Asset Pricing with Markovian Productivity Growth

Asset Pricing with Markovian Productivity Growth Volker Böhm, Tomoo Kikuchi, and George Vachadze October 2006

Discussion Paper No. 558 Department of Economics Bielefeld University P.O.Box 10 01 31 D-33501 Bielefeld Germany

Asset Pricing with Markovian Productivity Growth ∗ ¨ hm, Tomoo Kikuchi, and George Vachadze Volker Bo

†

Department of Economics Bielefeld University, P.O.Box 10 01 31 D-33501 Bielefeld, Germany October 2006 Discussion Paper No. 558

Abstract The paper studies the influence of productivity growth on asset prices in an aggregate macroeconomic growth model under different consumption hypotheses. The production side of the economy is given by a random growth model with a competitive labor market and a constant dividend pay out ratio. For an isoelastic technology with multiplicative production shocks this implies a random dynamical system for the rate of capital accumulation with a unique asymptotically stable random fixed point for a large class of mean reverting Markov processes of productivity growth. The paper examines the implications of the stationary process of productivity growth on asset prices under rational expectations for two consumption scenarios: overlapping generations with two period lives vs. the representative infinitely lived optimizing agent. We show the existence of the minimum state representation rational expectations equilibrium in both cases and apply the projection method in order to numerically approximate the equilibrium asset price map. An extensive numerical study shows that the two scenarios imply a number of diverse effects regarding asset prices, rates of return, and equity premia. Keywords: Asset pricing, economic growth, equity premium JEL classification: G12, E44, F43

∗

This paper was written as part of the project ”International Financial Markets and Economic Development of Nations” supported by the Deutsche Forschungsgemeinschaft under contract Bo. 635/12-1. † E-mail addresses: [email protected], [email protected], [email protected]

1

1 Introduction

1

Introduction

It has been one of the challenging question of macroeconomics to identify the major determinants of asset valuations in growing economies. In particular it has often been argued on the basis of empirical evidence that productivity growth, earnings, dividends, and the growth rate of output are good factors explaining the development of asset prices, for example, Fama & French (1988, 1989) and Fama (1990). Available recent US data suggest that productivity and industrial share prices move together while their respective growth rates show distinct differences in volatility. The theoretical question whether there are significant linkages between productivity growth and asset prices or asset returns, and why these act in a positive way may be subdivided into the two issues about the correlation of the levels and the variance along the orbits. The conjecture points to a specific form of interaction between the real factor markets and investment behavior and the pricing and valuation of paper assets which serve as a device for optimizing intertemporal consumption programs. Any affirmative answer to these questions in a frame work of rational expectations would necessarily exclude the possibility of rational bubbles, implying that expectations effects play a minor role in the long run. Macroeconomic theory uses two distinct models to explain intertemporal allocations, the model with overlapping generations of consumers (OLG) and the model with a representative consumer (RA). For both models different variants exist with a more or less explicitly formulated asset market structure. The challenge taken up in the present paper consists in developing a common properly extended growth model for the two consumption scenarios with exogenous Markovian productivity shocks and ask the comparative question whether one or both models can explain the observed positive relationships. The paper analyzes an extension of the standard growth model with random technological shocks augmented by explicit markets for paper assets (equity/shares) and bonds to describe the associated prices and market clearing. For the two consumption scenarios OLG and RA, the long run development of asset prices and returns under rational expectations will be examined in a fully parameterized class of models and their results will be derived and compared. In this way the approach extends the existing models in the OLG tradition (such as (as in Huberman (1984), Böhm & Wenzelburger (2002), Böhm, Kikuchi & Vachadze (2006)), Huffman (1986a, 1986) )) and those of optimal growth theory (as in Mehra & Prescott (1980, 1985) ) with asset market theory (as in Böhm & Chiarella (2005)). The purpose of the paper is two fold. On the one hand, a satisfactory and robust answer to the theoretical question posed above is sought. On the other hand, it will be demonstrated that such issues can be investigated with explicit numerical methods to generate ’generic’ and robust time series scenarios under two alternative consumption hypotheses. Combining computational methods from two different areas – approximations of functional equations and iterative procedures from random dynamical systems theory – allows a full parametric analysis of the two most commonly used models in financial 1

2

2 The Model

macroeconomics. Thus, in addition to the theoretical answer the paper offers a testing ground for comparative time series analysis in a fully stochastic environment. As such this may also open up possibilities of using experimental (theoretical) data scenarios and compare the predictive power of a given set of models using a variety of statistical tests.

2

The Model

Consider the typical structure of an aggregate economy evolving in discrete time with a production sector represented by a single infinitely lived firm producing a homogeneous output using a standard neoclassical technology with capital and labor as inputs. Capital is embodied, depreciates at a constant non stochastic rate, and there is no inventory holding. Both input factors are fully employed in each period. Output in each period is subject to a random productivity shock. After the productivity shock occurs, the firm pays wages according to the marginal product rule as income to consumers and a constant proportion/share of its random profits as dividends to share holders, while the remainder is invested in capital formation. Consumers have intertemporal preferences and choose optimal consumption plans buying equity or shares (paper assets) and bonds in associated frictionless markets, which operate in every period. All markets are competitive, agents are price takers with symmetric information. The two scenarios for the consumption sector which will be investigated consist either of overlapping generations of consumers with two period lives and consumption in both periods or of a representative agent optimizing over an infinite horizon. For simplicity, it will be assumed that there is no population growth. The central issues are 1) to investigate the impact of productivity growth on asset prices and returns and 2) to compare predicted asset returns and risk premia under the two scenarios.

2.1

Random Productivity and Capital Accumulation

Let the technology of a firm be given by a Cobb-Douglas production function, implying the produced output in each period to be given by the isoelastic per capita production function y = f (A, k) = Ak θ , (2.1) where A > 0 is the Hicks neutral productivity parameter, k is the capital intensity, and θ ∈ (0, 1) is the production elasticity1 . Wages being payed according to the marginal product rule, implying w(A, k) = f (A, k) − kfk (A, k). (2.2) 1

To minimize notation we suppress time indices whenever possible. Variables without time subscript refer to an arbitrary time period t, while subscripts 1 and −1 refer to periods t + 1 and t − 1 respectively.

