A. Time and Uncertainty

1. The model is a finite-horizon general equilibrium model, with time t = 0 ,…, T. Uncertainty is represented by a tree of date-events or states s ∊ S, including a root s = 0. Each state s ≠ 0 has an immediate predecessor s^*, and each non-terminal node s ∊ S\S_T has a set S (s) of immediate successors. Each successor τ ∊ S(s) is reached from s via a branch σ ∊ B(s); we write τ = sσ. We denote the time of s by the number of nodes t(s) on the path from 0 to s ^*.

B. Financial Contracts and Collateral

1. A financial contract (A, C) consists of both a promise, A, and collateral backing it, C. Collateral consists of durable goods, which will be called assets. The lender has the right to seize as much of the collateral as will make him whole once the loan comes due, but no more.

Suppose there is a single storable consumption good c and k = 1,…, K assets which pay dividends ${\mathit{d}}_{\mathit{s}}^{\mathit{k}}$ in each state s. We take the consumption good as numeraire and denote the price of asset k in each state as ${\mathit{p}}_{\mathit{s}}^{\mathit{k}}$. We will focus on one-period non-contingent contracts. Contract ${\mathit{j}}_{\mathit{s}}^{\mathit{k}}$ is of the form $(\mathit{j}\cdot {\tilde{1}}_{\mathit{s}},1_{\mathit{k}})$, where ${\tilde{1}}_{\mathit{s}}∊{\mathit{R}}^{\mathit{S}\mathit{\left(}\mathit{s}\mathit{\right)}}$ stands for the vector of ones with dimension equal the number of successors of s and 1_k stands for one unit of asset k. Hence, contract ${\mathit{j}}_{\mathit{s}}^{\mathit{k}}$ promises j units of consumption good in each successor state of s and the promise is backed by one unit of asset k. Contract ${\mathit{j}}_{\mathit{s}}^{\mathit{k}}∊J_{\mathit{s}}^{\mathit{k}}$ where ${\mathit{J}}_{\mathit{s}}^{\mathit{k}}$ is the set of all contracts at state s that use as collateral one unit of asset k. Finally, ${\mathit{J}}_{\mathit{s}}={\cup }_{\mathit{k}}{\mathit{J}}_{\mathit{s}}^{\mathit{k}}$ and ${\mathit{J}}_{\mathit{s}}\mathit{=}{\mathit{\cup }}_{\mathit{s}\mathit{∊}\mathit{S}\mathit{\backslash }{\mathit{S}}_{\mathit{T}}}{\mathit{J}}_{\mathit{s}}$.

The price of contract ${\mathit{j}}_{\mathit{s}}^{\mathit{k}}$ in state s is ${\mathit{\pi }}_{\mathit{s}}^{\mathit{jk}}$. An investor can borrow ${\mathit{\pi }}_{\mathit{s}}^{\mathit{jk}}$ today by selling contract ${\mathit{j}}_{\mathit{s}}^{\mathit{k}}$ in exchange for a promise of j tomorrow. Since the maximum a borrower can lose is his collateral if he does not honor his promise, the actual delivery of contract ${\mathit{j}}_{\mathit{s}}^{\mathit{k}}$ in states τ ∊ S(s) is $\mathit{\text{min}}\mathit{\{}\mathit{j}\mathit{,}{\mathit{p}}_{\mathit{\tau }}^{\mathit{k}}\mathit{+}{\mathit{d}}_{\mathit{\tau }}^{\mathit{k}}\mathit{\}}$. If the collateral is big enough to avoid default, the price of contract ${\mathit{j}}_{\mathit{s}}^{\mathit{k}}$ is given by ${\mathit{\pi }}_{\mathit{s}}^{\mathit{jk}}\mathit{=}\mathit{j}/(1+{\mathit{r}}_{\mathit{s}})$, where r_s is the riskless interest rate (and hence does not depend on the asset used as collateral).

The margin requirement ${\mathit{m}}_{\mathit{s}}^{\mathit{jk}}$ associated to contract ${\mathit{j}}_{\mathit{s}}^{\mathit{k}}$ in state s is given by

Leverage associated to contract ${\mathit{j}}_{\mathit{s}}^{\mathit{k}}$ in state s is the inverse of the margin, $1/m_{\mathit{S}}^{jk}$ and the Loan to Value (LTV) associated to contract ${\mathit{j}}_{\mathit{s}}^{\mathit{k}}$ in state s is $1-m_{\mathit{S}}^{jk}$.

