parameter ν_g governs qualitative and quantitative influence of government consumption for private consumption.

Equation (5.8) indicates household gains utility not from private consumption Cj,t but from the above composite consumption ˜C_j,t. Thus, household wants to smooth intertemporally the composite consumption from the Euler equation.

Suppose that government increases consumption. If the parameterν_g is negative, an increase in government consumptionG^c_t leads to a decrease in the composite consumption ˜Cj,t. Then, house-holds will increase private consumption C_j,t to keep the composite consumption constant along with the intertemporal consumption smoothing condition (Strictly speaking, an increase in gov-ernment consumption raises the marginal utility of private consumption at the present period).

Thus, a negative νg causes a cyclical comovement between private consumption and government consumption, which is the so-called Edgeworth complementarities (hereafter, EC).¹ If ν_g is posi-tive, a counter-cyclical comovement is shown, which implies government consumption is substitutes to private consumption. If νg is zero, there is no comovement, thus government consumption is independent of private consumption.

Examples on the EC are government spending to Medicare and education service. Fiorito and Kollintzas (2004) empirically investigate the complementarities between government consumption and private consumption in the Euro area, and they find that the government spending to the merit goods such as Medicare and education service becomes a complement to private consumption.²

1On the functional specification of the composite consumption, ˜Ct, we can consider more general functional form:

C˜t=h

φ^cC_t^θc+ (1−φ^c) (G^c_t)^θci1/θ^c

. Ifθ^c→1, then ˜Ct→φ^cCt+ (1−φ^c)G^c_t (linear function). On the other hand, ifθ^c→0, then ˜Ct→C_t^φc(G^ct)^1−φc (Cobb-Douglas type function). In specifying ˜Ctas the CES aggregator, however, we face a difficulty to identify two structural parameters, i.e. θ^candφ^c. In fact, Coenen et al. (2013) specifies ˜Ct as the CES aggregator, but they calibrate the private consumption share,φ^c, in CES aggregator due to the difficulty to identify it. Following Ni (1995), Iwata (2013) and Feve et al. (2013), we specify the bundled consumption, ˜Ct, as the linear function since we can easily recognize whether the government consumption is complements or substitutes to the private consumption from the sign of the parameter,νg.

2They also find government spending for general public goods (i.e. national defense, public security service, etc.) is not a compliment to private consumption. See also Sakai et al. (2015).

5.2. MODEL 193 Productive public capital

Public capital accumulated by government investment is assumed to improve the productivity of private firms. An intermediate good firm j produces a differentiated good Yj,t (j ∈ [0,1]), using capital ˜K_j,t and labor input H_j,t:

Y_j,t =_tK˜_j,t−1^α (z_tH_j,t)^1−α−z_t⁺Θ (5.9)

t stands for neutral technology shock, zt stands for labor-augmented technology, z_t⁺ stands for a scaling variable, α is capital income share and Θ is fixed cost.

There are two types of capital in this economy. One is the effective private capital u_j,tKj,t−1

where uj,t is the capital utilization rate. The other is the public capital K_t−1^g accumulated by the government.

K˜j,t−1 = K_t−1^g αg

(utKj,t−1)^1−αg (5.10)

α_g is the marginal productivity of public capital for the bundled capital, ˜Kj,t−1. It should be noted that the productivity of the public capital for output can be expressed as α×αg from (5.9) and (5.10).

Ifα_g is positive, then public capital accumulation by the government operates positive externali-ties, which takes the form of an exogenous increase in the productivity of private firms. Improvement of productivities via government investment causes a reduction of marginal costs of private firms.

Therefore, if α_g >0, we call K_t^g productive public capital (hereafter, PPC).³ Finally, public capital is accumulated by government investment as follows:

K_t^g = (1−δg)K_t−1^g +ζ_t^g,iGⁱ_t. (5.11)

δg is the depreciation rate of public capital and ˆζ_t^g,i is government investment specific technology shock.

The remaining parts of the model are in line with the standard medium-scale DSGE model (e.g.

SW 2007), embedding nominal price and wage rigidities, investment adjustment cost, monetary policy rule, and so on. Our model consists of 49 equations and 15 structural shocks. The entire model is described in Appendix. Tables 5.1 and 5.2 report endogenous variables and structural shocks.

Now, we turn to the illustration on effects of non-wasteful fiscal expansions on unemployment.

