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A business-cycle model with
monopolistically competitive firms
and Calvo wages: lessons for
Bulgaria (1999–2018)
Aleksandar Vasilev
University of Lincoln, Brayford Campus, Lincoln, UK
Abstract
Purpose –The author augments an otherwise standard business-cycle model with a rich government sector
and adds monopolistic competition in the product market and rigid prices, as well as rigid wages a la Calvo
(1983) in the labor market.
Design/methodology/approach –This specification with the nominal wage rigidity, when calibrated to
Bulgarian data after the introduction of the currency board (1999–2018), allows the framework to reproduce
better observed variability and correlations among model variables and those characterizing the labor market
in particular.
Findings –As nominal wage frictions are incorporated, the variables become more persistent, especially
output, capital stock, investment and consumption, which help the model match data better, as compared to a
setup without rigidities.
Originality/value –The computational experiments performed in this paper suggest that wage rigidities are
a quantitatively important model ingredient, which should be taken into consideration when analyzing the
effects of different policies in Bulgaria, which is a novel result.
Keywords Business cycles, Monopolistic competition, Rigid (Calvo) prices, Rigid (Calvo) nominal wages
Paper type Research paper
1. Introduction
As shown in Vasilev (2009), the standard real-business-cycle (RBC) model does not capture
well model dynamics for Bulgaria. In other words, an important aspect from the real world is
missing from the model setup. One explanation for the model failure along the labor market
dimension could be the way product markets are modeled; In other words, the perfect-
competition assumption that imposed everywhere in the firm problem in the RBC model
might be too restrictive for a transition economy such as Bulgaria. Instead, as demonstrated
in Lozev et al. (2011) and Paskaleva (2016), imperfections in the product and factor markets
are observed in Bulgaria, together with the presence of price- and wage rigidities, and those
patterns should be taken as stylized facts in theoretical models.
In light of this evidence, in this paper we take all those phenomena seriously and
incorporate those rigidities in our modelingstrategy, which effectively leads to the adoptionof
the New Keynesian (NK) approach [1], which departs from perfect competition in goods and
factors market; as a result, the prices of the factors of production no longer will equal their
marginal products. In addition, instead of a stand-in firm, there will be imperfect competition in
the intermediate goods firms, and the differentiated intermediate goods will be then combined
into a final good, which is produced by a perfectly competitive firm [2]. We can then compare
Model with
competitive
firms and
Calvo wages
JEL Classification —D43, D58, E32
© Aleksandar Vasilev. Published by Emerald Publishing Limited. This article is published under the
Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and
create derivative works of this article (for both commercial and non-commercial purposes), subject to full
attribution to the original publication and authors. The full terms of this licence may be seen at http://
creativecommons.org/licences/by/4.0/legalcode
The current issue and full text archive of this journal is available on Emerald Insight at:
https://www.emerald.com/insight/1859-0020.htm
Received 13 September 2020
Revised 23 October 2020
Accepted 17 November 2020
Journal of Economics and
Development
Emerald Publishing Limited
e-ISSN: 2632-5330
p-ISSN: 1859-0020
DOI 10.1108/JED-09-2020-0131
and contrast how a model with rigidities compares to the standard RBC model without
rigidities, when it comes to capturing the dynamics of aggregate labor market variables.
Another important difference from the RBC model is that in the setup in this model the
prices of production inputs depend on the elasticity of substitution between the differentiated
intermediate goods, which in turn reflects the market power of monopolistically competitive
firms to set prices. The distortion driven by the industry structure pushes the prices of labor
and capital below their marginal products. More specifically, as shown in Rotemberg and
Woodford (1995), the higher the market power of monopolistic firms, the higher the mark-up
and as a result, the greater the difference between the marginal product of labor and capital
and their prices [3]. In addition, with imperfect competition in the output market, a technology
shock affecting the marginal product of capital and labor leads to lower reaction of both the
real wage rate and the real interest rate (compared to the perfectly competitive environment).
As a result, the owners of the two factors of production will perceive a smaller effect from the
productivity shock and that will drive down the use of labor and capital and that would lead
to lower output at intermediate- and final-good level. In turn, the model no longer generates
efficient allocations in equilibrium.
Another important novelty in this paper, which distinguishes the model setup from the
standard NK model, would be the presence of rigidities in the wage determination process a la
Erceg et al. (2000),Canzoneri et al. (2005) and Christiano et al. (2005). The stickiness in nominal
wages is an important ingredient in the transmission of technology shocks and a rigidity
which could potentially generate employment and wage fluctuations similar to the ones
exhibited in Bulgarian data [4]. Importantly, relative to the rest of the NK literature, we allow
for the capital accumulation motive to work alongside nominal rigidities. In the absence of
restraint on capital (in the form of investment adjustment costs) and in the absence of active
monetary policy, which is the case of Bulgaria, the neoclassical mechanisms seems to
dominate quantitatively, with the nominal rigidities being only secondary in importance. In
other words, nominal rigidities –despite being a relevant feature of reality –are not the
primary factor behind the observed business-cycle fluctuations in Bulgaria.
