Chaos and Sector-Specific Externalities
WORKING PAPER SERIES* DEPARTMENT OF ECONOMICS ALFRED LERNER COLLEGE OF BUSINESS & ECONOMICS UNIVERSITY OF DELAWARE WORKING PAPER NO. 2007-17 CHAOS AND SECTOR-SPECIFIC EXTERNALITIES David R. Stockman
____________________________ *http://lerner.udel.edu/economics/workingpaper.htm .
© 2007 by author(s). All rights reserved.
Chaos and Sector-Specific Externalities David R. Stockman∗ Department of Economics University of Delaware Newark, DE 19716 August 3, 2007
Abstract Benhabib and Farmer (1996) explore the possibility of local indeterminacy in a twosector model with sector-specific externalities. They find that very small sector-specific externalities are sufficient for local indeterminacy. In this case, it is possible to construct sunspot equilibria where extrinsic uncertainty matters. In this paper, I provide a global analysis of their model revealing the existence of Euler equation branching. This branching allows for regime switching equilibria with cycles and chaotic behavior. These equilibria occur whether the “local dynamics” are determinate or indeterminate. Keywords: two-sector model, regime switching, global indeterminacy, cycles and chaos. JEL: E13, E32, E62.
I would like to thank the Lerner College of Business & Economics for its generous summer research support. E-mail: [email protected]. ∗
1
1
Introduction
Benhabib and Farmer (1996) explore the possibility of local indeterminacy in a two-sector model with sector-specific externalities. They find that with very small sector-specific externalities, the steady state is locally indeterminate and consequently sunspot equilibria are possible.1 This is an important paper for at least two reasons, both of which address criticism of Benhabib and Farmer (1994). First, Benhabib and Farmer (1994) require a high degree of increasing returns to scale have been deemed empirically implausible, whereas the two-sector model requires only mild externalities. the findings addresses one of the important issues in the local indeterminacy literature, namely are the parameter values needed for local indeterminacy are empirically reasonable. Second, the condition needed for indeterminacy in Benhabib and Farmer (1994) is for the labor demand curve to slope upwards more steeply than the labor supply curve. The two-sector model can deliver indeterminacy with conventional downward sloping labor demand curves.2 In this paper, I perform a global analysis of Benhabib-Farmer model and find that Euler equation branching exists for every parameterization of the model where the labor demand curves are downward sloping (the externality is not too large). The existence of Euler equation branching implies (generically) the existence of regime switching equilibria with cycles and chaotic behavior. Moreover, these equilibria occur whether the steady state is “locally” a saddle, sink or source. This paper joins the literature that has stressed the importance of global analysis in exploring possible equilibria in dynamic general equilibrium models.3 Benhabib and Perli (1994) illustrate the possibility of global indeterminacy with multiple balanced-growth paths in the endogenous growth model of Lucas. They extend the model to include a labor-leisure choice and illustrate that there are two balanced-growth paths for a given a level of physical and human capital, and the choice of labor can put the economy on either of these two equilibrium paths. Boldrin et al. (2002) develop a method for characterizing the global dynamics in the two-sector growth model. They find that global indeterminacy can arise and that the growth rate along an equilibrium trajectory can fluctuate chaotically. In a one-sector growth model with a production externality, Christiano and Harrison (1999) illustrate the possibility of deterministic and stochastic regime switching equilibria along with equilibria that appear chaotic. Furthermore, they discuss how a stabilization tax policy can support the efficient 1
See Benhabib and Farmer (1999) for a survey of the literature in macroeconomics on local indeterminacy and sunspots. 2 For more on externalities as a source of indeterminacy in a two-sector model, see Harrison (2001), Harrison and Weder (2002), Harrison (2003) and Herrendorf and Valentinyi (2006). 3 In addition to the papers discussed here, see also Hommes and de Vilder (1995), Michener and Ravikumar (1998), Benhabib et al. (2001), Guo and Lansing (2002), Coury and Wen (2002) and Medio and Raines (2007).
2
equilibrium allocation. Raines and Stockman (2007) provide necessary and sufficient conditions for the existence of Euler equation branching in a one-sector model with a production externality. They also provide sufficient conditions for the existence of chaos in models with Euler equation branching and prove that these conditions are almost always satisfied near a steady state equilibrium. Stockman (2007) extends the work of Schmitt-Groh´e and Uribe (1997) and illustrates that a balanced-budget rule induces aggregate instability regardless of the determinacy of the steady state. In particular, he provides sufficient conditions for Euler equation branching to be a generic quality under a balanced-budget rule. Consequently, regime switching equilibria involving cycles and chaotic behavior are possible. In the next section I briefly describe the model of Benhabib and Farmer (1996). I demonstrate Euler equation branching always occurs for this model in section 3. I discuss chaos for dynamic models in the plane with Euler equation branching in section 4 and apply these results to the two-sector model. In section 5, I consider two variants of Benhabib-Farmer model: sector-specific externalities from Harrison (2001) and imperfect investment substitutes from Herrendorf and Valentinyi (2006). I provide necessary and sufficient conditions for Euler equation branching to occur in the model of Harrison (2001) and illustrate that such branching will occur for reasonable parameter values. I also show that Euler equation branching does not occur in the model of Herrendorf and Valentinyi (2006). This result reinforces the important role of substitutability in generating multiple equilibria in the two-sector model.
2
Model
The model is the two-sector model in Benhabib and Farmer (1996), hereafter BF.4 Let K and L denote economy-wide aggregate capital and labor and µK and µL denote the fractions of K and L used in the consumption sector. There are two sectors in this economy. One produces a consumption good and the other an investment good. The technologies in each of these sectors are given by Ct = At (µK Kt )a (µL Lt )b , It = Bt [(1 − µK )Kt ]a [(1 − µL )Lt ]b ,
(1)
where a, b > 0 and a + b = 1. As BF note, with factor intensities identical across both sectors, the factor intensities can be shown to be equal, so we let µ = µK = µH . The scaling coefficients At and Bt are taken to be exogenous by the individual firms in each sector, but may depend on both aggregate and sector-specific factor inputs in the following way: ¯ aθ (¯ ¯ bθ K ¯ aσ L ¯ bγ , B = [(1 − µ ¯ aθ [(1 − µ ¯ bθ K ¯ aσ L ¯ bγ . A = (¯ µK K) µL L) ¯K )K] ¯L )L] 4
The discussion here is brief. See BF for more details.
3
(2)
A bar over the variable indicates an economy-wide average (a value taken as given by the individual firm). The parameter θ captures the sector-specific externality and σ and γ represent the aggregate externality. Output in the two sectors can be combined to give the following production possibilities frontier: Ct + (At /Bt )It = At Kta Lbt =: Yt .
(3)
Using the expressions for A and B, one can derive the following social production possibilities frontier: 1/ν
Ct
1/ν
+ It
α/ν
= Kt
β/ν
Lt ,
(4)
where v := (1 + θ) ≥ 1, α := a(1 + θ + σ) and β := b(1 + θ + σ). When v = 1 there are no sectoral externalities (v > 1 implies a sectoral externality). In solving the model, it is useful to define a variable
α/ν
St :=
Kt
β/ν
Lt
1/ν
Ct
,
(5)
which represents the reciprocal of the fraction of aggregate capital and labor going to the production of the consumption good (we then have S ≥ 1). Preferences are given by Z
∞
e
−ρt
0
L1+χ t dt, log Ct − (1 + χ)
(6)
with ρ < 0 < χ. The evolution of capital is standard K˙ = I − δK,
(7)
where 0 < δ < 1 is the depreciation rate. Benchmark parameter values are in Table 1. Table 1: Benchmark parameter values. Parameter:
a
b
δ
Value:
0.3
0.7 0.1 0.05 0.0
4
ρ
σ
γ 0.0
Equilibrium Conditions The equilibrium conditions in the model can be reduced to the following (using λ as a co-state variable): α/ν
Kt
St = bSt
β/ν
Lt
1/ν
Ct 1+χ = Lt ,
,
Ct (St − 1)ν−1 = 1/λt , (St − 1) K˙ t = − δKt , λt aSt+1 , λ˙ t = λt (ρ + δ) − Kt+1
(8) (9) (10) (11) (12)
along with the transversality condition limt→∞ e−ρt λt Kt = 0. As shown in BF, the steady state for this model is unique and can be solved for recursively: S¯ = ¯ = L ¯ = K C¯ = ¯ = λ
3
ρ+δ , ρ + (1 − a)δ ¯ 1/(1+χ) , (bS) ¯β ¯ 1 L (S − 1)ν 1−α , δ S¯ν ¯ αL ¯β K , S¯ν S¯ − 1 ¯ . δK
(13) (14) (15) (16) (17)
Euler Equation Branching
Given K and λ, one uses (8)–(10) to solve for C, S, and L. Note that (11) and (12) give a system K˙ = G(K, λ, S). λ˙
Equations (8)–(10) can be reduced to
˜bK α λ = H(S) := S ν−β/(1+χ) (S − 1)1−ν ,
(18)
where ˜b := bβ/(1+χ) . Given K and λ one uses (18) to solve for S. Proposition 1. Assume σ = 0, a + b = 1, a, b > 0, ν > 1, χ ≥ 0 and β < 1. Then there exists a unique 1 < S ∗ < ∞ with H 0 (S ∗ ) = 0 given by S ∗ = ν(1 + χ − b)/(1 + χ − β). Furthermore, H 0 (S) < 0 for S ∈ (1, S ∗ ), H 0 (S) > 0 for S ∈ (S ∗ , ∞), limS↓1 H(S) = +∞ and limS→∞ H(S) = +∞. 5
Proof. Given H : (1, ∞) → R, direct calculation gives H 0 (S) = [ν − β/(1 + χ)]H(S)/S + (1 − ν)H(S)/(S − 1). Note that for S > 1, H(S) > 0. Setting H 0 (S) = 0, one gets [ν − β/(1 + χ)](S − 1) = (ν − 1)S, which gives the unique value S ∗ = ν(1 + χ − b)/(1 + χ − β) > 1 that satisfies H 0 (S ∗ ) = 0. Since ν − β/(1 + χ) 1 − ν 0 H (S) = H(S) + S S−1
and H(S) > 0 for S ∈ (1, ∞), it is clear that H 0 (S) < 0 if and only if S ∈ (1, S ∗ ) and H 0 (S) > 0 if and only if S ∈ (S ∗ , ∞). Given ν > 1, χ ≥ 0 and β < 1, it follows that limS→1 (S − 1)1−ν = +∞. Since
limS→1 S ν−β/(1+χ) = 1 it follows that limS→1 H(S) = +∞. Since ν − β/(1 + χ) > ν − 1, it follows that limS→∞ H(S) = +∞.
