Market distortions and local indeterminacy:

a general approach

Teresa Lloyd-Braga

1

, Leonor Modesto

2∗

and Thomas Seegmuller

3

1

Universidade Católica Portuguesa (UCP-FCEE) and CEPR

2

Universidade Católica Portuguesa (UCP-FCEE) and IZA

3

Paris School of Economics and CNRS

May 13, 2009

Abstract

We provide a general methodology to study the role of market

distortions on local indeterminacy and bifurcations. We extend the

well-known Woodford (1986) model to account for market distortions,

introducing general speciﬁcations for three crucial functions: the real

interest rate, the real wage and the workers’ oﬀer curve. The elastic-

ities of these three functions play a key role on local dynamics and

allow us to identify which types of distortions are the most powerful

for indeterminacy.

Most of the speciﬁc market imperfections considered in the related

literature are particular cases of our general framework. Comparing

them we obtain several equivalence results in terms of indeterminacy

mechanisms. We also provide examples of distortions that illustrate

new results. Furthermore we show that, for an elasticity of substitu-

tion between inputs around unity, indeterminacy requires a minimal

degree of distortions. However, the degree of labor market distortions

compatible with that requirement is empirically plausible.

JEL classiﬁcation: C62, E32.

Keywords: Indeterminacy, endogenous ﬂuctuations, market imperfections,

externalities, imperfect competition, taxation.

∗

Corresponding Author: correspondence should be sent to Leonor Modesto, Univer-

sidade Católica Portuguesa, FCEE, Palma de Cima, 1649-023 Lisboa, Portugal. e-mail:

lrm@fcee.ucp.pt.

1

1 Introduction

In this paper we develop a methodology to study and fully characterize the

role of market distortions on the occurrence of local indeterminacy and bi-

furcations. Several papers have been studying the eﬀects of certain speciﬁc

market distortions (linked to externalities, imperfectly competitive markets,

or government intervention) on local dynamics,

1

but a systematic analysis

within a general uniﬁed framework, able to compare the importance of dif-

ferent types of distortions as a route to indeterminacy and bifurcations, is

still missing. In order to do that we introduce a general framework, able to

account for market distortions without specifying a priori their source, and

highlight the main channels through which indeterminacy occurs.

Although our methodology can be applied to any dynamic general equi-

librium model, the dynamic framework considered in this paper is based on

the perfectly competitive one sector model of a segmented asset economy of

Woodford (1986) and Grandmont et al. (1998).

2

Market distortions play a

role on the local stability properties of the steady state because they modify

the elasticities of three crucial functions that characterize our two dimen-

sional equilibrium dynamic system: the real interest rate, the real wage or

equivalently eﬀective consumption per unit of labor, and the generalized oﬀer

curve. We introduce general speciﬁcations for these elasticities that allow us

to recover most of the distortions on product, capital and labor markets, and

admit perfect competition as a particular case.

Focusing on not too weak values of the elasticity of capital-labor sub-

stitution,

3

we show that, in contrast to the perfectly competitive economy,

4

when there are market distortions, indeterminacy and bifurcations may oc-

cur in the presence of suﬃciently high capital-labor substitution and labor

supply elasticities. However, in some cases, indeterminacy is ruled out if the

individual labor supply elasticity becomes arbitrarily large, implying that,

by imposing an inﬁnitely elastic labor supply, one may obtain a wrong idea

of the dynamic implications of some distortions. We ﬁnd that distortions

aﬀecting the real interest rate do not play a major role. On the contray, even

(arbitrarily) small distortions on the oﬀer curve and/or eﬀective consumption

1

See the survey by Benhabib and Farmer (1999) and the bibliographic references in

Section 4.

2

This is a suitable framework for our purpose, since several papers have introduced

speciﬁc market distortons on product and factor markets in this model. These papers

provide examples to apply our general methodology (see Section 4).

3

Weak values of this elasticity are not empirically relevant (Hamermesh (1993), Duﬀy

and Papageorgiou (2000)).

4

Indeterminacy only occurs under perfect competition (Woodford (1986), Grandmont

et al. (1998)) for a weak capital-labor substitution.