2

2.1 Random Productivity and Capital Accumulation

3

The firm pays dividends as a constant fraction of operating profits, i. e. if 0 ≤ d ≤ 1 is the dividend pay-out ratio, then dividend payment in each period is given by D(A, k) = d [f (A, k) − w(A, k)] = d kfk (A, k).

(2.3)

The remaining part of profits is invested in physical capital, implying an investment function to be given by i(A, k) = (1 − d) kfk (A, k). (2.4) In order to discuss the general impact of productivity growth on the real and financial A1 variables in the economy, let us define productivity growth by a1 := ln , and make A the following assumption about its evolution. Assumption 2.1 The evolution of the productivity {at } is given by a1 = H(a, ε) = µ + αa + σε

(2.5)

where ε ∼ N (0, 1) is an i. i. d. standard normal random variable, and the parameters µ ∈ R, α ∈ (−1, 1), and σ ∈ R+ describe the mean, the speed of mean reversion, and the volatility of the productivity growth process. Thus, productivity growth is Markovian with time invariant conditional characteristics given by the parameters (µ, α, σ). Given the dividend pay out ratio d and a constant rate of capital depreciation 0 ≤ δ ≤ 1, the law of capital accumulation is given by the stochastic time one map k1 = (1 − δ)k + (1 − d)Aθk θ . (2.6) Let g1 := k1 /k denote the growth factor of capital (one plus the growth rate). Then, equation (2.6) implies that θ k k1 − (1 − δ)k (1 − d)A−1 θ θ g = = k−1 (1 − d)Aθ k − (1 − δ)k−1 =

g1 − (1 − δ) A−1 k1 − (1 − δ)k = e−a g . A k − (1 − δ)k−1 g − (1 − δ)

(2.7)

Solving equation (2.7) for g1 implies g1 = G(a, g) := (1 − δ) + ea g θ−1 (g − (1 − δ)).

(2.8)

Thus, capital growth is described by a family of strictly monotonically increasing and strictly concave maps (see Figure 2.1) G(a, ·) : R++ → R,

a ∈ R,

Therefore, g1 = G(a, g) := (1 − δ) + ea g θ−1 (g − (1 − δ)) a1 = H(a, ε) := µ + αa + σε 3

(2.9)

4

2.1 Random Productivity and Capital Accumulation

is a pair of random difference equations with additive i.i.d. noise with ε ∼ N (0, 1) and E log a < ∞. Hence, there exists an associated presentation as a random dynamical system in the sense of Arnold (1998). Let (Ω, F, P) denote the canonical representation of the stochastic process {at } and ϑ : Ω → Ω denote the left shift for any ω ∈ Ω. Then, (Ω, F, P, (ϑt )) together with the mapping G becomes a random dynamical system in the sense of Arnold (1998). 1

The mapping G has two fixed points g¯1 = 1 − δ and g¯2 = (ea ) 1−θ with g¯1 ≤ g¯2 if and only if a ≥ (1 − θ) ln(1 − δ). Hence, X := [1 − δ, ∞) ⊂ R+ is a forward invariant set and G(a, ·) : X → X is a family of contractions. Therefore, the long run development of G(a, g) 1.1 1 0.9 0.8 0.7 0.6 0.6

0.7

0.8

0.9

1

1.1

g

Figure 2.1: Graphs of G(a, ·) for a = −0.1, 0, 0.1 productivity growth is described by a unique asymptotically stable random fixed point. Specifically, one has the following lemma2 . Lemma 2.1 Given the isoelastic production function (2.1) and Assumption 2.1, 1. there exists a unique random fixed point (a unique stationary random variable) g∗ : Ω → R + , g∗ (ϑω) = G(a(ω), g∗ (ω)), almost surely (2.10) 2. which is asymptotically stable, i.e. on some set U ⊂ Ω × X lim ||gt (ω) − g∗ (ϑt ω)|| = 0 for all (ω, g0 (ω)) ∈ U.

t→∞ 2

see Schmalfuß (1996, 1998) and Böhm & Wenzelburger (2002) for details.

4

(2.11)

5

2.2 Growth of Productivity and Capital in the Long Run

2.2

Growth of Productivity and Capital in the Long Run

Lemma 2.1 combined with ergodicity implies that the limiting behavior of productivity and capital growth can be characterized completely by the statistical properties of the sample path of a single noise path of the perturbation alone. Therefore, for a full qualitative analysis for different values of the parameters, it is sufficient to consider a single time series of the above system. The following Figures display the typical numerical results of productivity and capital growth and the influence of the parameters. Figure (2.2) shows the long run behavior of a typical time series of productivity and capital growth along with some statistics. a g 0.1

1.04

0.05

1.0275

0

1.015

−0.05

1.0025

t

−0.1 100

125

150

175

t

0.99 100

200

(a) productivity growth 100

18.75

75

12.5

50

6.25

25

a −0.05

0

0.05

150

175

200

(b) capital growth

25

0 −0.1

125

0 0.99

0.1

g 1.0025

1.015

1.0275

1.04

(d) capital: T = 104

(c) productivity: T = 104

parameters: θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 statistics mean variance st. d. variation coefficient Sharpe ratio a 0.0101497 0.000400094 0.0200024 1.97073000 1.00737 g 1.0113500 0.000026521 0.0051519 0.00509407 198.248 Figure 2.2: Invariant behavior of productivity and capital

Role of noise and other parameters Figure (2.3) presents the numerical time series results of the system using the same noise sample path, describing the qualitative impact of the important parameters on the long run behavior. Each diagram contains the five time series associated with the five parameter values listed. All other parameters are kept at their benchmark level. Notice that the dividend payout ratio has no effect on growth rates (as it should be!). 5

6

2.2 Growth of Productivity and Capital in the Long Run

Also, a change in the mean reversion parameter α shows an almost negligible effect, since the series are almost identical numerically. The effects of the other parameters are of two kinds: production elasticity θ and the conditional mean µ of productivity creates a shift, a level effect on the time series (panels (a) and (d)), while the rate of depreciation δ and the conditional variance of productivity increase the conditional volatility of growth and the reversion without effecting the level (panels (b) and (f)). g g 1.15