Proposition 2. In this model, under the parameter restrictions given in Proposition 1, for a given value of K and λ there exists typically 0 or 2 solutions for S, i.e., Euler equation branching occurs. Proof. Proposition 1 implies that H|(1, S ∗ ] is monotonic and onto [H(S ∗ ), ∞) and H|[S ∗ , ∞) is monotonic and onto [H(S ∗ ), ∞). Therefore, for any (K, λ) satisfying ˜bK α λ > H(S ∗ ), there are exactly two solutions for S satisfying H(S) = ˜bK α λ. If ˜bK α λ < H(S ∗ ), there are no solutions. If ˜bK α λ = H(S ∗ ), there is only one solution. The intuition for Euler equation branching can be found in how the price of the investment good A/B changes as more resources are shifted from the consumption good to the investment good. Due to the sector specific externality, as S increases, the price of the investment good A/B = (S − 1)1−ν decreases. So the Frisch demand for the consumption good C = (A/B)(1/λ) is decreasing in S (note that when ν = 1 (no externality) A/B is constant so the Frisch demand for the consumption good does not depend on S). Typically, we will have the supply of the consumption good decreasing as S increases (more resources are being devoted to investment) will result in less of the consumption good, i.e., the supply of the consumption good is decreasing in S as well. Two decreasing curves can cross multiple times. See Figure 1. Note that none of this would be possible if A/B were constant (ν = 1). Of course, the fact that both curves are downward sloping does not imply that they must cross multiple times. However, in the BF model, for S close to 1, there is excess demand 6
Figure 1: Market for the consumption good.
1.5
1.45
1.4
1.35
1.3
S 1.25
1.2
Consumption Supply 1.15
1.1
1.05
1 0.8
Consumption Demand
0.9
1
1.1
1.2
1.3
1.4
1.5
Consumption
for the consumption good and as S → ∞ there is excess demand again implying that if the curves cross at all, they will cross multiple times. It is important to note that the existence of Euler equation branching does not depend on the “local” determinacy properties of the (unique) steady state in the model.
4
Chaos and Cycles
For all t, in equilibrium it must be that H(S ∗ ) ≤ ˜bKtα λt . There are two cases to consider: 1. H(S ∗ ) = ˜bKtα λt . In this case there is a unique equilibrium value for St = S ∗ . 2. H(S ∗ ) < ˜bKtα λt . In this case there are two equilibrium values for S with 1 < S1t < S ∗ < S2t < ∞.
7
Given the existence of Euler equation branching, the dynamics in this two-sector model when ˜bK α λ > H(S ∗ ) are given by a multi-valued dynamical system (MVDS): K˙ ∈ {G(K, λ, S1 ), G(K, λ, S2 )}. λ˙ Here I give some definitions for a MVDS. Let the state space X be a metric space with metric d and T := [0, ∞) our time index. The space of all possible orbits on X is denoted by W := {γ | γ : T → X}. Let F : X → 2X be a set-valued function. A dynamical system on X generated by F is a subset of W given by D := {γ ∈ W | γ˙ ∈ F (γ)}. Definition 1. D has a periodic orbit of length m > 0 if there exists an orbit γ ∈ D with γ(t) = γ(t + m) for all t ∈ T and there does not exist an n ∈ (0, m) with γ(t) = γ(t + n) for all t ∈ T . D has a periodic orbit of length m = 0 if there exists an orbit γ ∈ D with γ(t) = γ¯ for all t ∈ T . Definition 2. D has sensitive dependence on initial conditions if there exists a sensitivity constant δ > 0 such that for any given x ∈ X and neighborhood N (x), there exists orbits γ, σ ∈ D and m ≥ 0 such that γ(0) = x, σ(0) ∈ n(x) and d(γ(m), σ(m)) > δ. Definition 3. D has a dense set of periodic points if for any given x ∈ X and neighborhood N (x), there exists a periodic orbit γ ∈ D with γ(0) ∈ N (x). Definition 4. D is topologically transitive if for any (nonempty) open sets U, V ⊂ X, there exists an orbit γ ∈ D and n ∈ T with γ(0) ∈ U and γ(n) ∈ V . Definition 5. D is chaotic in the sense of Devaney (2003) if D is topologically transitive, has a dense set of periodic points and has sensitive dependence on initial conditions. To develop some intuition for why a MVDS will often exhibit chaotic behavior, consider the following example of a MVDS from Stockman (2007) generated by a linear function and a constant function. Example 1. Let X := R2 and H(x) := {Ax, b}, where A is a 2 × 2 matrix with no purely imaginary eigenvalues and b ∈ X. Stockman (2007) shows that such simple building blocks for a MVDS can generate rich dynamical behavior. He considers three cases for x∗ = 0 ∈ X under A: (1) saddle, (2) sink with complex eigenvalues and (3) sink with real eigenvalues. In all of these cases, there will typically exist an invariant closed set with a non-empty interior on which H will be chaotic. The next two theorems are from Stockman (2007). 8
Theorem 1 (saddle). Let X := R2 and H(x) := {Ax, b} where b ∈ X and A is a 2 × 2 matrix with real eigenvalues λ2 < 0 < λ1 and eigenvectors e1 and e2 . Without loss of generality, assume that A is diagonal and e1 and e2 are the canonical basis vectors. Then provided b 6= αe1 and b 6= βe2 , then the dynamical system generated by H restricted to one of the orthants of R2 is chaotic. Theorem 2 (sink or source – complex root). Let X := R2 and H(x) := {Ax, b} where b(6= ~0) ∈ X and A is a 2×2 matrix with complex eigenvalues λ1 and λ2 satisfying Re (λi ) 6= 0. Then the dynamical system generated by H is chaotic on R2 . Proof. See Stockman (2007) for the sink case. A similar argument can be made for a source with complex roots by simply reversing time. For the intuition on the saddle result, see Figure 2. Assume that the vertical axis and the horizontal axis are the unstable and stable manifolds of A (respectively) and the flow from b is running from “northwest” to “southeast.” The importance of the vector b not being a scalar multiple of either eigenvector is so that an integral curves generated by b will intersect those generated by A typically twice (or not at all). This system is chaotic on the “northeast” quadrant. Sensitive dependence of initial conditions is easy to see since every open set in this quadrant has a point with an orbit that diverges to (0, +∞) along with an orbit that converges to (0, 0). To get to (0, +∞) simply follow the integral curves generated by A. To get to (0, 0) simply follow the integral curve generated by b to the stable manifold of A and then follow the stable manifold to (0, 0). A dense set of periodic points follows since every point in the interior of this quadrant is part of a cyclic orbit. To see this, note that one can construct cyclic orbits that look like “half moons” using integral curves of A and the integral curves of b. In fact, there is an orbit connecting any two points in the interior of this quadrant. From this, topological transitivity follows. Next, I extend Stockman’s treatment of a sink with real roots to include repeated roots and for the weaker restrictions on the eigenvectors. Theorem 3 (sink or source – real root). Let X := R2 and H(x) := {Ax, b} where b ∈ X and A is a 2 × 2 matrix with real eigenvalues λ1 < λ2 < 0 or 0 < λ1 < λ2 and eigenvectors e1 and e2 . Without loss of generality, assume that A is diagonal and e1 and e2 are the canonical basis vectors. Then provided b 6= αe1 and b 6= βe2 , the dynamical system generated by H restricted to a cone in R2 is chaotic. If b = αe1 or b = βe2 or λ1 = λ2 6= 0, then the dynamical will be chaotic on a ray emanating from the origin. Proof. For the sink case with λ1 < λ2 and b 6= αe1 and b 6= βe2 , see Stockman (2007). The source case with with λ1 < λ2 and b 6= αe1 and b 6= βe2 is similar. See Figure 3 for integral curves from this system. 9
Figure 2: Saddle: integral curves from the MVDS given by x˙ ∈ {Ax, b}. 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Suppose λ∗ := λ1 = λ2 or b = αe1 or b = αe2 . In any of these cases, the system will be chaotic on the ray R := {µb | µ ≥ 0} if λ∗ < 0 or the ray R := {µb | µ ≤ 0} if λ∗ > 0. Consider the case where λ∗ < 0 (the other case is similar). First, I show there is a periodic orbit containing any two points in R. Let x = µ1 b ∈ R and y = µ2 b with µ2 > µ1 . Then let γ be the orbit running from x to y using the flow generated by b and then running from y to x using the flow generated by Ax. Thus there is a periodic orbit in R containing x and y. Topological transitivity follows immediately as well as a dense set of periodic points (all points are periodic). Let δ > 1. Let x ∈ R, N (x) ⊂ R be a neighborhood of x and y ∈ N (x). Then follow the integral curve γ from x generated by A. Note limt→∞ γ(t) = 0. Next, follow the integral curve σ from y generated by b. Note that limt→0 ||σ(t)|| = ∞. Therefore, there exists M < ∞ such that d(γ(M ), σ(M )) > δ. This simple family of MVDSs is useful as a way of understanding the behavior of nonlinear MVDSs as well. To see this, suppose one has a non-linear system x˙ ∈ {M (x), N (x)} with M (x∗ ) = 0 and N (x∗ ) 6= 0. Suppose further that x∗ is a hyperbolic point of M , i.e., DM (x∗ ) has no purely imaginary eigenvalues. Then in a neighborhood of x∗ , M (x) behaves like Ax where A = DM (x∗ ) and N (x) behaves like b where b = N (x∗ ). For this reason, one can expect that most non-linear MVDSs near a steady state will be chaotic as well. In fact, Raines and Stockman (2007) prove the following theorems about non-linear MVDSs. Let x˙ ∈ {f (x), g(x)} where f, g : R2 → R2 are continuous. Let T := [0, ∞) and define Γ := {γ : T → R2 | γ(t) ˙ ∈ {f (γ(t)), g(γ(t))}. 10
Figure 3: Sink or source with real roots: integral curves from the MVDS given by x˙ ∈ {Ax, b}. 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Let x∗ ∈ R2 with f (x∗ ) = 0 and g(x∗ ) 6= 0. Let A = Df (x∗ ) with eigenvalues λ1 , λ2 . Theorem 4 (sink or source). Suppose there exists > 0 such that x∗ is a sink (asymptotically stable) or a source (asymptotically unstable) for f on B (x∗ ). Then there is a close invariant R such that Γ|R is chaotic. Theorem 5 (saddle). Suppose λ1 < 0 and λ2 > 0 with g(x∗ ) 6= αe1 and g(x∗ ) 6= βe2 for all α, β ∈ R. Then there is a close invariant R such that Γ|R is chaotic. I now turn to the non-linear MVDS from the model and consider two cases for the steady state: local determinacy and local indeterminacy. We will see that in both cases the model can exhibit chaotic behavior. Example 2 (Local Indeterminacy). Let parameter values be set at baseline values (see Table 1) with χ = 1 and ν = 1.25. The steady state and eigenvalues from the log-linearization around the steady state are given by K = 1.4503, λ = 1.7238, L = 0.9354, C = 0.8204, S = 1.2500, Y = 1.0255, P = 1.4142, eigenvalues µ1 = −0.0179+0.2922i, µ2 = −0.0179−0.2922i. We see the steady state is locally a sink. The integral curves from the non-linear system are plotted in Figure 4. From this figure it is clear that on a significant region around the steady state the dynamics are chaotic. Example 3 (Local Determinacy). Let parameter values be set at baseline values (see Table 1) with χ = 2.0 and ν = 1.10. The steady state and eigenvalues from the loglinearization around the steady state are given by K = 2.1026, λ = 1.1890, L = 0.9565, 11
Figure 4: The steady state is locally a sink. Plotted are integral curves from both the low-S and high-S branches. The plotted integral curves from the high-S branch flow from the top left to the bottom right. The plotted integral curves for the low-S branch are flowing counter clockwise.
