An OLG Model with Stochastic Dynamics an

An OLG Model with Stochastic Dynamics and

Matching in the “Old Keynesian” Tradition

Marco Guerrazzi

∗

Department of Economics

University of Pisa

(Preliminary Draft: October, 2008)

Abstract

This paper proposes a dynamic 2-period OLG model inspired to the “Old Key-nesian” tradition. Specifically, following the microfoundation of theGeneral Theory

(1936) recently provided by Farmer (2006, 2007), we build a competitive search model in which nominal output and employment are driven by effective demand and prices are not sticky. In our theoretical proposal, old households consumption is financed by young households savings and investments are financed with realized profits. Moreover, we formalize the “animal spirits” hypothesis by assuming that the nominal expenditure in investments follows a stochastic process. Finally, cali-brating the model in order to match the first-moments of US data, we show that our framework can provide a rationale for the so-called Shimer’s (2005) puzzle, i.e., the relative stability of real wages in spite of the large volatility of labour market tightness.

JEL Classification: E12, E24, J63, J64

Keywords: OLG Model, Stochastic Dynamics, Old Keynesian Tradition, Demand-Constrained Equilibrium, Search Theory and Animal Spirits

∗_{Research Fellow, Department of Economics, University of Pisa, via F. Serafini n. 3, 56126 Pisa}

1

Introduction

In addressing some criticisms raised after the publication of the General Theory (1936), Keynes (1937) stated: “I’m more attached to the comparatively simple fundamental ideas which underlie my theory than to the particular forms in which I have embodied them, and I have no desire that the latter should be crystallized at the present stage of the debate. If the simple basic ideas can become familiar and acceptable, time and experience and the collaboration of a number of minds will discover the best way of expressing them”.

An influential evaluation of the circulation of the Keynesian legacy in the economic profession is given by Leijonhufvud (1966) inOn Keynesian Economics and the Economics of Keynes. By “Keynesian Economics” Leijonhufvud (1966) meant the interpretation of the General Theory (1936) incorporated into the IS-LM apparatus and more recently into the new-Keynesian paradigm. This kind of modeling has a Neo-classical core and deviations from the “natural” rate of unemployment are assumed to be the (optimal) reaction to different kind of rigidities.

In this paper we take a different perspective. Following the microfoundation of the

General Theory (1936) recently provided by Farmer (2006, 2007), we build a 2-period OLG competitive search model in which nominal output and employment are driven by effective demand and prices are not sticky. In our theoretical proposal, old households consumption is financed by young households savings and investments are financed with realized profits. Moreover, we formalize the “animal spirits” hypothesis by assuming that the nominal expenditure in investments follows a stochastic AR(1) process.

Finally, calibrating the model in order to match the first-moments of the US data, we show that our framework can provide a rationale for the so-called Shimer’s (2005) puzzle,

i.e., the relative stability of the real wage in spite of the large volatility of labour market tightness. In fact, this striking feature of the US business cycle can hardly be explained by using the standard Pissarides’s (2000) matching model.

The paper is arranged as follows. Section 2 describes the model. Section 3 provides some numerical simulations. Finally, section 4 concludes.

2

The Model

their labour services by tacking as given labour demand. Thereafter, young households decide their contribution to the pension scheme according to their inter-temporal prefer-ences.

On the other side, firms produce a perishable-homogeneous good by mean of the labour supplied by the young households and a fixed capital factor that accumulates according to the usual dynamic law. Labour demand is derived from the classical assumption of profit maximization. By contrast, in each firm, nominal expenditure in investments is assumed to follow a stochastic AR(1) process that formalizes the Keynesian “animal spirits” hypothesis. Finally, that nominal expenditure is financed by the firm’s profits. Therefore, the profit rate implicitly defines the real interest rate.

In our framework, the matching technology combines searching worker and recruiters and enters the model as an externality because we assume that moral hazard factors prevents the creation of markets for the search inputs. Therefore, even if agents are price-takers, the equilibrium allocation is not - in general - Pareto optimal.

Finally, we show how to tune an optimal fiscal policy that implements the social optimum level of employment.