We define the asset margin requirement for asset k, ${\mathit{m}}_{\mathit{s}}^{\mathit{k}}$, as the trade-value weighted average of ${\mathit{m}}_{\mathit{s}}^{\mathit{jk}}$ across all contracts actively traded in equilibrium that used asset k as collateral.⁴

C. Production

1. Each investor h has an endowment of the consumption good and labor, denoted by ${\mathit{e}}_{\mathit{s}}^{\mathit{h}}\mathit{∊}{\mathit{R}}_{\mathit{+}}$ and ${\mathit{l}}_{\mathit{s}}^{\mathit{h}}\mathit{∊}{\mathit{R}}_{\mathit{+}}$ in each state s ∊ S. We assume that the consumption good and labor are present at time 0, ${\mathrm{\sum }}_{h\in H}{e}_0^h>,{\mathrm{\sum }}_{h\in H}{l}_0^h>0.$.

Every agent has direct access to two types of constant-returns-to-scale production processes in the model: an inter-period and a within-period production. The inter-period production is a simple way to model consumption good durability in the economy. A unit of consumption warehoused in state s yields one unit of consumption in all successors states. There is no depreciation.

The second type of production, the within-period production, transforms labor, l, into a portfolio of assets to be chosen by the investor in the set ${\mathit{Z}}_{\mathit{s}}^{\mathit{h}}=\{({\mathit{z}}_{\mathit{s}}^{\mathit{1}},\dots ,{\mathit{z}}_{\mathit{s}}^{\mathit{K}})∊{\mathit{R}}_{\mathit{+}}^{\mathit{K}}\mathit{:}{\mathit{z}}_{\mathit{s}}^{\mathit{1}}+\dots +{\mathit{z}}_{\mathit{s}}^{\mathit{K}}\le {\mathit{l}}_{\mathit{s}}^{\mathit{h}}\}$. Any investor can use his ${\mathit{l}}_{\mathit{s}}^{\mathit{h}}$ units of labor to produce any combination of assets.

D. Utility

1. The von-Neumann-Morgenstern expected utility of each investor h ∊ H is characterized by a Bernoully utility, u^h, a discounting factor, δ^h and subjective probabilities, q^h. We assume that the Bernoulli utility function for consumption in each state s ∊ S; u^h : R₊ → R, is differentiable, concave, and monotonic. Agent h assigns subjective probability ${\mathit{q}}_{\mathit{s}}^{\mathit{h}}$ to the transition from s ^* to s, naturally q₀ = 1. Letting ${\mathit{q}}_{\mathit{s}}^{-\mathit{h}}$ be the product of all ${\mathit{q}}_{\mathit{s}\prime }^{\mathit{h}}$ along the path from 0 to s, we have

E. Budget Set

1. Given asset and contract prices $\left(\right({\mathit{p}}_{\mathit{s}}^{\mathit{k}}\mathit{,}{\mathit{\pi }}_{\mathit{s}}^{\mathit{jk}}),\mathit{s}∊\mathit{S},{\mathit{j}}_{\mathit{s}}^{\mathit{k}}∊{\mathit{J}}_{\mathit{s}}^{\mathit{k}})$, each agent h ∊ H decides what assets to produce, z_s, consumption, c_s, warehousing, w_s, asset holdings, y_s, and contract sales (borrowing) and purchases (lending), ${\mathit{\varphi j}}_{\mathit{s}}^{\mathit{k}}$, in order to maximize utility (2) subject to the budget set defined by

In each state s, expenditures on consumption and warehousing minus endowments and storage, plus total expenditures on assets minus asset holdings carried over from the last period and asset output from the within-period technology, can be at most equal to total asset deliveries plus the money borrowed selling contracts, minus the payments due at s from contracts sold in the previous period.⁵ Within-period production is feasible. Finally, those agents who borrow must hold the required collateral.

Let us emphasize two important things. First, notice that there is no sign constraint on ${\mathit{\varphi j}}_{\mathit{s}}^{\mathit{k}}$: a positive (negative) ${\mathit{\varphi j}}_{\mathit{s}}^{\mathit{k}}$ indicates the agent is selling (buying) contracts or borrowing (lending) ${\mathit{\pi }}_{\mathit{s}}^{\mathit{jk}}$. Second, notice that we are assuming that short selling of assets is not possible. This assumption, however, is not crucial for the results in the paper as we discuss in Section 3.8.

