WHEN DEMAND CREATES ITS OWN SUPPLY: SAVING SINKS Christophe Chamley Boston University and PSE November 5, 2011 Abstract The mechanism by which aggregate supply creates the income that generates a demand equal to that income (called Say s Law, may not work in a general equilibrium with decentralized markets, production made to order and private bonds. Full employment is an equilibrium, but convergence to that state is slow. A self-fulfilling precautionary motive to accumulate private bonds (equal to zero in the aggregate may set the economy on a path towards a steady state with unemployment. This equilibrium is a sink from which no path emerges to restore full employment. Keywords: Say s law, paradox of thrift, private debt, credit constraint, unemployment trap. JEL codes: E00, E10, E21, E41 I am grateful for the comments by participants of the April 2010 General Equilibrium workshop at Yale University, seminars at Boston University, the Booth School in Chicago the Paris School of Economics, the Bank of Italy, Humboldt University, the Stockholm school of economics, and, in the order they were received, by Domingo Cavallo, Larry Kotlikoff, Bart Lipman, Veronica Guerrieri, Guido Lorenzoni, Ruilin Zhou and Herakles Polemarchakis.
1 Introduction In the recent global crisis, the demand for precautionary saving and the reduction of consumption have played an aggravating role 1. Uncertainty about employment and income generates a motive for saving. The lower demand for goods may generate a self-fulfilling equilibrium with unemployment and individuals uncertainty. This mechanism is proven here in a model of general equilibrium with decentralized markets for goods and an economy-wide market for bonds. The mechanism invalidates the common view of Say s Law that the aggregate supply (that is, the capacity to supply goods creates its matching demand and that full employment is the natural state of economies 2. A central issue in the analysis is that individuals do not consume their own production. That feature obviously does not per se prevent the full employment of resources, as we know from general equilibrium. An analytical model that addresses the problem of Say s Law should indeed have an equilibrium with full employment. In such an equilibrium, all individuals realize their transactions according to perfect foresight and there is no unused capacity of production. The model presented here has at least one other equilibrium where output falls short of capacity. The structural feature that individuals do not consume their own production bites when individuals face individual uncertainty. The economy-wide individual uncertainty has an aggregate impact and that aggregate impact generates the individual uncertainty because individuals are heterogenous and meet each other randomly. In the model, all uncertainty is endogenous. There is no exogenous aggregate shock 3. There is no aggregate uncertainty and individuals have perfect foresight on the future path of unemployment. When individuals are uncertain about the future sales of their own product, they may attempt to save as a precaution. That motive arises because of an individual credit constraint 4. For clarity s sake, in the present model, there is no physical capital, no 1 As of this writing, consumers cut spending in June by the most in nearly two years and saved at a faster rate, signs that reinforced the economy s sluggishness (Wall Street Journal, August 2, 2011. 2 What Say (1803 really meant is probably different from the current interpretation. He was not familiar with general equilibrium. 3 For uncertainty with aggregate shocks, see Bloom (2009. 4 The present model is in the line of Ayagari (1994, but Ayagari assumed individual exogenous productivity shocks. There are no such shocks here. 1
money, and goods are not storable. The impact of saving on the demand for goods through investment is obviously important, but that effect should be the subject of another work. Assets are bonds, and, in most of the paper, the net supply of bonds is zero. The assets of some are the debts of others. In each period, there is a clearing house that alleviates the use of money. In Sweeney and Sweeney (1977, which is discussed at the end of the paper 5, an insufficient quantity of money generates the unemployment of resources in an economy without credit. That study may give the impression that unemployment arises because people are chasing a fixed quantity of money is a kind of musical chair game. That seems to be the wrong approach to the problem of Say s law. Hence, the present focus on an economy where the aggregate supply of assets is zero and where the assets of some are the liabilities of others. In such an economy, a partial relaxation of the individual credit constraints is equivalent to a positive supply of assets and that equivalence is discussed in Section 7. There is no money as asset, but there a unit of account, as there can be in any economy without money as means of exchange 6. In any equilibrium, the price of any good is endogenously determined. We will focus on equilibria where all goods have the same price in terms of the unit of account. That price level is irrelevant for the real allocation of resources. The property that a uniform change of the prices of goods or of the wage rate has no impact on allocation of resources confirms the model-free view of Keynes that a reduction of the wage rate would not be a solution to the problem of unemployment. The dynamics of the equilibria with and without full employment are not symmetric. In a steady state with full employment, nothing restrains agents from increasing saving and reducing demand. The economy can shift at any time towards a self-fulfilling path to lower output. By contrast, in the low equilibrium, agents at their credit limit (and there must exist some, otherwise the precautionary motive would not be rational cannot increase their demand for goods even if they expect a higher level of activity during a transition to full employment. With some parameters of the economy, the weight of these agents prevents a take-off towards full employment and the economy cannot escape from a low employment trap. 5 See also Guerrieri and and Lorenzoni (2009. 6 In Egypt, such units of account (châts without means of exchange were used before 2600 BC, well before the introduction of physcial money, (Daumas, 1977. 2