1.15

1.1

1.1

1.05

1.05

1

1

t

0.95 100

125

150

175

100

(a) role of θ = 0.1, 0.3, 0.5, 0.7, 0.9

125

150

175

200

(b) role of δ = 0.05, 0.10, 0.15, 0.20, 0.25

g

g

1.15

1.2

1.1

1.125

1.05

1.05

1

0.975

t

0.95 100

t

0.95

200

125

150

175

t

0.9

200

100

125

150

175

200

(c) role of d = 0.05, 0.15, 0.25, 0.35, 0.45

(d) role of µ = −0.04, −0.02, 0, 0.02, 0.04

g

g

1.15

1.15

1.1

1.1

1.05

1.05

1

1

t

0.95 100

125

150

175

t

0.95

200

100

(e) role of α = −0.04, −0.02, 0, 0.02, 0.04

125

150

175

200

(f) role of σ = 0.005, 0.01, 0.015, 0.025, 0.03

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 2.3: Role of parameters on capital growth

6

7

3 Asset Pricing with OLG Consumers

3

Asset Pricing with OLG Consumers

Consider the standard OLG framework. Each generation of consumers lives for two consecutive periods. They supply one unit of labor and receive wage income only in the first period, but they consume in both periods. The asset market of the economy offers two paper assets to consumers in order to optimize intertemporal consumption streams: a discount bond and a paper asset (equity/share). Shares are traded in each period and share holders receive a dividend payed by the firm. Thus, purchasing x units of shares at price p in any period implies a random cum-dividend return p1 + d1 in the following period. Purchasing b units of the discount bond at price q implies one unit of return in the following period. Therefore, the consumption when young and old (cy , co ) are restricted by cy + xp + bq ≤ w

and

co1 ≤ x(p1 + d1 ) + b.

(3.1)

Intertemporal preferences are assumed to be time separable with constant relative risk aversion and described by the CRRA utility function u(c) :=

c1−γ − 1 1−γ

(3.2)

with discount factor 0 < β < 1. Young consumers maximize u(cy ) + βEu(co1 ),

(3.3)

subject to (3.1). Joining the OLG consumption sector and the production sector of the previous section into a competitive system with market clearing, one obtains the description of an economy E whose production sector is described by the three parameters (θ, δ, d), the capital share, the rate of capital depreciation, and the dividend pay-out ratio, the consumption sector by two parameters (β, γ), the consumer’s time discount and relative risk aversion, and by three parameters of the noise process (µ, α, σ), the mean, speed of the mean reversion, and the volatility of productivity growth.

3.1

Rational Expectations Equilibria

It is useful to describe the state of the economy at any period t by the four dimensional vector ω = (A, k, a, g), whose evolution is governed by the Markovian process induced by the stochastic difference equations (2.9) with A1 = Aea1 and k1 = kg1 . Lemma 2.1 assures that the random system converges to a unique (asymptotically stable) stationary solution. Therefore, it is natural to define the rational expectations equilibria as the associated MSV solutions describing the supporting asset prices under optimal behavior of OLG consumers. Definition 3.1 A rational expectations equilibrium of the economy is a pair of price functions (p(ω), q(ω)) and a pair of consumption functions (cy (ω), co (ω)) such that 7

8

3.1 Rational Expectations Equilibria

• for the given pair of equity and bond price functions, (p(ω), q(ω)), young and old consumers’ consumption plans satisfy the first order optimality conditions p(ω)u′ (cy (ω)) = βE [u′ (co (ω1 ))(p(ω1 ) + d(ω1 ))]

(3.4)

q(ω)u′ (cy (ω)) = βE [u′ (co (ω1 ))].

(3.5)

• for the given pair of consumption functions, (cy (ω), co (ω)), equity and bond markets clear, i.e. cy (ω) = w(ω) − p(ω)

and

co (ω) = p(ω) + d(ω);

(3.6)

Equity shares are assumed to be in positive net supply (normalized to 1) while bonds are in zero net supply. Therefore, under market clearing portfolio purchases by young consumers are always (x, b) = (1, 0) in each period and state. With these simplifications one obtains a substantial simplification of the system of functional equations required in definition 3.1 Let s(ω) denote the young consumer’s saving propensity out of wages. Then, the asset price function has the multiplicative form p(ω) = w(ω)s(ω), with cy (ω) = w(ω)(1−s(ω)) and co (ω) = p(ω) + d(ω) = w(ω)(s(ω) + π) where π is a constant given by π=

d(A, k) dAθk θ dθ = = . θ w(A, k) A(1 − θ)k 1−θ

(3.7)

Then, Assumption (3.2) on preferences implies that the Euler equation (3.4) can be rewritten Z p(ω) (p(ω1 ) + d(ω1 ))1−γ dF (ω1 |ω) (3.8) γ = β (w(ω) − p(ω)) R

where ω1 is next period’s state and F (ω1 |ω) is the one period state transition distribution function. To obtain the so called minimum state variable solution (see Wenzelburger (2006)), we divide both sides of equation (3.8) by [w(ω)]1−γ and obtain 1−γ Z s(ω) w(ω1 ) 1−γ dF (w1 |w). (3.9) = β (s(ω1 ) + π)) (1 − s(ω))γ w(ω) R Since the wage ratio is given by w(ω1 ) A1 k1θ = = ea1 g1θ . w(ω) A kθ

(3.10)

the minimum state variable solution of the integral equation (3.9) depends only on the pair (a, g). Analogously, equations (3.5) and (3.6) imply that the discount bond price also depends only on the pair (a, g) and is given by γ γ Z w(ω) 1 − s(ω) dF (ω1 |ω), (3.11) q(ω) = β s(ω1 ) + π w(ω1 ) R with the wage ratio as given in (3.10). 8

