Self-fulfilling Expectations in an OLG Model with Credit Market Imperfection

Transcription

1 Self-fulfilling Expectations in an OLG Model with Credit Market Imperfection Tomoo Kikuchi National University of Singapore Department of Economics Singapore George Vachadze Bielefeld University Department of Economics Germany June 2, 2009 The authors acknowledge financial support of the German Research Foundation (DFG) Bo. 635/2 and the National University of Singapore for the Academic Research Fund Financial Market and Globalization. Corresponding author. Fax: (65) ,

2 Abstract We study Matsuyama s (Econometrica, 72, p , 2004) model of credit market imperfection when agents have a forward looking saving behavior. Multiple equilibria arise whenever the credit market is sufficiently imperfect. This paper is a one dimensional illustration of an overlapping generations economy how self-fulfilling expectations can bring the economy out of steady states and also cause endogenous cycles. We derive a sufficient condition for forward equilibrium paths to converge to a unique fractal attractor independent of initial conditions. JEL classification: O; O6; E32 Keywords: Credit market imperfection; Multiple equilibria; Self-fulfilling beliefs; Endogenous cycles; Fractal attractor 2

3 Introduction It is well known that that credit market imperfection might generate multiple steady states (e.g. [7], [], and [9]). Mookherjee and Ray [7] show that multiple steady states arise in a non-convex model with forward looking saving behavior of agents when capital market is imperfect. Whether agents make forward looking saving decisions or not, these models predict that the economy converges to one or another steady state depending on initial conditions. Alternatively, there are models of external economies which analyze the importance of expectations for selecting the equilibrium path to one or another steady state (e.g. [6], [9], and [8]). In these models when self-fulfilling expectations matter, coordination failure of expectations is the main cause of underdevelopment of the economies. The choice of one equilibrium out of many can be viewed as a complementarity game. For example, in attempt to optimize own choice, each agent must predict the choice of other agents. Matsuyama [0] (p.245) states From a model with multiple equilibria or multiple steady states, one can construct fluctuating equilibrium paths by letting the players in the economy play different equilibria in different periods. A jump from one equilibrium to another is coordinated by sunspots, a random variable with no intrinsic effect on the economy. Here, sunspots is viewed as coordination devices upon which agents coordinate their decisions on extrinsic uncertainty, market uncertainty not transmitted through the funda- Agents in these models do not have a forward looking saving decision. 3

4 mentals. 2 Motivation of such models with sunspots equilibrium is usually to explain volatility of market outcomes or business cycles in contrast to models with self-fulfilling expectations, which predict long run stationary outcomes. Matsuyama [] extends the Diamond overlapping generations model with two period lives to include credit market imperfection. To avoid multiplicity of steady states in autarky, agents in his model are assumed to consume only in the second period and simply save their entire wage income. This makes capital stock in autarky to converge to the unique steady state independently of the imperfect credit market. 3 This present paper introduces first period consumption into the Matsuyama model in order to analyze the effect of credit market imperfection on dynamics of the autarky economy. Intertemporal consumption decision leads to a saving function dependent on the interest rate. Savings play a dual role in this model. On the one hand, savings provide the supply of capital. On the other hand, in the presence of credit market imperfection, savings allow more agents to start investment projects by weakening the borrowing constraint. Therefore, the demand for credit and the interest rate may increase simultaneously. The resulting non-monotonic saving function generates multiplicity 2 See [20] for an extensive survey on sunspot equilibrium. 3 The imperfection generates a mechanism whereby diversity is endogenously generated when economies with inherently identical characteristics interact in the international financial market. This symmetry breaking phenomenon is particularly clear in [] because the steady state for the world economy is symmetric when all countries operate in autarky. 4

