macro_mini_thesis

Cost and Benefits of Job Security Programs ∗
Sebastian Flennerhag†
, Matteo Zambanini‡
June 2015
Abstract
A ubiquitous element of labor market institutions in developed countries are their many
policies geared towards greater job security. This need arose during the deplorable
condition of workers during the industrial revolution and the periods after it and were
almost certainly warranted. Today, however, the rationale for most of these rules are
taken for granted despite the fact that their effects on modern economies have rarely
been studied. In this paper we aim to evaluate how labor protection policies affect
macro-economic variables in a closed economy. We assess whether these policy regimes
are effective given their intended outcome and whether their full effects generate a
welfare enhancing economic environment. Our research will be focused on two main
measures which affect respectively rigidity in the labor market and rigidity on the level
of wages. We use an Infinitely Lived agents framework with perfectly competitive
markets, and find that while the policies achieve their directly intended outcome, they
are on the aggregate not welfare enhancing, although the difference is slight and some
caution should be exercise.
Keywords: Job Security, Wage Rigidity
∗Term paper, 5309 Dynamic Macroeconomic Analysis, Stockholm School of Economics
†22273
‡92844
1

Contents
Introduction 3
Job Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Minimum Wages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Model 4
Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Solving the Model 6
The Firm’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
The Household’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Policies 12
Job Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Wage Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Calibration 13
Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Value Function Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Evaluation 18
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Short Term Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Long-Term Eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Welfare Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Extensions 25
Indivisible Labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Further Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Concluding Remarks 32
References 32
Appendix A 33
Appendix B 33
Appendix C 34
2

Introduction
Labour law have always been a central theme in the political and economical policy in
all developed countries. Many politicians found their programs on these themes in order
to raise consensus, as these acts have a easy-to-understand and direct effect on the life of
everyone. Labor law started to gain importance during the first industrial revolution, back
in the 19th century, when a new class of workers moved from the countryside to the city
in the hope of higher employment opportunities and better working conditions. However,
deplorable conditions, long and exhausting working hours, low salaries, child labour and
high risk of injuries or death forced governments to take action and step in to protect the
weak laboring class. Starting with the British ”Health and Morals of Apprentices Act” in
1802, different kind of reforms were adopted by all governments to help better conditions for
workers. Nowadays, labour law consists in a collection of laws which regulate all the possible
aspects of a working contract, from regulation of the relationship between the individual
employee and the employer, to regulation concerning trading unions.
This paper focuses on policies that are aimed at the individual’s employment terms and
minimum wage, while abstracting from policies on living wages, working hours, health and
safety conditions as well as discrimination laws etc.
Job Security
The main objective of this kind of regulation is to prevent employers from firing employees
without any legitimate motive. Even if it is desirable to secure minimum levels of employ-
ment stability and transparency, this branch of labor law has been used in many developed
countries as a macroeconomic tool to adopt counter-cyclical measures and maintain the un-
employment rate under a desired level. In Europe, for instance, it is common to find very
articulated requirements for firms when it comes to fire people and in most cases, when
the number of employees dismissed reaches certain thresholds, additional rights and veto
power are given to trade unions and governments. Even if these measures ensure stabil-
ity to individuals, it has been argued that this rigid system makes the labor factor much
more expensive to firms which will make investment decisions taking the increased cost into
account. A question is then whether these costs are unduly high or whether the added
security to household outweighs these costs. Moreover, another critique against this policy
relates the average unemployment period of an individual who has lost his job or has just
entered the job market. As the ability to make employees redundant decreases, the risk of
employment increases and so a consequence of such policies can be reduced hiring, which
in turn translates into longer time in unemployment for those that are made redundant.
In addition, as a consequence of potentially longer unemployment periods, many currently
employed workers are disincentivized from leaving their current position to find a new job.
3

Finally, a cause for concern relates the fact that longer unemployment periods reduce the
human capital accumulation of individuals who will be less productive as a consequence.
Minimum Wages
The intuitive appeal of these types of policies is fairly straight-forward; its prevents employ-
ers from exploiting a potential power asymmetry, and safe-guards employees from volatility
in what is typically their primary income source. While this policy can be beneficial from
the employees point of view, the effects on firm’s investment decision is not so clear cut and
indeed it would seem that such a policy is rather detrimental. For instance, during a down-
turn in the economy, the demand for factor inputs such as labor is reduced which according
to standard economic theories of supply and demand should result in lowered factor price,
i.e. wages. However, with a minimum wage this may be impossible, and consequently the
price on labor is kept artificially high despite the lower demand, lowering it even further.
This suggest that this policy may be one with a populist undertone, it benefits the majority
(the workforce) at the minority’s expense (those employed in the margin). Presently, we
shall investigate both these policy regimes formally.
Model
Presentation
We develop a one-sector general equilibrium model with endogenous labor and infinitely
lived agents. Households maximize their expected utility over time taking prices as given,
subject to their budget constraint. Their budget require them to balance their income
sources with consumption and savings.
maximize
{ct, t}∞
t=0
E0U =
∞
t=0
βt
u(ct, t)
subject to ct = (1 + ρt)at + wt t − at+1.
a(0) = a0 given.
I
Where u(·, ·) is a utility function that is strictly concave in both input arguments and satisfy
the Inada conditions. a denotes assets while wt and ρt are the supply-side factor prices that
household face. ct and t denotes consumption and labor choices respectively and β is the
time-preference discount factor. For each period, there is a perfectly competitive production
market, where each firm has an identical technology. There is a continuum of firms indexed
on the interval [0, 1] such that they have a unit mass. Each firm ω ∈ [0, 1] take prices as
4

given and optimize their current-period profits;
maximize
{ ω
t ,kω
t }
max {Atf(kω
t , ω
t ) − wt
ω
t − rtkω
t , 0}
subject to
ω
t ∈ [˜ω
t−1, 1]
A(t) = At given.
II
Hence, firms choose to produce if their profits are at least 0. At each point in time they face
a lower bound ˜ω
t on the amount of labor they can hire. Typically, this is set to 0. Later,
we will define a policy that determines this lower bound as a function of the labor choice in
the previous period. The functions involved are f(·, ·) which is some production technology
that is homogeneous of degree 1, increasing and strictly concave in all inputs and satisfy
the Inada conditions. Note that the price of the good does not feature in this model, as we
assume perfect competition and normalize the price to 1. Similar to above, wt, rt are the
demand side-factor prices. Note that i) the capital of the firm is as of yet not identical to
the consumer’s assets and ii) the price on asset and capital are two different parameters.
We shall use these facts to establish the no-arbitrage condition from which we can retrieve
Solow’s law of motion for capital. Lastly At is a stochastic variable that follows a Markov
Chain. We let A denote the set of permissible states of technology, with n such discrete
states;
At ∈ {A1, A2, ..., An} = A.
We shall frequently use the notation j ∈ A as a short-hand for Aj ∈ A. Furthermore, we
define a Markov Chain in the usual manner and construct its associated matrix. First, the
first order probability condition is
P(At+1 = Aj|A0, A1, ...At) = P(At+1 = Aj|At).
Given these, the time t conditional probability distribution entails
j∈A
pi,j = 1, i ∈ A.
We summarize this information in the transition matrix,
Θ = [pi,j],
P(At) ∈
t
Θ = Θt
.
(1)
5

To tie the production and consumption side together, we impose aggregation rules such that
the market prices clears the supply and demand side of factor input;
No Arbitrage Condition: ρt = rt − δ
Total Output Aggregation:
1
0
Atf(kω
t , ω
t )dω = AtF(Kt, Lt)
Asset and Capital Aggregation:
1
0
aj
t dj = Kt
Labor Aggregation:
1
0
j
t dj = Lt
Feasibility Constraint:
1
0
(cj,t + ij
t )dj ≤ AtF(Kt, Lt).
III
Where ij
t denotes the investments of individual j, implicitly defined through the budget
constraint in (I). We include it here for completeness sake and make use of it to retrieve
the LOM of capital. Finally, the functional forms we use are the familiar
f(kt, t) ≡ kα
t
1−α
t , α ∈ (0, 1) (F.1)
u(ct, t) ≡ log (ct) + θ log (1 − t), θ ∈ (0, 1). (F.2)
Solving the Model
In this section, we study the conditions under which an equilibrium exists and we show that
the lower bound on capital does change the demand of resources, but the zero profit condition
of the unconstrained firm remains. Consequently, production will is feasible under the
constraint, and we show that with an auxiliary wage restriction the decentralized economy
is isomorphic to the Social Planners Problem.
The Firm’s problem
First, let us introduce a shadow firm that is unconstrained in its maximization;
maximizeω
t ,kω
t
Atf(kω
t , ω
t ) − wt
ω
t − rtkω
t .
Given the Inada conditions, the solution is in the interior with the usual FOC;
Atfk = rt
Atf = wt
6