9

3.2 Numerical Methods

3.2

Numerical Methods

The numerical procedures to obtain point wise approximations for the price and consumption functions are quite involved, but nevertheless straightforward. Let L(s) denote the following functional Z θ(1−γ) L(s)(ω) := β (s(ω1 ) + π))1−γ ea1 (1−γ) g1 dF (w1 |w) (3.12) R

where a1 = H(a, ε1 ) and g1 = G(a, g). In order to solve the functional equation (3.9), write f (s(ω)) = L(s)(ω), (3.13) where f (x) =

x . (1 − x)γ

We construct the state space, discretize it, and initialize s0 (ω) := 0. At each collocation point we solve f (sn ) = L(sn−1 )(ω) for sn . For the numerical interpolation of sn (ω), we use Chebyshev polynomials defined on the interval [−1, 1] as Tj (x) = cos(j ∗ arccos(x)) for j = 0, 1, 2, . . . . Since the state space does not belong to the unit square [−1, 1] × [−1, 1], we apply the transformations h1 (a) = 2

g−g a−a − 1 and h2 (g) = 2 − 1, a−a g−g

(3.14)

where [a, a] × [g, g] is the rectangle to which the (a, g) process belongs with high prob s(a, g) ability. We approximate the unknown function f (a, g) = ln ∈ (−∞, ∞) 1 − s(a, g) by the polynomial function fˆ(a, g) =

N N X X

cij Ti (h1 (a))Tj (h2 (g)).

(3.15)

i=0 j=0

For collocation points we use zeros of the Chebychev polynomial TN +1 given by 2m + 1 xm = cos π for m = 0, 1, ..., N. (3.16) 2N + 2 In order to solve for the unknown coefficients cij , i, j = 0, 1, ..., N , we require the integral equation to hold exactly at (N + 1) × (N + 1) collocation points. The algorithm is implemented in C++. After initializing Chebychev coefficients as zero, we solve the equation f (sn ) = L(sn−1 )(ω) for sn at each collocation points. Finally, we interpolate the Chebychev polynomials and update the values of cij . We continue the process until achieving sufficiently small uniform error in the Euler equation. 9

10

4 Asset pricing with a Representative Consumer

Summarizing, the numerical procedure generates the point wise solutions of the relevant state variables under the MSV solution with rational expectations as functions of the pair of random variables (a, g). Since these converge asymptotically to a unique random fixed point (stationary solution), the associated functions of savings, asset and bond returns yield as values the stationary rational expectations solution of the associated market clearing process under price taking for any seed. Since the underlying stochastic process is unique and ergodic, its statistical properties are obtainable from the data generated by any chosen seed of random perturbations. Thus, the numerical investigation of the limiting properties of one orbit is sufficient to provide a complete description of the full model. The numerical results shown below are calculated for the same seed of random productivity to allow for a meaningful comparison of the influence of different parameters. For the standard parameter set (given in the table of Figure 2.2) the long run behavior (PDF’S) with the statistical characteristics is calculated for iterations of length T = 104 is which are shown in Figure/Table 3.1. 12

15

9

11.25

6

7.5

3

3.75

p/p−1

0 0.9

0.95

1

1.05

1.1

R−r −0.05

0

0.05

0.1

(b) Equity premium

(a) Asset price growth

R−r p/p−1

0 −0.1

Statistics OLG mean variance st. d. variation coefficient Sharpe ratio 0.00092956 0.000522618 0.2286080 24.593200 0.478091 1.01473 0.000416287 0.0204031 0.0201068 50.2244 Figure 3.1: Long run PDF’s and Correlation; T = 104

4

Asset pricing with a Representative Consumer

Consider an infinitely lived consumer who maximizes expected discounted utility (∞ ) X E β t u(ct ) t=0

where current and next period consumptions ct ≥ 0 and ct+1 ≥ 0 are subject to the budget constraint c t + x t p t + b t q t ≤ wt

and

ct+1 ≤ xt (pt+1 + dt+1 ) + bt , 10

t = 0, 1, . . .

11

4.1 Rational Expectations Equilibria

in every period, while xt is the number of equity shares purchased in the equity market at price pt , bt is the purchase of units of discount bonds at price qt , and pt+1 + dt+1 is next period’s cum-dividend payment. Thus, the consumption sector is represented again by two parameters (β, γ).

4.1

Rational Expectations Equilibria

Given the structure of the economy, a rational expectations equilibrium is defined in the same way as in the OLG case as the MSV solution of the supporting price functions satisfying optimality and feasibility along an entire orbit. Definition 4.1 A rational expectations equilibrium of the economy E is a pair of functions, (p(ω), q(ω)) for assets and bonds, and a consumption allocation function c(ω) such that • for given consumption function c(ω), pair of equity and bond price functions, (p(ω), q(ω)), satisfy the following optimality conditions p(ω)u′ (c(ω)) = βE [u′ (c(ω1 ))(p(ω1 ) + d(ω1 ))]

(4.1)

q(ω)u′ (c(ω)) = βE [u′ (c(ω1 ))].

(4.2)

• consumption function, c(ω), satisfies the feasibility constraint, i.e. c(ω) = w(ω) + d(ω); Since equity shares are in positive net supply normalized to unity and bonds are in zero net supply in every period, market clearing implies that in each temporary equilibrium young agents portfolio purchases are (x, b) = (1, 0) in each period/state. Let s denotes the asset price to wage ratio, then the consumer’s consumption plan is given by c(ω) = w(ω)+d(ω) = w(ω)(1+π), where the constant π is defined in equation (3.7). Assumption (3.2) on agent’s preferences implies that the Euler equation (4.1) can be rewritten Z p(ω) p(ω1 ) + d(ω1 ) (4.3) γ = β γ dF (ω1 |ω) (w(ω) + d(ω)) R (w(ω1 ) + d(ω1 )) where ω1 is next period’s state and F (ω1 |ω) is the one period state transition distribution function. Dividing both sides of equation (4.3) by [w(ω)]1−γ one obtains s(ω) = β

Z

R

(s(ω1 ) + π)

w(ω1 ) w(ω)

1−γ

dF (w1 |w).

(4.4)

Since the wage ratio is given by w(ω1 ) A1 k1θ = = ea1 g1θ , θ w(ω) Ak 11

(4.5)

12

4.2 Numerical Method

it follows that the minimum state variable solution of the integral equation (4.4) depends only on the pair (a, g). Analogously equations (4.2), and (4.5) imply that the discount bond price depends only on the pair (a, g) and is given by γ Z w(ω) q(ω) = β dF (ω1 |ω). (4.6) w(ω1 ) R where the wage ratio is given in equation (4.5). Thus, one finds, as in the case with overlapping generations of consumers, that the consumption and pricing process with a representative consumer is given by a list of deterministic function inducing random variables linked directly to the same productivity and growth process.