F. Collateral Equilibrium

1. A Collateral Equilibrium in this economy is a set of asset prices and contract prices, production and consumption decisions, and financial decisions on assets and contract holdings $\left(\right(p,\pi ),{(z^h,c^h,w^h,y^h,{\varphi }^h,)}_{h∊H})∊{(R_{+}^K\times R_{+}^{J_s})}_{s∊S\setminus S_T}\times {(R_{+}^{SK}\times R_{+}^S\times R_{+}^{SK}\times {\left(R^{J_s}\right)}_{s∊S\setminus S_T})}^H$ such that ∀s

1. $\sum _{h∊H}(C_s^h+w_s^h-e_s^h-w_{s*}^h)=\sum _{h∊H}{y}_{s*}^h{d}_s$

2. $\sum _{h∊H}(y_s^h-y_{s*}^h-z_s^h)=0$

3. $\sum _{h∊H}{\varphi }_{j_s^k}^h=0,{\forall \mathit{j}}_{\mathit{s}}^k∊J_s$

4. $\begin{array}{l}(z^h,c^h,w^h,y^h,{\varphi }^h)∊B^h(p,\pi ),\forall h\\ (z,c,w,y,\varphi )∊B^h(p,\pi )\Rightarrow U^h\left(c^h\right),\forall h\end{array}$

Markets for consumption, assets and promises clear in equilibrium and agents optimize their utility in their budget set. As shown by Geanakoplos and Zame (1997), equilibrium in this model always exists under the assumptions we have made so far.

A. A One-Asset Baseline Example

1. In this section we assume that there is only one asset. Throughout the paper we consider assets and projects as synonyms.

Suppose there are three periods, t = 0, 1, 2. The single asset, Y, delivers only at the final period. We assume that state 0 has two successors U, for up, and D, for down, representing good and bad news respectively. Each of these states s ∊ {U, D} has at most two successors sU and/or sD, at which the asset pays 1 or R < 1, respectively.⁶ Figure 4 depicts a tree consistent with this description.

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Asset payoff description.

Citation: IMF Working Papers 2010, 206; 10.5089/9781455205370.001.A001

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U can be interpreted as good news since we assume that

i.e., the probability of full payment after U is higher than after D.

In this example the set of states is S ⊆ {0, U, D, UU, UD, DU, DD}.

There is a continuum of heterogenous agents indexed by h ∊ H = [0, 1]. The only source of heterogeneity is in subjective probabilities, ${\mathit{q}}_{\mathit{s}}^{\mathit{h}}$. The higher the h, the more optimistic the agent is about the future. Whenever $h>\mathit{h}\prime ,q_U^h>q_U^{\mathit{h}\prime }$ and $q_{sU}^{\mathit{h}}>q_{sU}^{\mathit{h}\prime }$ for $\mathrm{s}∊\{U,D\}$, provided s has two successors.

Agents are risk neutral and do not discount the future. They start at t = 0 with an endowment of 1 unit of the consumption good and 1 unit of labor. More formally, $U^h=\sum _{s∊S}{\mathit{q}}_{\mathit{s}}^{\mathit{-}\mathit{h}}{c}_s,e_0^h=1$ and ${\mathit{e}}_{\mathit{s}}^{\mathit{h}}=0$, s ≠ 0, and ${\mathit{l}}_{\mathit{0}}^{\mathit{h}}=1$ and ${\mathit{l}}_s^{\mathit{h}}=0$, s ≠ 0.

In this baseline economy with one asset it is clear that in equilibrium every investor will transform his labor into one unit of the asset at time 0.

A more subtle conclusion is the following result regarding leverage:

Proposition 1: In this economy, in which every node has at most two successors states, the only contract j_s traded in equilibrium is the one which promises j_s = min_τ∊S(s){p_τ + d_τ}.

Proof: See Geanakoplos (2003), Fostel-Geanakoplos (2010).

In every state, the only contract actively traded is the one promising the minimal payoff in the future. Equilibrium default is endogenously ruled out and the contract will trade at the riskless interest rate r_s. All contracts will be priced in equilibrium, but only one will be actively traded.

As discussed before, leverage is endogenously determined in equilibrium. In particular, the proposition derives the conclusion that the only contract traded in equilibrium is the one given by the Value at Risk equal zero rule assumed by many other papers in the literature.