Low−S Branch Integral Curve
1.76
1.75
1.74
1.73
High−S Branch Integral Curves
1.72
1.71
1.7
1.69
Steady State 1.68
1.67
1.66
1.4
1.45
1.5
12
1.55
1.6
C = 0.9661, S = 1.2500, Y = 1.2076, P = 1.1487, eigenvalues µ1 = 0.3922, µ2 = −0.2985. We see the steady state is locally a saddle. The integral curves from the non-linear system are plotted in Figure 5. We see a similarity to the integral curves depicted in Figure 2 generated by a linear function and a constant function. From this figure it is clear that on a significant region near the steady state the dynamics are chaotic. Figure 6 contains the stable/unstable manifold of the linear approximation around the saddle steady state along with the direction of the vector field of the low-S branch at the steady state. The stable/unstable manifold divides the state space into four quadrants (Q1–Q4). From the linear approximation, chaos is expected in Q1 and this is exactly what one sees for the non-linear system in Figure 5. This illustrates how, in the case of a saddle, one can use information from the linear approximation not only to establish the existence of chaos, but to numerically locate a region of chaos for the nonlinear model as well.
13
Figure 5: The steady state is locally a saddle. Plotted are integral curves from both the low-S and high-S branches. The plotted integral curves from the high-S branch (those associated with the local saddle) flow from the top left to the bottom right. The plotted integral curves for the low-S branch are flowing from the bottom right to the top left.
1.32
1.3
1.28
High S Branch Integral Curves 1.26
1.24
1.22
High S Branch Integral Curves Low S Branch Integral Curves
1.2
Steady State 1.85
1.9
1.95
2
2.05
14
2.1
2.15
2.2
2.25
2.3
Figure 6: Stable/unstable manifold from the linear approximation around the steady state. Included is the direction of the flow on the low-S branch. The stable/unstable manifold divides the state space into four quadrants labeled Q1–Q4. Given the direction of the flow on the low-S branch and the position of the stable/unstable manifolds, chaos is expected in Q1.
0.2
0.15
0.1
Flow Direction on Low−S Branch
Q2 0.05
Q1
0
Q4
Unstable Manifold Q3 −0.05
Stable Manifold −0.1
−0.15
−0.3
−0.2
−0.1
0
15
0.1
0.2
0.3
0.4
5
Extensions
In this section, I explore two extensions of the BF model: (1) sector-specific externalities with more general preferences and (2) investments in each sector are imperfect substitutes.
5.1
Sector-Specific Externalities
Harrison (2001) explores the role of sector-specific externalities in generating indeterminacy in a two-sector model. She does this by allowing (1) different degrees of externalities in the consumption sector and the investment sector (BF set these equal to each other) and (2) more general preferences over consumption (BF consider only log utility). Here, I briefly describe a continuous-time version of her model. Preferences are given by 1−σ Z ∞ L1+χ −1 t −ρt Ct − dt, e 1−σ (1 + χ) 0
(19)
with σ, χ > 0 and ρ < 0 (with utility of consumption given by log(C) for σ = 1). The evolution of capital is standard. The budget constraint is given by Ct + pt It ≤ rt Kt + wt Lt . The household seeks to maximize (19) subject to K˙ t = (1/pt )[wt Lt + rt Kt − Ct ] − δKt . The Hamiltonian is given by Ht =
L1+χ Ct1−σ − 1 − t + λt [(1/pt )(wt Lt + rt Kt − Ct ) − δKt ]. 1−σ (1 + χ)
Optimality conditions: Ct−σ = λt /pt , Lχt = λt wt /pt , λ˙ t = λt (ρ + δ − rt /pt ),
(20) (21) (22)
along with a transversality condition limt→∞ e−ρt λt Kt = 0. In the consumption sector, the firm maximizes profits given the following technology: α α β lC,t , fC,t := (KC,t LβC,t )θC kC,t
where lower-case letters denote firm values and upper-case letters denote economy-wide averages. In the investment sector, the firm maximizes profits given the following technology: α β θI α β fI,t := (KI,t LI,t ) kI,t lI,t .
16
The factor inputs receive their marginal products: pt αfI,t αfC,t = kc,t kI,t βfC,t pt βfI,t = = . lc,t lI,t
rt =
(23)
wt
(24)
Aggregate factor inputs must satisfy KC,t + KI,t = Kt , LC,t + LI,t = Lt . In equilibrium, taking into account that the firm level of inputs must equal the economywide averages, we have fC,t = (µt Ktα Lβt )1+θC ,
(25)
fI,t = [(1 − µt )Ktα Lβt ]1+θI
(26)
where
KC,t LC,t = . Kt Lt The price of the investment good (measured in the consumption good) is given by µt :=
µθt C pt = (Ktα Lβt )θC −θI . θ I (1 − µt )
(27)
(28)
Combining the equilibrium conditions, an equilibrium in the model must satisfy the following conditions (in addition to the transversality conditions): L1+χ = λt β(1 − µt )θI (Ktα Lβt )1+θI , t (pt /λt )σ = (µt Ktα Lβt )1+θC , pt =
µθt C (1 − µt
)θ I
(Ktα Lβt )θC −θI ,
λt αfI,t λ˙ t = λt (ρ + δ) − , (1 − µt )Kt " 1/σ # f 1 pt C,t K˙ t = − − δKt . pt µt λt
(29) (30) (31) (32) (33)
With Kt as the state and λt as the co-state, one uses equations (29)–(31) to solve for Lt , pt and µt . Let τ := 1 + χ − β(1 + θI ). Assuming τ 6= 0, one can use (29) to solve for L as a function of K, λ and µ: L(K, λ, µ) := [λβK α(1+θI ) ]1/τ (1 − µ)θI /τ =: l(K, λ)(1 − µ)θI /τ , 17
where l(K, λ) := [λβK α(1+θI ) ]1/τ . Using equation (30), one can solve for P as a function of K, λ and µ: P (K, λ, µ) := λ[µK α L(K, λ, µ)β ]
1+θC σ
.
Substituting for L(K, λ, µ) and simplifying gives P (K, λ, µ) := p(K, λ)µ
1+θC σ
(1 − µ)
β(1+θC )θI τσ
,
where p(K, λ) := λK
α(1+θC ) σ
l(K, λ)
β(1+θC ) σ
.
Using P (K, λ, µ) and L(K, λ, µ) in equation (31) one gets P (K, λ, µ) =
µθ C (K α L(K, λ, µ)β )θC −θI . (1 − µ)θI
The right-hand side of this equation simplifies to g(K, λ)µθC (1 − µ)
βθI (θC −θI ) −θI τ
,
where g(K, λ) := K α(θC −θI ) l(K, λ)β(θC −θI ) . Substituting for P (K, λ) and simplifying, one gets M (µ) := µa (1 − µ)b = where
g(K, λ) =: N (K, λ), p(K, λ)
β 1 + θC − σθC 1 + θC − σθC and b := θI 1 + + θI . a := σ τ σ
(34)
(35)
We see there is Euler equation branching in this model iff given K > 0 and λ > 0 there exists more than one solution to µ satisfying equation (34). Note that M (µ) > 0 for µ ∈ (0, 1) and 0
M (µ) = M (µ)
a b − µ 1−µ
.