2.1

The Households’ Sector

For sake of simplicity, we assume that for each generation there is large number of identical households. Thereafter, the representative household chooses current (ct) and future (ct+1)

consumption by solving the following problem:

max

ct,ct+1

cβ_tc1−β

t+1 0< β <1

s.to

(1)

ptct+st=wtLt (2)

pt+1ct+1 =st

1 +ie_t+1 (3)

where pt is the current price level, st is current saving, wt is the current nominal wage,

Lt is current employment, pt+1 is the next period price level and iet+1 is the (expected)

nominal return on the inter-generational transfer.

In each period, the young household is endowed with a single unit of time Ht that it

devotes inelastically to the search activity. Of the searching workers, Lt are the ones that

Lt+Ut=Ht = 1 for all t (4)

Current employment and labour supply are linked by the following expression:

Lt =htHt (5)

where the hiring effectiveness htis taken as given by the individual household. Obviously,

its expression is given by

ht =

Lt

Ht

=Lt (6)

where Lt is aggregate employment while Ht is aggregate labour supply.

Notice that (6) conveys a typical “thick market” externality, i.e., the higher the ag-gregate employment rate, the easier to find a job. See, for example, Diamond (1982).

Solving (3) for st and substituting in (2), we derive an inter-temporal budget

con-straint. Hence,

ct+

ct+1

1 +re t+1

= wt

pt

Lt (7)

where re

t+1 is the (expected) real return on the inter-generational transfer.

The maximization of (1) subject to (7) leads to the following solutions:

ct =β

wt

pt

Lt and ct+1 = (1−β)

wt+1

pt+1

Lt+1 (8)

Notice that the inter-temporal preferences in (1) deliver a constant marginal propensity to consume lower than one.

2.2

The Productive Sector

For symmetry with the households’ sector, we assume that in the economy there is a large number of identical firms. Thereafter, in each period, the representative firm produces an homogeneous perishable good (Yt) by mean of the following Cobb-Douglas production

function:

Yt =AtKtαX1− α

t 0< α <1 (9)

where At is a productivity shock,Kt is the current stock of capital and Xtis the fraction

The log of productivity shock is assumed to follow a stochastic AR(1) process. Hence,

lnAt= lnµ+ξlnAt−1+ηt (10)

where µ is a positive constant (drift), ξ measures the persistence of productivity shocks and ηt∼N

0, σ_η2 is a stochastic disturbance.

Given (9), the representative firm solves the following problem:

max

whereVtis the fraction of workers devoted to recruiting andvtis the exogenous recruiting

efficiency.

Current employment and the fraction of workers devoted to recruiting are linked by the following expression:

Lt =vtVt (13)

where the recruiting efficiency vt is taken as given by the individual firm1. Obviously, its

expression is given by

vt=

Lt

Vt

(14)

where Vt is the fraction of workers allocated to recruiting by all firms.

Given (12) and (13) the problem of the firm simplifies to

max

Notice that vt acts as an externality in the firm’s problem. In other words, the firm

chooses employment by taking the recruitment efficiency as given. The FOC for Lt is the following:

(1−α)ptYt

Lt

=wt (16)

1

In other words, the firm knows that Vtrecruiters can hirevtVt workers of whom Xtwill be used in

In each period, the level of investments is not derived from (rational) optimization. Following Farmer (2006), the Keynesian “animal spirits” hypothesis is formalized by as-suming that the nominal expenditure in investments (It) follows an exogenous stochastic

AR(1) process2_{. Hence,}

It=κ+ρIt−1+ǫt (17)

where κis a positive constant (drift), ρmeasures the persistence of the exogenous invest-ment sequences and ǫt∼N(0, σǫ2) is a stochastic innovation to beliefs3.

The capital stock evolves according to the following dynamic law:

Kt+1 =

It

pt

+ (1−δ)Kt (18)

where δ∈(0,1) is the depreciation rate of capital.

In the remainder of the paper, we will assume that the stock of capital is the only state variable of the model. In other words, we will assume that at the end of each period all the labour force is fired and (possibly) re-hired at the beginning of the next.