When the parameters are such that the economy can move in both directions between full employment and unemployment, the rates of convergence are not of the same order of magnitude. In a transition to unemployment, demand falls abruptly, (individuals are free to save, and the rate of unemployment increases rapidly. In the other direction, towards full employment, the convergence rate is asymptotically zero because individuals can reduce their credit balances only gradually as they make sales, and sales are hampered by the credit constrained agents. The issues that are addressed in this paper are central in macroeconomics and hence not new. In policy circles, the positive impact on private consumption on growth during a recession is a triviality (e.g., footnote 1, which is not been matched by the number of formal analyses. The coordination of aggregate demand and supply has been discussed by John Law (1705, Robertson (1892, and, of course, Keynes (1936, among others. In the late 1960s, Leijhonhufvud (1968 emphasized again, with no analytical model, that in an economy where goods are traded with money, Say s Law may not hold. The coordination of demand and supply was analyzed by Diamond (1982 in a model of search where individuals exchange their production for a good produced by another agent, which they consume. The externality in exchanges is generated by the structural feature of search. Any equilibrium is inefficient compared to the first-best. When production is high, meeting opportunities are high and the incentive to produce is also high. With low production, the incentive to produce would be as low as going to an empty singles bar. Despite the title of that paper, supply and demand effects are not separated. The multiplicity of equilibria arises essentially from the search externality, and the model cannot deal with the critical role of saving which is socially unproductive in the aggregate, as argued by Keynes. The model is presented in Section 2. The feature that agents do not consume their own production is represented by a matching structure 7. Agents are heterogenous in their utility of consumption. Any agent in any period has a randomly determined need for consumption that is either high or low. In the latter case, under some conditions, the agent will prefer saving over consumption. In the matching structure, the agent who saves generates an individual uncertainty for sellers. For simplicity, goods are assumed to be indivisible. (That assumption is removed in Section 7. 7 The paper owes an important intellectual debt to Green and Zhou (1998, 2002 and Zhou (1999. However, they focus on the use of money as an asset when agents lack double-coincidence of wants. They emphasize that there is a welfare-ordered continuum of the equilibrium value of the price level. There is no such property in the present model. 3
Individual behavior is analyzed in Section 3. Agents with a high need for consumption always consume unless they are against their credit limit. Agents with a low need accumulate savings up to some level. We can therefore analyze the equilibria according to that consumption function. When all unconstrained agents consume, we have, by a definition, a high regime. A special case is an equilibrium in the steady state with full employment. When agents with a low need save up to some level N (endogenously determined, there is a low regime. After setting the conditions for the optimal consumption function, the argument is developed in three steps, each with two propositions. In Section 4, the equilibrium in the high regime is analyzed in two steps. First, the regime is shown to converge to full employment. Second, after the path of unemployment is determined, the consumption function is shown to be optimal if the level of the aggregate debt is not too high. Convergence is slow, with a rate asymptotically equal to zero. In Section 5, the same two-step analysis is carried for the low regime where agents save up to one unit. The convergence to a steady state with unemployment is asymptotically strictly positive. When high and low consumer types have the same mass, convergence occurs in one period. In that steady state, the consumption function with accumulation up to one unit of bonds is optimal if the utility of consumption of the high need agents is not too high. Section 6 analyzes conditions under which there is a path from full employment to a steady state with unemployment. In that state, there is no equilibrium that can increase the level of consumption. Section 7 provides extensions and shows the robustness of the properties. Section 7.1 shows that there can be multiple steady states with different levels of precautionary savings (N > 1. Sections 7.2 and 7.3 show that the model with a higher credit limit is equivalent to a model with a positive level of aggregate assets a credit limit of 1. (Outside money is a special case. In Section 7.4, the indivisibility assumption is removed. In Section 7.5, finally, the matching structure is extended such that the total population is divided into two parts with overlapping demands and production. In the concluding section, given the abstract structure of the model, the policy remarks are suggestive and brief. Technical proofs are in the Appendix. 4