(36)
Definition 6. The function M is single-caved if there exists a unique µ∗ ∈ (0, 1) such that M 0 (µ∗ ) = 0 and either (1) M 0 (µ1 ) < 0 and M 0 (µ2 ) > 0 or (2) M 0 (µ1 ) > 0 and M 0 (µ2 ) < 0 for all µ1 ∈ (0, µ∗ ) and µ2 ∈ (µ∗ , 1). It is single-caved up if it is single-caved with M 0 (µ) < 0 for µ ∈ (0, µ∗ ). It is single-caved down if it is single-caved with M 0 (µ) > 0 for µ ∈ (0, µ∗ ). Lemma 1. M (µ) defined in (34) is single-caved on (0, 1) if and only if a, b > 0 or a, b < 0, otherwise M is monotonic. Moreover, if a, b > 0, M is single-caved down and if a, b < 0 then M is single-caved up. 18
Proof. Suppose M is single-caved. Then there exists a unique µ∗ ∈ (0, 1) such that M 0 (µ∗ ) = 0. This implies a/µ∗ = b/(1 − µ∗ ). Since µ∗ , (1 − µ∗ ) > 0, it must be that either a, b > 0 or a, b < 0. Now assume a, b > 0 or a, b < 0. In this case, there is a unique solution to M 0 (µ∗ ) = 0 given by 0 < µ∗ := a/(a + b) < 1. If a, b > 0 then M 0 (µ) > 0 for µ ∈ (0, µ∗ ) and M 0 (µ) < 0 for µ ∈ (µ∗ , 1), so M is is single-caved down If a, b < 0 then M 0 (µ) < 0 for µ ∈ (0, µ∗ ) and M 0 (µ) > 0 for µ ∈ (µ∗ , 1), so M is single-caved down. If a ≥ 0 and b ≤ 0, then H 0 (µ) ≥ 0 and so is monotonic. If a ≤ 0 and b ≥ 0, then H 0 (µ) ≤ 0 and so is monotonic. If the function M is single-caved, then it also satisfies the following properties. Lemma 2. If a, b > 0 then lim M (µ) = lim M (µ) = 0,
µ→0
µ→1
and if a, b < 0 then lim M (µ) = lim M (µ) = +∞.
µ→0
µ→1
The following proposition follows from Lemmas 1 and 2. Proposition 3. Let a and b be defined by equation (35). There is Euler equation branching in this model iff ab > 0. Discussion. The externality in the investment sector is essential since b = 0 if θI = 0. However it is not sufficient. To see this, let θC > 0 and σ > (1 + θC )/θC so a < 0. Then b > 0 for all θI > 0 if and only if (β − 1 − χ)/β < a. If 0 < θC < (1 + χ − β)/β, then this conditions is automatically satisfied. If θC > (1 + χ − β)/β, then b > 0 iff 1 + θC 1 + θC 0 and b > 0, so Euler equation branching exists. Consider the case a, b > 0. If θI = 0, then b = 0, so we must have θI > 0. If θC = 0, then a > 0 puts no additional restrictions on σ. If θC > 0, then σ < (1 + θC )/θC . Note that β b = θI 1 + (a + θI ) . τ There are two cases to consider: τ > 0 and τ < 0. If τ > 0, then b > 0. If τ < 0, then b > 0 if and only if θC > (1 + χ − β)/β and 1 + θC < σ. θC + (β − 1 − χ)/β 19
(37)
However, θC > (1 + χ − β)/β, require θC > 0 so we must have σ
0. If θC = 0, then a > 0, so we must have θC > 0. If θC > 0 then a < 0 if and only if σ > (1 + θC )/θC . For b < 0 there are again two cases: τ > 0 and τ < 0. Suppose τ > 0, then b < 0 iff θC > (1 + χ − β)/β and (37) holds. Suppose τ < 0, then b < 0 iff θC ≤ (1 + χ − β)/β or θC > (1 + χ − β)/β and σ
(1 + χ − β)/β since 1 + θC 1 + θC 0; (P2) If θC > 0, then σ < (1 + θC )/θC ; (P3) τ := 1 + χ − β(1 + θI ) > 0; or (N1) θI > 0; (N2) θC > 0 and σ > (1 + θC )/θC ; (N3) If τ := 1 + χ − β(1 + θI ) > 0, then (37) and θC > (1 + χ − β)/β; (N4) If τ := 1 + χ − β(1 + θI ) < 0 and θC > (1 + χ − β)/β, then (38). We see (P1)–(P3) will be satisfied for modest externalities in the consumption sector, plausible values of σ and conventional downward-sloping labor demand curves (β(1+θI ) < 1). If there is no externality in the consumption sector (θC = 0), then β(1+θI ) < 1 will guarantee Euler equation branching for any σ, θI > 0. 20
5.2
Imperfect Substitutes
Herrendorf and Valentinyi (2006) study a variation of the two-sector model where investment in the consumption sector is an imperfect substitute for investment in the investment sector. They show analytically that the steady state is locally determinate for every plausible parameterization of the model. Note that these results are essentially the opposite of the model analyzed by Benhabib and Farmer (1996). Euler equation branching will also be sensitive to this attribute of the model. In particular, I will show that there is no Euler equation branching near the steady state in this model. Preferences are given by U (C, Lc , Lx ) := log C −
(Lc + Lx )1+χ , 1+χ
where C is consumption and Lc and Lx represent labor devoted to the consumption and investment sectors. The household’s budget constraint is given by Ct + Pct Xct + Pxt Xxt = πct + πxt + wct Lct + wxt Lxt + rct Kct + rxt Kxt . a 1−a Production in the consumption sector is given by At kct lct . Production in the investment b 1−b sector is given by Bt kxt lxt . This investment good is then allocated to either the consumption
sector xct or the investment sector xxt according to b 1−b f (xct , xxt ) = Bt kxt lxt ,
where f is C 2 , non-negative, homogeneous degree 1, increasing in both arguments and strictly quasi-convex. The laws of motion for the capital stocks are K˙ ct = Xct − δc Kct and K˙ xt = Xxt − δx Kxt . It is assumed that Xct , Xxt ≥ 0 so that installed capital is sector specific. Externalities in each sector are given by θ (1−a)
θc a c At = kct lct
θ (1−b)
θx b x and Bt = kxt lxt
,
with θx , θc ≥ 0. Let α1 := (1 + θc )a, α2 := (1 + θc )(1 − a), β1 := (1 + θx )b and β2 := (1 + θx )(1 − b). The household’s first order conditions are Pct /Ct = µct ,
(39)
Pxt /Ct = µxt ,
(40)
wct /Ct = (Lct + Lxt )χ ,
(41)
wxt /Ct = (Lct + Lxt )χ ,
(42)
µ˙ ct ≤ µct (ρ + δc ) − rct /Ct ,
(43)
µ˙ xt ≤ µxt (ρ + δx ) − rxt /Ct ,
(44)
21
along with transversality conditions for the capital stocks. Profit maximization in the consumption sector implies rct = aAt Kcta−1 L1−a and wct = (1 − a)At Kcta L−a ct ct .
(45)
Profit maximization in the investment sector implies b−1 1−b rxt /Pxt = bBt Kxt Lxt /f (xct , xxt ), b−1 −b wxt /Pxt = bBt Kxt Lxt /f (xct , xxt ),
(46) (47)
Proposition 5. In this model, near the steady state there is no Euler equation branching. Proof. Given the state [Kct , Kxt ] and co-state [µct , µxt ] all that needs to be shown is that the other endogenous variables are uniquely determined. This is done in a way similar to Herrendorf and Valentinyi (2006). The household FOCs imply Pct /Pxt = µct /µxt . This condition along with the investment good firm’s FOCs imply fc /fx = µct /µxt . Since f is strictly quasi-convex and homogeneous, one can use this condition to solve for the ratio of the investment goods as a function of the ratio of these multipliers: xct /xxt = g(µct /µxt ). In the consumption sector, since C = AKca L1−a and labor receives its marginal product, c we have wct = (1 − a)Ct /Lct . This condition along with the household’s FOC for labor in the consumption sector gives Lct (Lct + Lxt )χ = (1 − a).
(48)
In the investment sector, a similar condition can be found 1=
β1 β2 −1 Lxt (1 − b)µxt Kxt . χ (Lct + Lxt ) fx (g(µct /µxt ), 1)
Using (48) and (49), one can solve for equilibrium labor in the investment sector 1 " # 1−β β1 2 1−b µxt Kxt Lct Lxt = . . 1 − a fx (g(µct /µxt ), 1)
(49)
(50)
Note that Lxt is increasing in Lct (assuming 0 < β2 < 1). From this expression we can solve for the unique equilibrium Lct : Lct =
1−a . (Lct + Lxt )χ
With χ > 0, the right-hand side is strictly decreasing in Lct and approaches +∞ as Lct ↓ 0 and approaches 0 as Lct → +∞. Therefore there is a unique equilibrium Lct . This pins down Lxt and the other endogenous variables as well. 22
This proposition reinforces the main point of Herrendorf and Valentinyi (2006), namely that the ability of the two-sector model to generate multiple equilibria depends critically on whether or not the investments in each sector are perfect substitutes.
6
Conclusion
In this paper, I show that Euler equation branching occurs in the two-sector model of Benhabib and Farmer (1996) and Harrison (2001), but not in the model of Herrendorf and Valentinyi (2006). However, when such branching occurs, there will typically exist regime switching equilibria with chaotic behavior that occur whether the “local” dynamics are determinate or indeterminate.
References Benhabib, J., Farmer, R. E. A., 1994. Indeterminacy and increasing returns. Journal of Economic Theory 63, 19–41. Benhabib, J., Farmer, R. E. A., 1996. Indeterminacy and sector specific externalities. Journal of Monetary Economics 37, 397–419. Benhabib, J., Farmer, R. E. A., 1999. Indeterminacy and sunspots in macroeconomics. In: Taylor, J., Woodford, M. (Eds.), Handbook of Macroeconomics. Vol. IA. North-Holland, New York, NY, pp. 387–448. Benhabib, J., Perli, R., 1994. Uniqueness and indeterminacy: on the dynamics of endogenous growth. Journal of Economic Theory 63, 113–142. Benhabib, J., Schmitt-Groh´e, S., Uribe, M., 2001. The perils of Taylor rules. Journal of Economic Theory 96, 40–69. Boldrin, M., Nishimura, K., Shigoka, T., Yano, M., 2002. Chaotic equilibrium dynamics in endogenous growth models. Journal of Economic Theory 96, 97–132. Christiano, L. J., Harrison, S. G., August 1999. Chaos, sunspots and automatic stabilizers. Journal of Monetary Economics 44 (1), 3–31. Coury, T., Wen, Y., 2002. Global indeterminacy and chaos in standard rbc models. Manuscript, Cornell University.