In equilibrium the risk-less interest rate must be zero: r_s ≤ 0 because agents do not discount the future, and the presence of the perfect warehousing technology prevents r_s < 0.

By proposition 1, buying 1 unit of Y on margin at state s means: selling a promise of min_τ∊S(s)[p_τ + d_τ] using that unit of Y as collateral, and paying (p_s – min_τ∊S(s) [p_τ + d_τ]) in cash. The Loan to Value (LTV) of Y at s is,

If s ∊ {U, D} has only one successor sU, then s must be good news and so s = U. Moreover, every agent will agree on ${\mathit{q}}_{\mathit{U}}^{\mathit{h}}\mathit{=}{\mathit{q}}_{\mathit{UU}}^{\mathit{h}}=1$ and so in equilibrium we must have p_U = d_UU = 1 and therefore LTV_U = 1/1= 1 = 100%. Analogously, if s ∊ {U, D} has only one successor sD, then $\mathit{s}\mathit{=}\mathit{D}\mathit{,}{\mathit{q}}_{\mathit{sD}}^{\mathit{h}}\mathit{=}{\mathit{q}}_{\mathit{DD}}^{\mathit{h}}\mathit{=}\mathit{1}\mathit{,}{\mathit{p}}_{\mathit{D}}\mathit{=}{\mathit{d}}_{\mathit{DD}}\mathit{=}\mathit{R}$ and therefore LTV_U = R/R = 100%. If s ∊ {U, D} has two successors then R < p _s < 1 and hence LTV_s = R/p_s < 100%. Thus, when volatility post s ∊ {U, D} is zero (because there is only one successor of s), LTV_s = 100%, whereas when volatility post s ∊ {U, D} is positive, LTV_s < 100%.

B. Equilibrium

1. Let us describe the system of equations that characterizes the equilibrium. Because of linear utilities and the continuity of utility in h and the connectedness of the set of agents H = [0, 1], at each state s there will be a marginal buyer, h_s, who will be indifferent between buying or selling Y. All agents h > h _s will buy all they can afford of Y, i.e., they will sell all their endowment of the consumption good and borrow to the max using Y as collateral. On the other hand, agents h < h_s will sell all their endowment of Y and lend to the more optimistic investors. Equating expenditures and revenues provides us with the first three equations in our system.

At s = 0 aggregate revenue from sales of the asset is given by p₀.⁷ On the other hand, aggregate expenditure on the asset is given by (1 – h₀)(1 + p₀) + p_D. The first term is total income (endowment plus revenues from asset sales) of buyers h ∊ [h₀, 1]. The second term is borrowing, which from proposition 1 is p_D. Equating we have

Let s ∊ {U, D} have two successors sU and sD. Total revenue from asset sales must equal total expenditure on asset purchases. This gives us

The first term on the RHS is the income after debt repayment of those holding the asset from period 0. The second term is the income of the new buyers h ∊ [h_s, h₀], carried over from period 0. The last term is new borrowing. Notice that because at s the original buyers h ∊ [h₀, 1] can only borrow R, which is less than the p_D they owe, they will not be able to roll over all their loans without selling some assets. Hence, h_s < h₀, i.e. the marginal buyer must go down. If s has just one successor then it does not matter who the marginal buyer is because they all agree and any one agent can buy all the assets since leverage is 100%.

The next equations state that the price at s ∊ {U, D} is equal to the marginal buyer’s valuation of the asset’s future payoff.

The last equation equates the marginal utility to h ₀ of one dollar to the marginal utility of using one dollar to purchase Y at s = 0:

This last equation needs further explanation. Notice that payoffs on both sides of the equation are weighted by the ratio $\mathit{(}{\mathit{q}}_{\mathit{sU}}^{{\mathit{h}}_{\mathit{0}}}\mathit{/}{\mathit{q}}_{\mathit{sU}}^{{\mathit{h}}_{\mathit{s}}}\mathit{)}$ for s ∊ {U D}. If agent h ₀ reaches state s ∊ {U, D} with a dollar he will want to leverage his wealth to the max to purchase Y.⁸ This will result in a gain per dollar of

$\frac{q_{sU}^{h_0}(1-R)}{p_s-R}=\frac{q_{sU}^{h_0}(1-R)}{q_{sU}^{h_s}1+q_{sD}^{h_s}R-R}=\frac{q_{sU}^{h_0}}{q_{sU}^{h_s}}$

Hence the marginal utility of a dollar at time 0 is given by the probability of reaching U times the dollar times the marginal utility given above plus the analogous expression for reaching D. This explains the RHS of equation (8).⁹ The LHS has exactly the same explanation once we realize that the best action for the h₀ at s ∊ {U, D} is to sell the asset and use the cash to buy it on margin. If s has a unique successor, then $\mathit{(}{\mathit{q}}_{\mathit{sU}}^{{\mathit{h}}_1}\mathit{/}{\mathit{q}}_{\mathit{sU}}^{{\mathit{h}}_{\mathit{s}}}\mathit{)}=1$ and the same equations applies.