2.3

Matching

Now we describe how searching workers find jobs in the economy as a whole. Specifically, the aggregate searching technology is described by the following constant returns-to-scale matching function:

Lt=H γ tV

1−γ

t 0< γ <1 (19)

The expression in (19) suggests that aggregate employment is the result of the match-ing between searchmatch-ing unemployed workers and recruiters employed by the firms. There-fore, in contrast to the matching framework developed by Pissarides (2000), we are as-suming that vacancies are posted by using labour instead of output.

Given (4), (19) simplifies to

Lt=V

1−γ

t (20)

2

In the analysis that follows, nothing would be changed by assuming that a share of It is consumed

by entrepreneurs.

3

The expression in (20) describes how aggregate employment is related to the aggregate recruiting effort of all the firms. Given (4), a simple manipulation of (20) allows also to derive a decreasing relationship between aggregate recruiters and unemployed workers which provides a version of the well-known Beveridge curve. Specifically,

Ut = 1−V

1−γ

t (21)

where Ut is the aggregate unemployment rate. A graphical outlook is given in figure 1.

1

1 t

V

t

U

Figure 1: The Beveridge Curve

Notice that our version of the Beveridge curve crosses the horizontal (vertical) axis when all the workers are allocated to production (employed).

If we consider a situation of “symmetric” equilibrium, i.e, a situation in which Lt =

Lt and Vt = Vt, it becomes possible to express the recruiting efficiency as function of

aggregate employment. Specifically,

vt=L

−1−γγ

t (22)

In contrast to (6), (22) conveys a “thin market” externality, i.e., the recruiting ef-ficiency of the firm is higher (lower), the lower (higher) is aggregate employment. See, Diamond (1982).

2.4

The Social Optimum

simultaneously the production and the matching technologies by internalizing the exter-nality in (22). Given the formal structure of the model, in each period the Pareto-optimal allocation is defined by the level of the employment that maximizes the sum of the real wage bill and realized profits in real terms, i.e, the level of Lt that maximizes the real

amount of production for given levels of the productivity shock and the stock of capital4_.

Hence,

where πt are profits in real terms.

The FOC for the social planner problem is given by

(1−α)Yt

Therefore, the social-optimal level of employment is simply the following:

LS = (1−γ) 1−γ

γ for all t (26)

Notice thatLS depends only on the parameter of the matching technology. Obviously,

US = 1−LS provides the social-optimal unemployment rate. A graphical outlook is given

in figure 2.

Figure 2 allows to clarify some features of the production and matching technolo-gies. Obviously, whenever Lt = 0 there is no production because no worker is employed.

However, there is no production even when Lt = 1. In fact, in this case, the aggregate

recruiting efficiency is equal to one. See eq. (22). Therefore, it will be optimal to allo-cate all the employees to recruiting and no worker is alloallo-cated to production activities. Total output is at its maximum level whenever Lt = LS. In fact, any additional

em-ployed worker would not produce additional output but he would be simply emem-ployed in recruiting additional recruiters without improving the resulting allocation.

4

Yt

Lt

1 YS

(

)

γ

γ γ

−

−

1

1

Figure 2: The social optimum

2.5

Aggregate Demand and Supply

Given the national account identity, aggregate demand in nominal terms (ADt) is given

by the sum of aggregate nominal consumption (Ct) and nominal investment expenditure

(It). Hence,

ADt=Ct+It (27)

Aggregate nominal consumption is given by the sum of the oldcO t

and young cY t

household’s consumption evaluated at the current price (pt). Hence,

Ct=pt

cO_t +cY_t =wtLt (28)

Normalizing the nominal wage rate to unity (wt= 1, all t), aggregate demand in

nom-inal terms becomes

ADt=Lt+It (29)

ASt=

1

1−αLt (30)

The equilibrium condition for the good market,i.e,ADt=ASt, provides the following

solutions for the nominal output and the level of employment:

p∗

tYt∗ =

1

αIt (31)

L∗

t =

1−α

α It (32)

Obviously,U∗

t = 1−L∗t provides the corresponding rate of (actual) unemployment. A

graphical outlook is given in figure 3.

ptYt

LS 1

YS

t

I

α

α

−

1

_Lt

It

ADt

ASt

Figure 3: Aggegate demand and supply

Given the erratic nature of the nominal investment expenditure, there is no guarantee that the realized employment level coincides with the social optimum stated in (26). For example, the graph in figure 3 depicts a situation in which actual employment is lower than the social-optimal level so that the economy is experiencing inefficient unemployment. Obviously, the model is also able to capture situations of over-employment in which