23
Devaney, R. L., 2003. An introduction to chaotic dynamical systems, 2nd Edition. Westview Press, Boulder, Colorado. Guo, J.-T., Lansing, K. J., 2002. Fiscal policy, increasing returns, and endogenous fluctuations. Macroeconomic Dynamics 6, 633–664. Harrison, S. G., 2001. Indeterminacy in a model with sector-specific externalities. Journal Of Economic Dynamics & Control 25 (5), 747–764. Harrison, S. G., 2003. Returns to scale and externalities in the consumption and investment sectors. Review of Economic Dynamics 6 (4), 963–976. Harrison, S. G., Weder, M., 2002. Tracing externalities as a source of indeterminacy. Journal of Economic Dynamics and Control 26, 851–867. Herrendorf, B., Valentinyi, A., 2006. On the stability of the two-sector neoclassical growth model with externalities. Journal of Economic Dynamics and Control 30 (8), 1339–1361. Hommes, C., de Vilder, R., 1995. Sunspot equilibria in an implicitly defined overlapping generations model. working paper, 1–20. Medio, A., Raines, B., 2007. Backward dynamics in economics. the inverse limit approach. Journal of Economic Dynamics and Control, 31, 1633–1671. Michener, R., Ravikumar, B., 1998. Chaotic dynamics in a cash-in-advance economy. Journal of Economic Dynamics and Control 22, 1117–1137. Raines, B., Stockman, D. R., 2007. Chaos and euler equation branching in a model with increasing returns to scale. Manuscript, University of Delaware, 1–22. Schmitt-Groh´e, S., Uribe, M., 1997. Balanced-budget rules, distortionary taxes, and aggregate instability. Journal of Political Economy 105, 976–1000. Stockman, D. R., 2007. Balanced-budget rules: chaos and cycles. Manuscript, University of Delaware.
24
____________________________ *http://lerner.udel.edu/economics/workingpaper.htm .
© 2007 by author(s). All rights reserved.
Chaos and Sector-Specific Externalities David R. Stockman∗ Department of Economics University of Delaware Newark, DE 19716 August 3, 2007
Abstract Benhabib and Farmer (1996) explore the possibility of local indeterminacy in a twosector model with sector-specific externalities. They find that very small sector-specific externalities are sufficient for local indeterminacy. In this case, it is possible to construct sunspot equilibria where extrinsic uncertainty matters. In this paper, I provide a global analysis of their model revealing the existence of Euler equation branching. This branching allows for regime switching equilibria with cycles and chaotic behavior. These equilibria occur whether the “local dynamics” are determinate or indeterminate. Keywords: two-sector model, regime switching, global indeterminacy, cycles and chaos. JEL: E13, E32, E62.
I would like to thank the Lerner College of Business & Economics for its generous summer research support. E-mail: [email protected]. ∗
1
1
Introduction
Benhabib and Farmer (1996) explore the possibility of local indeterminacy in a two-sector model with sector-specific externalities. They find that with very small sector-specific externalities, the steady state is locally indeterminate and consequently sunspot equilibria are possible.1 This is an important paper for at least two reasons, both of which address criticism of Benhabib and Farmer (1994). First, Benhabib and Farmer (1994) require a high degree of increasing returns to scale have been deemed empirically implausible, whereas the two-sector model requires only mild externalities. the findings addresses one of the important issues in the local indeterminacy literature, namely are the parameter values needed for local indeterminacy are empirically reasonable. Second, the condition needed for indeterminacy in Benhabib and Farmer (1994) is for the labor demand curve to slope upwards more steeply than the labor supply curve. The two-sector model can deliver indeterminacy with conventional downward sloping labor demand curves.2 In this paper, I perform a global analysis of Benhabib-Farmer model and find that Euler equation branching exists for every parameterization of the model where the labor demand curves are downward sloping (the externality is not too large). The existence of Euler equation branching implies (generically) the existence of regime switching equilibria with cycles and chaotic behavior. Moreover, these equilibria occur whether the steady state is “locally” a saddle, sink or source. This paper joins the literature that has stressed the importance of global analysis in exploring possible equilibria in dynamic general equilibrium models.3 Benhabib and Perli (1994) illustrate the possibility of global indeterminacy with multiple balanced-growth paths in the endogenous growth model of Lucas. They extend the model to include a labor-leisure choice and illustrate that there are two balanced-growth paths for a given a level of physical and human capital, and the choice of labor can put the economy on either of these two equilibrium paths. Boldrin et al. (2002) develop a method for characterizing the global dynamics in the two-sector growth model. They find that global indeterminacy can arise and that the growth rate along an equilibrium trajectory can fluctuate chaotically. In a one-sector growth model with a production externality, Christiano and Harrison (1999) illustrate the possibility of deterministic and stochastic regime switching equilibria along with equilibria that appear chaotic. Furthermore, they discuss how a stabilization tax policy can support the efficient 1
See Benhabib and Farmer (1999) for a survey of the literature in macroeconomics on local indeterminacy and sunspots. 2 For more on externalities as a source of indeterminacy in a two-sector model, see Harrison (2001), Harrison and Weder (2002), Harrison (2003) and Herrendorf and Valentinyi (2006). 3 In addition to the papers discussed here, see also Hommes and de Vilder (1995), Michener and Ravikumar (1998), Benhabib et al. (2001), Guo and Lansing (2002), Coury and Wen (2002) and Medio and Raines (2007).
2
equilibrium allocation. Raines and Stockman (2007) provide necessary and sufficient conditions for the existence of Euler equation branching in a one-sector model with a production externality. They also provide sufficient conditions for the existence of chaos in models with Euler equation branching and prove that these conditions are almost always satisfied near a steady state equilibrium. Stockman (2007) extends the work of Schmitt-Groh´e and Uribe (1997) and illustrates that a balanced-budget rule induces aggregate instability regardless of the determinacy of the steady state. In particular, he provides sufficient conditions for Euler equation branching to be a generic quality under a balanced-budget rule. Consequently, regime switching equilibria involving cycles and chaotic behavior are possible. In the next section I briefly describe the model of Benhabib and Farmer (1996). I demonstrate Euler equation branching always occurs for this model in section 3. I discuss chaos for dynamic models in the plane with Euler equation branching in section 4 and apply these results to the two-sector model. In section 5, I consider two variants of Benhabib-Farmer model: sector-specific externalities from Harrison (2001) and imperfect investment substitutes from Herrendorf and Valentinyi (2006). I provide necessary and sufficient conditions for Euler equation branching to occur in the model of Harrison (2001) and illustrate that such branching will occur for reasonable parameter values. I also show that Euler equation branching does not occur in the model of Herrendorf and Valentinyi (2006). This result reinforces the important role of substitutability in generating multiple equilibria in the two-sector model.
2
Model
The model is the two-sector model in Benhabib and Farmer (1996), hereafter BF.4 Let K and L denote economy-wide aggregate capital and labor and µK and µL denote the fractions of K and L used in the consumption sector. There are two sectors in this economy. One produces a consumption good and the other an investment good. The technologies in each of these sectors are given by Ct = At (µK Kt )a (µL Lt )b , It = Bt [(1 − µK )Kt ]a [(1 − µL )Lt ]b ,
(1)
where a, b > 0 and a + b = 1. As BF note, with factor intensities identical across both sectors, the factor intensities can be shown to be equal, so we let µ = µK = µH . The scaling coefficients At and Bt are taken to be exogenous by the individual firms in each sector, but may depend on both aggregate and sector-specific factor inputs in the following way: ¯ aθ (¯ ¯ bθ K ¯ aσ L ¯ bγ , B = [(1 − µ ¯ aθ [(1 − µ ¯ bθ K ¯ aσ L ¯ bγ . A = (¯ µK K) µL L) ¯K )K] ¯L )L] 4
The discussion here is brief. See BF for more details.
3
(2)
A bar over the variable indicates an economy-wide average (a value taken as given by the individual firm). The parameter θ captures the sector-specific externality and σ and γ represent the aggregate externality. Output in the two sectors can be combined to give the following production possibilities frontier: Ct + (At /Bt )It = At Kta Lbt =: Yt .
(3)
Using the expressions for A and B, one can derive the following social production possibilities frontier: 1/ν
Ct
1/ν
+ It
α/ν
= Kt
β/ν
Lt ,
(4)
where v := (1 + θ) ≥ 1, α := a(1 + θ + σ) and β := b(1 + θ + σ). When v = 1 there are no sectoral externalities (v > 1 implies a sectoral externality). In solving the model, it is useful to define a variable
α/ν
St :=
Kt
β/ν
Lt
1/ν
Ct
,
(5)
which represents the reciprocal of the fraction of aggregate capital and labor going to the production of the consumption good (we then have S ≥ 1). Preferences are given by Z
∞
e
−ρt
0
L1+χ t dt, log Ct − (1 + χ)
(6)
with ρ < 0 < χ. The evolution of capital is standard K˙ = I − δK,
(7)
where 0 < δ < 1 is the depreciation rate. Benchmark parameter values are in Table 1. Table 1: Benchmark parameter values. Parameter:
a
b
δ
Value:
0.3
0.7 0.1 0.05 0.0
4
ρ
σ
γ 0.0
Equilibrium Conditions The equilibrium conditions in the model can be reduced to the following (using λ as a co-state variable): α/ν
Kt
St = bSt
β/ν
Lt
1/ν
Ct 1+χ = Lt ,
,
Ct (St − 1)ν−1 = 1/λt , (St − 1) K˙ t = − δKt , λt aSt+1 , λ˙ t = λt (ρ + δ) − Kt+1
(8) (9) (10) (11) (12)
along with the transversality condition limt→∞ e−ρt λt Kt = 0. As shown in BF, the steady state for this model is unique and can be solved for recursively: S¯ = ¯ = L ¯ = K C¯ = ¯ = λ
3
ρ+δ , ρ + (1 − a)δ ¯ 1/(1+χ) , (bS) ¯β ¯ 1 L (S − 1)ν 1−α , δ S¯ν ¯ αL ¯β K , S¯ν S¯ − 1 ¯ . δK
(13) (14) (15) (16) (17)
Euler Equation Branching
Given K and λ, one uses (8)–(10) to solve for C, S, and L. Note that (11) and (12) give a system K˙ = G(K, λ, S). λ˙
Equations (8)–(10) can be reduced to
˜bK α λ = H(S) := S ν−β/(1+χ) (S − 1)1−ν ,
(18)
where ˜b := bβ/(1+χ) . Given K and λ one uses (18) to solve for S. Proposition 1. Assume σ = 0, a + b = 1, a, b > 0, ν > 1, χ ≥ 0 and β < 1. Then there exists a unique 1 < S ∗ < ∞ with H 0 (S ∗ ) = 0 given by S ∗ = ν(1 + χ − b)/(1 + χ − β). Furthermore, H 0 (S) < 0 for S ∈ (1, S ∗ ), H 0 (S) > 0 for S ∈ (S ∗ , ∞), limS↓1 H(S) = +∞ and limS→∞ H(S) = +∞. 5
Proof. Given H : (1, ∞) → R, direct calculation gives H 0 (S) = [ν − β/(1 + χ)]H(S)/S + (1 − ν)H(S)/(S − 1). Note that for S > 1, H(S) > 0. Setting H 0 (S) = 0, one gets [ν − β/(1 + χ)](S − 1) = (ν − 1)S, which gives the unique value S ∗ = ν(1 + χ − b)/(1 + χ − β) > 1 that satisfies H 0 (S ∗ ) = 0. Since ν − β/(1 + χ) 1 − ν 0 H (S) = H(S) + S S−1
and H(S) > 0 for S ∈ (1, ∞), it is clear that H 0 (S) < 0 if and only if S ∈ (1, S ∗ ) and H 0 (S) > 0 if and only if S ∈ (S ∗ , ∞). Given ν > 1, χ ≥ 0 and β < 1, it follows that limS→1 (S − 1)1−ν = +∞. Since
limS→1 S ν−β/(1+χ) = 1 it follows that limS→1 H(S) = +∞. Since ν − β/(1 + χ) > ν − 1, it follows that limS→∞ H(S) = +∞.