2 The model Goods, demand and production There is a continuum of goods and agents indexed on a circle by i [0, 1. Time is discrete. Goods are perishable from one period to the next and indivisible. (The case of divisible goods is considered in Section 7.4. In any period, agent i is endowed with the capacity to produce one unit of good i, at no cost. As in any macroeconomic model, agents consume goods produced by others. In a standard Walrasian model with complete markets, production and consumption take place according to plan after the auctioneer has found the equilibrium prices between supply and demand for consumption. A fundamental feature of the present model is that there is no central institution to coordinate production and consumption. The complexity of the coordination of consumption and production of different goods in a modern economy with decentralized markets is modeled by the following structure. At the beginning of a period, each agent i can either place an order for the consumption of some good, or save. The order for the consumption good is placed with an intermediary who delivers by purchasing, one for one, good j that is produced by agent j where j is determined by a matching function j = φ t (i from [0, 1 to [0, 1. Without loss of generality, that matching function is defined by i + ξ t, if i + ξ t < 1, φ t (i = i + ξ t 1, if i + ξ t 1, where the random variables ξ t (0, 1 that are i.i.d., with a uniform density 8 on [0, 1. The key feature is that goods are made to order and when an agent places an order for consumption at the beginning of the period, he does not know whether an order will be made for his production. The intermediary is a shortcut to model the overlapping feature of consumption and production in an economy. In Section 7.5, it is shown that the properties of the model hold when the structure is extended such that agents are divided in two groups of equal size in each day (period: morning agents consume in the morning and produce in the afternoon, and afternoon agents do the reverse. For practical purpose, agent i, if he consumes, consumes good j = φ t (i. When he decides to consume or save at the beginning of the period, he does not know whether an agent k = φ 1 t (i has ordered his production of good i. There is no search in 8 One could use other matching ( functions φ t provided that they satisfy the property that for any subset H of [0, 1, µ(h = µ φ t(h, where µ is the Lebesgue-measure on [0, 1. The property is required for a uniform random matching of all agents. (1 5
the model in order to avoid the issue of physical externalities generated by a search process. Utilities Agents are heterogenous in their consumption propensity. In each period, an agent is either of a high or low type. A type is revealed at the beginning of the period and is defined by a random variable θ it {0, 1} that is independently distributed across agents and periods. The probability of the high type is exogenous and equal to α, (0 < α < 1, which is known by all agents. The utility function of all agents is linear with a higher slope for the high type. Since utility can be defined up to a constant, we can reduce the level of utility of the high type by a constant scalar. Let x i,t be the consumption of individual i in period t. The welfare of individual i is the discounted sum of expected utilities of consumption from all future periods: [ ] W i = E β t u(x i,t, θ i,t, with u(x, θ = (1 + θcx θc, (2 t 0 where c is a parameter and β = 1/(1 + ρ is the discount factor between periods. The utility function u(x, θ is represented 9 in Figure 1. Financial sector In decentralized markets, all goods are traded at some price against a unit of account (UA. There is no outside money in the economy but there is a financial sector. The financial sector may be made of a large number of institutions. For simplicity, we will refer to a unique representative financial institution. The financial institution acts as a clearing house for the period s transactions (in UA. At the end of the period, any individual has a position with the financial institution. That position may be positive, zero or negative. A positive position is an amount of bonds (deposits that are carried from this period to the next and are measured in UA. The motive for accumulation of bonds between periods will be the precaution motive, as in Ayagari (1994. However, in Ayagari the shocks are on the individual production and are exogenous, while here, all individual 9 The assumption of a finite capacity and a linear utility function could also be replaced by an linear production cost γ with no capacity limit and a utility function with a slope of 1 or 1 + c for x 1 and γ for x 1 with γ < γ. 6
( θ = 0 ( θ = 1 Figure 1: Utility functions for the high type (θ = 1 and the low type (θ = 0. uncertainty is generated by trading uncertainty, which is endogenous and can be at different levels depending on the equilibrium. A negative position with the financial institution means a debt that is carried to the next period. There is an upper bound on an individual s liability. That credit constraint is motivated by standard constraints imposed on the recoverability of debts. The liability constraint has to be satisfied with probability one at the end of the period. During the period, the liability of the individual may be higher than the constraint because the financial institution is acting as a clearing house. Bonds are the only means of saving from one period to the next. Their rate of return is fixed at zero. One could also view the bonds as inside money. Note that the aggregate net wealth is equal to zero: the aggregate of the debt is equal to the aggregate of the bonds. The case of a positive aggregate wealth is considered in Section 7.3. Prices Prices between the goods and the unit of account (UA are determined by price posting. A transaction is made if the posting of the seller is not strictly higher than that of the buyer. In equilibrium, all prices are identical and normalized to 1 by an argument in two steps. First, assume that all sellers post the same price that is normalized to 1. If a seller deviates and posts a lower price, he cannot attract more sales because of the 7