Given these, under the aggregation rules of (III), the decentralized economy coincide with
the SPP. Let the equilibrium choice of capital and labor from this problem be (¯kt, ht). Our
original problem, (II), is a static nonlinear optimization problem with a strictly concave
goal-function and a linear inequality constraint. The Karusch-Kuhn-Tucker FOC are both
necessary and sufficient. We form the Lagrangian1
and derive FOCs;
max
kω
t , ω
t
L = Atf(kω
t , ω
t ) − wt
ω
t − rtkω
t − µ(˜ω
t − ω
t )
Lk = Atfk − rt = 0
⇒ Atfk = rt (2)
L = Atf − wt + µ = 0
⇒ Atf = wt − µ (3)
µ ≥ 0, ˜ω
t−1 − ω
t ≥ 0, µ(˜ω
t−1 − ω
t ) = 0.
Note that (2) is the usual marginal productivity of production. (3) however give rise to
a non-standard system than requires some attention. Suppose the unconstrained problem
yields ht > ˜t. Then, complimentary slackness imply µ = 0 and we retrieve the standard
marginal productivity equation. Note that, at these levels, optimal input choices yield zero
profits since the production function is homogeneous of degree 1;
Atf(kt, t) = fkkt + f t = rtkt + wt t.
By Euler’s Homogeneous Function Theorem. Thus, a firm is indifferent between entering
and not and with uncountably infinite number of firms, the mass of entrants is again infinite
and indexed over [0, 1]. However, suppose instead that ht < ˜t. In this case, the constraint
is binding and holding capital fixed at ¯kt, (3) imply that
Atf˜ < wt ⇒ µ > 0.
However, (2) is still a condition of an optimal point, and at ˜t this is not satisfied for
the control ¯kt, as this choice was optimal for ht. Now, since f(·, ·) is strictly concave,
fk(¯kt, ˜t) > fk(¯kt, ht). This follows from the fact that the Hessian matrix must be negative
definite for a strictly concave function and sincef2
k,k < 0, we must have f2
k, > 0. Therefore,
we find;
Atfk(¯kt, ˜t) > rt.
1The Inada conditions on f(·, ·) and u(·, ·) exclude corner solutions of the form t = 0, t = 1, kt = 0.
7

From this fact we can deduce the optimal constrained capital choice, since f2
k,k < 0, the firm
responds by increasing capital, which in turn means that the marginal rate of productivity
of labor increases. This is a crucial feature of the model for an equilibrium to exists. To
illustrate the importance of this question, note from the profit function that (3) and (2)
imply
πt = Atf(kt, ˜t) − rtkt − wt
˜t
= Atf(kt, ˜t) − Atfkk − (Atf˜ + µ)˜t = 0 − µ˜t.
(4)
By Euler’s Homogeneous Function Theorem. Compare this result to the original problem
in (II), unless µ = 0 there will be no production in the economy and so the economy is
undefined. However, we noted that capital increases as a response to a constrained choice of
labor, which in turn increases Atf˜, implying that µ goes to 0. Formally, by Euler’s theorem,
the derivative of a k-degree homogeneous function is itself homogeneous by degree k − 1. In
our case, this means that fi is homogeneous of degree 0. Now consider the unconstrained
optimal solution, where Atfk = rt. Let v be the fraction ˜/ht of additional labor required in
the constrained problem. Then, for (2) to hold, capital must increase by the same fraction;
rt = Atfk(¯kt, ht) = Atfk(v¯kt, vht) = Atfk(kt, ˜). (5)
Where the first equality stems from the unconstrained FOC, the second uses Euler’s theorem,
the third uses the definition of v. Altogether we find that the optimal constrained choice
of capital is kt = v¯kt, that is, an increase by the same fraction. Equation (5) establishes
that the increase in capital is proportional to the policy-induced increase in labor. It then
follows;
Atf˜(kt, ˜t) = Atf˜(v¯kt, ˜t) = Atf (v¯kt, vht) = Atf (¯kt, ht) = wt. (6)
Note that this result implies µ = 0, and hence the profit function in (4) equal 0. Thus, by
(5) and (6) the solution to the constrained maximization problem is structurally symmetric
to the unconstrained solution. In particular, the solution to (II) is given by
Atfk = rt (2 )
Atf = wt. (3 )
Note that these are the same conditions as under an unconstrained economy. We conclude
that the firm’s problem in (II) has an identical solution to its unconstrained counterpart,
albeit with different factor inputs. Since the base-line infinite-horizon model is isometric to
the Social Planner’s Problem, this must also be the case for the model in (I)-(III). Invoking
the aggregation rules in (III), we find the representative firm; all firms behave identically
and the production market has a unit measure. Thus the product market is characterized
by a stand-in firm.
8

The Household’s problem
In what follows, standard results will be offered without derivation, in order to avoid an
unnecessarily dense discussion. We refer the reader to Appendix A for derivations of value
functions. Beginning in (I), we note that (III) implies that at = kt, and that ω
t = t. This
implies that we can rewrite the budget constraint in (I) as2
at+1 = (1 + ρt)at + wt t − ct
⇔
kt+1 = (1 − δ)kt + rtkt + wt t − ct. (7)
Now, before implementing this equation as a constraint on the consumer, we must note
crucially that in so doing we are importing the constraint of the firm, as that is the relevant
constraint for kt and ω
t . Now, the implication would be that not only are firms restricted
in their firing policy, households are restricted in their ability to voluntarily quit. This
naturally, would be a rather deplorable consequence of the model and so we must constrain
the economy in such a way as to avoid it. Now, providing job security hinges crucially on
households actually wanting to stay in employment, and so two natural ways to incorporate
this feature are i) to make the constraint on the firm conditional on the optimal choice of
the consumer or ii) set the wage such that the constrained labor choice of the firm coincide
with a voluntary supply of labor by the households. In this paper, we shall opt for ii) for two
reasons. Firstly, condition i) implies that the firm’s problem and the consumer’s must be
solved independently which complicates things considerably. Secondly, in reality the policy
safeguards against discrete drops in labor, from full employment to nil. Typically, a person
on the margin of employment would very strongly like to stay in employment rather than
forced exit. Given that we use a stand-in household and interpret the labor choice as a
fraction of population in employment, this strong desire is omitted as the labor function
takes a much smoother shape. Consequently it is reasonable to model the economy such
that labor should be kept high, but the wage set in such that the supply of labor is a
voluntary choice from the point of view of the household. While a more complex model
might be able to better capture these features, we argue that our approach has a natural
interpretation and serves well as a first approach to the subject matter. Now, let us study
how the household in (I) behaves taking prices as given. Substituting the constraint into
2Note that the production feasibility constraint in (III) together with a = k yields the LOM of capital:
kt+1 = ct + (1 − δ)kt − rtkt − wt t
= [f(kt, t) − it] + (1 − δ)kt − rtkt − wt t
= [rtkt + wt t − it] + (1 − δ)kt − rtkt − wt t
= (1 − δ)kt + it.
9

the utility function gives
u(at, at+1, t),
From which the associated value function is
V (at, Ai) = max
at+1, t



u(at, at+1, t, Ai) + β


j∈A
pi,jV (at+1, Aj)





