Remaining Framework of the DSGE model

This subsection describes the remaining structure of our DSGE model in Section 4.3.

Household Sector

There is a continuum of members in the household where the total population measures to one.

Within the household, there are fractions of f^E entrepreneurs, f^F financial intermediaries (or

“bankers”), and 1−f^E −f^F workers. Entrepreneurs engage in a business where they produce intermediate goods and transfer the net worth back to the household when they exit from the business. Now, each financial intermediary manages a bank where it accepts the deposits from the household sector and lend to entrepreneurs. When financial intermediaries exit from their business, they also transfer their net worth back to the household sector. Finally, remaining fraction of the members of the household become workers. Workers supply labor input to earn wage and they transfer their wage earnings to the household each period. Within the household, each member shares the risk perfectly.

The representative household maximizes her expected discounted sum of utility over time and their objective function is specified as follow;

Et

X∞ i=0

βⁱχ^c_t+i

"

(ct+i−hCt+i−1)^1−σc

1−σ^c −χ^L_t+i(lt+i)^1+σL 1 +σ^L

#

(4.57) where β is the discount rate, h is the habit persistence,σ^c is the inverse of intertemporal elasticity of substitution, ct is final goods consumption, Ct−1 represents the external habit formation, σ^L is the inverse of Frisch labor supply elasticity andltis the supply of aggregate labor by workers. Now, there are two structural shocks embedded in the function. χ^c_t represents an intertemporal preference shock, whileχ^L_t represents labor disutility shock relative to consumption.

Next, turning to the budget constraint of the representative household, they make a deposit, b_t, at periodt and earn real interest rate,R_t/π_t+1, next period whereR_t is risk-free gross nominal interest rate at period tand πt+1 is gross inflation rate at periodt+ 1. In addition, the household pays lump sum tax ofτtto the government. Now, they receive a lump-sum transfer of wage incomes from workers which is expressed asR1

0 w_t(x)l_t(x)dx, where w_t(x) andl_t(x) are real wage and labor supply by individual workerx, respectively.¹⁶ Finally, the household earns the combined dividend of Ξ^div_t from retailers, earns the net transfer of Ξ^E_t from entrepreneurs, and the net transfer of Ξ^F_t from bankers each period. Thus, the representative household’s budget constraint at period t can be expressed as, in real terms, as follow, ,

c_t+b_t= Rt−1

πt

bt−1−τ_t+ Ξ^div_t + Ξ^E_t + Ξ^F_t. (4.58)

Consumption and Deposit Decision The first-order conditions (hereafter, FOCs) of the house-hold with respect to ct andbt as follows;

ζ_t^H =χ^c_t(ct−hct−1)^−σc (4.59)

ζ_t^H =βE_tζ_t+1^H R_t

π_t+1. (4.60)

where ζ_t^H is Lagrangian multiplier associated with the budget constraint. (4.59) is the FOC of consumption which equates the marginal utility of consumption to the shadow price of the final goods. (4.60) is the FOC of deposit decision.

16Here, the real wage set by workerxis defined as wt(x)≡Wt(x)/Pt, whereWt(x) stands for the nominal wage set by workerxandPt stands for the price index of final goods. The formulation ofWt(x) andPt will be described later in this section.

4.9. APPENDIX 183 Wage Setting Decision by Workers Following Erceg, et al. (2000), each worker indexed by x ∈ [0,1] supplies differentiated labor input, lt(x), monopolistically and sells this service to the labor union who is perfectly competitive.¹⁷ Each worker sets his nominal wage according to Calvo style sticky price setting where fraction θ^w of the entire workers cannot freely adjust the wages at their discretion. For fraction θ^w of workers, the partial indexation of the nominal wage is assumed.¹⁸ Due to the perfect risk-sharing assumed in the model, each worker maximizes the objective function (4.57) by choosing the amount of individual labor supply,l_t(x), while taking the amount of consumption,ct, as given. Under this setting, (1−θ^w) fraction of workers maximize their objective function by setting the nominal wage,Wft, such that

Et

X∞ i=0

βⁱ(θ^w)ⁱ

"

fWt

Pt+i

Pt−1+i

Pt−1

ι^w

χ^c_t+i(ct+i−hct+i−1)^−σc−(1 +ψ^w)χ^c_t+iχ^L_t+i(lt+i(x))^σL

#

lt+i(x) = 0.

(4.61) The law of motion of the aggregate wage index can be shown to be as follow,

W_t^−1/ψw =θ^w

"

Wt−1

Wt−1

Wt−2

ι^w#−1/ψ^w

+ (1−θ^w)fW_t^−1/ψw. (4.62)

Finally, the real wage index in the economy is defined as wt≡Wt/Pt. Capital Production Sector

Capital producers are identical, perfectly competitive, and risk neutral. They purchase i^k_t units of final goods from the retailer, convert them to i^k_t units of capital goods, and combine them with existing capital stock, (1−δ)k_t, to produce new capital stock, k_t+1. Capital producers will, then, sell off new capital stock to entrepreneurs in a perfectly competitive manner. Capital producers have linear production technology in converting final goods to capital goods. In addition, they will incur quadratic investment adjustment cost when they change the production capacity of capital goods from previous period. Each capital producer maximizes the expected discounted cash flow with respect toi^k_t. ¹⁹ The FOC is given by

qt= 1 A^k_t



1 +κ i^k_t i^k_t−1 −1

! i^k_t i^k_t−1 +κ

2 i^k_t i^k_t−1 −1

!2

−β κ A^k_t+1

i^k_t+1 i^k_t −1

! i^k_t+1 i^k_t

!2

. (4.63)

17The labor union transforms labor services to an aggregate labor input, lt using the Dixit and Stiglitz type aggregator function. The factor demand function forlt(x) is given bylt(x) = (Wt(x)/Wt)^{−(1+ψw)/ψw}lt where ψ^w is the wage markup,Wt(x) is the nominal wage set by workerxandWtis the aggregate nominal wage index which is given asWt=h

R1

0 Wt(x)^−1/ψwdxi−ψw

.