Overall, allowing for nominal wage rigidities in the model improves the model
performance against data, and in addition, this extended setup marginally dominates the
standard RBC model framework without wage frictions, e.g. Vasilev (2009,2019). Therefore,
the computational experiments performed in this paper for Bulgaria in the period 1999–2018
[5] suggest that Calvo wages are a quantitatively important model ingredient, which should
be taken into consideration when analyzing the effects of different policies. This is a
contribution in itself, as this is the first dynamic general equilibrium model with Calvo wages
done for Bulgaria, which –following Canova (2007) –has been subjected to a variety of
statistical tests. Overall, micro-founded theoretical dynamic general equilibrium models are
therefore to be considered as very important devices in the macro modelers’toolboxes, as
those setups provide the necessary disciplining of data and allows researchers to
discriminate between different alternative explanation, as well as break any observational
equivalence problems, such as the ones pertaining to labor market dynamics [6].
The rest of the paper is organized as follows: section 2 describes the model framework and
defines the decentralized monopolistically competitive equilibrium system. Section 3
discusses the calibration procedure, and section 4 presents the steady-state model solution.
Sections 5 proceeds with the out-of-steady state dynamics of model variables and compared
the simulated second moments of theoretical variables against their empirical counterparts.
Section 6 concludes the paper.
2. Model setup
There is a continuum of ex-ante identical one-member households distributed uniformly on
the unit interval and indexed by j. Final output is obtained through the aggregation of
JED
intermediate good outputs, in an environment of perfect competition and can be used for
household consumption, investment or government purchases. In contrast, in the
intermediate goods sector, there is monopolistic competition with free entry, which means
that each intermediate good is produced by a single monopolistic firm, which has market
power and sets the price of the particular good they produce at a mark-up above their
marginal cost. Lastly, the government is levying taxes on consumption, labor and capital
income in order to finance spending on government purchases and lump-sum government
transfers.
2.1 Households’problem
Household jmaximizes the expected discounted utility, which is of the form [7]
U¼EtX
t¼0
∞
βt(c1−
σ
t
1
σ
h1þw
t
1þw);(1)
where Etis the expectations operator as of period t,0 <β<1 denotes the discount factor, ctis
consumption of household jin period t,htare the hours worked by household jin period t,
σ
is
the relative risk aversion parameter (and the inverse of the intertemporal elasticity of
substitution parameter), and parameter wdenotes the curvature of the function capturing the
disutility of hours worked.
The household starts with a unit endowment of time in each time period, and a positive
endowment of physical capital, k0, in period 0, which is rented to the firm at the nominal rental
rate Rt, that is, before-tax capital income equals Rtkt. Therefore, each household can decide to
invest in capital to augment the capital stock, which evolves according to the following law of
motion:
ktþ1¼itþð1δÞkt;(2)
where 0 <δ<1 is the depreciation rate of physical capital.
In addition to the rental income, each household owns an equal share of the final-good-
producing firm and thus has a legal claim to the firms’nominal profit, Πt. Household j’s period
tbudget constraint is then
Pt½ð1þ
τ
cÞctþktþ1ð1δÞkt¼ð1
τ
yÞ½WthtþRtktþΠtþPtgt
t;(3)
where Ptis the aggregate price index,
τ
c;
τ
yare consumption and income tax rate, respectively,
and gt
tare per household government transfers.
The problem faced by each household is then to choose fct;ht;ktþ1;Wtg∞
t¼0to maximize
utility subject to budget constraint [8]. The first-order conditions for the allocations are as
follows:
ct:c−
σ
t¼λtPtð1þ
τ
cÞ(4)
ht:hw
t¼λtWtð1
τ
yÞ(5)
ktþ1:λtPt¼βEtλtþ1½ð1δÞPtþ1þð1
τ
yÞRtþ1(6)
TVC :lim
t→∞βtλtktþ1¼0:(7)
The interpretation of the optimality conditions above is standard: the first states that at the
margin, optimal consumption is characterized by the balance between the benefit of extra
consumption utility and the cost in terms of shadow price of wealth. The second equation
balances the disutility of extra work and the benefit in terms of extra income, weighted by
Model with
competitive
firms and
Calvo wages
consumption utility. The third equation, the so-called Euler equation, describes how capital
should be allocated in any two congruent periods. The last condition, the “Transversality
condition,”is a boundary constraint, in order to rule out explosive solution paths.