matching process and he makes less on a sale. A price strictly below 1 is strictly dominated. Given the uniform posting of sellers, a buyer cannot have a reservation price below 1. Such a reservation price would be equivalent to a strategy of no consumption. The first step shows that prices are rigid downwards. In the second step, given the posting of 1 by sellers, buyers have a reservation price not strictly higher than 1. Given the posting of 1 by the buyers, no seller posts a price higher than 1. The equilibrium price is therefore 1. The second step can be reinforced by omitting the price posting of buyers and showing that for some parameter values, if a seller increases his price, he lowers his expected value from a sale 10. The first best The first best allocation maximizes a social welfare function. Without loss of generality, that function can be taken as the sum (integral of the agents utilities. Since goods are perishable, optimization applies to each period independently of the others and we can omit the time subscript. Let x i (θ be the consumption i for type θ. From (2, the first best allocation maximizes the function J = 1 0 ( αu(x i (1; 1 di + (1 αu(x i (0; 0 di, with x i {0, 1}. (3 The optimal allocation is obviously x i (θ i = 1. The maximum level of utility from consumption is achieved when consumption takes place in every period. In this allocation all agents produced up to capacity. There is no unemployment. Lemma 1. In the first best allocation, each agent consumes one unit in each period and there is no unemployment. The first best is an equilibrium according to the definition that is given in the next section. 3 Individual behavior in the equilibrium We assume without loss of generality that the individual s credit limit is set to one unit of UA. That limit is increased in Section 7.2. 10 For example, this property can be shown as an exercise for the steady state of the low regime in Section 5. Green and Zhou (2002 make the stronger assumption of the double posting auction by seller and buyer. Their model has outside money and the continuum of equilibrium prices in that model has been the subject of some debate. This issue is not relevant here 8
3.1 The structure of equilibria As all transactions are zero or one, the distribution of the holdings of bonds is discrete with the support contained in K = { 1, 0, 1, 2,...}. An agent with a wealth k is said to be in state k. When k = 1, he is in debt. Let Γ(t be the vector of the distribution of agents at the beginning of period t across states: Γ(t = (γ 1 (t, γ 0 (t,..., where γ k (t is the mass of agents in state k. It will be shown later that in any equilibrium, the support of the distribution is bounded and Γ(t is a vector of finite dimension. The population has a total mass equal to 1 and the net aggregate amount of wealth is zero: γ k (t = 1, kγ k (t = 0. (4 k 1 k 1 Let π(t be the fraction of agents who do not demand goods in period t. The total demand is 1 π(t and 1 π(t = x i,t di, (5 where x i,t is the demand of agent i in period t. Because of random matching, the probability that an agent makes no sale is period t is equal to π(t, which will be called the rate of unemployment. Let ω i,t be the bond balance of agent i at the end of period t. That balance evolves such that ω i,t+1 = 1 + ω i,t x i,t, with probability 1 π(t, (6 ω i,t+1 = ω i,t x i,t, with probability π(t. Let π t = {π(τ} τ t be a path of unemployment rates for periods τ t. Suppose perfect foresight on that path. The consumption function of an agent in period t depends only on his state (his savings in bonds, his type (low or high and the path of future unemployment rates π t. Given the consumption decisions and the unemployment rate in period t, the distribution of bonds in period t+1 is deterministic. That distribution determines π(t + 1 and the consumption functions in that period. From period to period, the path of the unemployment rate is deterministic, and rational agents who know the structure of the model can compute that path. The assumption of perfect foresight is therefore justified. 9
Definition of an equilibrium An equilibrium is defined by an initial distribution of bonds Γ(0 that satisfies (4, a path of unemployment rates π 0 = {π(τ} τ 0, a consumption function x t = x(ω t, θ t, π t, and the evolution of the distribution of assets that is determined by (6. The path of unemployment rates satisfies (5 and the consumption function maximizes, in any period t, the utility function of E[ β τ t u(x t, θ t ], τ t for a given balance ω t at the beginning of period t and type θ t, subject to the accumulation constraint(6, the credit constraint ω i,t 1, and under perfect foresight about the path of unemployment after period t, π(t = {π(τ} τ t. 3.2 Household optimization of consumption Let us first solve the optimization problem of the high type agents. Their behavior is simple: as shown in the next result, they consume whenever they can. Lemma 2. In any period of an equilibrium, a high-type agent consumes if he is not credit constrained. The property is intuitive. Suppose an agent of the high type saves today. He incurs a penalty c. The best use of that savings is to consume it in the future when he is also a high type and he is credit constrained. Because of discounting, the expected value of the future penalty is smaller than the penalty today. The agent is better off by not saving. The proof in the Appendix follows the intuitive argument. It follows from Proposition 2 that the dynamics of the economy are driven by the behavior of the low-type agents. Since, in an equilibrium, the path π t is known with perfect foresight, we omit it from the notation. The optimization problem of an agent is standard. For a given path of future unemployment rates, let V k (t denote the utility of an agent in state k (with a balance k, at the end of period t, after consumption and sales have taken place in period t. We have the Bellman equations [ ] V 1 (t 1 = βe u(0, θ + π(tv 1 (t + (1 π(tv 0 (t, ( ] V k (t 1 = βe [max x {0,1} u(x, θ + π(tv k x (t + (1 π(tv k x+1 (t, k 0, (7 10
where expectations are taken with respect to the preference shock θ. On the first line, the utility of a credit constrained agent (in state 1 does not depend on his consumption during the period, since he cannot consume. A low-type agent who is not credit constrained (in state k 0 consumes in period t if his utility of consumption is greater than the value of one more unit of saving. If he consumes, his balance increases with probability 1 π(t from k to k + 1 and his expected utility of holdings at the end of the period is π(tv k 1 (t + (1 π(tv k (t. If he saves instead, his expected utility is π(tv k (t + (1 π(tv k+1 (t. The difference between the two expected utilities can be called here, by an abuse of notation, the marginal utility of savings ( ζ k (t π(t V k (t V k 1 (t ( + (1 π(t V k+1 (t V k (t. (8 Hence, consumption is optimal for an agent of the low type and in state k if and only if 11 ζ k (t 1. (9 For any fixed t, the function V k (t is increasing in k. It is bounded by the discounted value of consumption in all future