5 of equilibrium. 4 The dual role of saving induces complementarity between individual and aggregate saving. We show that both complementarity and high elasticity of saving with respect to the interest rate are necessary for multiplicity of equilibrium. When multiple equilibria exist, the situation is a typical case of sunspot equilibrium where extrinsic uncertainty matters and the forward dynamics is ill-defined. We make use of the theory of the so called random iterated function systems (RIFS) to obtain well defined forward orbits in such cases. 5 Suppose that multiple equilibria exist. As agents are assumed to be symmetric, the correlation between individual and aggregate savings in equilibrium is unity. This implies that optimistic expectations of high aggregate savings leads to high individual savings and pessimistic expectations of low aggregate savings leads to low individual savings. Hence, expectations are self-fulfilling. We derive parameter conditions for which both multiple equilibria and multiple steady states occur. When multiple rational expectations equilibria exist at steady states, the economy can jump out of the steady states through coordinating expectations. In such situation, not only initial conditions or history but also coordination of expectations determine the path of the economy. Moreover, we show that endogenous cycles can arise through switching between optimistic and pessimistic expectations. Finally, we show under certain conditions that the forward equilibrium 4 The multiplicity is neither a result of non-convex technology nor external economies in production but of indivisibility of investment and imperfect credit market. 5 See [3], [4], [5], and [6]. 5

6 paths converge to a fractal set. The structure of the paper is as follows. Section 2 formulates the model and derives the necessary and sufficient conditions for multiple equilibria. Section 3 defines and analyzes the forward orbit of the economy. Conditions which generate endogenous cycles are derived. Remaining proofs can be found in Appendix. For demonstration purpose, all figures are made using constant relative risk aversion utility function and Cobb- Douglas production function. It is easy to show that all assumptions made in this paper are satisfied for these functions. 2 Set up As a first step, we describe the investment and saving behavior of the agents separately. of the economy. Every agent i Then, we connect the two to define the equilibrium 0, lives for two periods, supplying one unit of labor in the first period and consuming in both periods. Successive generations have unit mass. At time t, production combines the current stock of capital k t supplied by the old with the unit quantity of labor supplied by the young. 6 The resulting per-capita output is f k t, where f : is 2 and satisfies f 0 0, f k 0 f k, lim k f k 0, and lim k 0 f k. Factor markets are competitive, paying young agents the wage W k t : f k t k t f k t, and old agents 6 Capital may be either human or physical. It depreciates fully between periods, so capital stock is equal to investment. 6

7 a gross return on capital given by f k t. After production and the distribution of factor payments, the old consume and exit the model, while the young take their wage earnings and make saving and investment decisions. 2. Investment behavior When investing, young agents can either lend in the credit market at gross interest rate r t, or run a discrete indivisible project, which takes one unit of the consumption good in period t and returns R units of the capital good in period t. Let s t be the aggregate savings in the economy. The interest rate adjusts so that savings is equal to investment, and hence capital stock evolves according to k t Rs t. () If all young agents start projects, then the capital stock at t is R. This leads to the resource constraint 0 k t R. (2) The gross rate of return on the project, measured in units of the consumption good, is R f k t. whenever Thus investors are willing to start the project r t R f k t. (3) We refer to this inequality as the profitability constraint. 7

8 Suppose that s t for all t. Then, the resource constraint (2) is never binding. We will later restrict our parameters such that this holds. Then, young agents who start projects must borrow s t at rate r t. 7 As a result, their obligation at t is given by r t s t. Against this obligation, 0, of their expected borrowers can only credibly pledge a fraction earnings R f k t. The borrowing constraint is therefore r t s t R f k t. (4) The parameter can be interpreted as a measure of credit market imperfection, with higher values corresponding to lower imperfection. Equilibrium implies that the mass of agents who start projects (and hence k t ) increases until one of the constraints either (3) or (4) binds. From this reasoning we obtain r t r s t : min s t R f Rs t, R f Rs t (5) where we make use of the capital accumulation law (). r t r s t is the interest rate consistent with the optimal investment behavior commonly referred to as the investment curve. It follows from (5) that the borrowing constraint is binding and the investment curve may be non-monotonic when s t 0,. 8 This means that the interest rate consistent with the optimal investment behavior can be increasing with respect to the aggregate savings. This is because savings 7 We denote individual savings as s i t. Since all agents are symmetric and have unit mass, s i t s t. 8 For s t! ", #, r$&% s t' R 2 f$ $(% Rs t'*) 0. 8