Proposition 2. In this model, under the parameter restrictions given in Proposition 1, for a given value of K and λ there exists typically 0 or 2 solutions for S, i.e., Euler equation branching occurs. Proof. Proposition 1 implies that H|(1, S ∗ ] is monotonic and onto [H(S ∗ ), ∞) and H|[S ∗ , ∞) is monotonic and onto [H(S ∗ ), ∞). Therefore, for any (K, λ) satisfying ˜bK α λ > H(S ∗ ), there are exactly two solutions for S satisfying H(S) = ˜bK α λ. If ˜bK α λ < H(S ∗ ), there are no solutions. If ˜bK α λ = H(S ∗ ), there is only one solution. The intuition for Euler equation branching can be found in how the price of the investment good A/B changes as more resources are shifted from the consumption good to the investment good. Due to the sector specific externality, as S increases, the price of the investment good A/B = (S − 1)1−ν decreases. So the Frisch demand for the consumption good C = (A/B)(1/λ) is decreasing in S (note that when ν = 1 (no externality) A/B is constant so the Frisch demand for the consumption good does not depend on S). Typically, we will have the supply of the consumption good decreasing as S increases (more resources are being devoted to investment) will result in less of the consumption good, i.e., the supply of the consumption good is decreasing in S as well. Two decreasing curves can cross multiple times. See Figure 1. Note that none of this would be possible if A/B were constant (ν = 1). Of course, the fact that both curves are downward sloping does not imply that they must cross multiple times. However, in the BF model, for S close to 1, there is excess demand 6
Figure 1: Market for the consumption good.
1.5
1.45
1.4
1.35
1.3
S 1.25
1.2
Consumption Supply 1.15
1.1
1.05
1 0.8
Consumption Demand
0.9
1
1.1
1.2
1.3
1.4
1.5
Consumption
for the consumption good and as S → ∞ there is excess demand again implying that if the curves cross at all, they will cross multiple times. It is important to note that the existence of Euler equation branching does not depend on the “local” determinacy properties of the (unique) steady state in the model.
4
Chaos and Cycles
For all t, in equilibrium it must be that H(S ∗ ) ≤ ˜bKtα λt . There are two cases to consider: 1. H(S ∗ ) = ˜bKtα λt . In this case there is a unique equilibrium value for St = S ∗ . 2. H(S ∗ ) < ˜bKtα λt . In this case there are two equilibrium values for S with 1 < S1t < S ∗ < S2t < ∞.
7
Given the existence of Euler equation branching, the dynamics in this two-sector model when ˜bK α λ > H(S ∗ ) are given by a multi-valued dynamical system (MVDS): K˙ ∈ {G(K, λ, S1 ), G(K, λ, S2 )}. λ˙ Here I give some definitions for a MVDS. Let the state space X be a metric space with metric d and T := [0, ∞) our time index. The space of all possible orbits on X is denoted by W := {γ | γ : T → X}. Let F : X → 2X be a set-valued function. A dynamical system on X generated by F is a subset of W given by D := {γ ∈ W | γ˙ ∈ F (γ)}. Definition 1. D has a periodic orbit of length m > 0 if there exists an orbit γ ∈ D with γ(t) = γ(t + m) for all t ∈ T and there does not exist an n ∈ (0, m) with γ(t) = γ(t + n) for all t ∈ T . D has a periodic orbit of length m = 0 if there exists an orbit γ ∈ D with γ(t) = γ¯ for all t ∈ T . Definition 2. D has sensitive dependence on initial conditions if there exists a sensitivity constant δ > 0 such that for any given x ∈ X and neighborhood N (x), there exists orbits γ, σ ∈ D and m ≥ 0 such that γ(0) = x, σ(0) ∈ n(x) and d(γ(m), σ(m)) > δ. Definition 3. D has a dense set of periodic points if for any given x ∈ X and neighborhood N (x), there exists a periodic orbit γ ∈ D with γ(0) ∈ N (x). Definition 4. D is topologically transitive if for any (nonempty) open sets U, V ⊂ X, there exists an orbit γ ∈ D and n ∈ T with γ(0) ∈ U and γ(n) ∈ V . Definition 5. D is chaotic in the sense of Devaney (2003) if D is topologically transitive, has a dense set of periodic points and has sensitive dependence on initial conditions. To develop some intuition for why a MVDS will often exhibit chaotic behavior, consider the following example of a MVDS from Stockman (2007) generated by a linear function and a constant function. Example 1. Let X := R2 and H(x) := {Ax, b}, where A is a 2 × 2 matrix with no purely imaginary eigenvalues and b ∈ X. Stockman (2007) shows that such simple building blocks for a MVDS can generate rich dynamical behavior. He considers three cases for x∗ = 0 ∈ X under A: (1) saddle, (2) sink with complex eigenvalues and (3) sink with real eigenvalues. In all of these cases, there will typically exist an invariant closed set with a non-empty interior on which H will be chaotic. The next two theorems are from Stockman (2007). 8
Theorem 1 (saddle). Let X := R2 and H(x) := {Ax, b} where b ∈ X and A is a 2 × 2 matrix with real eigenvalues λ2 < 0 < λ1 and eigenvectors e1 and e2 . Without loss of generality, assume that A is diagonal and e1 and e2 are the canonical basis vectors. Then provided b 6= αe1 and b 6= βe2 , then the dynamical system generated by H restricted to one of the orthants of R2 is chaotic. Theorem 2 (sink or source – complex root). Let X := R2 and H(x) := {Ax, b} where b(6= ~0) ∈ X and A is a 2×2 matrix with complex eigenvalues λ1 and λ2 satisfying Re (λi ) 6= 0. Then the dynamical system generated by H is chaotic on R2 . Proof. See Stockman (2007) for the sink case. A similar argument can be made for a source with complex roots by simply reversing time. For the intuition on the saddle result, see Figure 2. Assume that the vertical axis and the horizontal axis are the unstable and stable manifolds of A (respectively) and the flow from b is running from “northwest” to “southeast.” The importance of the vector b not being a scalar multiple of either eigenvector is so that an integral curves generated by b will intersect those generated by A typically twice (or not at all). This system is chaotic on the “northeast” quadrant. Sensitive dependence of initial conditions is easy to see since every open set in this quadrant has a point with an orbit that diverges to (0, +∞) along with an orbit that converges to (0, 0). To get to (0, +∞) simply follow the integral curves generated by A. To get to (0, 0) simply follow the integral curve generated by b to the stable manifold of A and then follow the stable manifold to (0, 0). A dense set of periodic points follows since every point in the interior of this quadrant is part of a cyclic orbit. To see this, note that one can construct cyclic orbits that look like “half moons” using integral curves of A and the integral curves of b. In fact, there is an orbit connecting any two points in the interior of this quadrant. From this, topological transitivity follows. Next, I extend Stockman’s treatment of a sink with real roots to include repeated roots and for the weaker restrictions on the eigenvectors. Theorem 3 (sink or source – real root). Let X := R2 and H(x) := {Ax, b} where b ∈ X and A is a 2 × 2 matrix with real eigenvalues λ1 < λ2 < 0 or 0 < λ1 < λ2 and eigenvectors e1 and e2 . Without loss of generality, assume that A is diagonal and e1 and e2 are the canonical basis vectors. Then provided b 6= αe1 and b 6= βe2 , the dynamical system generated by H restricted to a cone in R2 is chaotic. If b = αe1 or b = βe2 or λ1 = λ2 6= 0, then the dynamical will be chaotic on a ray emanating from the origin. Proof. For the sink case with λ1 < λ2 and b 6= αe1 and b 6= βe2 , see Stockman (2007). The source case with with λ1 < λ2 and b 6= αe1 and b 6= βe2 is similar. See Figure 3 for integral curves from this system. 9