Lt > LS. In this case, a reduction of employment would be Pareto-improving because it

Moreover, (4) and (32) impose a precise upper bound for the stochastic process followed by the nominal investment expenditure5_{. Specifically,}

0It<

α

1−α for all t (33)

Finally, notice that in our demand-driven search model the equivalent of the Hosios’s (1990) condition operates as a constraint on the realization of It. Specifically, whenever

It= ₁₋α_αLS the resulting allocation is Pareto-optimal.

2.6

Demand Constrained Equilibrium

Now we can provide a formal definition of equilibrium based on the building blocks de-scribed above. Following Farmer (2006, 2007), we use the term “Demand Constrained Equilibrium” (DCE) in order to describe a competitive search model closed by a balance material condition.

Definition 1 (Demand Constrained Equilibrium) For each It∈ [0,₁α

−α), At>0, Kt>0

a symmetric DCE is given by

(i) a price level p∗

t

(ii) a production plan {Y∗

t , Vt∗, Xt∗, L∗t, Ut∗}

(iii) a consumption allocation cOt, cYt

(iv) a pair {h∗

t, v∗t}

with the following properties:

• Feasibility:

Y∗

t =At(Kt)α(Xt∗)

1−α ₍₃₄₎

L∗

t =Xt∗+Vt∗ (35)

cO_t +cY_t + It

p∗

t

=Y∗

t (36)

U∗

t = 1−L∗t (37)

5

• Consistency with the optimal choices of households:

• Consistency with the optimal choices of firms:

(1−α)Y

• Search market equilibrium:

h∗

Notice that in a DCE all the variables are expressed in money wage units,i.e.,w∗

t = 1

for all t, so that _p1∗

t represents the real wage rate. Moreover, notice that It→lim1−αα

Y∗

A graphical description of a DCE is given in figure 4.

t

Figure 4: A demand constrained equilibrium

2.7

Stochastic Dynamics

Given (18), (32), (35), (41) and (44), the dynamics of our model economy is given by the following stochastic first-order difference equation:

Kt+1 =αAtKtα

t

K

1 +

t

K

Kˆ

°

45

Figure 5: The phase line

The stochastic steady-state solution is given by

K =α

δAt

1 1−α

1−α

α It 1−

1−α

α It

γ 1−γ

(46)

In our model the realizations of the stochastic processes that summarize, respectively, the productivity shocks and the “animal spirits” hypothesis contribute to determine the steady-state equilibrium6_.

2.8

Optimal Fiscal Policies

A DCE can be characterized by any level of employment in the closed interval [0,1]. However, we shown that a omniscient-benevolent social planner who can operate simulta-neously the production and the matching technologies would always choose the level LS.

Therefore, unless the realization of the stochastic process that describes the “animal spir-its” of investors is consistent with the social planner solution, a DCE will be alternatively characterized by over or under employment.

In order to provide a remedy for those sub-optimal situations, we augment the model with a public sector responsive for fiscal policies. Specifically, we assume that the young household nominal income is taxed at the proportional tax rate τt and subsidized with

a lump-sum transfer Tt. In this case, having in mind the normalization of the nominal

6

wage rate (wt= 1, all t), the aggregate demand in nominal terms becomes equal to the

following expression:

ADt= (1−β)Lt+β((1−τt)Lt+Tt) +It (47)

The existence of a public sector can raise some problematic issues. On the one hand, government spending leads to the creation of public goods. In the other hand, deficit spending raises the issue of the discharge of the public debt. In order to avoid this problems, we focus only on balanced budged fiscal policies. Therefore, we consider policies in which the tax rate and the lump-sum transfer are linked in the following way:

Tt=τt

Lt

1−α (48)

If we assume that the public authority that designs the fiscal policy knows the optimal level of employment in (26) and is able to observe the actual realization of It, it becomes

possible to derive a pair τS t , TtS

that implements the social-optimal level of employment with a balanced budget fiscal policy. Specifically, solving (48) for τt and substituting in