9 play a dual role. On the one hand, savings provide the supply of capital, which lowers the return on capital and therefore the interest rate. On the other hand, savings allow more agents to start investment projects by weakening the borrowing constraint. This causes the demand for credit to increase. If the latter effect is stronger, savings increase with the interest rate. Non-monotonicity of the investment curve, shown in Figure, 9 implies possible multiplicity of equilibrium in the credit market even for monotonic saving schedule. r t PSfrag replacements s t Figure : Investment curve 9 All figures are made using the additive separable utility function u% c ',+ βu% c 2' c where u% c' : - γ.. γ with the coefficient of relative risk aversion parameter γ % 0, # and the Cobb-Douglas production function f % k' : k α with constant capital share α % 0, '. 9

10 / 2.2 Saving behavior When the borrowing constraint is binding, the young agents strictly prefer borrowing to lending. In this case, the equilibrium allocation in the credit market involves credit rationing where the fraction s t of young agents are denied credit. Since all agents are identical, the probability that an agent i can obtain credit when the borrowing constraint is binding, is s t. The young make their saving decision before the allocation of credit and, therefore, their second period consumption is random and its expected value is / c 2 s t R f 0 Rs t r t s i t 2 3 s t r t s i t s t R f 0 Rs t r t r t s i t. (6) When the profitability constraint is binding, the young must be indifferent between lending and borrowing and therefore the expected consumption in the second period is 0 / c 2 r t s i t. (7) Let the preference of agents be additive separable across time. Furthermore, let us assume that agents are risk neutral. This implies that the certainty equivalent of random consumption must equal its expected value. Therefore, the utility function takes the following form: u c v 0 In this case, the second period consumption is deterministic but we denote it in terms of expectations for notational convenience. c 2 0

11 / / where u and v are both 2, strictly increasing, strictly concave, and satisfy Inada conditions. When individual agents make their saving decisions, they take the aggregate savings and the interest rate as given. Then, their maximization problem takes the form: max s i465 0,W7 k t8:9 u W k t ; s i t v c 2. The first order condition is u W k t ; s i t < r t v c 2. (8) The solution s i t >= s t, r t to (8) and r t a function of aggregate savings. If s t r s t define individual saving as,, we obtain that dsi provided that the elasticity of intertemporal substitution γ v c :? ds 0 c8 v@ 7 c8 is less than one, an assumption that we make below. Therefore, individual and aggregate savings are substitutes when the borrowing constraint is not binding. When the borrowing constraint is binding, however, individual and aggregate savings may become complementary for some s t, i.e., the saving curve may have an upward sloping segment as shown in Figure Equilibrium For a given W k t, equilibrium is a solution to s i t A= s t, r s t 2 and s i t s t. When individual and aggregate savings are substitutes, optimal individual saving is a decreasing function of aggregate savings and therefore

12 s i t PSfrag replacements B s t Figure 2: Individual saving as a function of aggregate saving there exists a unique equilibrium. However, when they are locally complementary, there may exist multiple solutions as shown by the intersection of s i t C= s t, r s t 2 and s i t s t in Figure 2. In this case, there exist multiple pairs of individual and aggregate savings s i t, s t which are consistent with rational expectations and equilibrium is indeterminate. By the symmetry of individuals it is always trivially true that low individual savings correspond to low aggregate savings and high individual savings to high aggregate savings. In the following we first establish necessary conditions for multiple equilibria. Let ε f@ k :D capital. Assumption 2.. k k8 f@ 7 k8 denote the elasticity of marginal product of 2

