Market distortions and local indeterminacy: A general approach

TL;DR: In this paper, the role of market distortions on the emergence of indeterminacy and bifurcations is investigated, and it is shown that distortions in the capital market do not play a major role.

About: This article is published in Journal of Economic Theory.The article was published on 2014-05-01 and is currently open access. It has received 7 citations till now. The article focuses on the topics: Indeterminacy (literature) & Elasticity of substitution.

Summary

1 Introduction

• In this paper the authors develop a methodology to study and fully characterize the role of market distortions on the occurrence of local indeterminacy and bifurcations.
• These papers provide examples to apply their general methodology (see Section 4).
• 3Weak values of this elasticity are not empirically relevant (Hamermesh (1993), Duffy and Papageorgiou (2000)).
• On the other hand, the authors also discuss the degrees of specific market distortions required for the occurrence of indeterminacy under empirically plausible values (i.e. around one) of the elasticity of capital-labor substitution.
• The authors confirm, focusing on capital income taxation, that distortions on the capital market do not, per se, promote indeterminacy.

2 The model

• The authors framework extends the perfectly competitive Woodford model to take into account market imperfections.
• In the cases of productive externalities, imperfect competition in the product market or with consumption, labor or capital taxation, the real interest rate and/or the real wage relevant to consumers’ decisions are no longer equal to the marginal productivities of capital and labor at the firm level.
• Empirical studies also show that the wage bill is increasing in labor.
• Hence, the occurrence of indeterminacy and bifurcations in their framework is due to the existence of market distortions, mainly through their effects on αi,j, which are more relevant than βi,j when inputs are not weak substitutes in production.

3 The role of market distortions on local dy-

• The authors then analyze the occurrence of indeterminacy and bifurcations by studying how T and D evolve in the space (T,D) as some relevant parameters of the model are made to vary continuously in their admissible range, according to the geometrical method developed in Grandmont et al. (1998).
• To locate these values in the plane (T,D), three lines are relevant .
• The first condition, necessary for indeterminacy when D′1 (σ) < 0, can only be met with distortions.
• Take εγ ∈ [1,∞) as the bifurcation parameter with Hopf, flip and transcritical bifurcation values εγH εγF and εγT , respectively given in (23),(24) and (25).18 (a) Consider that αKK takes either nonpositive or positive values.
• 18Assumptions 3 and 4 are satisfied in all the examples considered in the related literature, and simplify considerably their analysis.

3.1 Discussion of the results

• By inspection of Tables 1 and 2, the authors see that, when capital and labor are sufficiently substitutable in production, indeterminacy and bifurcations occur in the presence of market distortions.
• 20 Indeterminacy requires a critical lower bound on the elasticity of substitution between capital and labor (σ), which may be different across configurations.
• ΣH1 is no longer a necessary condition for indeterminacy, indeterminacy also requires a lower bound for σ.
• Indeterminacy also requires a critical upper bound (either ǫγH or ǫγT ) on ǫγ, depending on the configuration considered.
• This is a new interesting result that will be illustrated and further discussed in the economic examples provided in Section 4.

3.1.1 Distortions on the ̺ and Ω functions

• The authors start with the case where distortions affecting only the ̺ and Ω functions are present (αΓ,i = βΓ,i = 0), as it happens, for example, when they only have product or capital market distortions (see Section 4).
• Then, only configurations (i)− (iv) can be obtained (see (19)), where, as seen above, σ > σH1 is a necessary condition for indeterminacy.
• Let us now discuss in detail the role of distortions affecting the ̺ function.
• For αK,K > 0, given the necessary condition αL,L > θαK,K, indeterminacy requires a positive value for αL,L.

3.1.2 Distortions on the Ω and Γ functions

• The authors consider now distortions that only affect both Ω and Γ, i.e. the intertemporal trade-off condition of workers, as it happens, for instance, when they have labor market imperfections (see Section 4).
• These last two configurations require that αΓL < α ∗ ΓL as shown in (21).
• Thus, indeterminacy is possible when the only distortion is an arbitrarily small negative αΓ,L.
• While general distortions on the real interest rate do not seem to play a major role on the occurrence of indeterminacy, distortions on the generalized offer curve and on the effective consumption seem to help the occurrence of indeterminacy.
• This would require infinitely large elasticities of private labor supply and of substitution between inputs.

4 Applications

• The authors now proceed by applying their general methodology and results to several examples that provide microeconomic foundations for the model developed above.
• The strategy used to analyze each example is the following.
• The authors also discuss the minimal degree of distortions required for indeterminacy to occur with an empirically plausible value of the elasticity of capital-labor substitution (around 1).
• The authors will see in particular that indeterminacy emerges under new configurations (those of Table 1 and Table 2), and that for some sets of distortions’ parameters indeterminacy requires a value for ǫγ bounded away from 1.