periods. Therefore, the increments V k (t V k 1 (t are arbitrarily small when k is sufficiently large and the previous condition (9 is satisfied. We have the following Lemma (proven in the Appendix. Lemma 3. There is a fixed N such that in any equilibrium and any period, any agent with a balance that is at least equal to N consumes. One can choose N such that β N+1 (1 + c/(1 β < 1. Since a sale is at most for one unit, any agent with savings greater than N does not increase his savings. The following property follows immediately. Lemma 4. Assume that the support of the initial bond distribution is bounded by N 0. Then, in any period of an equilibrium, the support of that distribution is bounded by Max(N 0, N. Without loss of generality we will assume in the rest of the paper that N 0 N and that, in any equilibrium for any period, the support of the distribution of bonds is bounded by N. 11 In the standard optimization problem with continuous consumption, the optimal rule compares the marginal utility of consumption (here equal to one and the marginal value of accumulated savings. Here, consumption is discrete and the marginal utility of savings is replaced by a weighted average of the increments of utilities when the balance increases by one unit from k, (with probability 1 π(t when no sale is made in the period, and to k (with probability π(t. 11
Lemma 5. In a steady state, an optimal consumption for a low-type agent, x(ω, 0, satisfies the following property: there is some N 1 such that x(ω, 0 = 0 if ω < N, x(ω, 0 = 1 if ω N. From the previous result, we define the N-consumption function as the function by which low types consume if and only if their balance at the beginning of the period is at least equal to N. We extend this definition beyond the steady states to any period t where the threshold N t depends on the period. From (9, we have the next result. Lemma 6. The N t -consumption function is optimal in period t if and only if ζ k (t 1 for k N t 1, ζ k (t 1 for k > N t 1, ( with ζ k (t = π(t V k (t V k 1 (t (10 ( + (1 π(t V k+1 (t V k (t. From the previous result, we can distinguish two regimes of consumption. In the first, all non credit constrained agents consume: the consumption function of the low type agents is the 0-consumption function. That regime is called the high regime. In a low regime, low-type agents accumulate some savings and have an N t -consumption function with N t 1. The method of analysis is in two steps: the first determines the dynamics of the bond distribution and the unemployment rate in a given regime; the second step uses the path of the unemployment rate that is known with perfect foresight to determine the optimality of consumption and whether the path of the economy is an equilibrium. 3.3 High and low regimes Suppose that in some period t, no agent holds more than one unit of asset. If all agents have an N-consumption function with N = 0 or N = 1, no one with one unit of savings saves, and, since the earning in a period is at most one, no one holds more than one unit of savings in period t + 1. The distribution of bonds in any period t is defined by the vector Γ(t = (γ 1 (t, γ 0 (t, γ 1 (t, where γ k (t is the mass of agents in state k at the beginning of period t. Because the total mass of agents is 1, and that of net wealth is 0, γ 1 (t + γ 0 (t + γ 1 (t = 1, γ 1 (t = γ 1 (t. (11 12
The number of degrees of freedom is therefore reduced to one, and the analysis of the dynamics can be characterized by one variable, which will be chosen as the amount of debt, B(t = γ 1 (t = γ 1 (t. The precautionary motive is generated by the parameter c, and increases with its level. We will see that, when the value of c is in some bounded interval, it is not optimal to save more than one unit. When c increases beyond that interval, the target level of saving rises above 1, which is intuitively obvious and is observed in the numerical examples of Section 7.1. However, the dynamics cannot be expressed analytically by the evolution of one variable. One has to rely on numerical results. In order to have an analytical treatment, in the next three sections, we focus on the cases of high and low regimes with individual savings bounded by 1. (That bound will not exogenous. Let x {0, 1} be the consumption of a low-type agent in state 0: in the 0-consumption function, that agent consumes, x = 1; in the 1-consumption function, that agent saves, x = 0. That is the only difference between the high and low regimes when no agent holds more than one unit of savings. Therefore, the Bellman equations (7 become ( V 1 = β α ( c + πv 1 + (1 πv 0 + (1 α (πv 1 + (1 πv 0. ( V 0 = β α (1 + πv 1 + (1 πv 0 + (1 α (x + πv x + (1 πv 1 x. (12 V k = β (1 + πv 0 + (1 πv 1, for k 1, with x = 0 in the low regime and x = 1 in the high regime. We are now equipped for the analysis of the two regimes. 4 The high regime By definition of the high regime, all non credit constrained agents consume. The unemployment rate is equal to the fraction of agents who do not consume, that is the fraction of agents in state 1. Each of these agents has a debt of one and their total mass is γ 1 (t. We define here the aggregate debt, or debt, as the mass of liabilities in the economy, B(t = γ 1 (t. We have π(t = B(t. (13 13
B(t+1 0 B(0 1/2 1 B(t Figure 2: Dynamics of the aggregate debt (equal to the mass of credit constrained agents in the high regime with three states. 4.1 Dynamics The evolution of the bond distribution Γ(t = (γ 1 (t, γ 0 (t, γ 1 (t, and of the unemployment rate are determined by Γ(t + 1 = H(π t.γ(t, with π t = γ 1 (t, (14 and the transition matrix π(t π(t 0 H(π = 1 π(t 1 π(t π(t. (15 0 0 1 π(t For example in the first line, agents are in state 1 at the end of period t: either because they were in state 1 at the beginning of the period, could not consume, and made no sale, hence the term H 11 = π(t; or because they were in state 0 at the beginning of period t, consumed, and made no sale, hence the term H 12 = π(t. Since π(t = B(t, using (11, (14 and (15, ( B(t + 1 = B(t 1 B(t. (16 The evolution of B(t is represented in Figure 2. Because each bond is the asset of an agent and the liability of another, the total amount of bonds is bounded by 1/2. For any B(0, the value of B(t converges to 0 which defines the full employment steady state. That steady state is globally stable under the individuals behavior of the high regime. Whether the path is an equilibrium will be analyzed below. Convergence to full employment is slow: the rate of convergence tends to zero as unemployment tends to zero. The debt is gradually reduced as agents repay that 14