13 IIIH J KIII (a) k f k is non-decreasing or equivalently ε f@ F 0, (b) cv c is non-decreasing or equivalently γ v F 0, Proposition 2.. If Assumption 2. is satisfied, then, (a) complementarity between individual and aggregate saving (b) sufficiently large elasticity of saving are necessary for multiple equilibria. Next we establish necessary and sufficient conditions for existence and multiplicity of equilibrium. Given (), (8), and s i t s t, the equilibrium saving rate s t is the solution to where Γ s : u W Rs tg ; s t < of s t, which we denote G s t Γ s t if s t u W Rs tg ; s t < Γ s t if s t s tg Ψ s t : 0,, R f Rs v0 Rs f Rs 2. Note that (9) defines s tg as a function where Ψ 0 S 0 and Ψ s UT 0 if s to W R V W u G R f R WL M s t 7 u@ 8 L N L s t Γ7 s t8&oqp R if s t 0, (9) (0) WL s t 7 u@ 8 L 7 Γ7 s t8r8 R if s t,,. 2 Let R be the solution v R f R 2, which is equivalent to Since 0 X lim sy 0 s f$ % s' X f % 0' 0, lim sy 0 s f$ % s' 0. Then, Ψ% 0' 0 follows from the properties of f, u, and v. 2 Γ% s' is non-decreasing when Assumption 2. is satisfied. 3

14 Ψ. If Assumption 2. is satisfied, then the left hand side is nonincreasing. As W is increasing, if there exists a solution R, it is unique. We assume that W R Z [ u G R f R v R f R 2. In other words, we consider only R \ 0, R or Ψ ]T. As seen in the next proposition, this assumption corresponds to Assumption (A) in Matsuyama [] and ensures that s t, so that agents need to borrow s t in the competitive credit market to start the project and therefore the resource constraint (2) is never binding along the forward orbit as has been assumed. Now, for R W 0, R, continuity of Ψ ensures that there always exists at least one equilibrium pair s tg, s t F 0, <^ 0, for all t. Let G : R be a sufficiently small, which is defined in Lemma 4. in Appendix and s c be a critical point such that Ψ s c _ 0. The following proposition establishes that for given initial conditions there may exist multiple equilibria if the credit market is sufficiently imperfect. Proposition 2.2. If Assumption 2. is satisfied, then for any R [ 0, R, there exist three equilibria if and only if ` 0, G and s ` Ψ, Ψ s c 2, otherwise, there exists a unique equilibrium. 3 Dynamics Backward dynamics is defined by s t Ψ s t. Whether the forward dynamics is well defined depends on invertibility of Ψ. Proposition 2.2 implies that for any R [ 0, R there exists a sufficiently small G such that 4

15 IIIIIIIIH J KIIIIIIII h Ψb s cc Ψb a c PSfrag replacements Ψb sc PSfrag replacements Ψb sc Ψb s cc PSfrag replacements Ψb sc sh Ψb cc shs Ψb a c Ψb a c sh s c a s s c a s s c a s (a) Unique steady state s L (b) Unique steady state s H (c) Multiple steady states s L ) s M ) s H Ψ d Figure 3: Different configurations of Ψ function 0 and thus Ψ is not invertible. The theory of so-called random iterated function systems (RIFS) provides a framework to define forward 0, there exists orbits for ill-defined forward system. 3 When Ψ e a unique s c such that Ψ is monotonically increasing on 0, s c f, and is monotonically decreasing on s c,. Let Ψ G L, ΨG M denote the inverse of Ψ on the intervals 0, s c, s c,, and, respectively. Then for any s 0 F 0,, we can define the RIFS s t Ψ G L s t if s t g 0, Ψ 2 Ψ G L s t, Ψ G M s t, Ψ G H s t ji if s t Ψ, Ψ s c Ψ G H s t if s t g Ψ s c,, and ΨG H () with non-negative probabilities h p L, p M, p H attached to Ψ G L s t, Ψ G M s t, Ψ G H s t ji and p L p M p H when s t Ψ, Ψ s c. The system () randomizes over multiple solutions by attaching probability to them. 3 See [3], [4], [5], and [6]. 5