We have a system of six equations, described by expressions (5)-(8), and six unknowns: marginal buyers and asset prices at s = 0, U, D.

C. Projects

1. Suppose there are two different projects, variations of the baseline example discussed above. These projects are exactly the same in terms of final asset payoff distribution. To fix ideas, suppose that the probability of final good output 1 is

The only difference between the two projects is in the way information is revealed in the intermediate period. More precisely, projects can differ in the post-volatility induced by news in the intermediate period. By post-volatility we mean the final payoff volatility conditioned on reaching a particular node or state.

D. Pro-Cyclical Leverage

1. There is only one project that gives rise to pro-cyclical leverage and we describe it in figure 5.

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BV Project.

Citation: IMF Working Papers 2010, 206; 10.5089/9781455205370.001.A001

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The probabilities in the tree satisfy equations (3) and (9). If state U is reached in the second period, uncertainty is completely resolved since the asset pays for sure 1 at the end. Leverage at U is 100%. However, if D is reached, uncertainty remains. In fact, D is bad news, but of the sort that not only decreases the expected asset payoff compared with U but also increases final payoff volatility. This project represents the situation in which each piece of bad news is not very informative and induces high future volatility. We call it “Post-Bad News Volatility” project, BV.¹⁰

We solve the system of equations described in section 3.2 to find the equilibrium in this project. Table 1 shows equilibrium prices, marginal buyers and leverage for R = .2. It is easy to check that this is indeed an equilibrium, i.e investors are maximizing and markets clear.

Table 1:

BV Equilibrium.

Table 1:

BV Equilibrium.

	0	U	D
Price, ps	0.95	1.00	0.69
Marginal Buyer, hs	0.87	1.00	0.62
Leverage, LTVs	0.73	1.00	0.29

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Table 1:

BV Equilibrium.

	0	U	D
Price, ps	0.95	1.00	0.69
Marginal Buyer, hs	0.87	1.00	0.62
Leverage, LTVs	0.73	1.00	0.29

The first observation is that the price of Y falls from 0 to D, from .95 to .69, a fall of 27%. The marginal buyer at t = 0, h = .87, thinks at the beginning that there is a probability of 1.69% of reaching the disaster state DD, but once D is reached this probability rises to 13%. This would imply a fall in the price of only 9%. So why is the crash of 27% so much bigger than the bad news of 9%? There are three reasons for the crash.

First, as we just saw, is the presence of bad news. The second reason is that after bad news, the leveraged investors lose all their wealth: the value of the asset at D is exactly equal to their debt, so they go bankrupt. Therefore even the topmost buyer at D is below the marginal buyer at 0. Third, with the arrival of bad news, leverage goes down (margins go up), from LTV ₀ = .73 to LTV _D = .3, so more buyers are needed at D than at 0. Thus the marginal buyer at D is far below the marginal buyer at 0: h _D = .62 < : 87. The asset falls so far in price at D because every agent values it less and because the marginal buyer is so much lower.

The main result of this exercise is that the BV project endogenously generates pro-cyclical leverage. With bad news, leverage goes down and with good news leverage goes up. Why is this? As mentioned before, bad news not only decreases expected asset payoff in the future, but increases future volatility as well and good news reduces the volatility. By equation (4) an increase in volatility increases endogenous margin requirements and lowers leverage in equilibrium. This phenomenon was called the Leverage Cycle by Geanakoplos (2003) and extended further to many assets and adverse selection by Fostel-Geanakoplos (2008).

E. Counter-Cyclical Leverage

1. Every other project gives rise to counter-cyclical leverage because p_U > p_D and hence LTV _U = R/p _U < R/p _D = LTV_D. We concentrate on the simplest example, which we call “Post-Good News Volatility” project, GV, defined by the following tree depicted in figure 6.

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GV Project.