4.2

Numerical Method

Since the structural forms of the implicit inverse demand functions in both cases are similar, the numerical procedures are analogues. Let L(s) denote now the functional Z θ(1−γ) L(s)(ω) = β (s(ω1 ) + π) ea1 (1−γ) g1 dF (w1 |w) (4.7) R

where a1 = H(a, ε1 ) and g1 = G(a, g). In order to solve the functional equation (4.4), s(ω) = L(s)(ω)

(4.8)

we proceed as follows. We construct the state space, discretize it, and initialize s0 (ω) := 0. At each collocation point we solve, sn = L(sn−1 )(ω), for sn , For numerical interpolation of the sn (ω) function, we use Chebyshev polynomials (of the first kind), defined on the [−1, 1] interval as Tj (x) = cos(j ∗ arccos(x))

for

j = 0, 1, 2, ..., .

Since the state space does not belong to the unit square [−1, 1] × [−1, 1], we apply the same transformations as before g−g a−a − 1 and h2 (g) = 2 − 1. (4.9) h1 (a) = 2 a−a g−g We approximate the unknown function f (a, g) = ln (s(a, g)) ∈ (−∞, ∞) by the polynomial function N X N X ˆ f (a, g) = cij Ti (h1 (a))Tj (h2 (g)), (4.10) i=0 j=0

and use for collocation points zeros of the Chebychev polynomial TN +1 given by 2m + 1 π for m = 0, 1, ..., N. (4.11) xm = cos 2N + 2

After initializing Chebychev coefficients as zero, we determine sn = L(sn−1 )(ω) at each collocation points. Finally, we interpolate the Chebychev polynomials and update the values of cij . We continue the process until achieving sufficiently small uniform error in the Euler equation. 12

13

5 Comparing OLG and RA

5

Comparing OLG and RA

Comparing the correlation patterns productivity growth shows perfect positive correlation with asset price growth and with the equity premium. However, the predictive power of productivity growth is much stronger in the OLG model. In addition, productivity growth is negatively correlated with the bond price and the riskless rate, while asset price growth and savings are negatively correlated. Moreover, the long run asset price inflation p/p−1 has almost identical PDF’s with the same mean and almost the same variance (see Figure 5.1). There is also no distinguishable difference in autocorrelations. Thus, the predictive power of both models for the asset price process is almost the same from a statistical point of view. Given the numerical complexity, general Statistics: OLG vs. RA T = 104 , θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 mean variance s.d. var. coefficient Sharpe ratio R − r: OLG 0.00092956 0.000522618 0.2286080 24.593200 0.478091 R − r: RA 0.01780370 0.000207059 0.0143895 0.808234 1.932210 p/p−1 :OLG 1.01473 0.000416287 0.0204031 0.0201068 50.2244 p/p−1 :RA 1.01472 0.000248853 0.0155837 0.0153576 65.7559 15

15

11.25

11.25

7.5

7.5

3.75

3.75

0 −0.1

R−r −0.05

0

0.05

0.1

0 −0.1

R−r −0.05

(a) R − r :OLG 12

9

9

6

6

3

3

p/p−1

0 0.95

1

0.05

0.1

1.05

1.1

(b) R − r : RA

12

0.9

0

1.05

1.1

p/p−1

0 0.9

0.95

1

(d) p/p−1 :RA

(c) p/p−1 :OLG

Figure 5.1: Statistics: OLG versus RA in the long run bifurcation diagrams have not been calculated in order to study the role of parameters on the stationary solutions. Instead, as in the previous section, the orbits of the main 13

14


a

a

0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

p/p−1

−0.1 0.9

0.95

1

1.05

p/p−1

−0.1

1.1

0.9

0.95

(a) OLG

1.05

1.1

0.05

0.1

(b) RA

a

a

0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

−0.1 −0.1

1

R−r −0.05

0

0.05

0.1

−0.1 −0.1

R−r −0.05

(c) OLG

0

(d) RA

Figure 5.2: Correlation: OLG versus RA

variables are shown in one diagram for five different values of a parameter (from the lowest to the highest value by the coloring sequence red-green-blue-yellow-turquoise). This allows to distinguish in particular whether the parameter induces an effect on the level or on the volatility of the particular variable. The deviations are taken one by one from the standard set of parameters given in Table 2.2. When comparing the return processes of the two models, however, significant structural and statistical differences appear. These are caused by the non linear interplay of the bond price, the riskless interest rate and the stochastic return on assets together with the structural differences of the first order conditions of the two optimizing problems. Notice that these should be considered second order implications caused by the features of the consumption models but induced by the same stochastic process of incomes, dividends, and asset prices. For these effects the role of the parameters becomes important leading to diverse effects on time series and on some statistical features. It seems completely out of reach to track down the functional relationships of these effects for either model. Yet, the availability of the numerical solutions of the MSV in both cases iterations of the stochastic process allow a systematic qualitative analysis of the differences of the two models. Three of the most striking ones are presented here. One major difference appearing for the standard set of parameters is that the RA model predicts a mean risk premium R − r about twenty times that of the OLG model, where the latter is only 0.0009. The variances, however, are not that different (see Figure 5.1). 14

15


Second, increases of the capital share parameter θ induce a dramatic increase of the volatility of the risk premium in the OLG model while it causes only a moderate increase of the volatility in the representative agent model (see Figure 5.3 panels (a) and (b). As θ becomes large the production function becomes less curved inducing the same effect for the time one map of growth. This implies a large volatility of capital growth and associated incomes and dividends. At this point it cannot be explained why this causes such a large impact on the volatility of the risk premium in the OLG model and only a small one in the representative agent model. Third, consumer discounting β has no R−r R−r 0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

t

−0.1 100

125

150

175

t

−0.1

200

100

(a) OLG: θ = 0.1, 0.3, 0.5, 0.7, 0.9

175

200

R−r

0.1

0.5

0.05

0.375

0

0.25

−0.05

0.125

t

−0.1 125

150

175

t

0

200

100

125

150

175

200

(d) RA: β = 0.70, 0.80, 0.85, 0.90, 0.95

(c) OLG:β = 0.70, 0.80, 0.85, 0.90, 0.95

R−r

R−r

0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

t

−0.1 100

150

(b) RA: θ = 0.1, 0.3, 0.5, 0.7, 0.9

R−r

100

125

125

150

175

t

−0.1

200

100

(e) OLG:µ = −0.04, −0.02, 0, 0.02, 0.04

125

150

175

200

(f) RA: µ = −0.04, −0.02, 0, 0.02, 0.04

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 5.3: Role of parameters on Risk premium: OLG versus RA