16 For any R 0, R the state space is taken to be 0,. 4 or Ψ ST, () induces a forward invariant set and 3. Steady states From (0), the steady state s is defined by the solution to s Ψ s. We first establish the existence of steady states. Let γ u c :k c8 u@ 7 denote c8 the elasticity of intertemporal substitution, σ f f@l7 k : k8 W7 k8 f 7 k8 W@ 7 k8 the elasticity k of substitution between capital and labor, and ε f f@l7 k : k8 f 7 k8 the elasticity of production. Assumption 3.. σ f 0 dt ε f 0 and 0 γ u 0 γ v 0 Under these assumptions we obtain the following elementary result. Lemma 3.. If Assumption 3. is satisfied, then, Ψ 0 < 0. If Assumptions 2. and 3. are satisfied, then for any R m 0, R and 0,, continuity of Ψ ensures that there always exists at least one steady state. Note that Ψ 0 n 0 also ensures the instability of zero under (). 5 We will now restrict our attention to a class of production and utility functions that satisfy certain properties. They guarantee the existence of at most three steady states and hence simplify our exposition. 4 We exclude zero from the state space in order to rule out trivial steady states.. 5 ε Since σ f o f ε fp, if σ f % 0'rq ε f % 0', ε f % 0'r) s ε fpet 0u ) given ε fp % 0'rq 0. Therefore, Assumption 2. implies the condition on non-vanishing labor share lim ky 0 ε f % k' ]! 0, ' in Wendner [2]. 6

17 Assumption 3.2. (a) σ f k vt k R f 7 k8 (b) 0 sup γ u c inf γ v c The assumption (a) is equivalent to the condition on minimum elasticity of substitution in Wendner [2] that is required to show the existence of a unique and globally stable steady state when R and. 6 We can now state the main result of this section. Proposition 3.. If Assumptions 2., 3., 3.2 are satisfied, then for any R 0, R and g 0,,. there exist at most three steady states. 2. for F 0, G, (a) there exists a unique steady state s L W 0,, if and only if Ψ. (b) there exists a unique steady state s H m,, if and only if s c T Ψ s c. (c) there exist three steady states 0 s L s M s H, if and only if T Ψ and s c Ψ s c. Proposition 3. (a), (b), and (c) corresponds to Figure 3 (a), (b), and (c). 6 Wendner [2] considers the economy with positive population growth rate n q 0. 7

18 3.2 Self-fulfilling expectations When individuals make their saving decisions, they take aggregate savings as given. In case of multiple equilibria, there exist multiple aggregate savings that are consistent with rational expectations. The choice of aggregate savings in equilibrium is made outside the market mechanism. As all individuals are assumed to be symmetric in our model, they share the same expectations. The following proposition states conditions for which there exist multiple equilibria at steady states s L and s H. Proposition 3.2. If Ψ has three steady states s L s M s H and Ψ d s L and s H Ψ s H, then for any R g 0, R and g 0, G, (a) there exist two s t strictly greater than s L solving s L d s L. Ψ s t if Ψ (b) there exist two s t strictly less than s H solving s H Ψ s t if s H Ψ s c The proof of Proposition 3.2 is trivial and therefore omitted. Proposition 3.2 (a) corresponds to Figure 4(a) where there exist three possible rational expectations equilibria at s L. This implies that expectations of high aggregate savings, which we call optimistic, can bring the economy out of the low steady state s L. Proposition 3.2 (b), on the other hand, corresponds to Figure 4(b) where there exist three possible rational expectations equilibria at s H. This implies that pessimistic expectations of low aggregate savings can divert the economy out of the high steady state s H. Both 8