debt, but in order to repay, they have to make sales. That process takes time. The property of the slow process of debt reduction through high demand and sales probably holds in other, more complex, models and one can conjecture that it is generic. The present model is sufficiently simple such that the dynamics can be represented by the single variable of the total B(t and its rate of convergence. We can state the next result 12. Proposition 1. In the high regime (when all non credit constrained agents consume, if the support of the initial distribution of bonds is bounded by 1, the economy converges to the full-employment steady state. The rate of convergence of the unemployment rate is asymptotically equal to zero. 4.2 Equilibrium Using the Bellman equations (12 with x = 1, the marginal utilities of savings ζ k (t in state k, defined (8, satisfy the equations of backward induction ( ζ 0 (t = β π(t(1 + αc + (1 π(tζ 0 (t + 1, ( ζ k (t = β π(tζ k 1 (t + 1 + (1 π(tζ k (t + 1, for k 1. Assume that the economy is in a high regime for all periods t 0. By (16, that path depends only on the initial level of the debt, B(0. By repeated iterations of (17, and using π(t = B(t, the marginal utility of savings for an agent with zero balance, ζ 0 (0, is a function of the initial debt, B, such that ( ζ(b; c = (1 + αcβ B(0 + t 0 βt+1 (1 B(0... (1 B(tB(t + 1, with B(t + 1 = B(t(1 B(t. Since B is the mass of agents that are credit constrained, B 1/2. We can expect that a higher level of B in the first period induces higher rates of unemployment on the path of the high regime and therefore a higher value of ζ(b; c in the first period of that regime. That intuitive mechanism is confirmed by the following result. Lemma 7. In the high regime, the utility of a unit of additional savings is a function of the debt B, ζ(b; c, as defined in (18. B [0, 1/2], and ζ(0 = 0. (17 (18 It is continuous, strictly increasing in 12 It is an exercise to show that the convergence result holds for any bounded initial support. 15
From (18, the value of ζ(b; c increases linearly with the cost parameter c. Define c such that ζ(1/2; c = 1. If c c, the marginal utility of savings is always smaller than 1 and the high regime is an equilibrium. The interesting case arises when the cost parameter is greater than c. From the previous Lemmata, we have a characterization of the global stability of the full employment steady state. Proposition 2. If c c, with ζ(1/2; c = 1 and ζ( ; c defined in (18, the full employment equilibrium is globally stable. From any level of aggregate debt, the high regime is an equilibrium. If c > c, there exists B(c < 1/2 such that from an initial level of debt B, the high regime is an equilibrium if and only if B B(c. The result characterizes the domain of the aggregate debt from which the full employment steady state can be reached through a high regime path. In Section 6, the case B > B(c will be a saving trap. 5 The low regime Consider now the dynamics of the aggregate debt under the 1-consumption function where low-type agents consume if they have at least one unit of savings. The only difference from the previous case is that low-type agents with no savings do not consume and save instead. The support of the distribution of bonds is bounded by 1 in the first period and therefore in every period, and the dynamics are analyzed through the distribution vector Γ(t of dimension 3. 5.1 Dynamics The evolution of Γ(t is now determined by π απ 0 Γ(t + 1 = L(π(t.Γ(t, with L(π = 1 π a π, (19 0 b 1 π and a = (1 απ + α(1 π, b = (1 π(1 α. (20 16
In any period t, the agents who do not consume are either credit constrained (of a mass equal to γ 1 (t, or of low type and with no savings (of mass (1 αγ 0 (t. Hence, π(t = γ 1 (t + (1 αγ 0 (t. Since the debt is equal to B(t = γ 1 (t = γ 1 (t and γ 0 (t = 1 γ 1 (t γ 1 (t, the unemployment rate is a linear function of B(t: π(t = 1 α (1 2αB(t. (21 Using (19, B(t + 1 = γ 1 (t + 1 = bγ 0 (t + (1 πγ 1 (t and one finds the evolution of B(t: B(t + 1 = P (B(t, with P (B = (1 2α 2 B 2 + (1 2α 2 B + α(1 α. (22 The polynomial P (B has its maximum at B = 1/2, with P (1/2 < 1/2. Hence, there is a unique value B (0, 1/2 such that P (B = B. For any initial value B(0, the sequence B(t + 1 = P (B(t converges monotonically to the fixed point B, as represented in Figure 3. The figure represents the special case of the path that begins at the full-employment steady state with B(0 = 0. The amount of debt increases over time to its steady state level. Note that when comparing with the high regime, convergence is fast : the rate of convergence stays asymptotically at some strictly positive level. For α = 1/2, the economy gets to the steady state in one period. When the economy is at full employment and switches to the low regime, in period 0, the fraction of agents who do not consume is that of the low-type agents. It is equal to 1 α and the reduction of consumption is equal to the unemployment rate π(0 = 1 α. The evolution of the unemployment rate after period 0 depends on α. When the fraction of high type agents, α, is relatively large, that is α > 1/2, the unemployment rate increases over time with the debt B(t (see (21. When α < 1/2, the fraction of high-type, who always consume, is relatively small. In the first period of a switch to the low regime, demand falls by a large amount and the unemployment rate jumps up. As more agents accumulate the desired level of savings to consume, unemployment decreases over time. This discussion is summarized in the next result. 17