In addition, since labor can be a differentiated product among households, this implies
that households have some market power when setting wages. As in Christiano et al. (2005)
and Canzoneri et al. (2005), each household supplies differentiated labor services in a market
structure of monopolistic competition. These labor services are rented to a representative
firm that aggregates these different types of labor hjinto a single labor input H.AsinJunior
(2016), the labor-aggregating firm is assumed to use the following constant elasticity of
substitution (CES) technology:
Ht¼0
B
B
@Z1
0
h
ψ
W−1
ψ
W
j;tdj1
C
C
A
ψ
W
ψ
W−1
;(8)
where
ψ
Wis the elasticity of substitution between differentiated labor services and hj;tis the
amount of differentiated labor hours supplied by household j. Each type of labor hours jis
paid for with a nominal wage Wj;t. The problem of the labor-aggregating firm is then to
maximize its static profit:
max
hj;t
WtHtZ1
0
Wj;thj;tdj (9)
subject to the aggregation constraint above, or
max
hj;t
Wt0
B
B
@Z1
0
h
ψ
W−1
ψ
W
j;tdj1
C
C
A
ψ
W
ψ
W−1
Z1
0
Wj;thj;tdj:(10)
The first-order condition for each type of differentiated labor services is
Wt
ψ
W
ψ
W10
B
B
@Z1
0
h
ψ
W−1
ψ
W
j;tdj1
C
C
A
ψ
W
ψ
W−1−1
ψ
W1
ψ
Wh
ψ
W−1
ψ
W
−1
j;tWj;t¼0 (11)
or,
Wt0
B
B
@Z1
0
h
ψ
W−1
ψ
W
j;tdj1
C
C
A
1
ψ
W−1
h−1
ψ
W
j;tWj;t¼0 (12)
Next, noting that we can express aggregate hours as follows:
H
1
ψ
W
t¼0
B
B
@Z1
0
h
ψ
W−1
ψ
W
j;tdj1
C
C
A
1
ψ
W−1
;(13)
JED
so we can replace it in the following equation:
WtH
1
ψ
W
th
−1
ψ
W
j;tWj;t¼0:(14)
After some algebra, we can express the demand for hjas follows:
hj;t¼HtWt
Wj;t
ψ
W
(15)
Substitute now this expression back into aggregate hours to solve for the aggregate wage
rate as a function of household-specific wage rates
Wt¼0
@Z1
0
W1−
ψ
W
j;tdj1
A
1
1−
ψ
W
(16)
In terms of the wage rigidity, in each period, 1 −θWhouseholds, chosen independently and at
random, optimally define/set their wages in nominal terms. The remaining households, θW,
follow a wage stickiness rule a la Calvo (1983) and keep the same wage level as the previous
period, or, Wj;t¼Wj;t−1. In particular, the 1 −θWfraction of households that can choose wage
levels in period tknows that, even setting optimal nominal wage W
j;tfor the period, it faces a
θN
Wprobability of these wages remaining fixed for Nfuture period. When household jchooses
W
j;tto solve the following problem [9]
max
W
j;t
EtX
t¼0
∞
ðβθWÞt"...h1þw
t
1þwλtð...Wj;thtÞ#(17)
s.t.
ht¼HtWt
Wj;t
ψ
W
(18)
Substituting the expression into the objective function
max
W
j;t
EtX
t¼0
∞
ðβθWÞt"...1
1þwHtWt
Wj;t
ψ
W1þw
λt...W
j;tHtWt
Wj;t
ψ
W#(19)
After some algebraic manipulations, and using that λt¼c−
σ
t=Pt, we can derive the expression
for the optimal wage equation set by household j:
W
j;t¼
ψ
W
ψ
W1EtX
t¼0
∞
ðβθWÞtc
σ
thw
tPt(20)
As 1 −θWfraction of households chooses the same nominal wages, W
j;t¼W
t, and the mass
of remaining households, θW, set their wage equal to the nominal wage observed in the
previous period. Thus, the aggregate nominal wage can be expressed as
W1−
ψ
W
t¼ZθW
0
W1−
ψ
W
t−1dj þZ1
θW
W1−
ψ
W
tdj
W1−
ψ
W
t¼hjW 1
ψ
W
t1iθW
0þW1
ψ
W
t1
θW
Model with
competitive
firms and
Calvo wages
W1−
ψ
W
t¼θWW1−
ψ
W
t−1þð1θWÞW1−
ψ
W
t;
hence the aggregate nominal wage rule is:
Wt¼hθWW1−
ψ
W
t−1þð1θWÞW1−
ψ
W
ti1
1−
ψ
W(21)
2.2 Firms
The modeling approach of the industry structure in the setup follows Dixit and Stiglitz (1977),
with a continuum of differentiated goods. In turn, these differentiated goods are then
aggregated into a single final goods, which is consumed by the households. Each firm
produces a single intermediate good that the final producer then uses as an input in the
production of the final good via a CES output aggregator function. The final good producer
takes prices as given, while intermediate good producers have power over setting their own
prices.