5.2.3 Effects of Non-wasteful Fiscal Expansions for Unemployment

How do non-wasteful fiscal expansions affect unemployment? Here, we intuitively explain mecha-nisms that non-wasteful fiscal expansions may bring additional channels for improvements of un-employment.

Suppose that the economy is in a steady state at the initial period. Real wage, aggregate em-ployment, and aggregate labor supply are determined atA,H and L, respectively. Unemployment

3Several ways of introducing productive public capital are suggested by previous studies. Coenen et al. (2013) specifies the capital production function as a CES aggregator. Iwata (2013) specifies an increasing return to scale production function of output such thatyt=t(utkt−1)^αH_t^1−α(k_t^g)^αg. The specification in this paper is the constant returns to scale production function (5.9) and the Cobb-Douglas type capital production function (5.10) because of the difficulty in identifying the parameterαg in the estimation.

U is depicted as the difference between Land H. In addition, for the sake of simplicity of illustra-tion, real wage is assumed to be a constant at A at least in the short-term due to nominal wage and price rigidities.

Figure 5.2 (b) shows effects of fiscal expansions without EC and PPC (“wasteful” government spending) on unemployment. The standard story goes as follows: Fiscal expansions create aggregate demand, which induces an increase in labor demand. This effect is depicted by the shift of the labor demand curve to the right, which is shown as (i) in Figure 5.2 (b). But forward-looking households will decrease consumption because of anticipation of future tax increases (negative wealth effect).

The decrease in private consumption lowers labor demand, which is depicted by the shift of labor demand curve to the left (shown as (ii) in Figure 5.2(b)). Thus, the effect of increase in aggregate demand will be partly cancelled out by the negative wealth effect. In addition, the decrease in private consumption also causes an incentive to work more (an increase in labor supply), which is shown by the shift of labor supply curve to the right (shown as (iii) in Figure 5.2 (b)). Meanwhile, real wage adjustment is sluggish due to nominal rigidities. Here, real wage is assumed to remain at A. As a result, aggregate employment is determined atH⁰, aggregate labor supply is determined at L⁰ and unemployment is determined by the difference L⁰ and H⁰. If the increase in labor demand dominates the increase in labor supply, then fiscal stimuli decrease unemployment. Otherwise, fiscal stimuli increase unemployment.

Now, we consider effects of non-wasteful fiscal expansions on unemployment in Figure 5.2 (c).

Under “non-wasteful” government spending, several channels are added to the previous story: Fiscal expansions create aggregate demand. Forward-looking households will decrease private consumption from the negative wealth effect. The decrease of private consumption causes an increase in labor supply. Up to this point, effects of fiscal stimuli to unemployment are the same as the previous story.

Suppose that the parameterν_g in equation (5.8) is negative, which corresponds to the case with EC. Then, an increase in government consumption stimulates private consumption because of EC.

The additional channel of EC to private consumption is depicted by the shift of labor demand to the right, which brings an improvement in unemployment. This channel is shown as (iv) in Figure 5.2(c). Furthermore, from equation (5.6), the increase in private consumption leads to a decrease of labor supply under nominal rigidities: The increase of private consumption decreases marginal utility of consumption ϕ^c_t. This raises the RHS in (5.6), that is, MRS between labor supply and consumption. On the other hand, real wage, the LHS in (5.6), is fixed due to nominal rigidities.

Thus, to recover the equality of (5.6), members must work less (a decrease of labor disutility). This channel is shown as (v) in Figure 5.2 (c).

Suppose that the parameterα_gin equation (5.10) is positive, which corresponds to the case with PPC. Then, an increase in government investment improves the productivities of private firms, which implies a decrease in marginal costs of private firms. Forward-looking private firms set their prices by taking future marginal costs into account under nominal stickiness. Thus, accumulation of PPC delivers a decrease in inflation, which triggers monetary easing policy. Therefore, the “crowd-out”

effect will be weakened, which stimulates both private consumption and private investment. These effects are realized by the shift of labor demand to the right (shown as (iv) in Figure 5.2 (c)), which improves unemployment. It is worth noting that the channel through PPC has relatively longer effects than the channel through EC: Due to price adjustment sluggishness, the effect through the decrease in future inflation will be delayed but long-lasting.

Through the channels of non-wasteful fiscal expansions, aggregate labor supply and aggregate employment are determined at L⁰⁰ and H⁰⁰, respectively. As a result, non-wasteful government

5.3. ESTIMATION METHOD 195