15

16

6 Summary and Conclusions

effect on the risk premium in the OLG model, while it has a negative influence on the mean (see panels (c) and (d) of Figure 5.3). Thus, there exists a value for θ such that the mean risk premium of the two models coincide. Since the parameter β has slightly different interpretations in the two models, it would be necessary to discuss any issue of empirical comparability taking into account all other parameters and their empirical comparability as well.

6

Summary and Conclusions

The comparison of the two important macro economic models of asset pricing was carried out for a general stochastic growth model with a mean reverting productivity process. While the process describing the technology shocks is quite general, allowing a wide range of endogenous income processes, the investment of firms was chosen to be independent of consumer savings decisions. In contrast to other models which display significant interactions of income and risk aversion on real capital formation,3 this feed back from preferences and risk aversion on real investment is absent in the present model. In other words, real growth is uncoupled from asset valuation, implying that the latter exerts no feed back on the former. Within this scenario it has been shown that the two models generate statistically the same growth process of asset prices, i. e. from a time series point of view the properties of asset pricing in both models are completely indistinguishable. All time series show significant mean reversion and volatility clustering for low values of the Markovian reversion parameter. In both models all asset market data are uniquely and perfectly correlated with productivity growth, but essentially uncorrelated to capital growth, implying that productivity would be a perfect predicting device if it were observable. However, the two models show significant differences in the role of the parameters on time series and their statistical properties of asset market characteristics like returns, bond prices, and the risk premium. Whether these may lead to a sharper criterion to distinguish between the two models is an open question to be studied in more detail with the same tools as used here.

7

Appendix

The following figures 7.1, 7.2, 7.3, 7.4 and 7.5 show the correlation of variables as well as the role of parameters on variables in OLG Model and figures 7.6 7.7, 7.8, 7.9 and 7.10 in RA Model.

3

see for example Böhm, Kikuchi & Vachadze (2005, 2006), Böhm & Vachadze (2006)

16

17

7 Appendix

g

a

1.05

0.1

1.0313

0.05

1.0125

0

0.9938

−0.05

0.975 −0.1

p/p−1 −0.05

0

0.05

0.1

−0.1 −0.02

(a) Capital growth vs asset price growth

s 0.492

0.05

0.4919

0

0.4917

−0.05

p/p−1 −0.05

0

0.05

0.17

p/p−1

−0.1

0.1

0.9

(c) Asset price growth vs savings

0.95

1

1.05

1.1

(d) Asset price growth vs productivity

a

a

0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

q 0.9448

0.9465

0.9483

−0.1 −0.1

0.95

(e) Bond price vs productivity

R−r −0.05

0

0.05

0.1

(f) Risk premium vs productivity

s

a

0.4922

0.1

0.492

0.05

0.4919

0

0.4917

−0.05

0.4915 −0.1

0.1225

a 0.1

−0.1 0.943

0.075

(b) Equity return vs productivity

0.4922

0.4915 −0.1

R 0.0275

a −0.05

0

0.05

−0.1 0.0525

0.1

(g) Savings vs productivity

r 0.0544

0.0563

0.0581

(h) Riskless rate vs productivity

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 7.1: Correlation in OLG Model 17

0.06

18

7 Appendix

s

s

0.5

0.5

0.425

0.4875

0.35

0.475

0.275

0.4625

t

0.2 100

125

150

175

t

0.45

200

100

(a) θ = 0.1, 0.3, 0.5, 0.7, 0.9

s 0.4875

0.475

0.475

0.45

0.4625

0.425

t

0.45 125

150

175

200

t

0.4

200

100

(c) d = 0.05, 0.15, 0.25, 0.35, 0.45

125

150

175

200

(d) β = 0.70, 0.80, 0.85, 0.90, 0.95

s

s

0.55

0.5

0.525

0.4875

0.5

0.475

0.475

0.4625

t

0.45 125

150

175

t

0.45

200

100

125

150

175

200

(f) µ = −0.04, −0.02, 0, 0.02, 0.04

(e) γ = 0.5, 1.0, 3.0, 7.0, 11.0

s

s

0.5

0.5

0.4875

0.4875

0.475

0.475

0.4625

0.4625

t

0.45 100

175

s 0.5

100

150

(b) δ = 0.05, 0.10, 0, 15.0.20

0.5

100

125

125

150

175

t

0.45

200

100

(g) α = −0.04, −0.02, 0, 0.02, 0.04

125

150

175

(h) σ = 0.005, 0.01, 0.015, 0.025, 0.03

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 7.2: Role of parameters on savings in OLG Model 18

200

19

7 Appendix

r

r

10

0.22

7.5

0.21

5

0.2

2.5

0.19

t

0 100

125

150

175

t

0.18

200

100

(a) θ = 0.1, 0.3, 0.5, 0.7, 0.9

r 0.375

0.21

0.25

0.2

0.125

0.19

t

0 125

150

175

200

t

0.18

200

100

(c) d = 0.05, 0.15, 0.25, 0.35, 0.45

125

150

175

200

(d) β = 0.70, 0.80, 0.85, 0.90, 0.95

r

r

0.23

0.3

0.215

0.25

0.2

0.2

0.185

0.15

t

0.17 125

150

175

t

0.1

200

100

125

150

175

200

(f) µ = −0.04, −0.02, 0, 0.02, 0.04

(e) γ = 0.5, 1.0, 3.0, 7.0, 11.0

r

r

0.22

0.22

0.21

0.21

0.2

0.2

0.19

0.19

t

0.18 100

175

r 0.22

100

150

(b) δ = 0.05, 0.10, 0, 15.0.20

0.5

100

125

125

150

175

t

0.18

200

100

(g) α = −0.04, −0.02, 0, 0.02, 0.04

125

150

175

200

(h) σ = 0.005, 0.01, 0.015, 0.025, 0.03

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 7.3: Role of parameters on risk less interest rate in OLG Model 19