2.2.1 Final goods production sector (retail). The functional form chosen for the aggregation
technology is
Yt¼0
B
@Z1
0
Y
ψ
−1
ψ
j;tdj1
C
A
ψ
ψ
−1
;(22)
where Ytis aggregate output (the product of the retailer) in period t, and Yj;t,j∈½0;1is the
output of intermediate (wholesale) good j, and
ψ
>1 denotes the elasticity of substitution
between the differentiated wholesale goods.
Let Pj;tdenote the nominal price of wholesale good j, the price of each wholesale good is
taken as a given by retail firms. The problem faced by each retail firm is then to
max
Yj;t
PtYtZ1
0
Pj;tYj;tdj (23)
s.t. (22), or when plugging that expression back into the objective function, to
max
Yj;t
Pt0
B
@Z1
0
Y
ψ
−1
ψ
j;tdj1
C
A
ψ
ψ
−1
Z1
0
Pj;tYj;tdj (24)
Taking the first-order condition and after some algebra we obtain
Yj;t¼YtPt
Pj;t
ψ
:(25)
In other words, the individual demand is proportional to aggregate demand and inversely
proportional to relative price level. Now substitute this expression back into aggregate output
to obtain the expression for the price of the final (retail) goods in term of the prices of the
intermediate goods:
Pt¼0
@Z1
0
P1−
ψ
j;tdj1
A
1
1−
ψ
;(26)
which is also the aggregate price index.
JED
2.2.2 Intermediate goods production sector (wholesale). As pointed out earlier, each
wholesale firms sell their differentiated goods to the stand-in retail (final-goods) firm.
Intermediate-good producers will be assumed to possess some market power and will have
some power in setting the price of their product (facing a downward-sloping demand for their
product). In addition, it will be assumed that fixed (entry or period) costs do not exist [10].
Since the retailer has constant-returns-to-scale technology, its marginal cost is independent of
quantity produced. Furthermore, the marginal cost function coincides with the average cost
function, and total cost equals the product of marginal cost time quantity. Net, the retail firm’s
problem can be split in two parts. In the first, the prices of capital and labor are taken as given,
and the firm minimizes total cost subject to the production function (the technology
constraint), or
min
ht;kt
WthtþRtkt(27)
s.t
Yt¼Atk
α
th1−
α
t(28)
The first-order conditions are
ht:ð1
α
Þ
μ
tAtk
α
th−
α
t¼Wt(29)
kt:
αμ
tAtk
α
−1
th1−
α
t¼Rt;(30)
where
μ
tis the Lagrange multiplier attached to the constraint. With
μ
t¼MCj;tthe equation
above become
ht¼ð1
α
ÞMCj;t
Yj;t
Wt
(31)
kt¼
α
MCj;t
Yj;t
Rt
(32)
The expressions above are the optimal demand for the two inputs (capital and labor) by each
wholesale firm. Deriving the total and marginal cost function can be done from the dual
problem–the profit maximization one:
max
ht;kt
π
t¼Atk
α
th1−
α
tPj;tWthtRtkt(33)
FOCs:
kt:Rt¼
α
Yj;t
kt
(34)
ht:Wt¼ð1
α
ÞYj;t
ht
(35)
After some algebra, we arrive at the following expression: [11]
MCj;t¼1
AtWt
1
α
1−
α
Rt
α
α
(37)
Next, the second stage of the problem of the wholesale firm jis optimally setting the price of
its product. This firm decides how much to produce in each period a la Calvo (1983). More
specifically, in each period, each wholesale firm has a θprobability of keeping the price of its
Model with
competitive
firms and
Calvo wages
good unchanged in the next period ðPj;t¼Pj;t−1Þand a 1 −θprobability of optimally setting
its price. Therefore, the problem of a wholesale firm jthat is able to reset the price of its good is
max
P
j;t
EtX
t¼0
∞
ðβθÞtP
j;tYj;tTCj;tþi(38)
s.t. demand constraint, or
max
P
j;t
EtX
t¼0
∞
ðβθÞtP
j;tYtPt
Pj;t
ψ
YtPt
Pj;t
ψ
MCj;tþi(39)
The first-order condition (after some algebra)
P
j;t¼
ψ
ψ
1EtX
t¼0
∞
ðβθÞtMCj;t(40)
Note that all wholesale firms that fix their price have the same mark-up on the same marginal
cost. Thus, in all periods, P
j;tis the same price for all the 1 −θfirms that set their prices.
Following the argument in Junior (2016), the aggregate price level is
P1−
ψ
t¼Z
θ
0
P1−
ψ
t−1dj þZ
1
θ
P1−
ψ
tdj
P1−
ψ
t¼jP1
ψ
t1θ
0þjP1
ψ
t1
θ
P1−
ψ
t¼θP1−
ψ
t−1þð1θÞP1−
ψ
t
Pt¼θP1−
ψ
t−1þð1θÞP1−
ψ
t1
1−
ψ
(41)
Note that since there is a continuum of intermediate good producers, and the share that can
reset its price (and the group that cannot) is chosen randomly, regardless of when each firm
last altered its price. As a result, the distribution of prices among firms does not change
between periods.