18The lagged inflation indexation is specified asWt(x) = (Pt−1/Pt−2)^ιwWt−1(x) where ι^w controls the degree of nominal wage indexation to past inflation rate.

19The profit function for each capital producer at periodtcan be expressed as follows, Et

∞

X

i=0

βⁱ (

qt+ii^kt+i− 1 A^k_t+i

"

i^kt+i+κ 2

i^kt+i

i^k_t+i−1 −1 ²

i^kt+i

#)

where A^k_t is the investment-specific technology shock common across all capital producers and κ is the investment adjustment cost parameter. Finally, aggregate capital accumulation equation is given by

kt+1 =i^k_t + (1−δ)kt. (4.64)

Retailing Sector

Retailersz∈[0,1] purchase intermediate goods from the entrepreneur at perfectly competitive price and resale them monopolistically in the retail market.²⁰ We assume Calvo type sticky price setting for the retailer where, for any given periodt, fraction θ^p of the entire retailers cannot freely revise their prices. Further,θ^p fraction of the retailers who did not receive a ‘signal of price change’ will partially index their nominal prices to lagged inflation rate of price index.²¹ Under this setting, for (1−θ^p) fraction of the retailers who received a ‘price changing signal’ at period t, they maximize their expected discounted sum of profits by setting the nominal price, ˜p_t, such that

Et

X∞ i=0

βⁱ(θ^p)ⁱ

"

e pt

Pt+i

Pt−1+i

Pt−1

ι^p

−

−1

p^mc_t+i

#

yt+i(z) = 0. (4.65)

From the definition of aggregate price index, the law of motion ofP_tcan be shown to be as follow, (Pt)¹⁻=θ^p

"

Pt−1

Pt−1

Pt−2

ι^p#1−

+ (1−θ^p)˜p¹⁻_t . (4.66)

The Rest of the Economy

In closing the model, we describe the rest of the model structure here. The central bank is assumed to follow a standard Taylor-type monetary policy rule,

Rˆ_t=ρ^RRˆt−1+ (1−ρ^R)h

µ^ππˆ_t+µ^yYˆ_ti

+ε^R_t (4.67)

where ρ^R controls the magnitude of interest smoothing, µ^π is Taylor coefficient in response to inflation gap, µ^y is Taylor coefficient in response to output gap, and ε^R_t is i.i.d. monetary policy shock.

The government budget constraint is simply specified as

g_t=τ_t. (4.68)

The government expenditure, g_t, is financed solely by lump-sum tax, τ_t. In our model, we simply assume that the government expenditure to follow stochastic AR(1) process.

Finally, the market clearing condition for final goods is given as follow,

Y_t=c_t+i^k_t +g_t. (4.69)

20The demand function for retail goods sold by retailerzis given byyt(z) = (Pt(z)/Pt)⁻Yt, whereYtis aggregated final goods,pt(z) is nominal price of retail goodsyt(z), Pt is aggregate price index of final goods, andis the price elasticity of retail goods. Specifically, aggregated final goods, Yt, and the aggregate price index, Pt, are given as follows;Yt≡h

R1

0 yt(z)^(−1)/dzi/(−1)

andPt≡h R1

0 pt(z)^(−1)/dzi/(−1)

.

21The lagged inflation indexation is specified aspt(z) = (Pt−1/Pt−2)^ιppt−1(z) whereι^pcontrols for the magnitude of price indexation to past inflation rate.

4.9. APPENDIX 185 Structural Shocks in the Model

There are eight structural shocks in the model, each of them having a specific economic interpreta-tion as below. Except for monetary policy shock, all of the structural shocks are assumed to follow AR(1) stochastic processes whereρ is for the AR(1) coefficients for respective structural shocks.

TFP shock : Aˆ_t=ρ^AAˆt−1+ε^A_t Preference shock : χˆ^c_t =ρ^cχˆ^c_t−1+ε^c_t Labor supply shock : χˆ^L_t =ρ^Lχˆ^L_t−1+ε^L_t Investment specific technology shock : Aˆ^K_t =ρ^KAˆ^K_t−1+ε^K_t

Government spending shock : ˆgt=ρ^Ggˆt−1+ε^G_t Monetary policy shock : ε^R_t

Corporate net worth shock : ˆγ_t^E =ρ^Eγˆ_t−1^E +ε^E_t Bank net worth shock : ˆγ_t^F =ρ^Fˆγ_t−1^F +ε^F_t

Notice that each stochastic disturbance ε_t including monetary policy shock is assumed to follow time varying volatility using SV model as mentioned in Section 4.2.

Chapter 5

Impacts of Government Spending on Unemployment

5.1 Introduction

Chapter 5 examines the quantitative effect of government spending on unemployment in Japan.

The question is quite simple: Does government spending improve unemployment, if so, how big is it?

Note that Chapter 5 revised the paper based on “Impacts of Government Spending on Unem-ployment: Evidence from a Medium-scale DSGE Model,” (joint with Hasumi, R.), the discussion paper of Economic and Social Research Institute (ESRI Discussion paper series 329, 2016).

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