B t+1 α(1 α 0 B* 1/2 B t Figure 3: Dynamics of the fraction of liquidity-constrained agents in the low regime Proposition 3. If the support of the initial distribution of bonds is bounded by 1, the low regime with the 1-consumption function converges to a steady state with unemployment and debt. (i The rate of convergence is asymptotically strictly positive. For α = 1/2, convergence takes one period. (ii From an initial position of no debt, in the low regime, the rate of unemployment increases with time if α > 1/2, and decreases with time after a large initial jump if α < 1/2. 5.2 Equilibrium Assume first that the economy is in the steady state of the low regime with the 1- consumption function: low-type agents with a savings balance of 0 prefer to save and agents with a balance of at least 1 consume. We follow the same method of analysis as for the high regime. Using the Bellman equations (12 with x = 0 in the low regime, 18
and the definition of ζ k (t in (8, ( ζ 0 (t = β π(tα(1 + c + b(t + a(tζ 0 (t + 1, ( ζ 1 (t = β π(t(1 α + π(tαζ 0 (t + 1 + (1 π(tζ 1 (t + 1, ( ζ k (t = β π(tζ k 1 (t + 1 + (1 π(tζ k (t + 1, for k 2. (23 with a(t = α(1 π(t + π(t(1 α, b(t = (1 π(t(1 α. From Lemma 6, the conditions for the low regime are ζ 0 (t 1, ζ k (t 1 for k 1. The steady state In the steady state, omitting the time argument, ζ 0 = β πα(1 + c + b, 1 βa ζ 1 = β (1 απ + απζ 0, 1 β(1 π ζ k = λ k 1 ζ 1, with λ = βπ < 1 for k 2. 1 β(1 π The condition ζ 0 1 is equivalent to παc/ρ 1, which has an intuitive interpretation. A low regime is an equilibrium in the steady state only if low type agents have an incentive to save when their balance is zero. (24 The value of saving is equal to the discounted value of the stream of payments, in each future period, where the payment is equal to the product of the penalty c, the probability of being a high type and the probability of not making a sale in that period. When that discounted value is greater than 1, which is the value of consumption, saving is optimal. Let π be the unemployment rate in the steady state with the 1-consumption function. The condition ζ 0 > 1 is equivalent to c > c 1 = ρ π α. (25 The values of ζ 0 and ζ 1 in (24 are linear increasing functions of c, with ζ 1 < 1 if ζ 0 = 1. Hence there is a threshold c 1 such that if c [c 1, c 1 ], ζ 0 1 ζ 1. If c > c 1, the penalty of no consumption in the high type is sufficiently strong to accumulate more than one unit of savings. This case will be analyzed in Section 7.1. From the previous discussion and since the variables on the dynamic path are continuous functions of the initial condition, we have the following result. 19
Proposition 4. Let π and B be the rate of unemployment and the level of the debt in the steady state of the low regime with the 1-consumption function (Proposition 3. (i The steady state is an equilibrium if and only if ρ/(απ = c 1 < c < c 1, for some value c 1. (ii Under this condition, there exists an open interval containing B such that if B is in that interval, the low regime with the 1-consumption function is an equilibrium. The second part of the proposition holds by a continuity argument: if c (c 1, c 1, in a neighborhood of the steady state, ζ 0 (t > 1 > ζ 1 (t and the dynamic path defines an equilibrium in a neighborhood of B. The steady state of that low regime is locally stable. The interesting issue, to which we now turn, is the global stability, in particular whether an economy at full employment, in which no agent holds debt, can be the starting point of a self-fulfilling equilibrium path with a low regime. 6 Transitions and traps We now show that for some parameter values, the economy at full employment can at any time move to a situation of unemployment but that the reverse may not be true. 6.1 Transition from full employment to unemployment Assume that the economy is at full employment. To simplify the discussion, α = 1/2. Under the parameter conditions in Proposition 4, (c (c 1, c 1, the steady state under the 1-consumption function is an equilibrium. Since the transition path from full employment to that steady state takes one period, that transition is an equilibrium. Because paths are continuous functions of the parameters, we can extend the equilibrium property when α belongs to an open interval containing 1/2. Proposition 5. If c belongs to the interval (c 1, c 1 defined in Proposition 4 with α = 1/2, then there is an open interval J containing 1/2 such that if α J, there is an equilibrium path with the 1-consumption function from full-employment to the steady state with unemployment. In the previous result, the interval J may not be small. First, when α 1/2, the convergence to the low regime steady state is relatively fast and the optimality conditions of the 1-consumption function are likely to depend mainly on the steady state 13. 13 For example, the transition from full employment with the 1- consumption function is an equilib- 20
More importantly, if the penalty c becomes large, the set of values (α, β for which there is an equilibrium transition to unemployment becomes larger but a technical issue arises because agents may want to save more than one unit. (As can be seen in the equations of (23, the marginal value of saving increases. There is no simple analytical expression of the dynamic path for an N t -consumption function with N t > 1, and one has to rely on numerical simulations which are outside of the scope of this paper 14. However, an analysis of the equilibrium steady states with savings of more than one unit is presented in Section 7.1. From the previous result, if the economy is at full employment with no debt at time 0, there are at least two equilibrium paths. In the first, unemployment is maintained for ever. In the second, self-fulfilling pessimism sets the economy on a path to unemployment 15. Why such an expectation shock may arise is not the subject of this paper. As in all literature on multiple equilibria, the first task is to analyze the structure of equilibria under common knowledge. The problem of the choice between equilibria is a different issue. A standard extension would include exogenous shocks that trigger a regime switch and would require another paper. 6.2 Saving trap When the economy switches from full employment to a path with unemployment, the expectation of unemployment is self-fulfilling because individuals can reduce their consumption. But the reverse is not symmetric. In an economy with unemployment, some individuals are credit constrained. If expectations shift toward optimism, these individuals cannot consume and are unaffected by expectations. They impose an inertia that may prevent the take-off to a path toward full employment. Proposition 2 showed that, for some parameters, there is no high regime equilibrium path that converges to full employment if the debt exceeds a critical level. Such a situation may occur in the steady state of the low regime with the 1-consumption function, as shown in the next result. rium if α = β = 0.8, c = 3.1. 14 A previous version of this paper set one as an exogenous upper-bound for an individual s accumulation of bonds. In that case, there is no technical issue about ζ k (t 1, k 1. 15 There is obviously a continuum of other equilibrium path when the switch between regimes is driven by a Poisson process. 21