2.3 Government
In the model setup, the government is levying taxes on labor and capital income, as well as
consumption in order to finance spending on government purchases and government
transfers. The government budget constraint is as follows: [12]
τ
cctþ
τ
yðwthtþrtktÞ¼gt
tþgc
t(42)
Tax rates and government consumption-to-output ratio would be chosen to match the
average share in data, and government transfers would be determined residually.
2.4 Stochastic process
Total factor productivity, At, is assumed to follow AR(1) processes in logs, in particular
ln Atþ1¼ð1
ρ
aÞln A0þ
ρ
aln Atþea
tþ1;
where A0>0 is steady-state level of the total factor productivity process, 0 <
ρ
a<1 is the
JED
first-order autoregressive persistence parameter and ea
t∼iidNð0;
σ
2
aÞare random shocks to
the total factor productivity progress. Hence, the innovations ea
trepresent unexpected
changes in the total factor productivity process.
2.5 Dynamic monopolistically competitive equilibrium (DMCE)
Given the processes followed by the stochastic process fAtg∞
t¼0, average tax rates f
τ
c;
τ
yg,
endowments k0∀j, the decentralized dynamic competitive equilibrium is a list of sequences
fct;it;htg∞
t¼0, a sequence of government purchases and transfers fgc
t;gt
tg∞
t¼0, price level
sequence fPtg∞
t¼0and prices fwt;rtg∞
t¼0such that (1) each household jmaximizes its utility
function subject to its budget constraint, (2) the representative final-good firm maximizes
profit; (3) intermediate-good firms maximize profit; (4) government budget is balanced in each
period and (5) all markets clear.
3. Data and model calibration
To calibrate the model to Bulgarian data, we will focus on the period after the introduction of
the currency board (1999–2018). Annual data on output, consumption and investment was
collected from National Statistical Institute (2019), while the real interest rate is taken from
Bulgarian National Bank Statistical database (2019). The calibration strategy described in
this section follows a long-established tradition in modern macroeconomics: first, the
discount factor, β¼0:982, is set to match the steady-state capital to output ratio in Bulgaria
over the period, which is k=y¼3:491. The labor share parameter,
α
¼0:429, was computed
as the average value of labor income in aggregate output over the period 1999–2018. The
depreciation rate of physical capital in Bulgaria, δ¼0:05, was estimated as the average
depreciation rate over the period 1999–2018. The curvature parameters of consumption and
leisure components in the household’s utility function are set to
σ
¼2 and w¼1:5 in order to
generate plausible value for aggregate labor supply elasticity. The average income tax rate
was set to
τ
y¼0:1, and the tax rate on consumption,
τ
c¼0:2, is set to their values over the
period 1999–2018. Next, due to the lack of more recent data, the elasticity of substitution
between differentiated intermediate goods was set to
ψ
¼5 to generate the average mark-up
of 25 percent estimated by Dobrinsky et al. (2006) for Bulgaria. Similarly, the elasticity of
substitution between differentiated labor services was set to
ψ
W¼7 to generate a wedge
between marginal utility of consumption and the marginal disutility of leisure of 17 percent,
which is the average inter-industry wage difference. The price stickiness parameter was set
to θ¼0:74 following Paskaleva (2016)’s estimate on the share of firms that do not change
prices. Similarly, as in Lozev et al. (2011) the wage stickiness parameter, θW¼0:68, was set to
the share of firms that set wages to the previous period wages [13]. Lastly, the process
followed by total factor productivity is estimated from the detrended series by running an AR
(1) regression and saving the residuals. Table 1 below summarizes the values of all model
parameters used in the paper.
4. Steady-state
Once the values of model parameters were obtained, [14] the steady-state equilibrium system
solved, the “big ratios”can be compared to their averages in Bulgarian data. The results are
reported in Table 2 below. The model matches consumption-to-output ratio by construction.
The investment and government purchases ratios are free variables and are also closely
approximated. The shares of income are also identical to those in data, which is an artifact of
the assumptions imposed on functional form (Cobb–Douglas) of the aggregate production
function. The after-tax return, net of depreciation, ~
r¼ð1−
τ
yÞr−δ, is also very closely
captured by the model.
Model with
competitive
firms and
Calvo wages
5. Out-of-steady state model dynamics
Since the model does not have an analytical solution for the equilibrium behavior of variables
outside their steady-state values, we need to solve the model numerically. This is done by log-
linearizing the original equilibrium (non-linear) system of equations around the steady-state.
This transformation produces a first-order system of stochastic difference equations. First,
we study the dynamic behavior of model variables to an isolated shock to the total factor
productivity process, and then we fully simulate the model to compare how the second
moments of the model perform when compared against their empirical counterparts. Special
focus is put on the cyclical behavior of labor market variables.