2

promote the occurrence of indeterminacy. However, indeterminacy can only

prevail for values of the elasticity of capital-labor substitution around one

(those considered empirically plausible), under a minimal level of distortions

in the oﬀer curve and/or in eﬀective consumption. Furthermore, distortions

modifying the oﬀer curve aﬀect signiﬁcantly the emergence of indeterminacy,

leading to new results.

To clarify these ﬁndings, we apply our general method to examples of spe-

ciﬁc distortions on the product, capital and labor markets, focusing on two

types of results. On one hand, we obtain several equivalence results in terms

of local dynamics and indeterminacy mechanisms. We ﬁnd that labor and

consumption taxation with a balanced budget are equivalent to consumption

externalities, sharing the same indeterminacy mechanism. Product market

imperfections (due to mark-up variability and taste for variety) can be seen

as particular cases of the framework with positive externalities in produc-

tion, and unemployment beneﬁts with eﬃciency wages can be recovered as

a particular case of an economy where the desutility of labor is negatively

aﬀected by labor externalities. On the other hand, we also discuss the de-

grees of speciﬁc market distortions required for the occurrence of indetermi-

nacy under empirically plausible values (i.e. around one) of the elasticity of

capital-labor substitution. We conﬁrm, focusing on capital income taxation,

that distortions on the capital market do not, per se, promote indeterminacy.

Under output market distortions, indeterminacy may emerge, but requires

parameters conﬁgurations at odds with empirical evidence, i.e., too strong

positive externalities in production or high markups. On the contrary, under

labor market distortions (unions, eﬃciency wages, unemployment beneﬁts,

externalities in preferences), indeterminacy and bifurcations emerge for em-

pirically plausible distortions. Hence, labor market distortions are the most

relevant for indeterminacy.

The rest of the paper is organized as follows. We present our general

framework in Section 2, study the role of distortions on local dynamics in

Section 3, and apply our results to examples with speciﬁc market distortions

in Section 4. Section 5 provides concluding remarks. Proofs and technical

details are provided in the Appendix.

2 The model

Our framework extends the perfectly competitive Woodford model to take

into account market imperfections. To ease the presentation we begin with

a brief exposition of this model.

According to the perfectly competitive economy studied by Woodford

3

(1986) and Grandmont et al. (1998), in each period t ∈ N

∗

, a ﬁnal good is

produced under a constant returns to scale technology AF (K

t−1

, L

t

), where

A > 0 is a scaling parameter, F (K, L) is a strictly increasing function, con-

cave and homogeneous of degree one in capital, K > 0, and labor, L > 0.

From proﬁt maximization, the real interest rate ρ

t

and the real wage ω

t

are re-

spectively equal to the marginal productivities of capital and labor, i.e. ρ

t

=

AF

K

(K

t−1

, L

t

) ≡ Aρ(K

t−1

/L

t

) and ω

t

= AF

L

(K

t−1

, L

t

) ≡ Aω(K

t−1

/L

t

).

There are two types of inﬁnitely-lived consumers, workers and capitalists.

Both can save through two assets, money and productive capital. However,

capitalits are less impatient than workers and do not supply labor, whereas

workers face a ﬁnance constraint which prevents them from borrowing against

their wage earnings. Focusing on equilibria where the ﬁnance constraint is

binding and capital is the asset with the greatest return, it follows that

only workers hold money (they save all their wage income in money), and

capitalists hold the entire stock of capital. As in Woodford (1986), the be-

haviour of the representative worker can be summarized by the maximization

of U

C

w

t+1

/B

− V (L

t

) subject to the budget constraint P

t+1

C

t+1

= w

t

L

t

,

where P

t

is the price of the ﬁnal good and w

t

the nominal wage at period t,

C

w

t+1

≥ 0 the worker’s consumption at period t + 1, B > 0 a scaling para-

meter, V (L) the desutility of labor in L ∈ [0, L

∗

] and U(C

w

/B) the utility

of consumption.

5

The solution of this problem is given by the intertemporal

trade-oﬀ between future consumption and leisure:

ω

t+1

L

t+1

/B = γ(L

t

) (1)

where γ(L) is the usual oﬀer curve and C

w

t+1

= ω

t+1

L

t+1

at the monetary

equilibrium, with a ﬁxed constant amount of money in the economy.