.
The first order condition with respect to labor is
∂V
∂
= u (at, at+1, t) = 0.
This equation pins down the optimal choice of labor, given prices and the two-period asset
path. However, the above equation is also an implicit function for the wage required given
some asset path and labor choice. Now, suppose the firm is constrained to keep ˜t in the
payroll. We can then solve for the wage that makes ˜t the unconstrained optimal choice
of labor for the consumer. To retrieve the wage it is necessary to impose some functional
forms. Hence, substituting for u(·, ·) from (F.2) we obtain;
wt
(1 + ρt)at + wt
˜t − at+1
−
θ
1 − ˜t
= 0.
Rearranging for wage gives
wt = θ
(1 + ρt)at − at+1
1 − ˜t(1 − θ)
.
Note that the above equation is increasing in ˜t as ˜t, θ ∈ (0, 1). Consequently, in the
case the firm-wise demand for labor is higher, so would the required wage be. Hence our
minimum wage condition becomes,
wt ≥ θ
(1 + ρt)at − at+1
1 − ˜t(1 − θ)
.
Finally, imposing the market clearing conditions in (III) gives the general equilibrium con-
straint as
wt ≥ θ
(1 − δ)kt + rtkt − kt+1
1 − ˜t(1 − θ)
≡ ω(kt, kt+1). (M)
For brevity, we define ω(·, ·) as the right hand side of (M). Let us pause here and study the
consequences of this constraint. From the firms problem we know that a constrain labor
choice increases demand for capital. Now, in a general equilibrium, the capital at time t is
actually through households investment in previous period. Consequently, if the constraint
is binding, the only variable in (M) is kt+1, which increases with ˜t. This has a very intuitive
10

appeal; we saw from the firm’s problem that the constraint induced increased demand for
capital, which if capital remains fixed or adjust only marginally means that rental rates
increase. This in turn would make it more attractive to hold capital, which in turn induce
greater savings.
Finally, by the solution to the firm’s problem as captured in (2 )-(3 ) we can substitute
in (7) in the households problem. Under (M) this results in an economy where the firm is
potentially constrained in their choice of labor while the household is not. Since consumer’s
have identical preferences and initial assets, having a unit mass of households imply that
the mass of household reduce to a stand in one. Through the isomorphism of the baseline
model with the SPP, the general equilibrium of the decentralized economy in (I)-(III) is
characterized by
maximize
{kt+1, t}∞
t=0
E0U = E0
∞
t=0
βt
u(ct, t)
subject to
ct = (1 − δ)kt + rtkt + wt t − kt+1
wt = Atf ≥ ω(kt, kt+1|˜t)
rt = Atfk
t ∈ [˜t, 1] = Lt
At ∈ A
P(At+1) = Θt
k(0) = k0, h(0) = h0, A(0) = A0 given.
(8)
Note that this is a problem with two control variables, kt+1, t, and three state variables,
kt and t−1 and the stochastic At. By substituting to constraints into the goal function,
noting that (M) cannot be substituted for, the sequence problem of (8) is:
maximize
{kt+1}∞
t=0∈R∞
{ t}∞
t=0∈L∞
t
E0U = E0
∞
t=0
βt
u (kt, kt+1, t, At|At−1, Θ)
subject to
Atf ≥ ω(kt, kt+1|˜t).
IV
11

Using functional forms, we find the final value function as3
V (kt, ˜t, At)At∈A = maximize
{kt+1}∞
t=0∈R∞
{ t}∞
t=0∈L∞
t



log (1 − δ)kt + Atkα
t
1−α
t − kt+1
+ θ log (1 − t) + β


j∈A
pi,jV (kt+1, ˜t+1, Aj)





subject to Atf ≥ ω(kt, kt+1|˜t−1).
V
(V) embodies the solution the our model and we solve it numerically using the Value Iteration
Algorithm. This completes the characterization of the model, and for the detailed oriented
reader, further notes on the Value Function, the Markov Chain and the Value Function
Algorithm can be found in appendices A-C respectively.
Policies
Job Security
We formally model job security as a lower bound on the labor choice of firms that is a
function of the labor supply in the previous period. This captures the idea that while firms
are able to let employee’s go, there is a rigidity to amount of firing firms undertake in a given
period. In Sweden firms are typically required to reallocate employees to other departments
rather than outright firing them, and usually firing someone requires either exceptionally
poor performance or the closing of a division. Formally, we impose the following lower
bound of labor constraint on firms;
˜t = λ ω
t−1, λ ∈ (0, 1), ∀t (P.1)
Consequently, the inequality constraints on firms in (II) becomes
t ≥ λ t−1.
Wage Security
Additionally, we contrast this policy with a different policy regime under which the wage
instead has a lower bound, but the amount of labor is variable. Here job security is more
directly oriented towards the work-force, as those who remain in it during a low-turn earn a
3See Appendix A for a general derivation.
12

higher wage. However, as we shall see, the amount of labor under this policy is considerably
lower than in an unimpeded general equilibrium. Formally, we relax the lower bound con-
straint on the firm, essentially converting (II) into its unconstrained counterpart. However,
we impose a minimum wage by modifying the market clearing wage,
wt ≥ ˜w ∀t. (P.2)
Now, recall that in a general equilibrium, (3 ) implies that the full characterization of this
regime is
wt = Atf ≥ ˜w.
By setting ¯w at the average wage, say, we capture the idea of a real downward rigidity.
Here this rigidity takes the form of an absolute value in our model, which imply that the
policy is intended to insulate households from low-level wages and also reduce the downward-
variability of their earnings which at face value seems like an attractive feature.
Calibration
Parameter Values
The focus of this paper is on the effect of policy regimes when the economy is hit by
stochastic shocks. As such, the two most important set of parameters to consider are the
probability transition matrix, and the policy regime parameters. The remaining parameters
are set at values that are typically found in the literature, where most have been borrowed
from McCandlesss (2008) who in turn has used the estimations of Hansen (1985) for the
underlying, or in his terminology, ”deep” parameters. From McCandless (2008; p. 60) we
set the annual parameter values. However, as we are interested in short-term shocks, we
use geometric averages to convert the time-variant parameter values to quarterly data, as
summarized in table (1). Now, for constructing a Markov Chain, we wish to relate our
Parameters Annual Quarterly
β 0.98 0.99
δ 0.1 0.025
α 0.35 0.35
θ 0.5 0.5
Table 1: Underlying Parameter Values
findings to empirical data to the extent possible. Consequently, rather than constructing a
stochastic environment arbitrarily, we use Tauchen’s method (Tauchen, 1986) to elicit the
13

transition matrix. As a first pass, we initially limited the number of feasible states to 2, but
we found that using Tauchen’s method the estimated transition matrix was too persistent
with too narrow technology states. As a consequence most of the stochastic features of the
model disappeared. We therefore followed Tauchen who reports from his simulations that
a higher number of states is a more judicious choice for an accurate estimation, especially
with high persistence in the AR(1) process. Hence, at a considerable expense in computing
time we followed this advice and increased the number of states to 9. As a brief overview
of the concepts involved, the underlying technology sequence follows an AR(1) process. In
particular, we let the process have a mean of 1 and a stochastic element that follows4
zt+1 = γzt + t, γ ∈ (0, 1), ∼ WN(0, σ ).
Hence the technology state at a given time is At = 1 + zt. The parameters γ and σ are
estimated from productivity data, for instance by regressing a Cobb-Douglas production
function on GDP. Due to time constraints, we will refrain from estimating this process
ourselves, and instead implement the estimation results from McCandless (2008; pp. 90-
110). Letting the mean of A equal 1, the above equation means that At is distributed around
1, and for a suitable choice of error distribution cannot go negative. To find an estimate for
γ, the Cobb-Douglas production function in (F.1) is estimated on US quarterly data with the
production function parameter values above. McCandless reports the γ-value estimated by
Hansen (1985), as this is a rather straigthforward affair of estimating the log-version of the
Cobb-Douglas production function. Hansen finds that γ = 0.95 and that the standard error
for output (in log form) is 0.0176. Using these estimation, McCandless elicit the implied
standard deviation of the error term in the expression above, given γ = 0.95. Doing so, he
finds σ = 0.0032. Tauchen’s method is essentially based on approprately partitioning the
set of possible technology states over a probability distribution, and in a sense discretize the
AR(1) sequence. Using the values above we implement Tauchen’s method using Flodén’s
Matlab routine (Flodén, 2008). Table 2 reports the estimated technology states, and table
3 reports the estimated transition matrix.
A1 A2 A3 A4 A5 A6 A7 A8 A9
0.969 0.977 0.985 0.992 1.000 1.008 1.015 1.023 1.031
Table 2: Technology States
4For further details see Tauchen (1986).
14