5.1 Impulse response analysis
This subsection documents the impulse responses of model variables to a 1% surprise
innovation to technology. The impulse response function (IRFs) are presented in Figure 1 on
the next page against the IRFs from a model without nominal price and wage rigidities [15].
As a result of the one-time unexpected positive shock to total factor productivity, output
increases. This expands the availability of resources in the economy, so consumption,
investment and government consumption also increase upon impact. As a result of the
increase in productivity, the real interest rate increases as well, and households increase their
capital accumulation. Wages also increase in real terms more with Calvo wages and prices.
Importantly, compared to the perfectly competitive case, the effects over aggregate output is
smaller upon impact, as demonstrated in Junior (2016) and Torres (2013), since the
inefficiencies produced by imperfect competition at intermediate level itself reduce the effect
of the technology shock. In addition, with nominal wage frictions, model variables become
Parameter Value Description Method
β0.982 Discount factor Calibrated
α
0.429 Capital share Data average
1−
α
0.571 Labor share Calibrated
δ0.050 Depreciation rate on physical capital Data average
w1.500 Curvature, disutility of work Set
σ
2.000 Curvature, utility of consumption Set
h0.333 Share of time spent working Data average
τ
c0.200 VAT/consumption tax rate Data average
τ
y0.100 Average tax rate on income Data average
θ0.740 Price stickiness parameter Data average
θW0.680 Wage stickiness parameter Data average
ψ
5.000 Elasticity of substitution, intermediate goods Calibrated
ψ
W7.000 Elasticity of substitution, differentiated labor Calibrated
ρ
a0.701 AR(1) parameter, total factor productivity Estimated
σ
a0.044 st.dev, total factor productivity Estimated
Variable Description Data Model
c=yConsumption-to-output ratio 0.674 0.674
i=yInvestment-to-output ratio 0.201 0.175
gc=yGovernment cons-to-output ratio 0.159 0.151
wh=yLabor income-to-output ratio 0.571 0.571
rk=yCapital income-to-output ratio 0.429 0.429
~
rAfter-tax net return on capital 0.056 0.057
Table 1.
Model parameters
Table 2.
Data averages and
long-run solution
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more persistent, especially output, capital stock and consumption. The “adjustment
stickiness”of wages also causes households’labor supply to initially increase more than
in other models, e.g. the standard RBC and NK models without wage frictions, as shown in
Junior (2016). Physical capital also varies more due to its complementarity with labor.
Over time, as the effect of the shock waves, the return on capital decreases, which drives
down investment and capital accumulation back to their old steady-state values. The other
model variables return to their old values in a monotone manner as well as the effect of the
one-time surprise innovation in technology dies out.
5.2 Simulation and moment-matching
We will now simulate the model 10,000 times for the length of the data horizon. Both empirical
and model simulated data are detrended using the Hodrick and Prescott (1980) filter. Table 3
on the next page summarizes the second moments of data (relative volatilities to output and
contemporaneous correlations with output) versus the same moments computed from the
model-simulated data at annual frequency. We compare side by side the moments from a
model with nominal rigidities versus a model with no rigidities and perfect competition
(“benchmark model”). The two models match quite well the absolute volatility of output.
However, both models slightly overestimates the variability in consumption and more
substantially that of investment [16]. Still, the model is qualitatively consistent with the
finding the consumption varies less than output, and investment varies more than output. By
construction, government spending in the model varies as much as in data.
Figure 1.
Impulse responses to a
1% surprise
innovation in
technology
Model with
competitive
firms and
Calvo wages
With respect to the labor market variables, the variability of employment predicted by the
model with Calvo wages is much closer to that in data, compared to the benchmark model
without rigidities; however, the variability of wages in the Calvo-wage model is much higher
than that in data. Next, in terms of contemporaneous correlations, the model with Calvo-
wages slightly over-predicts the pro-cyclicality of the main aggregate variables—
consumption and government consumption. This, however, is a common limitation of this
class of models. In addition, the model with Calvo wages is a bit better than the alternative.
Along the labor market dimension, the contemporaneous correlation of employment with
output is of the right sign, but the model predicts it to be quite strong, while in data the linear
relationship is more moderate. With wages, the model predicts strong cyclicality, while wages
in data are acyclical. The same is true with the contemporaneous correlation between
productivity and hours. Again, the Calvo model is marginally better than the alternative
setup without rigidities along this dimension of data. In the next subsection, we investigate
the dynamic correlation between labor market variables at different leads and lags, thus
evaluating how well the model matches the phase dynamics among variables. In addition, the
autocorrelation functions (ACFs) of empirical data, obtained from an unrestricted VAR (1) are
put under scrutiny and compared and contrasted to the simulated counterparts generated
from the model.