Citation: IMF Working Papers 2010, 206; 10.5089/9781455205370.001.A001

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These probabilities also satisfy equations (3) and (9), that is, every agent h thinks the terminal probabilities of 1 and R are the same for GV as for BV. If D is reached, all uncertainty is resolved given that the asset pays for sure the low dividend R, and leverage is 100%. However, if U is reached uncertainty remains and leverage falls: investors can still borrow R but the price is higher. This GV project represents the situation in which each piece of good news, as opposed to bad news as in the BV project, is not very informative and induces high future volatility.

We solve the system of equations described in section 3.2 to find the equilibrium in this project. Table 2 shows equilibrium prices, marginal buyers and leverage for R = .2. It is easy to check that this is indeed an equilibrium, i.e investors are maximizing and markets clear.

Table 2:

GV Equilibrium.

Table 2:

GV Equilibrium.

	0	U	D
Price, ps	0.89	0.94	0.20
Marginal Buyer, hs	0.63	0.63	0.63
Leverage, LTVs	0.22	1.21	1.00

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Table 2:

GV Equilibrium.

	0	U	D
Price, ps	0.89	0.94	0.20
Marginal Buyer, hs	0.63	0.63	0.63
Leverage, LTVs	0.22	1.21	1.00

In equilibrium, the asset price collapses from .89 all the way to .2 given the imminent nature of the disaster once D has been reached. It goes up at U to .94. The marginal buyer at t = 0 and t = U is the same, so optimists roll-over their debt once they reach U.

F. Why BV is So Different from GV?

1. The main findings from Sections 3.4 and 3.5 are the following:

• (a) The initial price is higher in the BV project (.95) than in the GV project (.89).

• (b) Initial leverage is higher in the BV project (LTV = .73) than in the GV project (LTV = .22).

• (c) Leverage is pro-cyclical in the BV project and counter-cyclical in the GV project.

Why is this the case?

First, the reason why the BV project is more valuable than the GV project is because it can be leveraged more at the beginning. A higher borrowing capacity implies that all the assets in the economy can be afforded by fewer investors, so that the marginal buyer is more optimistic. This naturally raises the project’s price.

Second, BV can be leveraged more at time zero due to the type of bad news. By proposition 1 the maximum agents can promise is the worse case scenario in the immediate future, i.e, the price of the project after bad news. But in the BV project the price does not fall as much precisely because bad news is less informative and volatile. By contrast, bad news in the GV project is very informative, lowering the promise in equilibrium.

Third, as explained before, the cyclical properties derived in each project are a direct consequence of the difference in volatility between the projects. In BV bad news induces future volatility, lowering leverage, while in GV good news induces volatility, lowering leverage.

G. BV vs GV: Long Run Analysis

1. Having completely characterized the equilibrium in the two projects, considered as separate economies, we wonder if these results hold when we consider longer horizons. With this in mind, we extend our previous examples for an N horizon economy. We maintain the same terminal probabilities for outcomes 1 and R, independent of N, with constant probabilities of up throughout each tree. The BV and GV projects are described in figure 7. In the BV project, as before, the imminent occurrence of the bad final outcome R is pushed until the very end, and bad news comes in small drops with an associated higher future volatility. On the other hand, in the GV project, good news, instead of bad news, has the property of revealing little information and inducing high volatility. We calculate the equilibrium for each project separately. The complete system of equations that characterizes the equilibrium in each project is described in detailed in Appendix 1. They are the natural (though not obvious) extension of the three period case. The prices and leverage are noted at some of the nodes for N = 10 in figure 7, complete equilibrium information is presented in Appendix 1.

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Prices and leverage for BV and GV projects, N=10 periods.

Citation: IMF Working Papers 2010, 206; 10.5089/9781455205370.001.A001

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Figure 7 shows that the results of previous sections hold even in longer horizon economies. The price of the BV project is higher than the GV project and leverage is pro-cyclical in the BV project and counter-cyclical in the GV project. In fact, the longer the horizon the bigger the gap in initial prices.

H. Arrow-Debreu Equilibrium

1. In order to help understand why BV is more valuable than GV we also calculate the Arrow-Debreu equilibrium for each project. It is evident that every agent will wait until the last period to consume. In each case there are three terminal states. The difference is that in the BV project the good event (where the dividend is 1) is partitioned into two states, UU and DU, whereas in the GV project the bad event (in which the dividend is .2) is partitioned into two states, UD and DD. See figure 8.

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Projects and payoff event partitions.