A1 A2 A3 A4 A5 A6 A7 A8 A9
A1 0.764 0.235 0.001 0.000 0.000 0.000 0.000 0.000 0.000
A2 0.059 0.741 0.200 0.001 0.000 0.000 0.000 0.000 0.000
A3 0.000 0.075 0.757 0.168 0.000 0.000 0.000 0.000 0.000
A4 0.000 0.000 0.093 0.767 0.140 0.000 0.000 0.000 0.000
A5 0.000 0.000 0.000 0.115 0.770 0.115 0.000 0.000 0.000
A6 0.000 0.000 0.000 0.000 0.140 0.767 0.093 0.000 0.000
A7 0.000 0.000 0.000 0.000 0.000 0.168 0.757 0.075 0.000
A8 0.000 0.000 0.000 0.000 0.000 0.001 0.200 0.741 0.059
A9 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.235 0.764
Table 3: Technology Transition Matrix
Value Function Iteration
In order to hone in on a suitable range for the Value Function Iteration algorithm, we need
to define discrete grids for initial capital stocks and possible lower thresholds. In order to
do so, we start by characterizing the steady state values of the economy given the mean
technology parameter. Since the final result in (V) is similar to the model described in
McCandless (2008, pp.63-64), we omit derivations of FOC as they are identical apart from
the additional nonlinear constraint5
. For the purpose of finding a focal point for the grid, we
fix At at the mean of the technology process which is set to 1. We proceed in the ordinary
fashion by taking derivatives of the value function with respect to labor and one-period-
ahead capital. Note that labor at time t does not feature into the value function explicitly
for t + 1, hence we need not consider V t
(kt+1). On the other hand, kt+1 does feature in the
equation and using the envelope theorem on V (kt+1, t, At+1) we find the two sets of FOC,
together with the wage constraint as;
i)
(1 − α)Atkα
t
−α
t
(1 − δ)kt + Atkα
t
1−α
t − kt+1
−
θ
1 − t
= 0
ii) −
1
(1 − δ)kt + At+1kα
t
1−α
t − kt+1
+ β
(1 − δ) + αAt+1kα−1
t+1
1−α
t+1
(1 − δ)kt+1 + At+1kα
t+1
1−α
t+1 − kt+2
= 0
iii) (1 − α)Atkα
t
−α
t − ω(kt, kt−1) ≥ 0.
Focusing on the unconstrained case, we set ˜t sufficiently low to satisfy the unconstrained
solution to i)-ii). Imposing steady state-conditions (denoted by dropped subscripts) and
5Note in particular that while our policy does introduce an intertemporal link between t and ˜t+1, firm’s
does not take this link into account, and households are unconstrained. Thus this link is not part of the
optimizing behavior of the economy.
15

solving ii) for k yields
k =
1 − β(1 − δ)
αβ
1
α−1
. (9)
Note that the bracketed expression multiplying is a constant made up entirely of param-
eters. Furthermore, since δ < 1 and β < 1, it is also positive. To simplify notation, let
B denote the bracketed expression, i.e. B = [(1 − β(1 − δ)/αβ]. Substituting the above
expression for k into i) and rearranging yields
=
1 − α
(1 − δB−1)θ + (1 − α)
. (10)
These equations for k and l can then be used to find steady state values for consumption,
production and factor prices, reported in table 4. Given this information, we create a grid
k c y r w
21.806 0.634 1.642 2.187 0.035 2.242
Table 4: Steady State Values of Endogenous Variables
for capital spanning {20, 24} with equidistant nodes. Similarly, we use the steady state for
labor to construct a grid. However, we find just as in McCandles’s version of the basic
Hansen-RBC model that the variance of labor is very low. Therefore, to get an accurate
estimation, we use a rather narrow grid of potential lower states. We use equidistant nodes
between {0.62, 0.65}, noting that the optimal unconstrained choice is never lower than 0.63
for our economy. Furthermore, to get accurate estimates to conduct a numerical analysis,
we use 50 nodes for labor, while capital is not contained within a decimal range so we have a
somewhat courser grid with 25 nodes. We will later discuss in detail the consequences for our
analysis of this low variation. With respect to the minimum wage regime, we refrain from
running it on the minimum labor grid. In general, these two policies are incompatible in the
economy modeled here. To see this note that, since kt is given, if labor is also constrained,
wt cannot change. Hence if wt < w the control space is an empty set. Consequently, the
minimum wage regime is estimated using an identical procedure to the constrained labor
regime, with the modification that the minimum labor grid is a scalar set to 0.1. Naturally,
the nonlinear constraint is updated to that of (P.2). As we shall see, under a minimum
wage regime labor frequently dips well below the levels in even the unconstrained case.
Since all variables in this economy is real, one must ask to what extent a government can
keep real wages at a minimum level, as they are primarily in a position to influence nominal
wages. Here, we shall in general ignore such consideration due to time constraints, and
instead allow the government to impose a minimum-wage slightly below the steady-state
wage level from above. In particular we set ˜w = w − 0.04 = 2.02. We find that this leaves
the economy generally unconstrained, but forces firms to keep real wages high when the
16

economy is running into a persistent downturn. Lastly, it should be mentioned that we
could have replicated the the labor-constraint process for wages, which perhaps is a more
realistic notion. Under such a regime, the government cannot set an exact threshold, but
are instead in a position to impose some ”stickiness” on wages such that they decline more
slowly. However, as a first pass we shall refrained from such due to time constraints and
the computational costs of the labor-constrained regime. Indeed, running the VFI for the
constrained labor regime with the above parameter values takes 22.5 hours over 96 iterations
with a convergence tolerance of 0.01. Whether this level of precision is strictly needed may
be debated, but the extremely narrow range of is a caution against too coarse a grid. In
figure 1 we showcase the dynamics of the algorithm by plotting every sixth iteration where
for the case with the lowest boundary on labor and A = 1.
As can be seen in 1a, the value function is concave and converges over iterations.
Similarly, panel 1b capital is increasing in initial value and crosses the 45o
-degree line at
about 21. It is interesting to study how the optimal policy of labor behaves, first when
it is constrained (1c), and when the wage is constrained but labor unconstrained (1d).
By comparing the movements in the optimal choice of labor with that of capital. As the
optimal choice of capital increases, so does the choice of labor. Indeed, while the lower
bound is binding for the first iterations, as capital increases the constraint is gradually less
binding until it is irrelevant. Contrast this with a minimum wage; the effect is clearly to
reduce the amount of labor in the economy sufficiently for the marginal productivity to
equate the equilibrium wage rate. As the capital stock grows, the marginal productivity
of labor increases allowing for more labor until the the capital stock is sufficiently large
for the labor choice to be unconstrained. Clearly, this policy is of interest for those with
secure employments and against those who is employed in the margin, as the policy induce
volatility in employment.
17

18 19 20 21 22 23 24 25
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
(a) Value Function
19 20 21 22 23 24
19
20
21
22
23
24
(b) Capital
18 19 20 21 22 23 24 25
0.62
0.625
0.63
0.635
0.64
0.645
0.65
(c) Labor with Minimum Lower
Bound
18 19 20 21 22 23 24 25
0.52
0.54
0.56
0.58
0.6
0.62
0.64
(d) Labor w. Constrained Wage
Figure 1: The Value Function Iteration Algorithm
Evaluation
Simulation
Using the policy-functions from the VFI, we simulate an economy on a generated Markov
Chain for each policy regime as well as an unconstrained economy, the results of which
are displayed in figure 2. As can be seen from these figures, the simulated properties of
the various regimes follow each other rather closely, with the fixed-wage regime being the
most divergent regime. In particular, it would seem that the constraint on labor has a
very limited impact on the economy. This, as it turns out, is an artifact from the generally
low variance of labor, as can be seen in panel 2c. The unconstrained optimal choice of
labor is bounded within a range as small as 0.01 implying that the economy is supplying
labor almost in-elastically! Several reasons could explain this, here we offer a few. Labor
influences utility directly through a multiplier that is larger than one, the wage, given the
log-function of utility, small changes in labor can have very strong effects on utility, thus
discouraging household’s from inducing volatility in that variable. Similarly, small relative
changes in capital have stronger level effects on consumption, so volatility in capital may be
preferred to volatility in labor, here to the extent that labor is extremely stable. To better
understand the short and long term dynamics of the different economies with and without
the government interference, we will analyze briefly the mechanics and interrelation between
18