(47), provides and expression of aggregate demand in which the level of the lump-sum transfer is consistent with the balanced budget constraint. In this case, the equilibrium on the market for goods is given by

αβTt+It=

α

1−αLt (49)

Substituting (26) in (49) allows to derive the social-optimal level of the lump-sum transfer. Hence,

T_tS = αLS −It(1−α)

αβ(1−α) (50)

Finally, the social-optimal tax rate is given by

τ_tS = (1−α)T

S t

LS

(51)

The optimal fiscal policy operates as follows. Whenever the actual realization of It is

3

Some Computational Experiments: A Rationale

for the Shimer’s (2005) Puzzle

In recent years, a lot of computational efforts have been devoted to the explanation of the cyclical behavior of equilibrium unemployment and vacancies. Among the others, there is a very influential paper by Shimer (2005) that questions the predictive power of the standard Pissarides’s (2000) matching model by arguing that this framework cannot generate the observed business-cycle-frequency fluctuations in unemployment and job vacancies in response to shocks of plausible magnitude. In other words, the model would lack for an amplification mechanism for productivity shocks.

Using quarterly data from different sources, Shimer (2005) measures, inter-alia, the autocorrelation and the volatility of unemployment, job vacancies and real wages for the US economy in the period from 1951 to 2003. Some of his empirical findings are summarized in table 1.

lnU ln (vacancies) lnvacancies_U ln1_p Standard deviation 0.190 0.202 0.382 0.020 Quarterly autocorrelation 0.936 0.940 0.941 0.878 lnU 1 −0.894 −0.971 −0.408 Correlation matrix ln (vacancies) − 1 0.975 0.364

lnvacancies

U

− − 1 0.396

ln1

p − − − 1

Table 1: The Shimer’s (2005) puzzle

One of the most striking finding emphasized by Shimer (2005) is that the standard deviation of the vacancy-unemployment ratio is almost 20 times as large as the standard deviation of real wage rates over the period under examination. The so-called Shimer’s (2005) puzzle arises from the fact that the standard Pissarides’s (2000) matching model predicts that this two variables should have nearly the same volatility.

In the reminder of this section, we provide some computational experiments aimed at checking whether the model economy described in section 2 is able to provide a rationale for this puzzle, i.e., the relative stability of real wage rates in spite of the large volatility of labour market tightness7_{. Since in our competitive search model labour instead of output}

7

Our numerical experiments are carried out assuming that C _andI _{include goverment, net transfers}

is used to post job vacancies, our indicator of unfilled jobs will be given by the rate of recruiters (V) employed by the firms, while the labour market tightness will be given by the ratio recruiters-unemployment V

U

. Moreover, for the sake of comparability, the volatility of artificial series obtained with our model economy will be computed with same numerical procedure proposed by Shimer (2005). A short description of this procedure is given in Appendix.

3.1

The Calibration of the Model

Our model economy is calibrated in order to match the first-moments of US data. Specif-ically, we use the set of parameter values in table 2.

Parameter Description Value

α capital share 0.3

δ capital depreciation 0.025

γ matching elasticity of labour supply 0.9862

µ productivity drift 1

ξ productivity persistence 0.95

ση productivity variance 0.0076

κ investment drift 0.2018

ρ investment persistence 0.5

σǫ investment variance 0.0030

Table 2: The calibration of the model

The parameter values collected in table 2 comes from different sources. First, the values of α and δ are calibrated as in Kidland and Prescott (1982). Second, the value of

ρ is the one provided by Farmer (2006) for the US economy. Finally, the values of µ, ξ

and ση are calibrated as in Chang (2000).

The values of the other parameters have been chosen in the following way. First, the value ofγ delivers a social-optimum unemployment rate consistent with the historical US unemployment rate8_{. Second, the value ofκcombined withρdelivers an expected value of}

nominal investment expenditure consistent with the social-optimum unemployment rate9_.

Finally, the value of σǫ is consistent with the restriction on the It-process in (33).

8

Specifically, we computed the historical mean of the seasonally adjusted US unemploymet rate for the people of 16 years old and over provided by the Bureau of Labor Statistics (BLS). Series Id: LNS14000000.