20

7 Appendix

R−r

R−r

0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

t

−0.1 100

125

150

175

t

−0.1

200

100

125

(a) θ = 0.1, 0.3, 0.5, 0.7, 0.9

R−r 0.05

0.05

0

0

−0.05

−0.05

t

−0.1 125

150

175

t

−0.1

200

100

(c) d = 0.05, 0.15, 0.25, 0.35, 0.45

125

150

175

200

(d) β = 0.70, 0.80, 0.85, 0.90, 0.95

R−r

R−r

0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

t

−0.1 125

150

175

t

−0.1

200

100

125

150

175

200

(f) µ = −0.04, −0.02, 0, 0.02, 0.04

(e) γ = 0.5, 1.0, 3.0, 7.0, 11.0

R−r

R−r

0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

t

−0.1 100

200

R−r 0.1

100

175

(b) δ = 0.05, 0.10, 0, 15.0.20

0.1

100

150

125

150

175

t

−0.1

200

100

(g) α = −0.04, −0.02, 0, 0.02, 0.04

125

150

175

(h) σ = 0.005, 0.01, 0.015, 0.025, 0.03

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 7.4: Role of parameters on risk premium in OLG Model 20

200

21

7 Appendix

p/p−1

p/p−1

1.3

1.3

1.2

1.2

1.1

1.1

1

1

t

0.9 100

125

150

175

t

0.9

200

100

125

(a) θ = 0.1, 0.3, 0.5, 0.7, 0.9

p/p−1 1.2

1.2

1.1

1.1

1

1

t

0.9 125

150

175

t

0.9

200

100

(c) d = 0.05, 0.15, 0.25, 0.35, 0.45

125

150

175

200

(d) β = 0.70, 0.80, 0.85, 0.90, 0.95

p/p−1

p/p−1

1.3

1.3

1.2

1.2

1.1

1.1

1

1

t

0.9 125

150

175

t

0.9

200

100

125

150

175

200

(f) µ = −0.04, −0.02, 0, 0.02, 0.04

(e) γ = 0.5, 1.0, 3.0, 7.0, 11.0

p/p−1

p/p−1

1.3

1.3

1.2

1.2

1.1

1.1

1

1

t

0.9 100

200

p/p−1 1.3

100

175

(b) δ = 0.05, 0.10, 0, 15.0.20

1.3

100

150

125

150

175

t

0.9

200

100

(g) α = −0.04, −0.02, 0, 0.02, 0.04

125

150

175

200

(h) σ = 0.005, 0.01, 0.015, 0.025, 0.03

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 7.5: Role of parameters on Asset Price Inflation in OLG Model 21

22

7 Appendix

g

a

1.05

0.1

1.0313

0.05

1.0125

0

0.9938

−0.05

0.975 0.95

p/p−1 0.9875

1.025

1.0625

1.1

−0.1 −0.025

(a) Capital growth vs asset price growth

s 2.875

0.05

2.75

0

2.625

−0.05

p/p−1 0.9375

1

1.0625

(c) Asset price growth vs savings

a

0.05

0.05

0

0

−0.05

−0.05

q 0.96

0.97

0.98

p/p−1 0.9

0.1

0.95

−0.1 −0.1

0.99

1

1.05

1.1

(d) Asset price growth vs productivity

(e) Bond price vs productivity

R−r −0.05

0

0.05

0.1

(f) Risk premium vs productivity

s

a

3

0.1

2.875

0.05

2.75

0

2.625

−0.05

2.5 −0.1

0.125

−0.1

1.125

0.1

−0.1 0.95

0.0875

a 0.1

a

0.05

(b) Equity return vs productivity

3

2.5 0.875

R 0.0125

a −0.05

0

0.05

−0.1 0.0125

0.1

(g) Savings vs productivity

r 0.0219

0.0313

0.0406

(h) Riskless rate vs productivity

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 7.6: Correlations in RA Model

22

0.05

23

7 Appendix

s

s

25

3

18.75

2.875

12.5

2.75

6.25

2.625

t

0 100

125

150

175

t

2.5

200

100

(a) θ = 0.1, 0.3, 0.5, 0.7, 0.9

s 5.25

1.125

3.5

0.75

1.75

0.375

t

0 125

150

175

200

t

0

200

100

(c) d = 0.05, 0.15, 0.25, 0.35, 0.45

125

150

175

200

(d) β = 0.70, 0.80, 0.85, 0.90, 0.95

s

s

5

35

3.75

26.25

2.5

17.5

1.25

8.75

t

0 125

150

175

t

0

200

100

125

150

175

200

(f) µ = −0.04, −0.02, 0, 0.02, 0.04

(e) γ = 0.5, 1.0, 3.0, 7.0, 11.0

s

s

3

3.5

2.875

3.25

2.75

3

2.625

2.75

t

2.5 100

175

s 1.5

100

150

(b) δ = 0.05, 0.10, 0, 15.0.20

7

100

125

125

150

175

t

2.5

200

100

(g) α = −0.04, −0.02, 0, 0.02, 0.04

125

150

175

(h) σ = 0.005, 0.01, 0.015, 0.025, 0.03

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 7.7: Role of parameters on savings in RA Model 23

200

24

7 Appendix

r

r

0.5

0.05

0.375

0.0375

0.25

0.025

0.125

0.0125

t

0 100

125

150

175

t

0

200

100

(a) θ = 0.1, 0.3, 0.5, 0.7, 0.9

r 0.0375

0.0375

0.025

0.025

0.0125

0.0125

t

0 125

150

175

200

t

0

200

100

(c) d = 0.05, 0.15, 0.25, 0.35, 0.45

125

150

175

200

(d) β = 0.70, 0.80, 0.85, 0.90, 0.95

r

r

0.2

0.15

0.15

0.075

0.1

0

0.05

−0.075

t

0 125

150

175

t

−0.15

200

100

125

150

175

200

(f) µ = −0.04, −0.02, 0, 0.02, 0.04

(e) γ = 0.5, 1.0, 3.0, 7.0, 11.0

r

r

0.05

0.05

0.0375

0.0375

0.025

0.025

0.0125

0.0125

t

0 100

175

r 0.05

100

150

(b) δ = 0.05, 0.10, 0, 15.0.20

0.05

100

125

125

150

175

t

0

200

100

(g) α = −0.04, −0.02, 0, 0.02, 0.04

125

150

175

200

(h) σ = 0.005, 0.01, 0.015, 0.025, 0.03

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 7.8: Role of parameters on risk less interest rate in RA Model