variables in our model, in the three cases. We will start our analysis focusing on the first
case, the wage-constraint policy.
We shall compare two economies, one with a wage constraints and one without, and assume
they show the same variables’ initial levels. As a negative shock happens, firms in the un-
constrained case will see a reduction in their factors’ productivity and they will immediately
adjust their demand for these inputs reducing their amount and compensation (wages and
interest rates drop). However, in the constrained economy, this reduction may be infeasible,
leaving it with artificially high wage rate. Unable to put a downward pressure on wages,
the artifically high cost causes a further decline in labor demand. This kind of behavior
explains the sudden extreme dips in labor visible in panel 2c, as well as the generally higher
wage in panel 2f. From an economic perspective, this policy has some stark consequences
from a social conflict point of view; while those remaining in work enjoys a higher wage,
many loose their employment at the expense of those who remain with an artificially high
wage, in short the winner takes it all. However, wage rigidity has consequences also on the
other macro variables in the economy, starting with the level of capital. In fact, as labor
becomes a highly-costly input factor due to the minimum wage, demand for capital increases
as it becomes relatively cheap. Obviously, the effect of lower labor and higher capital leads
to a downward pressure on capital productivity and consequently on interest rates which
drop significantly in relative terms. Another consequence of the higher level of capital in
the economy can be found in the amount of labor and level of wages during good times.
As expected, higher capital means higher productivity for the labor factor and thus higher
compensation and higher employment rate during bullish periods.
A final interesting fact on capital is related to its volatility. It is clearly visible in 2b that
the capital level in a wage-constrained economy stabilizes at a higher value and evolves with
less significant swings. The reason behind this more stable path follows from how the in-
terest rate reacts during a negative turn; as labor drops sharply, the interest rate increases,
inducing higher investments during low turns which leaves the economy with more capital
in steady state. Hence a positive shock causes a lower relative shock as the larger capital
base need a lesser relative adjustment to facilitate the appropriate level change. Finally,
we conclude the analysis of the first policy looking at the output evolution over time: as
we can see, the production follows a path similar to the one of consumption, with higher
values in good time compared to the unconstrained case (due to larger accumulation of
assets and higher use of labor), and few sharp falls in bad times, following the trend in the
labor market.
We will move now our attention on the second policy which limits the firms’ ability to fire
their current workforce. We will start looking at the level of labor as this is the variable
directly influenced by the policy. As we can see, the evolution of this variable appears to
be smoothed compared to the base case. The reason is simple: firms which want to reduce
19

0 50 100 150 200 250 300 350 400 450 500
2.05
2.1
2.15
2.2
2.25
2.3
Wage Lambda=0.999 Lambda=0
(a) Output
0 50 100 150 200 250 300 350 400 450 500
21
21.2
21.4
21.6
21.8
22
22.2
22.4
22.6
22.8
23
(b) Capital
0 50 100 150 200 250 300 350 400 450 500
0.605
0.61
0.615
0.62
0.625
0.63
0.635
0.64
0.645
(c) Labor
0 50 100 150 200 250 300 350 400 450 500
1.54
1.56
1.58
1.6
1.62
1.64
1.66
1.68
1.7
(d) Consumption
0 50 100 150 200 250 300 350 400 450 500
0.033
0.0335
0.034
0.0345
0.035
0.0355
0.036
0.0365
0.037
(e) Interest Rate
0 50 100 150 200 250 300 350 400 450 500
2.16
2.18
2.2
2.22
2.24
2.26
2.28
2.3
2.32
2.34
2.36
(f) Wage
Figure 2: A Simulated Economy
the level of labor in their production during a recession, are only allowed to do it slowly.
The slow and delayed reduction in labor has a direct consequence on the speed of capital
reduction which in turn has a consequence on wages. All these three variables reduce pro-
gressively and stabilize at a higher level compared to the unconstrained case. In addition,
as expected, this higher level for all variables has a positive effect on the production and
consumption.
Predictable is also the behavior of the economy in this case when there is a positive techno-
logical shock. In fact, the higher level of capital leads to a steeper increase in labor, a higher
demand for this input factor and to a higher level of compensation, which in turn translates
into higher production and consumption in the short and long run. Even if this policy
increases aggregated consumption in the economy, we have to remember that it requires
agents to work more. We will come back to the discussion whether the policy is effectively
welfare enhancing or not later in this paper.
Short Term Dynamics
Since the Markov Chain that generated the policy functions is a discretization of an AR(1)
process for technology, we retrieve the original technology process and study how labor and
capital react (in relative terms) when technology receives a one-off shock from its steady
state at A = A5 = 1 to its most probable transitions, being At+1 = A6 = 1.008 for a
positive shock or At+1 = A4 = 0.992 for a negative shock. The results are depicted in
20

0 50 100 150
0.999
1
1.001
1.002
1.003
1.004
1.005
1.006
1.007
1.008
1.009
(a) Positive Shock
0 50 100 150
0.992
0.993
0.994
0.995
0.996
0.997
0.998
0.999
1
1.001
(b) Negative Shock
Figure 3: Shocks to Technology
figure 4. Several interesting features stand out. Beginning in the top panels, 4a and 4b, a
positive shock increases the productivity of production inputs. Labor immediately shoots
up while capital starts to be slowly accumulated. As new capital is used in the production,
labor becomes more and more productive and less amount of work is needed. However, as
technology comes back to its initial value both capital and labor slowly converge to their
initial long term equilibrium. It is interesting to notice the small differences among the three
economies: in the labor-constrained scenario, for example, labor seems to react less to the
shock (in relative terms) and adjust slowly in the transition period. The reason is that in
this economy the amount of labor used is higher than in the unconstrained case so, even if
the magnitude of the swing is bigger, in relative terms labor reacts less. The same way of
reasoning works also for capital in this scenario.
More trickier is the case of the wage-constrained economy which requires a a bit more at-
tention. As we have previously said, due to the presence of a minimum wage level, all the
downside wage risk is directly transferred to the labor so firms which have to pay too much
for this production input will respond with a sharp drop in the demand for it. For this
reason, we would expect extreme volatility but on the contrary, what we see is the opposite.
In fact, labor seems to react less to a positive shock and adjust towards its long term mean
faster than the other two cases. To explain this phenomenon we have first of all to clarify
that, given the ’normal’ initial state of the economy and the small technological shock con-
sidered, the wage-constraint is not binding. It is clear that in the extreme case, when wages
should drop below the minimum level, labor swings are wide. Our attention, though, is now
on the general case with a stable, normal economy. At this moment, we have to recall that
in equilibrium this economy tends to accumulate more capital and use more work (since it
is more productive). This initial situation explains why, given small technological shocks,
smaller variation of the two inputs are required to reach a new equilibrium. The net result
is less volatile variables.
An analogous case can be made for a negative shock. However, a new phenomena does
arise. In particular, looking at figure 4d, we can clearly see the direct effects of the two
21

policies. In fact, as expected, labor takes longer to adjust to a negative shock in a labor-
constrained economy. This delay leads to a lower level of labor adjustment and a faster
process of adjustment. Interesting to notice that the turning point is very close to the point
where the labor line crosses the labor line of the unconstrained economy (see dashed line).
In this case the policy achieved its purpose: people are not fired by firms as the negative
shocks happens, and as the economy comes back to its normal level the work force is already
deployed, capital stock is maintained high and recovery is faster. For the wage-constraint
policy, on the other hand, it is worth noting that the drop in labor with a negative shock
is much lower than the increase with a positive one. The reason is again due to the initial
capital level and its low volatility which keeps high the productivity of this input factor.
In conclusion, when studying relative effects of technology shocks, we see that labor-protection
policies achieve their goal, preventing the drop in employment during negative periods from
being too steep, stimulating a larger capital stock accumulation and hence maintaining a
higher level of employment during the recovery. Likewise, since a positive shocks causes
a temporary oversupply, the gradually re-settling keeps labor artificially high. A word of
caution, the comparison made here is purely in relative terms from an equal-sized shock.
As we shall shortly see, the effects in level-terms points to the fact that while the relative
effects of the policy seems attractive, in level terms the conclusion may sway in the opposite
direction.
22