Figure 2: Saddle: integral curves from the MVDS given by x˙ ∈ {Ax, b}. 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Suppose λ∗ := λ1 = λ2 or b = αe1 or b = αe2 . In any of these cases, the system will be chaotic on the ray R := {µb | µ ≥ 0} if λ∗ < 0 or the ray R := {µb | µ ≤ 0} if λ∗ > 0. Consider the case where λ∗ < 0 (the other case is similar). First, I show there is a periodic orbit containing any two points in R. Let x = µ1 b ∈ R and y = µ2 b with µ2 > µ1 . Then let γ be the orbit running from x to y using the flow generated by b and then running from y to x using the flow generated by Ax. Thus there is a periodic orbit in R containing x and y. Topological transitivity follows immediately as well as a dense set of periodic points (all points are periodic). Let δ > 1. Let x ∈ R, N (x) ⊂ R be a neighborhood of x and y ∈ N (x). Then follow the integral curve γ from x generated by A. Note limt→∞ γ(t) = 0. Next, follow the integral curve σ from y generated by b. Note that limt→0 ||σ(t)|| = ∞. Therefore, there exists M < ∞ such that d(γ(M ), σ(M )) > δ. This simple family of MVDSs is useful as a way of understanding the behavior of nonlinear MVDSs as well. To see this, suppose one has a non-linear system x˙ ∈ {M (x), N (x)} with M (x∗ ) = 0 and N (x∗ ) 6= 0. Suppose further that x∗ is a hyperbolic point of M , i.e., DM (x∗ ) has no purely imaginary eigenvalues. Then in a neighborhood of x∗ , M (x) behaves like Ax where A = DM (x∗ ) and N (x) behaves like b where b = N (x∗ ). For this reason, one can expect that most non-linear MVDSs near a steady state will be chaotic as well. In fact, Raines and Stockman (2007) prove the following theorems about non-linear MVDSs. Let x˙ ∈ {f (x), g(x)} where f, g : R2 → R2 are continuous. Let T := [0, ∞) and define Γ := {γ : T → R2 | γ(t) ˙ ∈ {f (γ(t)), g(γ(t))}. 10
Figure 3: Sink or source with real roots: integral curves from the MVDS given by x˙ ∈ {Ax, b}. 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Let x∗ ∈ R2 with f (x∗ ) = 0 and g(x∗ ) 6= 0. Let A = Df (x∗ ) with eigenvalues λ1 , λ2 . Theorem 4 (sink or source). Suppose there exists > 0 such that x∗ is a sink (asymptotically stable) or a source (asymptotically unstable) for f on B (x∗ ). Then there is a close invariant R such that Γ|R is chaotic. Theorem 5 (saddle). Suppose λ1 < 0 and λ2 > 0 with g(x∗ ) 6= αe1 and g(x∗ ) 6= βe2 for all α, β ∈ R. Then there is a close invariant R such that Γ|R is chaotic. I now turn to the non-linear MVDS from the model and consider two cases for the steady state: local determinacy and local indeterminacy. We will see that in both cases the model can exhibit chaotic behavior. Example 2 (Local Indeterminacy). Let parameter values be set at baseline values (see Table 1) with χ = 1 and ν = 1.25. The steady state and eigenvalues from the log-linearization around the steady state are given by K = 1.4503, λ = 1.7238, L = 0.9354, C = 0.8204, S = 1.2500, Y = 1.0255, P = 1.4142, eigenvalues µ1 = −0.0179+0.2922i, µ2 = −0.0179−0.2922i. We see the steady state is locally a sink. The integral curves from the non-linear system are plotted in Figure 4. From this figure it is clear that on a significant region around the steady state the dynamics are chaotic. Example 3 (Local Determinacy). Let parameter values be set at baseline values (see Table 1) with χ = 2.0 and ν = 1.10. The steady state and eigenvalues from the loglinearization around the steady state are given by K = 2.1026, λ = 1.1890, L = 0.9565, 11
Figure 4: The steady state is locally a sink. Plotted are integral curves from both the low-S and high-S branches. The plotted integral curves from the high-S branch flow from the top left to the bottom right. The plotted integral curves for the low-S branch are flowing counter clockwise.
Low−S Branch Integral Curve
1.76
1.75
1.74
1.73
High−S Branch Integral Curves
1.72
1.71
1.7
1.69
Steady State 1.68
1.67
1.66
1.4
1.45
1.5
12
1.55
1.6
C = 0.9661, S = 1.2500, Y = 1.2076, P = 1.1487, eigenvalues µ1 = 0.3922, µ2 = −0.2985. We see the steady state is locally a saddle. The integral curves from the non-linear system are plotted in Figure 5. We see a similarity to the integral curves depicted in Figure 2 generated by a linear function and a constant function. From this figure it is clear that on a significant region near the steady state the dynamics are chaotic. Figure 6 contains the stable/unstable manifold of the linear approximation around the saddle steady state along with the direction of the vector field of the low-S branch at the steady state. The stable/unstable manifold divides the state space into four quadrants (Q1–Q4). From the linear approximation, chaos is expected in Q1 and this is exactly what one sees for the non-linear system in Figure 5. This illustrates how, in the case of a saddle, one can use information from the linear approximation not only to establish the existence of chaos, but to numerically locate a region of chaos for the nonlinear model as well.
13
Figure 5: The steady state is locally a saddle. Plotted are integral curves from both the low-S and high-S branches. The plotted integral curves from the high-S branch (those associated with the local saddle) flow from the top left to the bottom right. The plotted integral curves for the low-S branch are flowing from the bottom right to the top left.
1.32
1.3
1.28
High S Branch Integral Curves 1.26
1.24
1.22
High S Branch Integral Curves Low S Branch Integral Curves
1.2
Steady State 1.85
1.9
1.95
2
2.05
14
2.1
2.15
2.2
2.25
2.3
Figure 6: Stable/unstable manifold from the linear approximation around the steady state. Included is the direction of the flow on the low-S branch. The stable/unstable manifold divides the state space into four quadrants labeled Q1–Q4. Given the direction of the flow on the low-S branch and the position of the stable/unstable manifolds, chaos is expected in Q1.
0.2
0.15
0.1
Flow Direction on Low−S Branch
Q2 0.05
Q1
0
Q4
Unstable Manifold Q3 −0.05
Stable Manifold −0.1
−0.15
−0.3
−0.2
−0.1
0
15
0.1
0.2
0.3
0.4
5
Extensions
In this section, I explore two extensions of the BF model: (1) sector-specific externalities with more general preferences and (2) investments in each sector are imperfect substitutes.
5.1
Sector-Specific Externalities
Harrison (2001) explores the role of sector-specific externalities in generating indeterminacy in a two-sector model. She does this by allowing (1) different degrees of externalities in the consumption sector and the investment sector (BF set these equal to each other) and (2) more general preferences over consumption (BF consider only log utility). Here, I briefly describe a continuous-time version of her model. Preferences are given by 1−σ Z ∞ L1+χ −1 t −ρt Ct − dt, e 1−σ (1 + χ) 0
(19)
with σ, χ > 0 and ρ < 0 (with utility of consumption given by log(C) for σ = 1). The evolution of capital is standard. The budget constraint is given by Ct + pt It ≤ rt Kt + wt Lt . The household seeks to maximize (19) subject to K˙ t = (1/pt )[wt Lt + rt Kt − Ct ] − δKt . The Hamiltonian is given by Ht =
L1+χ Ct1−σ − 1 − t + λt [(1/pt )(wt Lt + rt Kt − Ct ) − δKt ]. 1−σ (1 + χ)
Optimality conditions: Ct−σ = λt /pt , Lχt = λt wt /pt , λ˙ t = λt (ρ + δ − rt /pt ),
(20) (21) (22)
along with a transversality condition limt→∞ e−ρt λt Kt = 0. In the consumption sector, the firm maximizes profits given the following technology: α α β lC,t , fC,t := (KC,t LβC,t )θC kC,t
where lower-case letters denote firm values and upper-case letters denote economy-wide averages. In the investment sector, the firm maximizes profits given the following technology: α β θI α β fI,t := (KI,t LI,t ) kI,t lI,t .
16
The factor inputs receive their marginal products: pt αfI,t αfC,t = kc,t kI,t βfC,t pt βfI,t = = . lc,t lI,t
rt =
(23)
wt
(24)
Aggregate factor inputs must satisfy KC,t + KI,t = Kt , LC,t + LI,t = Lt . In equilibrium, taking into account that the firm level of inputs must equal the economywide averages, we have fC,t = (µt Ktα Lβt )1+θC ,
(25)
fI,t = [(1 − µt )Ktα Lβt ]1+θI
(26)
where
KC,t LC,t = . Kt Lt The price of the investment good (measured in the consumption good) is given by µt :=
µθt C pt = (Ktα Lβt )θC −θI . θ I (1 − µt )
(27)
(28)
Combining the equilibrium conditions, an equilibrium in the model must satisfy the following conditions (in addition to the transversality conditions): L1+χ = λt β(1 − µt )θI (Ktα Lβt )1+θI , t (pt /λt )σ = (µt Ktα Lβt )1+θC , pt =
µθt C (1 − µt
)θ I
(Ktα Lβt )θC −θI ,
λt αfI,t λ˙ t = λt (ρ + δ) − , (1 − µt )Kt " 1/σ # f 1 pt C,t K˙ t = − − δKt . pt µt λt
(29) (30) (31) (32) (33)
With Kt as the state and λt as the co-state, one uses equations (29)–(31) to solve for Lt , pt and µt . Let τ := 1 + χ − β(1 + θI ). Assuming τ 6= 0, one can use (29) to solve for L as a function of K, λ and µ: L(K, λ, µ) := [λβK α(1+θI ) ]1/τ (1 − µ)θI /τ =: l(K, λ)(1 − µ)θI /τ , 17
where l(K, λ) := [λβK α(1+θI ) ]1/τ . Using equation (30), one can solve for P as a function of K, λ and µ: P (K, λ, µ) := λ[µK α L(K, λ, µ)β ]
1+θC σ
.
Substituting for L(K, λ, µ) and simplifying gives P (K, λ, µ) := p(K, λ)µ
1+θC σ
(1 − µ)
β(1+θC )θI τσ
,
where p(K, λ) := λK
α(1+θC ) σ
l(K, λ)
β(1+θC ) σ
.