4.1 Examples with the same distortion on the real in-

• In the examples presented below, the same distortion affects both the real interest rate and the real wage, but the generalized offer curve coincides with the competitive one, Γ(K,L) = γ(L).
• 28They are usually justified by learning by doing or matching problems on labor market.
• Hence negative externalities illustrate the result, discussed in Section 3, that indeterminacy cannot occur when αK,i and αL,i are all negative.
• The authors will now emphasize that, for the analysis of local dynamics, many models with imperfect competition on the product market are, in fact, a particular case of the previous framework with positive productive externalities.
• The same happens when imperfect competition is associated with markup variability or with taste for variety, as the authors now show.

4.2 Examples with different distortions on the real in-

• This will allow us to emphasize that capital market distortions are not, per se, the most relevant ones for indeterminacy.
• The authors start with the case of capital taxation without public spending externalities in preferences, so that market distortions only appear in the function ̺(K,L).
• 38Notice that this implies that indeterminacy only occurs when υ > 0, i.e., consumption externalities are of the "keeping-up with the Joneses" type.
• Indeed, when there is only capital income taxation, the steady state is always a saddle.

4.3 Examples with distortions only on the generalized

• In the examples presented in this section distortions only affect the generalized offer curve.
• The authors can easily see that Assumptions 1 and 4 are ensured.
• This shows that it should be misleading to focus only on the case of an infinitely elastic labor supply when the authors want to fully study the role of some labor market distortions on the occurrence of indeterminacy.
• Therefore, this last model can be seen as a particular case of not too negative aggregate labor externalities in leisure utility.
• 44Each worker supplies one unit of labor with a labor desutility that depends on the level of effort.

4.4 Examples with distortions on the generalized offer

• L) and Ω(K,L) are affected by market distortions, while the capital market remains perfectly competitive.the authors.
• This case is new, and is able to illustrate most of the dynamic results exhibited in the general framework.
• Unions are able to set wages above a reservation wage, with a markup factor µ(K,L) = 1−αs(K/L) 1−s(K/L) ≥ 1, increasing in the bargaining power of unions (1−α) ∈ [0, 1).47 Employment is determined by the equality between the reservation wage and the marginal productivity of labor.
• 46See Grandmont (2008) for a more detailed discussion.
• Assuming that government expenditures (Gt) provide services that affect not only workers’ utility for consumption, but also their desutility of labor, the authors will be able to provide an illustration of configuration (vi).

5 Concluding remarks

• With their general analysis of the role of market distortions on local dynamics and the different examples of specific distortions presented above, the authors were able to they emphasize several interesting results, some of them already latent in previous works, but which are here confirmed, generalized and highlighted.
• 58 further research on this issue is welcome.
• Second, although the authors only discuss local deterministic indeterminacy and cycles, they may be able to construct stochastic sunspot cycles along indeterminacy and bifurcations 56Some recent works confirm the importance of labor market imperfections in explaining real business cycles data.
• If the authors estimate the relevant parameters of their general formulation, they will not be able to identify a particular source of specific distortions among those, which belonging to the same class, are observationally equivalent.

6.3 Local stability properties

• The authors main interest is to understand how the existence of market imperfections change the characterization of local stability properties in terms of the elasticity of substitution between capital and labor, σ, and of the elasticity of the private offer curve, εγ, while keeping s and θ constant at values satisfying Assumption 1.1.
• Most of the distortions considered in the literature satisfy this condition.

6.4.1 Derivation of Proposition 1

• And the half-line ∆1, that starts on (AC) and points upwards to the left, crosses (AB) above point B. See Figure 4. 61This last case does not appear if σH2 does not exist.
• With the help of geometrical arguments the authors can see that when σH2 > s−βLL 1+αLL exists, then σH2 > σH3. 69 However σH2 may be higher or lower than σF .
• To simplify the exposition the authors only present in Table 2 the results for this configuration assuming that σH2 > σF . 70 Configuration (vi) (S1 ∈ (SD, 1)) 71 Define σS2 as the critical value of σ such that the half line ∆ goes through point A.72.
• All these results are summarized in Table 2.

6.8 Existence of σH2

• To discuss the existence and uniqueness of σH2 , the authors consider first the configurations where S1 ∈ (0, 1), and then the remaining ones.
• Note that the equation ǫγH = ǫγT is a polynomial of degree 2, i.e. has at most two solutions.
• Note that in this particular case, ǫγT does not depend on σ.