0 50 100 150
1
1.001
1.002
1.003
1.004
1.005
1.006
1.007
Labor−Constrained Free Wage−Constrained
(a) Positive Shock; Capital
0 50 100 150
0.9995
1
1.0005
1.001
1.0015
1.002
1.0025
Labor−Constrained Labor Free Wage−Constrained
(b) Positive Shock; Labor
0 50 100 150
0.994
0.995
0.996
0.997
0.998
0.999
1
1.001
(c) Negative Shock; Capital
0 50 100 150
0.9975
0.998
0.9985
0.999
0.9995
1
1.0005
(d) Negative Shock; Labor
Figure 4: Impulse Response Functions
Long-Term Effects
We now turn to transitional dynamics. Rather than studying an impulse response, we
instead ”permanently” shift the technology state from A5 = 1 first to A4 and then back
to A5, to study how the system transitions from one state to another. Here, we keep the
variables in level terms to see aggregate effects on the economy. In figure 5 we can clearly
see the relationship among the variables in the three economies. In panel 5c we see the
slow reaction of labor in the labor-constraint policy which leads to an average higher level
of the same variable, as well as a lower level of assets. As a direct consequence, wages are
pushed down (due to higher work) and interest rates up (lower capital). On the opposite,
as we have already seen, capital levels are higher in the wage-constrained scenario, leading
to higher level of labor and wages, and lowering the average interest rate. A negative shock
has a negative effect on labor but, given its relatively higher productivity, the effect is less
pronounced than in the other two cases. As a final result, the policy seems to shift the
relative mass of income sources towards labor, while reducing the per-unit compensation
for holding assets. The policy seems to achieve its purpose being beneficial for the working
force at the expense of capital holders, who consist in the wealthy part of the population.
It is important to remember, though, that our model assumes all agents to be equal, with
23

0 20 40 60 80 100 120 140 160 180 200
21.3
21.4
21.5
21.6
21.7
21.8
21.9
22
(a) Capital
0 20 40 60 80 100 120 140 160 180 200
0.632
0.6325
0.633
0.6335
0.634
0.6345
0.635
0.6355
(b) Labor
0 20 40 60 80 100 120 140 160 180 200
2.205
2.21
2.215
2.22
2.225
2.23
2.235
2.24
2.245
2.25
(c) Wage
0 20 40 60 80 100 120 140 160 180 200
0.0346
0.0348
0.035
0.0352
0.0354
0.0356
0.0358
(d) Rent
Figure 5: Transitional Dynamics
same skills, same utility functions and same wealth. As a consequence, this kind of analysis
would require a more sophisticated model than the one presented in this section. We will
discuss the possible extensions which can be made to the basic model later in this paper.
Welfare Analysis
Up until this point, we have studied the general properties of the various regimes in a typical
simulation, their short-term behavior, and long term transition properties. It is now time to
tie the pieces together and study how the regimes impact the household’s expected lifetime
utility. To accomplish this, we let the economy adhere to the value function of (V) which
gives rise to the expected utility
U∗
= E0
∞
t=0
βt
u(c∗
t , ∗
t , At) .
Where c∗
, ∗
denote optimal control given some initial technology state . Since U∗
is non-
linear, solving the expectancy operator requires a probability density function over the
stochastic variable together with associated functional values. This is a rather intractable
problem and we shall instead resort to numerical methods and approximate the expect
24

utility through a Monte-Carlo simulation of technology sequences. Hence, given some policy
regime, we compute (Note the change of horizon of the discounted utility)
U∗
=
X
j=1
T
t=0 βt
u[c∗
t (j), ∗
t (j), At(j)]
X
. (11)
Where we let j denote the jth simulation of a technology sequence. Again, the issue of
low labor variation causes high computational costs. Since variance is so low, we need to
use 3-dimensional interpolation methods which drives up the computation time for a given
sequence. To retain some precision while keeping computational time down, we set the
utility sequence to T = 500 and X = 5000. Running this simulation for the three regimes
studied so far in this paper takes approximately 40 minutes. Table 5 reports the results.
As expected, the wage-constraint policy which lead to higher volatility in both consumption
and labor, is not welfare enhancing. It is interesting to note that the labor-constraint
policy is neither welfare enhancing. As we previously saw, this policy lead to a constantly
higher level of both consumption and labor and the net effect on the agents’ utility was
not straightforward. The result, as it turns out, is to the negative. Nevertheless, we should
caution that the Monte-Carlo simulation conducted here is underpowered, and a longer
simulation would be advisable. Further, our model studies only one agent, and perhaps a
more realistic welfare analysi would require several types of agents.
Labor-Constrained Unconstrained Minimum Wage
-0.65873 -0.65868 -0.6657
Table 5: Expected Lifetime Utility
Extensions
Indivisible Labor
In this section, we discuss some potential extension of the baseline model presented so far
as well as some tentative results. We shall focus on simple extensions that induce greater
volatility in labor and in particular consider two alternatives; firstly, we implement Hansen’s
basic indivisible labor model, as presented in McCandless (2008, pp. 112-121), secondly, we
consider an alternative policy-regime that does not directly constrain firms labor decision,
but manipulates their profit function in order to reach similar results. With respect to
the first, the modification is a straightforward one, rather than choosing its variable labor,
households sign a contract with firms in period x to work hx amounts in period t. A
stochastic process determines if the household is one of those that has to work, but if they
do not have to work a perfect unemployment insurance contract allow them a full wage
25

payment nonetheless. As for now, we remain agnostic as to how this insurance contract is
financed, and its implications for firm’s profits. Since individuals are identical over a unit
mass, we retain the stand-in household, and the representative households expected labor
in period t denotes the fraction of the population called to work, we denote this by
ht = ψth0.
The expected utility is
u(ct, ht) = log ct + ψtθ log (1 − h0) + (1 − ψt)θ log (1 − 0)
= log ct + Hht.
Where H = θ log (1 − h0)/h0 is a constant. Note that we made use of log 1 = 0 and
multiplied the second term by h0/h0 to retrieve ht. Now, since ht is the amount of labor
firms have, this is the relevant measure in the production function and budget constraint.
Substituting with ht, the firms problem is identical. As for the household, we note two
facts. Firstly, since ψ is an exogenous parameter to the household, once h0 is set they supply
labor if called to work. Hence, at time t each household supply labor inelastically and so a
minimum wage constraint is no longer a conceptually relevant condition6
. Secondly, the new
utility function means we must update the value function in (V). This gives the following
value function
V (kt, At)At∈A = maximize
{kt+1}∞
t=0∈R∞
{ht}∞
t=0∈H∞
t



log (1 − δ)kt + Atkα
t h1−α
t − kt+1
+ Hht + β


j∈A
pi,jV (kt+1, Aj)





.
V
To be able to make comparisons across models, we wish to parametrize ψt and h0 such that
the steady state value of the two models equate. Now, the only difference between them is
in the second sub-utility function. Consequently, the relevant difference will be with respect
to the expression for steady state labor. As for this model, we find this expression by taking
the derivative of the utility function with respect to labor.
wt
(1 − δ)kt + rtkt + wtht − kt+1
+ H = 0.
6In general, this omission could cause the intertemporal dynamics of capital to change, as the minimum-
wage constraint induced greater savings. However it is not clear what a comparable restriction would be in
the model with indivisible labor. Furthermore, we are primarily interested in generating a more plausible
volatility in labor. As such, we shall refrain from a more involved policy regime.
26