5.3 Auto- and cross-correlation
This subsection discusses the ACFs and cross-correlation functions (CCFs) of the major
model variables. The coefficients empirical ACFs and CCFs at different leads and lags are
presented in Table 4 against the simulated AFCs and CCFs. For the sake of brevity, we
present only results for the model with Calvo wages only. Following Canova (2007), this
comparison is used as a goodness-of-fit measure. As seen from Table 4 on the next page, the
model compares well vis-a-vis data. Empirical ACFs for output and investment are slightly
outside the confidence band predicted by the model, while the ACFs for total factor
productivity and household consumption are well-approximated by the model.
The persistence of labor market variables are also well-described by the model dynamics:
the ACFs unemployment and wages are close to the simulated ones until the third lag. Same
holds true for output and investment. The ACF for consumption is well-captured only until
the first lag. Overall, the model with persistence a la Calvo (1983) in nominal wages generates
the right persistence in model variables and is able to respond to the criticism in Nelson and
Plosser (1982),Cogley and Nason (1995) and Rotemberg and Woodford (1996), who argue that
Data
Model Benchmark model
(With calvo wages) (w/o rigidities)
σ
y0.05 0.05 0.05
σ
c=
σ
y0.55 0.92 0.84
σ
i=
σ
y1.77 5.25 2.36
σ
g=
σ
y1.21 1.00 1.00
σ
h=
σ
y0.63 0.53 0.29
σ
w=
σ
y0.83 1.49 0.81
σ
y=h=
σ
y0.86 1.49 0.81
corrðc;yÞ0.85 0.97 0.89
corrði;yÞ0.61 0.53 0.80
corrðg;yÞ0.31 1.00 1.00
corrðh;yÞ0.49 0.73 0.33
corrðw;yÞ0.01 0.94 0.96
corrðh;y=hÞ0.14 0.86 0.96
Table 3.
Business-cycle
moments
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this class of models do not have a strong internal propagation mechanism besides the strong
persistence in the TFP process. Furthermore, the Calvo nominal wage mechanism dominates
other non-Walrasian models such as Vasilev (2016,2017b). Next, as seen from Table 5 on the
next page, over the business cycle, in data labor productivity leads employment. The model
nominal wage persistence, however, cannot account for this fact. In this model, as well as in
the standard RBC model a technology shock can be regarded as a factor shifting the labor
demand curve, while holding the labor supply curve constant. Therefore, the effect between
employment and labor productivity is only a contemporaneous one. Still, the model with
nominal wage persistence a la Calvo (1983) is a clear improvement over the perfectly-
competitive labor market paradigm used in Vasilev (2009,2019).
6. Conclusions
We augment an otherwise standard DSGE model with a rich government sector and add
monopolistic competition in the product market and rigid prices, as well as rigid wages a la
Calvo (1983) in the labor market. This specification with the nominal wage rigidity, when
Method Statistic
K
0123
Data corrðut;ut−kÞ1.000 0.765 0.552 0.553
Model corrðut;ut−kÞ1.000 0.818 0.629 0.442
(s.e.) (0.000) (0.035) (0.063) (0.084)
Data corrðht;ht−kÞ1.000 0.484 0.009 0.352
Model corrðht;ht−kÞ1.000 0.818 0.629 0.442
(s.e.) (0.000) (0.035) (0.063) (0.084)
Data corrðyt;yt−kÞ1.000 0.810 0.663 0.479
Model corrðyt;yt−kÞ1.000 0.815 0.625 0.438
(s.e.) (0.000) (0.037) (0.067) (0.091)
Data corrðat;at−kÞ1.000 0.702 0.449 0.277
Model corrðat;at−kÞ1.000 0.814 0.624 0.437
(s.e.) (0.000) (0.038) (0.070) (0.096)
Data corrðct;ct−kÞ1.000 0.971 0.952 0.913
Model corrðct;ct−kÞ1.000 0.815 0.625 0.438
(s.e.) (0.000) (0.037) (0.067) (0.091)
Data corrðit;it−kÞ1.000 0.810 0.722 0.594
Model corrðit;it−kÞ1.000 0.816 0.624 0.434
(s.e.) (0.000) (0.038) (0.069) (0.095)
Data corrðwt;wt−kÞ1.000 0.760 0.783 0.554
Model corrðwt;wt−kÞ1.000 0.816 0.627 0.441
(s.e.) (0.000) (0.036) (0.065) (0.087)
Statistic
k
3210 1 2 3
Data corrðht;ðy=hÞt−kÞ0.342 0.363 0.187 0.144 0.475 0.470 0.346
Model corrðht;ðy=hÞt−kÞ0.034 0.033 0.029 0.960 0.030 0.034 0.035
(s.e.) (0.732) (0.643) (0.531) (0.098) (0.534) (0.645) (0.734)
Data corrðht;wt−kÞ0.355 0.452 0.447 0.328 0.040 0.390 0.57
Model corrðht;wt−kÞ0.034 0.033 0.029 0.960 0.030 0.034 0.035
(s.e.) (0.732) (0.643) (0.531) (0.098) (0.534) (0.645) (0.734)
Table 4.