19 PSfrag replacements stw sl sm sh PSfrag replacements stw sl sm PSfrag replacements sh Ψ stw sl sm sh a s c Ψ a s c a s c Ψb a c Ψb s cc st Ψb a c Ψb s cc Ψ st Ψb a c Ψb s cc st (a) Possible escape from s L (b) Possible fall from s H (c) Possible cycles Figure 4: Time one correspondence of saving rate cases imply that not only history or initial conditions but also expectations determine the equilibrium path that the economy follows. 3.3 Endogenous cycles In the following we demonstrate a case when endogenous cycles can arise in our model. 7 Let us consider the case where there exist multiple steady 7 When the forward dynamics is ill-defined, we can infer dynamics of such systems from studying the backward dynamics (e.g. [4], [5], and [3]). For example, we can analyze our backward dynamics when Ψ has three steady states s L ) s M ) " ) s H and Ψ% " 'xq s L and s H q Ψ% s H'. To show the existence of topological chaos we can make use of the well known theorem of Li and Yorke [8]. If Ψ is continuous and if there exists some s 0 such that Ψ% s 0'Z) s 0 ) Ψ 3 % s 0'v) Ψ 2 % s 0', then a cycle of period 3 exists, which implies the existence of topological chaos according to the definition of Li and Yorke. Then, there exists a RIFS associated with a unique fractal attractor such that for any initial state in % Ψ% " ', Ψ% s c '', the forward orbits obtained by () always stay in % Ψ% " ', Ψ% s c '' and tends to a unique fractal attractor or its subset (For proof see [3] p.9-20). 9

20 states and multiple equilibria both at s L and s H as in Figure 4(c). Since the steady states s L and s H are locally stable under the functions Ψ G L and ΨG H respectively, s L, s H is an invariant set induced by (). Hence, the saving rate s t can fluctuate indefinitely due to switching between optimistic and pessimistic expectations independent of its initial conditions. In such cases, only self-fulfilling expectations (and not history) matter for the path of the economy. The following proposition characterizes the invariant set obtained. Proposition 3.3. If Ψ has three steady states s L s M s H and Ψ Z s L and s H Ψ s H, then RIFS () has a unique and globally stable invariant distribution and the fractal attractor for any arbitrary s 0 g 0,. It is worthwhile to note that the existence and uniqueness of the attractor in Proposition 3.3 does not depend on the probability of selecting maps under the RIFS. In other words, the economy, independent of its initial conditions, fluctuates indefinitely as long as agents select each multiple expectations with a positive probability. 8 4 Appendix Proof of Proposition 2.. The proof of (a) is trivial, and therefore omitted. 8 The forward equilibrium paths converge to a fractal set, typically jumping erratically over the fractal (See [3]). 20

21 (b) Since s i A= s, r and r r s, When s s i s = s = r dr ds ln = ln r y = s lnz ln r d ln r d ln s{. (2),, = is monotonic. Therefore, we analyze = when s 0,. (5) implies that d ln r d ln s s s ε f@ Rs Z and (8) implies that = s lnz ln r γ v Rs f Rs 2 v ε f@ Rs s }. Therefore, the term in the bracket in (2) is bounded from above. This implies a unique equilibrium for sufficiently small elasticity of saving lnz ln r as si s will be arbitrarily small. Hence, sufficiently high elasticity of saving is a necessary condition for multiplicity of equilibria. Lemma 4.. Let R > 0, R then, lim s G Ψ s d 0.. If Assumption 2. is satisfied and > 0, G, Proof. (9) and (0) imply that u W R ~ Ψ s 2 s < s Γ s. (3) Differentiating both sides of (3) and rearranging terms we obtain RW RΨ s 2 Ψ s u 2 u G G s Γ s 2 2 s Γ s s Γ s }. (4) 2

22 Hence, lim s G Ψ s Z 0 if u * u G Γ 2 ƒ Observe that for any R 0, R G : R such that (5) is satisfied. Γ Γ. (5) there always exists a sufficiently small Lemma 4.2. If Assumption 2. is satisfied, then for any R 0, R and 0, G, there exists a unique s c g 0, which solves Ψ s c < 0. Proof. Γ s is non-decreasing when Assumption 2. is satisfied. This with properties of u and W implies that s R ~ WG ˆ s 3 u G s Γ s 2 }v has at most one local maximum on the interval 0,. Hence, there exists at most one critical point s c. Lemma 4. ensures the existence of a maximum on 0, for any R g 0, R and g 0, G. The proof of Proposition 2.2. First we show the necessary condition. At s, s i s ln = ln r ε f@ R 2 rš γ v R f R 2 Œ }. Hence, si s 0 for sufficiently large implying a unique equilibrium. In other words, sufficiently small is a necessary condition for multiplicity of equilibrium. 22