Imposing steady state conditions and substituting for k through (9) gives us h as
h =
1 − α
H(δB1/α − 1)
.
Since we wish to equate h with , we combine the above equation with the steady state
expression of in (10);
1 − α
δB1/α − 1
1
H
=
1 − α
(1 − δB−1)θ + (1 − α)
.
Note that H features in it. Rearranging for h0 gives;
h0
log (1 − h0)
= θ
δB1/α
− 1
(1 − δB−1)θ + (1 − α)
.
Since all terms on the right hand side are parameters, we can use numerical methods to find
h0, and estimate it to h0 = 0.9187 from which we retrieve ψ = 0.6900 and H = −1.3657.
Since we have chosen parameter values such that h = , it follows that the steady state value
of capital, and hence remaining variables, remain as in table 4. As seen in 6b, this model
produce considerably more volatility in labor. As such, we get good approximation with a
less dense grid. Therefore we halve the size of both grid and let the grid of initial capital be
constituted by 13 nodes while potential lower bounds on labor is constituted by 25 nodes.
The VFI was completed in 14 hours over 232 iterations7
. The model was also estimated un-
der a fixed wage regime in a similar manner to above, but not over the labor-constraint grid
and instead subject to the wage constraint in (P.2). We shall now undertake a brief analysis
of the differences of these models, with a primary interest in labor. In the following figures,
dashed lines represent the indivisible labor model, while solid lines represent the original
model. Figure 6 displays a simulation of both models under all regimes and the overall
impression is clear: the indivisible labor model generates much more volatility, especially in
labor. In particular, figure 6b hints at a much more efficient labor-constraint policy regime,
as the level of labor employment is almost universally higher than under an unconstrained
regime. This in turn has a material impact on capital; from 6a one sees that the capital
difference between the two regimes is now material. Furthermore, the capital stock is lower
during high periods, but higher during persistent lows and through the following upturns.
As a consequence, these periods with persistent low states of technology sees materially
higher output and consumption leaving the valuation of its welfare impact more ambiguous
than before.
7Note that the number of iterations almost doubled, which was caused by the reuse of the same tolerance
level. This was perhaps rather unnecessary. With divisible labor the value function is quite flat requiring
much higher precision. The greater curvature of the value function of the divisible labor model could have
been well approximated at a less strict tolerance level.
27

0 50 100 150 200 250 300 350 400 450 500
21
21.2
21.4
21.6
21.8
22
22.2
22.4
22.6
22.8
23
(a) Capital
0 50 100 150 200 250 300 350 400 450 500
0.6
0.61
0.62
0.63
0.64
0.65
0.66
(b) Labor
0 50 100 150 200 250 300 350 400 450 500
2
2.05
2.1
2.15
2.2
2.25
2.3
2.35
2.4
(c) Output
0 50 100 150 200 250 300 350 400 450 500
1.54
1.56
1.58
1.6
1.62
1.64
1.66
1.68
1.7
1.72
(d) Consumption
Figure 6: Comparison of Simulated Economies
Looking now at the wage-constraint policy, we see that implementing this extension volatil-
ity increases for labor as well as for capital. It is interesting to notice that production’s
volatility increases as its underlying input factors become more volatile, but consumption
becomes more stable. Having said that, consumption still remains much more volatile than
in the unconstrained case.
Turning now to short-run shocks analysis, the effect of indivisible labor is clearly visible. Be-
ginning with capital, a positive shock as in 7a seem to primarily produce a stronger reaction
with indivisible labor while the temporal pattern is qualitatively unaltered, which is reassur-
ing given that the minimum-wage constraint is not enforced in the indivisible labor model8
.
Intriguingly, while the timing of the maximum capital peak seems almost identical when the
shock is positive, it is slightly temporally delayed when the constraint is binding, which is
not the case under a divisible labor model. Not that if this was due to the minimum-wage
constraint, the effect should have been the opposite, as it induces higher savings, but the
model with the highest level of savings is the one without it. To further shed light on this
difference we now turn to the effects of a technology shock on labor in the various models,
as displayed in panels 7b and 7d. The first thing that jumps to our eyes is that the reaction
8Which hints at its irrelevance. In practice we have found in general be non-binding, i.e. the optimal
savings choice is greater than what the constraint would have implied.
28

0 50 100 150
1
1.001
1.002
1.003
1.004
1.005
1.006
1.007
1.008
1.009
0 50 100 150
0.998
1
1.002
1.004
1.006
1.008
1.01
1.012
1.014
0 50 100 150
0.991
0.992
0.993
0.994
0.995
0.996
0.997
0.998
0.999
1
1.001
(c) Negative Shock; Capital
0 50 100 150
0.988
0.99
0.992
0.994
0.996
0.998
1
1.002
1.004
(d) Negative Shock; Labor
Figure 7: Comparison of Impulse Response Functions
of labor to this shock is much stronger than before. Second and more important aspect,
the reaction patterns is no longer similar to the original model. Note in 7b that while the
original model produces similar spikes with- and without the labor constraint, the spike with
a labor constraint is now much lower. Furthermore, the delay is more protracted and while
the capital stock is lower for the first 25 periods, it is instead higher in the interval [25, 50]
which is not an effect we originally had. It is interesting to notice that, even if the initial
reaction is qualitatively identical (the turning point occurs when the unconstrained choice
has ”caught up” with the constraint), in the original model labor supply remained above the
unconstrained during the recovery while the relationship has now inverted; While the labor
constraint prevents a precipitous drop during the initial fall, it causes an overall lower level
of labor supply during the recovery. Now, this recovery is rather long and persistent and
more shocks may occur during this period. The effect that we can observe in a countinous
simulation is therefore limited to the immediate effect when the shock occurs. Hence, the
most important effect of the policy is the sharp increase in labor supply. This last piece of
the puzzle is clearly visible in figure 8 where we have shown the variables’ evolution through
time with their actual levels. Note here that while in the original model the level of capital
was typically lower in with the labor-constraint regime than in the base case, this is no
longer the case: while capital is at similar values during positive transitions, it is greater
29

0 20 40 60 80 100 120 140 160 180 200
21.3
21.4
21.5
21.6
21.7
21.8
21.9
22
0 20 40 60 80 100 120 140 160 180 200
0.626
0.628
0.63
0.632
0.634
0.636
0.638
0.64
0.642
0.644
Figure 8: Comparison of Transition Dynamics
during lower transition, pushing up labor as well. As a consequence productions is overall
greater in the economy, leading to higher consumption. From this follows an important
implication. While in general the labor-constraint policy seemed to have the desired effect
during a down-turn in the original model, one of the consequences was to suppress capital
growth during positive transitions. Now, on the contrary, this side-effect does not exist any
longer and through greater volatility in labor the policy-regime is more efficient in its work.
The final step now is to assess whether the policy may be welfare enhancing or not. In order
to do so, we will use the same Monte Carlo Simulation method used before. For the sake
of completeness, we have also computed the variance of relevant variables using Monte-Carlo
approximation of the expectancy operator; Var(Y ) =
X
j=1
T
t=1(Yt(j) − µj)2
/(T − 1) /X,
where Y is the objective variable and Y (j) is a generated sequence with mean µj. As before,
we set a time-horizon of T = 500 and 5000 iterations. The results are reported in table 6.
As can be seen, constraining labor is still not a welfare enhancing policy although the dif-
ferences are very minute. The minimum wage, on the other hand, is gaudily not welfare
enhancing. The reasons are probably related to the increased consumption’s volatility. In
fact, even is it reduces swings in capital stock, wage rigidity has very strong and direct
negative effects on the demand of labor. However, some caution should be taken when
evaluating these policies with this models, as we only consider one type of agent. With
heterogeneous agents, some more dependent on labor income that others, this policy may
be welfare enhancing for the intended sub-population. For instance, we have seen that the
policy is highly efficient as a volatility-reducing tool, reducing swings for both capital and
labor, and as a consequence for consumption as well.
30