Proposition 6. Assume that the economy is in a steady state with unemployment where agents save up to one unit. For any α, there exists β such that if β < β, there exists c such that the steady state is an equilibrium and from that initial position, the high regime path (that converges to full employment is not an equilibrium. The value of β in the previous result need not be very low. One verifies, for example that if α = 0.5, the debt and the unemployment rate is equal to 1/4 in the steady state (which is reached exactly within one period. When β = 0.8, that steady state is an equilibrium if c [1, 4]. Using the expression of ζ(1/4; c in (18, and a numerical computation, the high regime is not an equilibrium if c (2.84, 4. Assume now that the economy is in the steady state under the N 1 -consumption function. If there is an equilibrium path in which the rate of unemployment falls in the first period, the increase of consumption can only be generated by the low-type agents with no savings who switch from saving to consumption. In that case, the high regime is an equilibrium for period 0. The high regime is an equilibrium in period 0 if the marginal utility of saving for low-type agents with no savings, ζ 0 (0 in (8, is not strictly greater than 1. The inequality depends on the transition path after the first period. Fortunately, we do not have to compute that path. Lemma 8 in the Appendix shows that if the high regime is an equilibrium in period 0, then the high regime is an equilibrium for all the other periods. We have therefore the following result. Proposition 7. Under the conditions of Proposition 6, there is no equilibrium path where consumption increases in period 0. The previous result does not rule out theoretically the existence of an equilibrium path that begins by a lower level of consumption. On such a path, the support of the distribution of bonds is not bounded by 1 and the case is beyond the present analytical framework. However, the necessity of more debt in order to converge to the full employment steady state with no debt seems would be a bit strange 16. 16 In a previous version of this paper, the level of saving is bounded exogenously by 1. In that case, there is no technical difficulty and from 7, it follows immediately that there is no path from the steady state with unemployment towards full employment. 22
7 Extensions 7.1 High savings In the previous sections, agents save no more than one unit of bond because the cost of non consumption for the high type, c, is within some range [c 1, c 1 ]. When c > c 1, the precautionary motive induces agents to save more than 1. The main properties hold but the analysis cannot be done with algebra. For some insight, let us consider steady states. In an equilibrium steady state, the consumption function is an N-consumption where agents accumulate savings up to the level N (Proposition 5. Near the steady state, by continuity, the optimal consumption function will also be an N-consumption function. We now show that for any N, such a consumption function generates a steady state that will be shown to be an equilibrium for some value of c. Multiple steady states Without loss of generality, we can assume that the support of the bond distribution is { 1, 0, 1,..., N}. That distribution is defined by a column vector Γ(t = (γ 1 (t, γ 0 (t, γ 1 (t,..., γ N (t of dimension N + 2. In any period t of the low regime with an N-consumption function, the agents who consume are either of the high type with no constraint, α(1 γ 1 (t, or of the low type in state N, (1 αγ N (t. Since aggregate consumption is 1 π(t, the unemployment rate is ( π(t = 1 α(1 γ 0 (t + (1 αγ N (t. (26 Extending (19 to the dimension N + 2, the evolution of the distribution Γ(t is given by Γ(t + 1 = L(π(t.Γ(t, (27 where the transition matrix L(π of dimension (N + 2 (N + 2 is defined bys π απ 0 0... 0 1 π a απ 0... 0 0 b a απ... 0 L(π =.............., (28 0... b a απ 0 0... 0 b a π 0 0 0... b 1 π with a = (1 απ + α(1 π, b = (1 π(1 α. The matrix L(π has 1 as an eigenvalue of order 1 (Lemma 9 in the Appendix. Let w(π = (w 1 (π,..., w N (π be the eigenvector associated to the unit eigenvalue, and 23
define S N (π = N k= 1 kw k(π as the aggregate level of wealth associated to w(π. By assumption, the level of aggregate wealth is zero and S N (π = 0. One verifies easily that S(1 = 1 and S(0 = 1. The function S N (π is declining in π and continuous in π. Hence, there exists a value π N such that S N (π N = 0. This value of π N defines a steady state for the N-consumption function. The graphs of the function are represented in Figure 4 for different values of N. One verifies that the steady state unemployment rate rises with N: when agents accumulate a larger amount of precautionary savings, demand is lower and unemployment higher. Equilibria Consider a steady state with the N consumption function. Using the Bellman equations (7 in the steady state, using the notation v k = V k V k 1 for any k 0, and recalling the marginal utility of savings ζ k = πv k + (1 πv k+1, we have for N 2, v 0 = β (α(1 + c + (1 αζ 0 v k = β (αζ k 1 + (1 αζ k for 1 k N 1, v N = β (1 α + αζ N 1, v N+k = βζ N+k 1, for k 1, The vector ζ = (ζ 0,..., ζ N 1 satisfies the stationary equation 17 ζ = βq N (πζ + βr N (π, (29 where the matrix Q N (π and the vector R N (π are of dimension N: a b 0 0... 0 πα(1 + c πα a b 0... 0 0 0 πα a b... 0. Q N (π =........., R N (π =... 0. 0... 0 πα a b 0 0... 0 0 πα a b 17 On a transition with an N-consumption function, ζ(t = βq N (π(tζ(t + 1 + βr N (π(t. 24