Autocorrelations for
Bulgarian data and the
model economy
Table 5.
Dynamic correlations
for Bulgarian data and
the model economy
Model with
competitive
firms and
Calvo wages
calibrated to Bulgarian data after the introduction of the currency board (1999–2018), allows
the framework to reproduce better observed variability and correlations among model
variables and those characterizing the labor market in particular. These results suggest that
technology shocks seem to be the dominant source of economic fluctuations, but nominal
wage rigidities a la Calvo (1983), as well as the monopolistic competition in the product
market, might be important factors of relevance to the labor market dynamics in Bulgaria,
and such imperfections should be incorporated in any model that studies cyclical movements
in employment and wages. Therefore, the empirical findings that the theoretical setup with
Calvo wages fits data better, can be interpreted as a validation of the model and a rejection of
the model without nominal wage rigidities and perfect competition in the case of Bulgarian
data for the period 1999–2018. Overall, micro-founded theoretical dynamic general
equilibrium models are therefore to be considered as very important devices in the macro
modellers’toolboxes, as those setups provide the necessary disciplining of data and allows
researchers to discriminate between different alternative explanation, as well as break any
observational equivalence problems, e.g. in cases when similar impulse responses of model
variables are produced as a result of a technology shock, such as ones generated by an a-
theoretical VARs. As a result of the monopolistic competition the effects of technology shocks
on output are smaller upon impact, as compared to the perfectly competitive case, since the
inefficiencies produced by imperfect competition itself reduce the effect of the technology
shock. More specifically, the deviation from a perfectly competitive paradigm leads to an
inefficient allocation of labor and capital inputs (market failures), resulting in a lower
equilibrium level for the economy and lower effects from a productivity shock. As nominal
wage frictions are incorporated, the variables become more persistent, especially output,
capital stock and consumption. Still, the model suffers from some of the usual shortcomings
inherent in this class of DSGE models. As a suggestion for future research, the model might
be extended to accommodate other important (and real) frictions in the labor market, possibly
along the lines of Vasilev (2016,2017b,2018).
Notes
1. This modeling approach was initially developed by Rotemberg (1982),Blanchard and Kiyotaki
(1987) and Rotemberg and Woodford (1992,1995), among others.
2. In other words, the final stage is identical to the standard RBC model.
3. As pointed out in Torres (2013), this is a direct consequence of the assumption that the elasticity of
substitution between differentiated goods is strictly greater than unity.
4. In order to accommodate those, it will be also assumed that labor services are differentiated among
households. This assumption implies that households also possess certain market power in setting
their nominal wages. This assumption could be easily rationalized with the presence of labor unions,
as well as certain provisions in the Bulgarian Labor code, which protect workers’interests in labor
disputes with employers. Both explanations are empirically plausible in Bulgaria, as demonstrated
in Paskaleva (2016).
5. This period was chosen as it is a period of macroeconomic stability.
6. “Observational equivalence problems”occur in cases when similar impulse responses of model
variables are produced as a result of a technology shock, such as ones generated by an a-theoretical
unrestricted or structural VARs.
7. To simplify notation, we will suppress the jindex and use smaller case letters to denote individual
allocations and capital letters for aggregate quantities.
8. We postpone the discussion of optimal wage-setting until later.
9. As in Junior (2016), we keep only the relevant terms in the problem below.
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10. Actually, that assumption is not that relevant–we can have an equivalent representation with fixed
“period costs”and free entry, which leads the firm to set price above mc to cover the amount of the
fixed costs and have 0 profit. This is the representation utilized in Torres (2013).
11. Note that the expression below is consistent with the requirement that
MCj;t¼TCj;t=Yj;t:(36)
12. Given that there is a unit mass of households, individual and total allocations are identical.
13. Al those values are consistent with the values of those parameters in the literature for the US and
other EU countries.
14. The steady-state results for a model with no rigidities and perfect competition are very close and
thus not presented.
15. The results are insensitive to the degree of wage persistence, as captured by parameter
ψ
w.
16. This shortcoming of the models could be explained by structural factors in Bulgaria, such as
privatization of state assets and the short annual time series for Bulgaria. In addition, public
investment in infrastructure has been also substantial in the last few years due to the EU
accession funds.
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Further reading
Vasilev, A. (2015), “Welfare effects of at income tax reform: the case of Bulgaria”,Eastern European
Economics, Vol. 53 No. 2, pp. 205-220.
Vasilev, A.Z. (2017a), “VAT evasion in Bulgaria: a general-equilibrium approach”,Review of
Economics and Institutions, Vol. 8 No. 2, pp. 2-17.
Vasilev, A.Z. (2017c), “Business Cycle Accounting: Bulgaria after the introduction of the currency
board arrangement (1999-2014)”,European Journal of Comparative Economics, Vol. 14 No. 2,
pp. 197-219.
Corresponding author
Aleksandar Vasilev can be contacted at: alvasilev@yahoo.com
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