Citation: IMF Working Papers 2010, 206; 10.5089/9781455205370.001.A001

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To compute the Arrow-Debreu equilibrium, we guess that agents above h₁ buy only the Arrow security for state 1, agents h₁ > h > h₂ will buy only the Arrow security 2 and agents below h₂ will buy the Arrow security 3. Endowments in each state are the cash plus the asset dividends in each state.

As we can see in table 3 the price of the BV project is .55, higher than the price of the GV project, .48. Asset prices are given by the sum of the Arrow prices weighted by the asset dividend in each state.¹¹ The price of the good event is given by the sum of the first two Arrow prices, a total of .4332 in the BV project. In contrast, the price of the good event is given only by the first arrow price in the GV project, .3598. Of course, this difference makes the asset price higher in the BV project.

Table 3:

Arrow-Debreu equilibrium for BV and GB projects.

Table 3:

Arrow-Debreu equilibrium for BV and GB projects.

	BV Project	GV Project
p1	0.2848	0.3598
p2	0.1484	0.173
p3	0.5668	0.4672
Asset Price	0.5465	0.4878
h1	0.4789	0.2624
h2	0.2074	0.1915

View Table

Table 3:

Arrow-Debreu equilibrium for BV and GB projects.

	BV Project	GV Project
p1	0.2848	0.3598
p2	0.1484	0.173
p3	0.5668	0.4672
Asset Price	0.5465	0.4878
h1	0.4789	0.2624
h2	0.2074	0.1915

Why is the Arrow price of the good event worth more in the BV economy than in the GV economy, even though every agent attaches the same probability? Due to heterogenous priors, a finer partition of the good event allows agents to bet, increasing the Arrow price of the good event.

From the Arrow-Debreu equilibria we conclude that the gap in initial asset prices between BV and GV obtained in sections 3.4 and 3.5 does not rely on market incompleteness or the assumed short-selling constraints. The Arrow-Debreu equilibrium helps us understand why in collateral equilibrium leverage makes BV more valuable than GV. In the BV collateral economy, one can bet on a payoff of 1 by in effect buying the UU Arrow security via leverage at s = 0, or by warehousing at s = 0 and then leveraging at s = D, thus in effect buying a combination of UU and DU Arrow securities. In the GV collateral economy, one can only bet on the payoff of 1 via the UU Arrow security.

Though the gap in price between the complete markets BV and GV economies is just as big as in the collateral BV and GV economies, the absolute price level of the complete market economies is much lower. In the complete market economies pessimists can bet on the bad .2 outcome, whereas in the collateral equilibrium they cannot because of the short sale constraint.

A. Equilibrium Leverage

1. Before moving on to solve the model, let us go back to the question of endogenous leverage. Proposition 1 holds for the intermediate states s ∊ {UU, UD, DU, DD}, since for each asset there are at most two distinct successor payoff values. Hence, the only contract traded in all intermediate states is the one that prevents default in equilibrium as in section 3.

However, the situation is different at time 0 since there are four successor states in S(0) with three distinct successor payoff values for each asset¹², and therefore it is not possible to appeal to the result anymore. In fact, the following holds

Proposition 2: In this economy, two contracts are traded in equilibrium at time 0 for each asset: the one which promises ${\mathit{J}}_{\mathit{s}}^{\mathit{k}}={\mathit{p}}_{\mathit{DD}}^{\mathit{k}}$ and the one that promises ${\mathit{J}}_{\mathit{s}}^{\mathit{k}}={\mathit{p}}_{\mathit{DU}}^{\mathit{k}}$.

Proof: Fostel-Geanakoplos (2010).

For each asset two types of contracts will be traded: one that promises the worst-case scenario and another that promises the middle-case scenario. While the first one is risk-less as before, the second one is not since it defaults in the worst state. In this model, not only is there default in equilibrium, but also the same asset is traded simultaneously with different margin requirements by different investors. Araujo et.al. (2009) and Fostel-Geanakoplos (2010) show this in a two period model. We show in the following section that this is an equilibrium also in a dynamic setting for the first time. The dynamic setting is more difficult because the payoffs of the risky bonds are endogenous.