Figures

Table 2: Local stability properties and bifurcations

Table 1: Local stability properties and bifurcations

Frequently asked questions

Q1. What have the authors contributed in "Market distortions and local indeterminacy: a general approach" ?

The authors provide a general methodology to study the role of market distortions on local indeterminacy and bifurcations. 

Q2. What are the future works in "Market distortions and local indeterminacy: a general approach" ?

Empirical analysis on this issue is therefore an important direction for further research. 56 A possible explanation for these results may be linked to the fact that future expectations, which open the room for fluctuations driven by selffulfilling expectations, only affect the current decisions of consumers/workers, thus rendering distortions that affect the intertemporal trade-off of consumers/workers more important than those affecting the capital accumulation equation. Although some works have already considered some of these aspects,58 further research on this issue is welcome. Second, although the authors only discuss local deterministic indeterminacy and cycles, they may be able to construct stochastic sunspot cycles along indeterminacy and bifurcations 56Some recent works confirm the importance of labor market imperfections in explaining real business cycles data. 

Q3. What is the reason for the indeterminacy and bifurcations in their framework?

the occurrence of indeterminacy and bifurcations in their framework is due to the existence of market distortions, mainly through their effects on αi,j, which are more relevant than βi,j when inputs are not weak substitutes in production. 

Q4. What is the role of market distortions on the local stability properties of the steady state?

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 their two dimensional equilibrium dynamic system: the real interest rate, the real wage or equivalently effective consumption per unit of labor, and the generalized offer curve. 

Q5. In what configurations does indeterminacy emerge for > h1?

In configurations (ii).1 and (iii).1 indeterminacy emerges for σ > σH1 = s−(1−s−ψ)(η−µ)−θ(1−s)(1+µ)ψ(η−µ) , provided εγis sufficiently small. 

Q6. How do the authors obtain the local dynamics of the system?

By loglinearizing the system (3)-(4) around the normalized steady state,we obtain the local dynamics for K̂t = (Kt −K) /K and L̂t+1 = (Lt+1 − L) /L given by following equations:[ K̂t L̂t+1 ] = [ (1 + θε̺,K) θε̺,L ǫΓ,K−εΩ,K(1+θε̺,K)1+εΩ,LεΓ,L−θεΩ,Kε̺,L 1+εΩ,L ][ K̂t−1 L̂t ] ≡ [J ] [ Kt−1−K K Lt−L L ](6) Market distortions influence the local dynamics of the model, relatively to the perfectly competititive case, by modifying the elasticities εΩ,i, ε̺,i and εΓ,i. 

Q7. What is the methodology used in this paper?

Although their methodology can be applied to any dynamic general equilibrium 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). 

Q8. What is the significance of bifurcations in the case of a steady state?

Note also that bifurcations are quite relevant in explaining persistency of business fluctuations, since they appear when at least one eigenvalue crosses the unit circle. 

Q9. What is the first example of a perfectly competitive economy?

In the first example presented the authors consider a perfectly competitive economy where public expenditures, financed by variable taxation under a balanced budget rule, are introduced. 

Q10. What is the elasticity of the production function with respect to labor?

In what follows, the authors denote by εX,y the elasticity, evaluated at the steady state, of the function X = {̺,Ω,Γ} with respect to the argument y = {K,L}, while εγ − 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. 

Q11. What is the effect of distortions on the offer curve?

As seen above, indeterminacy is possible in the presence of arbitrarily small distortions affecting either effective consumption or the offer curve. 

Q12. What are the relevant factors for indeterminacy?

On the contrary, under labor market distortions (unions, efficiency wages, unemployment benefits, externalities in preferences), indeterminacy and bifurcations emerge for empirically plausible distortions. 

Q13. What is the definition of a perfect foresight intertemporal equilibrium?

A perfect foresight intertemporal equilibrium of the economy with market distortions is a sequence (Kt−1, Lt) ∈ R 2 ++, t = 1, 2, ...,∞, that for a given K0 > 0 satisfies:Kt = β [1− δ + ̺t]Kt−1 (3)(1/B)Ωt+1Lt+1 = Γt (4)where ̺t ≡ A̺(Kt−1, Lt), Ωt ≡ AΩ(Kt−1, Lt) and Γt ≡ Γ(Kt−1, Lt). 

Q14. What is the first example of a financial constrained framework?

The first example is based on Dufourt et al. (2008), where the Woodford finance constrained framework is extended to take into account the existence of involuntary unemployment (see also Lloyd-Braga and Modesto (2007)).