24

25

7 Appendix

R−r

R−r

0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

t

−0.1 100

125

150

175

t

−0.1

200

100

125

(a) θ = 0.1, 0.3, 0.5, 0.7, 0.9

R−r 0.05

0.375

0

0.25

−0.05

0.125

t

−0.1 125

150

175

100

(c) d = 0.05, 0.15, 0.25, 0.35, 0.45

0.05

0.05

0

0

−0.05

−0.05

t

−0.1 175

150

175

200

(d) β = 0.70, 0.80, 0.85, 0.90, 0.95

0.1

150

125

R−r

0.1

125

t

0

200

R−r

t

−0.1

200

100

125

150

175

200

(f) µ = −0.04, −0.02, 0, 0.02, 0.04

(e) γ = 0.5, 1.0, 3.0, 7.0, 11.0

R−r

R−r

0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

t

−0.1 100

200

R−r 0.5

100

175

(b) δ = 0.05, 0.10, 0, 15.0.20

0.1

100

150

125

150

175

t

−0.1

200

100

(g) α = −0.04, −0.02, 0, 0.02, 0.04

125

150

175

(h) σ = 0.005, 0.01, 0.015, 0.025, 0.03

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 7.9: Role of parameters on risk premium in RA Model

25

200

26

7 Appendix

p/p−1

p/p−1

1.3

1.3

1.2

1.2

1.1

1.1

1

1

t

0.9 100

125

150

175

t

0.9

200

100

125

(a) θ = 0.1, 0.3, 0.5, 0.7, 0.9

p/p−1 1.2

1.2

1.1

1.1

1

1

t

0.9 125

150

175

100

(c) d = 0.05, 0.15, 0.25, 0.35, 0.45

1.2

1.2

1.1

1.1

1

1

t

0.9 175

150

175

200

(d) β = 0.70, 0.80, 0.85, 0.90, 0.95

1.3

150

125

p/p−1

1.3

125

t

0.9

200

p/p−1

t

0.9

200

100

125

150

175

200

(f) µ = −0.04, −0.02, 0, 0.02, 0.04

(e) γ = 0.5, 1.0, 3.0, 7.0, 11.0

p/p−1

p/p−1

1.3

1.3

1.2

1.2

1.1

1.1

1

1

t

0.9 100

200

p/p−1 1.3

100

175

(b) δ = 0.05, 0.10, 0, 15.0.20

1.3

100

150

125

150

175

t

0.9

200

100

(g) α = −0.04, −0.02, 0, 0.02, 0.04

125

150

175

200

(h) σ = 0.005, 0.01, 0.015, 0.025, 0.03

θ = 0.3, δ = 0.1, d = 0.2, β = 0.99, γ = 2, µ = 0.01, α = 0.01, σ = 0.02 Figure 7.10: Role of parameters on Asset Price Inflation in RA Model

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REFERENCES

References Arnold, L. (1998): Random Dynamical Systems. Springer-Verlag, Berlin a.o. ¨ hm, V. & C. Chiarella (2005): “Mean Variance Preferences, Expectations ForBo mations, and the Dynamics of Random Asset Prices”, Mathematical Finance, 15(1), 61–97. ¨ hm, V., T. Kikuchi & G. Vachadze (2005): “Welfare and the Role of Equity in Bo an Economy with Capital Accumulation”, Discussion paper no. 547, Department of Economics, Bielefeld University, Bielefeld. ¨ hm, V., T. Kikuchi & G. Vachadze (2006): “On the Role of Equity for the Bo Dynamics of Capital Accumulation”, Discussion paper no. 551, Department of Economics, Bielefeld University, Bielefeld. ¨ hm, V. & G. Vachadze (2006): “Credit Risk and Symmetry Breaking Through Bo Financial Market Integration”, Discussion paper no. 555, Department of Economics, Bielefeld University, Bielefeld. ¨ hm, V. & J. Wenzelburger (2002): “Perfect Predictions in Economic Dynamical Bo Systems with Random Perturbations”, Macroeconomic Dynamics, 6(5), 687–712. Fama, E. (1990): “Returns and Real Activity”, Journal of Finance, 45, 1089–1107. Fama, E. & K. R. French (1988): “Dividend Yields and Expected Stock Returns”, Journal of Financial Economics, 22, 3–25. Fama, E. & K. R. French (1989): “Business Conditions and Expected Returns on Stocks and Bonds”, Journal of Financial Economics, 25, 23–49. Huberman, G. (1984): “Capital Asset Pricing in an Overlapping Generations Model”, Journal of Economic Theory, 33, 232–248. Huffman, G. (1986a): “Asset Pricing with Capital Accumulation”, Journal of Economic Review, 27(3), 565–581. Huffman, G. (1986b): “Asset Pricing with Capital Accumulation”, International Economic Review, 27(3), 565–582. Mehra, R. & E. C. Prescott (1980): “Recursive Competitive Equilibrium: The Case of Homogenous Households”, Econometrica, 48, 1365–1379. Mehra, R. & E. C. Prescott (1985): “The Equity Premium: A Puzzle”, Journal of Monetary Economics, 15(2), 145–161. Schmalfuß, B. (1996): “A Random Fixed Point Theorem Based on Lyapunov Exponents”, Random and Computational Dynamics, 4(4), 257–268.

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Schmalfuß, B. (1998): “A Random Fixed Point Theorem and the Random Graph Transformation”, Journal of Mathematical Analysis and Application, 225(1), 91–113. Wenzelburger, J. (2006): “Learning in Linear Models with Expectational Leads”, Journal of Mathematical Economics, 42, 854–884.

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