Model Divisible Indivisible
Policy P.1 Free P.2 P.1 Free P.2
Exp. Utility -0.65873 -0.65868 -0.6657 -36.3877 -36.3859 -36.3903
St. Deviation
Capital 0.3430 0.3550 0.2393 0.4071 0.4656 0.3803
Labor 0.0016 0.0018 0.0039 0.0051 0.0077 0.0086
Consumption 0.0204 0.0205 0.0257 0.0243 0.0254 0.0310
Table 6: Comparison of Expected Lifetime Utilities
Further Possibilities
While the models presented in this paper are simple, they lend themselves to extensions
in a variety of direction. First and perhaps most obvious, is to allow for a more forward
looking production market where firms anticipate that their hiring policy in period t affects
their manoeuvre-space in t+1. This extension can be easily implemented by converting the
firm’s maximization problem to an infinite horizon problem. In this way, we would have to
find its value function and, assuming the Second Welfare Theorem applies, import its FOC
to the consumer problem. The complication arises when one has to consider how the choice
of t affects the value function in t + 1, which we have ignored in this paper since firms do
not have any foresight. But with such foresight, one must specify what the functional form
is and hence add another dimensionality to the problem. On the other hand, the optimal
solution when the constraint is binding would still be the minimum choice of labor, at which
point such added dimensionality adds little of value.
Another interesting extension that we have hinted at is to study how various types of
households benefit from job-security policies. Perhaps the most important up-shot of this
paper is that capital is fundamentally affected by labor-market policies (in our models) and
so one should consider the behavior of those who’s income is primarily dependent on capital.
Moreover, given that the policy’s intended target is the less-endowed, this would also add a
dimension with respect to them.
Extensions in the policy directions are as many as imagination can conjure, where some
are more sound and relevant than others. A fairly straightforward extension that we had as
ambition to include was that of a ”firing cost function” which could be thought of as a policy
induced added cost. However this problem is not trivial, as it is not at all obvious what
such a function should look like. Moreover, if it never admits profits, one must take great
care in modeling the product market, lest no firms enter. In the end the task swelled to a
proportion beyond a simple extension, but nevertheless it should be an interesting exercise
to study how such a policy fares in comparison to the two studied here.
31

Concluding Remarks
In this paper we have analyzed the impact two typical job-protection policies may have on
the economy. We have developed a simple model of general equilibrium assuming perfectly
rational agents in a perfectly competitive market. The results we have found are very
interesting and they open challenging new questions to explore with further study. The first
result we found is related to the central importance of labor in the economy. In fact, we found
that the minimum wage policy, through the constraint in the competitive definition of work
compensation, increased incredibly the volatility of labor. As a consequence, production
and consumption volatilities increase, with a negative impact on the agents’ life-time utility.
On the contrary, the labor-constraint policy, as it directly intervened on the labor variable
limiting sharp reduction, lead to higher employment levels, slightly lower compensations but
higher and more stable consumption over the various simulation periods. The policy revealed
not to be welfare enhancing, though, due to the higher level of work (which enters with a
negative sign into the utility function). However, this results open the door to a new series
of analysis which could consider different types of agents, with different characteristics and
utility functions, in order to verify if this kind of policy, which primarily aims to protecting
the weakest portion of population, may be really welfare enhancing for its target population
sub-segment.
References
Acemoglu, Daron (2009). Introduction to Modern Economic Growth. Princeton, N.J: Prince-
ton University Press.
Flodén, Martin (2008). “A note on the accuracy of Markov-chain approximations to highly
persistent AR(1) processes”. In: Economics Letters 99.3, pp. 516 –520.
Hansen, Gary D. (1985). “Indivisible labor and the business cycle”. In: Journal of Monetary
Economics 16.3, pp. 309–327.
McCandless, George (2008). The ABCs of RBCs. Cambridge, Massachusetts and London,
England: Harvard University Press.
Tauchen, George (1986). “Finite state markov-chain approximations to univariate and vector
autoregressions”. In: Economics Letters 20.2, pp. 177–181.
32

Appendix A
We begin with the problem in equation (IV). Define the value function as;
V (kt, t−1, At) ≡ max
{kt+1+s}∞
t=0∈R∞
+
{ t+s}∞
t=0∈L∞
t
E0
∞
s=0
βs
u(kt+s, kt+1+s, t+s, At+1|Θ) . (A.1)
Now, since the arguments of V (·) where arbitrarily chosen, this holds for any set of initial
values. Since for all t, kt ∈ R+, ˜t ∈ [0, 1], t ∈ [˜t, 1], both the control- and state space
are non-empty convex sets. As for the goal-function, these controls map the goal function
into the a convex set in R. It’s strict concavity and differentiability on R− ∪ R+ ensures
the existence of the maximum of V (·) on the ranges our parametrization give rise to. For
further details see Acemoglu (2009, Ch. 15). Now, to derive the value function in (V), fix
some time t and initial states kt, ˜t, and At. Then
V (kt, ˜t, At)At∈A ≡ max
{kt+1+s}∞
t=0∈R∞
+
{ t+s}∞
t=0∈L∞
t
E0
∞
s=0
βs
u(kt+s, kt+1+s, ht+s, At+s|θ) , At = Ai
= max
{kt+1+s}∞
t=0∈R∞
+
{ t+s}∞
t=0∈L∞
t
u(kt, kt+1, t, At)+
βE0
∞
s=0
βs
u(kt+1+s, kt+2+s, ht+1+s, At+1|θ)
= max
kt+1∈R+
t∈Lt
{u(kt, kt+1, t, At) + βV (kt+1, At+1|θ)}
= max
kt+1∈R+
t∈Lt



u(kt, kt+1, t, At) + β


j∈A
pi,jV (kt+1, Aj)





.
(A.2)
Since the initial states were arbitrary, this holds for all feasible pairs.
Appendix B
Consider some discrete interval of feasible capital stocks, K = {kmin, k2, k3, ..., kN−1, kmax}.
For each of these, we wish to find the optimal control pair of kt+1, t, conditional on initial
state Ai. While not necessary in principle, we restrict kt+1 ∈ (x−, x+
) for some lower and
upper limit to speed up the algorithm. Recall that t ∈ (˜t, 1) = t. The economy in (V)
has three state variables, kt, ˜t and At. Thus, we must solve the Value Function for each
possible combination. This is where the issue of dimensionality rapidly makes itself known
as the computing cost increases exponentially as the set of each state variable increases. As
33

for ˜, we construct a an ordered grid of h = {hmin
0 , h2
0, .., hs−1
0 , hmax
0 }, with h0 = ˜) being
possible lower bounds. Fix an hs
0 and make an initial guess v0
j (i) = K(i) for each kt = K(i)
and for each Ai ∈ A. Let the matrix of initial guesses per possible lower bound be V 0
j .
Given the initial guess, for each initial state i solve;
V 1
j = T(V 0
j ) = max
t∈(h0,1)
kt+1∈(x−,x+)
u(ki
t, kt+1, t, Aj) + β


q∈A
pi, qV 0
q

 , kt∈K
s∈h .
Repeat for each case hs
0 ∈ h. To complete the algorithm, define an approximation tolerance
ζ and iterate until |V s+1
j − V s
| < ζ for each case s ∈ h. In general;
V r+1
j = T(V r
j ) = max
t∈(hs
0,1)
kt+1∈(x−,x+)
u(ki
t, kt+1, t, Aj) + β


q∈A
pi,qV r
q

 ,
i∈K
j∈Nn
r∈N0
s∈h
. (B.1)
Which converge on the vector of fixed points as r → ∞.
Appendix C
Set initial state A0 = Ai. Then each Aj ∈ A occurs with probability pi,j by (1). These
probabilities partition the probability space Ω = [0, 1]. Thus, in a repeated draw from Ω
each Aj occurs with a relative frequency of pi,j. Ordering the ith row of Θ cumulatively, the
sum of
j
1 pi,j act as an upper bound on the measure of the occurrence of each Aj, while
the sum
j−1
1 pi,j act as the lower bound. Applying this schema for each j we partition
Ω according to the measure of the relative frequency of each Aj conditional on At = Ai.
Now, by drawing a random number (from the uniform distribution) and comparing it to the
partitioning, we can determine At+1;
Step 1: Determine state of At.
Step 2: Conditional on At = Ai, define the vector P = [pi,jj∈A
as the i-conditional tran-
sition matrix. Order P in the inverse cumulative order9
. Now define the partitioning by
letting ϑ(j) =
j
1 pi,j −
j−1
1 pi,j.
Step 3: Draw a random number p from the uniform distribution
Step 4: Iteratively apply the logical rule ”if p ≤ ϑ(j) then At+1 = Aj”. If not satisfied,
repeat with j = j + 1 until the rule is satisfied.
Step 5: Repeat steps 1-4 for the full sequence of time T.
9Working with the cumulative order is also possible, but the inverse cumulative order is simpler to
implement.
34

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