The representative capitalist maximizes the log-linear lifetime utility func-

tion

∞

t=1

β

t

ln C

c

t

subject to the budget constraint C

c

t

+ K

t

= (1 − δ +

r

t

/P

t

)K

t−1

, where C

c

t

represents his consumption at period t, β ∈ (0, 1) his

subjective discount factor, r

t

the nominal interest rate and δ ∈ (0, 1) the

depreciation rate of capital. Solving the capitalist’s problem we obtain the

capital accumulation equation

K

t

= β [1 − δ + ρ

t

] K

t−1

(2)

A perfectly competitive intertemporal equilibrium is a sequence (K

t−1

, L

t

)

5

It is assumed that U

C

w

t+1

/B

is a continuous function of C

w

t+1

≥ 0, and C

r

, with

r high enough, U

′

> 0, U

′′

≤ 0 for C

w

t+1

> 0 , and −xU

′′

(x)/U

′

(x) < 1. Also, V (l) is a

continuous function for [0, L

∗

], and C

r

, with r high enough, V

′

> 0, V

′′

≥ 0 for (0, L

∗

).

We also assume that lim

L→L

∗

V

′

(L) = +∞, with L

∗

(the worker’s endowment) possibly

inﬁnite.

4

∈ R

2

++

, t = 1, 2, ..., ∞, given K

0

> 0, satisfying (1) and (2), where ω

t

=

Aω(K

t−1

/L

t

) and ρ

t

= Aρ(K

t−1

/L

t

).

6

We now present our general framework with market distortions, explain-

ing in a second step the main diﬀerences with respect to the perfectly com-

petitive case. We propose a more general equilibrium dynamic system, given

by (3)-(4) in Deﬁnition 1 below, where

t

represents the real interest rate rel-

evant to capitalists’ decisions, Γ

t

a generalized oﬀer curve, and Ω

t

L

t

eﬀective

consumption. In what follows, we denote by ε

X,y

the elasticity, evaluated at

the steady state, of the function X = {, Ω, Γ} with respect to the argument

y = {K, L}, while ε

γ

− 1 0 is the inverse of the elasticity of labor supply

of the representative worker with respect to labor, s ∈ (0, 1) the elasticity of

the production function with respect to capital, and σ > 0 is the elasticity

of capital-labor substitution of the representative ﬁrm, all evaluated at the

private level and at the steady state.

7

Deﬁnition 1 A perfect foresight intertemporal equilibrium of the economy

with market distortions is a sequence (K

t−1

, L

t

) ∈ R

2

++

, t = 1, 2, ..., ∞, that

for a given K

0

> 0 satisﬁes:

K

t

= β [1 − δ +

t

] K

t−1

(3)

(1/B)Ω

t+1

L

t+1

= Γ

t

(4)

where

t

≡ A(K

t−1

, L

t

), Ω

t

≡ AΩ(K

t−1

, L

t

) and Γ

t

≡ Γ(K

t−1

, L

t

). The

functions (K, L), Ω(K, L) and Γ(K, L) are positively valued and diﬀeren-

tiable as many times as needed for (K, L) ∈ R

2

++

, such that

ε

,K

= α

K,K

+

β

K,K

σ

−

1 − s

σ

, ε

,L

= α

K,L

+

β

K,L

σ

+

1 − s

σ

ε

Ω,K

= α

L,K

+

β

L,K

σ

+

s

σ

, ε

Ω,L

= α

L,L

+

β

L,L

σ

−

s

σ

ε

Γ,K

= α

Γ,K

+

β

Γ,K

σ

, ε

Γ,L

= α

Γ,L

+

β

Γ,L

σ

+ ε

γ

,

(5)

where α

i,j

∈ R and β

i,j

∈ R, for i = K, L, Γ and j = K, L, are parameters

independent of ε

γ

and σ

As under perfect competition, the dynamics of the economy with market

distortions are governed by a two dimensional system in capital and labor,

where the ﬁrst equation represents capital accumulation and the second one

6

See Grandmont et al. (1998) and Woodford (1986) for more details.

7

We consider the normalized steady state (K, L) = (1, 1) of the dynamic system (3)-(4),

whose existence is shown in Proposition 2 of Appendix 6.1.

5