Using P (K, λ, µ) and L(K, λ, µ) in equation (31) one gets P (K, λ, µ) =
µθ C (K α L(K, λ, µ)β )θC −θI . (1 − µ)θI
The right-hand side of this equation simplifies to g(K, λ)µθC (1 − µ)
βθI (θC −θI ) −θI τ
,
where g(K, λ) := K α(θC −θI ) l(K, λ)β(θC −θI ) . Substituting for P (K, λ) and simplifying, one gets M (µ) := µa (1 − µ)b = where
g(K, λ) =: N (K, λ), p(K, λ)
β 1 + θC − σθC 1 + θC − σθC and b := θI 1 + + θI . a := σ τ σ
(34)
(35)
We see there is Euler equation branching in this model iff given K > 0 and λ > 0 there exists more than one solution to µ satisfying equation (34). Note that M (µ) > 0 for µ ∈ (0, 1) and 0
M (µ) = M (µ)
a b − µ 1−µ
.
(36)
Definition 6. The function M is single-caved if there exists a unique µ∗ ∈ (0, 1) such that M 0 (µ∗ ) = 0 and either (1) M 0 (µ1 ) < 0 and M 0 (µ2 ) > 0 or (2) M 0 (µ1 ) > 0 and M 0 (µ2 ) < 0 for all µ1 ∈ (0, µ∗ ) and µ2 ∈ (µ∗ , 1). It is single-caved up if it is single-caved with M 0 (µ) < 0 for µ ∈ (0, µ∗ ). It is single-caved down if it is single-caved with M 0 (µ) > 0 for µ ∈ (0, µ∗ ). Lemma 1. M (µ) defined in (34) is single-caved on (0, 1) if and only if a, b > 0 or a, b < 0, otherwise M is monotonic. Moreover, if a, b > 0, M is single-caved down and if a, b < 0 then M is single-caved up. 18
Proof. Suppose M is single-caved. Then there exists a unique µ∗ ∈ (0, 1) such that M 0 (µ∗ ) = 0. This implies a/µ∗ = b/(1 − µ∗ ). Since µ∗ , (1 − µ∗ ) > 0, it must be that either a, b > 0 or a, b < 0. Now assume a, b > 0 or a, b < 0. In this case, there is a unique solution to M 0 (µ∗ ) = 0 given by 0 < µ∗ := a/(a + b) < 1. If a, b > 0 then M 0 (µ) > 0 for µ ∈ (0, µ∗ ) and M 0 (µ) < 0 for µ ∈ (µ∗ , 1), so M is is single-caved down If a, b < 0 then M 0 (µ) < 0 for µ ∈ (0, µ∗ ) and M 0 (µ) > 0 for µ ∈ (µ∗ , 1), so M is single-caved down. If a ≥ 0 and b ≤ 0, then H 0 (µ) ≥ 0 and so is monotonic. If a ≤ 0 and b ≥ 0, then H 0 (µ) ≤ 0 and so is monotonic. If the function M is single-caved, then it also satisfies the following properties. Lemma 2. If a, b > 0 then lim M (µ) = lim M (µ) = 0,
µ→0
µ→1
and if a, b < 0 then lim M (µ) = lim M (µ) = +∞.
µ→0
µ→1
The following proposition follows from Lemmas 1 and 2. Proposition 3. Let a and b be defined by equation (35). There is Euler equation branching in this model iff ab > 0. Discussion. The externality in the investment sector is essential since b = 0 if θI = 0. However it is not sufficient. To see this, let θC > 0 and σ > (1 + θC )/θC so a < 0. Then b > 0 for all θI > 0 if and only if (β − 1 − χ)/β < a. If 0 < θC < (1 + χ − β)/β, then this conditions is automatically satisfied. If θC > (1 + χ − β)/β, then b > 0 iff 1 + θC 1 + θC 0 and b > 0, so Euler equation branching exists. Consider the case a, b > 0. If θI = 0, then b = 0, so we must have θI > 0. If θC = 0, then a > 0 puts no additional restrictions on σ. If θC > 0, then σ < (1 + θC )/θC . Note that β b = θI 1 + (a + θI ) . τ There are two cases to consider: τ > 0 and τ < 0. If τ > 0, then b > 0. If τ < 0, then b > 0 if and only if θC > (1 + χ − β)/β and 1 + θC < σ. θC + (β − 1 − χ)/β 19
(37)
However, θC > (1 + χ − β)/β, require θC > 0 so we must have σ
0. If θC = 0, then a > 0, so we must have θC > 0. If θC > 0 then a < 0 if and only if σ > (1 + θC )/θC . For b < 0 there are again two cases: τ > 0 and τ < 0. Suppose τ > 0, then b < 0 iff θC > (1 + χ − β)/β and (37) holds. Suppose τ < 0, then b < 0 iff θC ≤ (1 + χ − β)/β or θC > (1 + χ − β)/β and σ
(1 + χ − β)/β since 1 + θC 1 + θC 0; (P2) If θC > 0, then σ < (1 + θC )/θC ; (P3) τ := 1 + χ − β(1 + θI ) > 0; or (N1) θI > 0; (N2) θC > 0 and σ > (1 + θC )/θC ; (N3) If τ := 1 + χ − β(1 + θI ) > 0, then (37) and θC > (1 + χ − β)/β; (N4) If τ := 1 + χ − β(1 + θI ) < 0 and θC > (1 + χ − β)/β, then (38). We see (P1)–(P3) will be satisfied for modest externalities in the consumption sector, plausible values of σ and conventional downward-sloping labor demand curves (β(1+θI ) < 1). If there is no externality in the consumption sector (θC = 0), then β(1+θI ) < 1 will guarantee Euler equation branching for any σ, θI > 0. 20
5.2
Imperfect Substitutes
Herrendorf and Valentinyi (2006) study a variation of the two-sector model where investment in the consumption sector is an imperfect substitute for investment in the investment sector. They show analytically that the steady state is locally determinate for every plausible parameterization of the model. Note that these results are essentially the opposite of the model analyzed by Benhabib and Farmer (1996). Euler equation branching will also be sensitive to this attribute of the model. In particular, I will show that there is no Euler equation branching near the steady state in this model. Preferences are given by U (C, Lc , Lx ) := log C −
(Lc + Lx )1+χ , 1+χ
where C is consumption and Lc and Lx represent labor devoted to the consumption and investment sectors. The household’s budget constraint is given by Ct + Pct Xct + Pxt Xxt = πct + πxt + wct Lct + wxt Lxt + rct Kct + rxt Kxt . a 1−a Production in the consumption sector is given by At kct lct . Production in the investment b 1−b sector is given by Bt kxt lxt . This investment good is then allocated to either the consumption
sector xct or the investment sector xxt according to b 1−b f (xct , xxt ) = Bt kxt lxt ,
where f is C 2 , non-negative, homogeneous degree 1, increasing in both arguments and strictly quasi-convex. The laws of motion for the capital stocks are K˙ ct = Xct − δc Kct and K˙ xt = Xxt − δx Kxt . It is assumed that Xct , Xxt ≥ 0 so that installed capital is sector specific. Externalities in each sector are given by θ (1−a)
θc a c At = kct lct
θ (1−b)
θx b x and Bt = kxt lxt
,
with θx , θc ≥ 0. Let α1 := (1 + θc )a, α2 := (1 + θc )(1 − a), β1 := (1 + θx )b and β2 := (1 + θx )(1 − b). The household’s first order conditions are Pct /Ct = µct ,
(39)
Pxt /Ct = µxt ,
(40)
wct /Ct = (Lct + Lxt )χ ,
(41)
wxt /Ct = (Lct + Lxt )χ ,
(42)
µ˙ ct ≤ µct (ρ + δc ) − rct /Ct ,
(43)
µ˙ xt ≤ µxt (ρ + δx ) − rxt /Ct ,
(44)
21
along with transversality conditions for the capital stocks. Profit maximization in the consumption sector implies rct = aAt Kcta−1 L1−a and wct = (1 − a)At Kcta L−a ct ct .
(45)
Profit maximization in the investment sector implies b−1 1−b rxt /Pxt = bBt Kxt Lxt /f (xct , xxt ), b−1 −b wxt /Pxt = bBt Kxt Lxt /f (xct , xxt ),
(46) (47)
Proposition 5. In this model, near the steady state there is no Euler equation branching. Proof. Given the state [Kct , Kxt ] and co-state [µct , µxt ] all that needs to be shown is that the other endogenous variables are uniquely determined. This is done in a way similar to Herrendorf and Valentinyi (2006). The household FOCs imply Pct /Pxt = µct /µxt . This condition along with the investment good firm’s FOCs imply fc /fx = µct /µxt . Since f is strictly quasi-convex and homogeneous, one can use this condition to solve for the ratio of the investment goods as a function of the ratio of these multipliers: xct /xxt = g(µct /µxt ). In the consumption sector, since C = AKca L1−a and labor receives its marginal product, c we have wct = (1 − a)Ct /Lct . This condition along with the household’s FOC for labor in the consumption sector gives Lct (Lct + Lxt )χ = (1 − a).
(48)
In the investment sector, a similar condition can be found 1=
β1 β2 −1 Lxt (1 − b)µxt Kxt . χ (Lct + Lxt ) fx (g(µct /µxt ), 1)
Using (48) and (49), one can solve for equilibrium labor in the investment sector 1 " # 1−β β1 2 1−b µxt Kxt Lct Lxt = . . 1 − a fx (g(µct /µxt ), 1)
(49)
(50)
Note that Lxt is increasing in Lct (assuming 0 < β2 < 1). From this expression we can solve for the unique equilibrium Lct : Lct =
1−a . (Lct + Lxt )χ
With χ > 0, the right-hand side is strictly decreasing in Lct and approaches +∞ as Lct ↓ 0 and approaches 0 as Lct → +∞. Therefore there is a unique equilibrium Lct . This pins down Lxt and the other endogenous variables as well. 22
This proposition reinforces the main point of Herrendorf and Valentinyi (2006), namely that the ability of the two-sector model to generate multiple equilibria depends critically on whether or not the investments in each sector are perfect substitutes.
6
Conclusion
In this paper, I show that Euler equation branching occurs in the two-sector model of Benhabib and Farmer (1996) and Harrison (2001), but not in the model of Herrendorf and Valentinyi (2006). However, when such branching occurs, there will typically exist regime switching equilibria with chaotic behavior that occur whether the “local” dynamics are determinate or indeterminate.
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