B. Procedure to Find the Equilibrium

B.2 Regimes

Next, we will guess a regime, consisting of a ranking of the marginal buyers and a description of what each agent buys in each node, in order to be able to define a system of equations. Once we get a solution we need to check: first, that ${\mathit{p}}_{\mathit{DU}}^{\mathit{X}}>{\mathit{p}}_{\mathit{DD}}^{\mathit{X}}$, so that prices are consistent with our guess about which bonds are risky and riskless on X, second, that ${\mathit{p}}_{\mathit{UU}}^{\mathit{Y}}>{\mathit{p}}_{\mathit{DU}}^{\mathit{Y}}$, so that prices are consistent with with our guess about which bonds are risky and riskless on Y, and finally, that each regime is genuine, i.e. all agents are maximizing with those choices.

1. We next describe the regimes at each node. Figure 10 shows a graphical illustration of them and of the equilibrium values of all marginal buyers.

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Equilibrium Regimes.

Citation: IMF Working Papers 2010, 206; 10.5089/9781455205370.001.A001

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• At s = 0

  $\mathit{h}>{\mathit{h}}_{\mathit{M}}^{\mathit{Y}}$ buy Y, sell X and promise ${\mathit{p}}_{\mathit{DU}}^{\mathit{Y}}\mathrm{.}{\mathit{h}}_{\mathit{M}}^{\mathit{Y}}>\mathit{h}>{\mathit{h}}_{\mathit{M}}^{\mathit{X}}$ buy X, sell Y and promise ${\mathit{p}}_{\mathit{DU}}^{\mathit{Y}}\mathrm{.}{\mathit{h}}_{\mathit{M}}^{\mathit{X}}>\mathit{h}>{\mathit{h}}_{\mathit{m}}^{\mathit{X}}$ buy X, sell Y and promise ${\mathit{p}}_{\mathit{DD}}^{\mathit{X}}\mathrm{.}{\mathit{h}}_{\mathit{m}}^{\mathit{X}}>\mathit{h}>{\mathit{h}}_{\mathit{m}}^{\mathit{Y}}$ buy Y, sell X and promise R. ${\mathit{h}}_{\mathit{m}}^{\mathit{Y}}>\mathit{h}>{\mathit{h}}^{\mathit{BY}}$ sell both assets and buy the BY bond (so lend in the risky market collateralized by Y). h^BY > h > h^BX sell all assets and buy the BX bond (so lend in the risky market collateralized by X). Finally, h < h^BX sell everything, hold risk-less securities (so lend in the risk-less markets).

• At s = UU

  $\mathit{h}>{\mathit{h}}_{\mathit{UU}}^{\mathit{Y}}$ buy Y and promise R. Below lend and buy X. ${\mathit{h}}_{\mathit{m}}^{\mathit{X}}>{\mathit{h}}_{\mathit{UU}}^{\mathit{Y}}>{\mathit{h}}_{\mathit{m}}^{\mathit{Y}}$.

• At s = DU

  All $\mathit{h}>{\mathit{h}}_{\mathit{M}}^{\mathit{X}}$ go bankrupt since they promise exactly what they own. $\mathit{h}>{\mathit{h}}_{\mathit{DU}}^{\mathit{X}}$ buy X and promise R. ${\mathit{h}}_{\mathit{DU}}^{\mathit{X}}>\mathit{h}>{\mathit{h}}_{\mathit{DU}}^{\mathit{Y}}$ buy Y and promise R. All $\mathit{h}<{\mathit{h}}_{\mathit{DU}}^{\mathit{Y}}$ lend. Finally, ${\mathit{h}}^{\mathit{BY}}>{\mathit{h}}_{\mathit{DU}}^{\mathit{X}}>{\mathit{h}}_{\mathit{DU}}^{\mathit{Y}}>{\mathit{h}}^{\mathit{BX}}$.

• At s = DD

  All $\mathit{h}>{\mathit{h}}_{\mathit{m}}^{\mathit{Y}}$ are out of business either because they default or they have no money left. $\mathit{h}>{\mathit{h}}_{\mathit{DD}}^{\mathit{X}}$ buy X and promise R. $\mathit{h}<{\mathit{h}}_{\mathit{DD}}^{\mathit{X}}$ lend. Finally, ${\mathit{h}}^{\mathit{BY}}>{\mathit{h}}_{\mathit{DD}}^{\mathit{X}}>{\mathit{h}}^{\mathit{BX}}$.

The system of equations is conceptually an extension of the system in section 3. In every state supply equals demand for all the securities. Also marginal buyers are determined by an indifference condition between investing in two different securities. As before, all marginal utility of a dollar invested in any security is weighted by the marginal utility of future actions in each state. The system is presented in Appendix 2.