9

3.2

Simulation Results

We start our computational experiments by allowing only for shocks to aggregate demand. In other words, using the set of parameters in table 2, we apply the Shimer’s (2005) procedure to the artificial series generated by our model economy under the assumption that it is hit only by demand shocks, i.e, At=At−1, for all t and ση = 0. The results are collected in table 3 (standard errors in parentheses).

lnU lnV lnV

U ln

1

p

Standard deviation 0.0476

(0.0031) 0(0..13700087) 0(0..50180477) 0(0.0143.0017)

Quarterly autocorrelation 0.9987

(0.0000) 0(0..98990014) 0(0..89510150) 0(0.9998.0000)

lnU 1 −0.9949

(0.0012) −(00..0049)9927 −(00..0091)9748

Correlation matrix lnV − 1 0.9957

(0.0044) 0(0.9503.0159)

lnV

U − − 1 0₍₀.9600_.0130)

ln1_p − − − 1

Table 3: Demand shocks

The numerical results for a model economy driven only by “animal spirits” suggest some interesting conclusions. First, our simple model is able to reproduce the ranking of the actual standard deviations of all the variables: labour market tightness (real wage) is the most (less) volatile series. Second, the model catches the actual sign of the correlation coefficients of all the variables. Specifically, notice the negative correlation between un-employment and recruiters (Beveridge curve). Third, the model with only demand shocks overstate real wage stickiness: the standard deviation of the labour market tightness is about 35 times the standard deviation of the real wage rate. Finally, the model tends to understate the volatility of unemployment and to overstate the autocorrelation of all the variables.

lnU lnV lnV

U ln

1

p

Standard deviation 0.0476

(0.0031) 0(0..13690087) 0(0..50120479) 0(0.0259.0030)

Quarterly autocorrelation 0.9987

(0.0000) 0(0..98990014) 0(0..89520149) 0(0.9998.0000)

lnU 1 −0.9950

(0.0012) −(00..0044)9928 −(00..0896)5386

Correlation matrix lnV − 1 0.9958

(0.0040) 0(0.5248.0881)

lnV

U − − 1 0₍₀.5303_.0888)

ln_p1 − − − 1

Table 4: Demand and Supply shocks

The simulations results for a model economy hit simultaneously by demand and supply shocks are straightforward. In our framework, productivity shocks affect only the volatility of the real wage rate10_{. Therefore, the value of the standard deviation of the real wage}

becomes larger and closer to empirical evidence. Taking a significance level of 95%, we do not reject the hypothesis that the simulated value is equal to the actual value. Moreover, notice that the ratio between the volatility of labour market tightness and real wage rates becomes very close to the value found by Shimer (2005).

4

Concluding Remarks

This paper aimed at providing a dynamic 2-period OLG model with an “Old Keynesian” flavour. Specifically, following the microfoundation of theGeneral Theory (1936) recently provided by Farmer (2006, 2007), we built a competitive search model in which nominal output and employment are driven by effective demand and prices are not sticky.

Finally, we shown that this simple competitive search model can provide a rationale for the so-called Shimer’s (2005) puzzle. In fact, calibrating the model in order to match the first-moments of US data, we shown that our framework can explain the relative stability of real wage rates in spite of the large volatility of labour market tightness.

10

5

Appendixes

5.1

The Shimer’s (2005) Procedure

The numerical procedure proposed by Shimer (2005) to measure the volatility of un-employment, vacancies, labour market tightness and real wages runs as follows. First, generate 1,212 observations for each of the involved variables, i.e., unemployment, re-cruiters, labour-market tightness and real wage rates. Second, throw away the first 1,000 observations for each simulated series in order to have 212 data points which correspond to the data for the 53 quarters from 1951 to 2003. Third, take the log of each series and compute their autocorrelation. Forth, detrend the log of the model-generated data by using the Hodrick-Prescott (HP) filter with a smoothing parameter set to 100,000. Fifth, compute the deviations of all the artificial series from their respective HP trend and take the respective standard errors and correlation. Sixth, repeat the previous steps for 10,000 times. Finally, take the mean and the standard deviation of all the variables obtained in the third and the fifth step. The MAT LAB 6.5 code that applies this procedure to the theoretical model developed in section 2 is available from the author upon request.

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