23 ~ The sufficient condition for multiplicity follows directly from Lemma 4.. Moreover, Lemma 4.2 implies that Ψ s has a unique critical point on 0, if 0, G. It follows that there exist three equilibria if and only if F 0, G and s t g Ψ, Ψ s c 2. The proof of Lemma 3.. From (9) and (0), u W RΨ s 2 s d G Γ s s for sufficiently small s. Taking logarithm of both sides and differentiating we obtain RΨ s W0 RΨ s 2 sψ s W RΨ s 2 Ψ s for sufficiently small s. Since kw@e7 k 8 ε f@ 0 γ v 0 ε f@ 0 2 γ u 0 ε f 7 k8 σ f 7 k8, it follows from σ f 0 UT ε f 0 (6) W7 k83ž that RΨ7 0 8 W@E7 RΨ7 08R8 W7 RΨ7 08R8. Moreover, 0 γ u 0 γ v 0 and ε f@ T 0 imply that the right hand side of (6) is greater than one. Since Ψ 0 v 0, it follows from (6) that Ψ is locally convex around zero and thus Ψ 0 0. Let Λ s : Γ7 s8 u@ 7 W7 Rs8 G s8. We will make use of the following result. Lemma 4.3. If Assumption 3.2 (a) and (b) are satisfied, then Λ s Z 0. Proof. We obtain sλ@ 7 s8 Λ7 s8 ε f@ Rs ; γ v Rs f Rs 2 ε f@ Rs 2 γ u W Rs s RsW@ 7 Rs8 G s W7 Rs8 G s. If Assumption 3.2 (a) is satisfied, RsW@l7 Rs 8 G s W7 Rs8 G s ε f@ Rs since RW@l7 Rs8 G G ε 7 Rs8 RW@l7 Rs8 G R7 f@l7 Rs8 G W@l7 Rs8R8 W7 Rs8 G s s W7 Rs8 G s f 7 Rs8 G W7 Rs8 R f 7 Rs8 W@ 7 Rs8 7 W7 Rs8 G s8 7 f 7 Rs8 G W7 Rs8R8 Rs R f 7 Rs8 σ f Rs ƒ 0. This with Assumption 3.2 (b) implies that Λ s d 0. 23

24 The proof of Proposition 3... From (9), Ψ has a fixed point on 0, if s Λ s since Λ s n. The left hand side can have at most one local minimum 0 from Lemma 4.3. Since the right hand side is constant, Ψ has at most two fixed points on 0,. The monotonicity of Ψ ensures at most one fixed point on,. 2. Notice that by construction of Ψ, Ψ s ZT Ψ, for all s F s c,. (a) If Ψ, then Ψ s T s on s c,. Monotonicity of Ψ on 0, s c and Ψ 0 r 0 ensure a unique steady state s L F 0,. (b) If s c T Ψ s c, then Ψ s Z s on 0, and Ψ ZT ensures a unique steady state s H F,. (c) s c Ψ s c is necessary and sufficient for s L to exist and T Ψ is necessary and sufficient for s H to exist. If s c Ψ s c and T Ψ, continuity of Ψ ensures that there exist three steady states 0 s L s M s H. The proof of Proposition 3.3. Suppose that Ψ has three steady states s L s M s H and Ψ [ s L s H Ψ s c. Then, by iterating () for any s 0? 0,, s t eventually hits and never leaves the set s L, s H ] D 0,. Since the images of s L, s H under () are disjoint and proper subsets of s L, s H, each map is contracting. This implies that () has a unique Cantor set or attractor with Lebesgue measure zero, independent of initial conditions. 9 9 The existence and uniqueness of the invariant measure can be established as in [4]. This implies that the invariant distribution is singular and its graph is the so called Devil s Staircase (See [] and [2]). 24

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