A Macroeconomic Model with Sudden Stops

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Master thesis in Stochastics and Financial Mathematics

A Macroeconomic Model with Sudden

Stops

Bart van der Paardt

Abstract

The article A Macroeconomic Model with a Financial Sector by M.K. Brunnermeier and Y. San-nikov studies the full equilibrium dynamics of an economy with financial frictions. Their model is characterized by nonlinear and asymmetric amplification effects. The economy spends most of its time in a prosperous region near the stochastic steady state, but it may degenerate to a volatile middle or even a depressed low region of the state space. We extensively discuss this model as the baseline model of this thesis. The economic variables are given by stochastic processes that are adapted to a Brownian motion that generates exogenous aggregate shocks to the economy. In the Markov equilibrium most of the economic variables depend solely on the state variable ηt, the experts’ wealth share. We ground the model in the theory of stochastic

integration, show that ηt has a reﬂecting boundary η∗, and check some of the implicit claims

that the model is built on. In the second part of this thesis we consider Sudden Stops. Emerg-ing economies are vulnerable to this phenomenon. We mathematically depict a Sudden Stop by adding an abrupt increase of the interest rate to the model. The impact of a Sudden Stop turns out to be state dependent. Generally, it reduces the price of capital and economic growth. Agents may anticipate the probability of a future Sudden Stop. Our model shows that the fear for a Sudden Stop, in itself, already deteriorates output.

supervised by Prof. dr. Peter J. C. Spreij Prof. dr. Sweder J. G. van Wijnbergen

Dr. Alex Clymo

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1 Introduction and related literature

This thesis is primarily based on a macroeconomic model by Brunnermeier and Sannikov (2014, hereafter BrunSan). They study the full equilibrium dynamics of an economy with ﬁnancial frictions. It models growth, the business cycle, the risk of negative shocks, and the possibility of crises. Important previous models of the macroeconomy with ﬁnancial frictions are based on log-linearised approximations around the steady state. BrunSan solve the full dynamics of the model using continuous-time stochastic processes.

The model has two types of agents: experts and households, where experts are more pro-ductive and less patient than households. The experts’ wealth share variable ηt plays an

important role. Together with the total amount of capital Kt, it describes the entire state of

the economy. Most other economic variables are a function of ηt. The economy is subject to

exogenous shocks that are mathematically expressed by a Brownian motion.

BrunSan’s model is characterized by some important phenomena. First, the system’s re-ception of shocks is nonlinear. If the economy is in a depressed state, shocks are strongly amplified, while the system quickly recovers from most shocks if it is near the steady state. Furthermore, the system’s reactions to shocks are asymmetric. At the steady state, positive shocks lead to more consumption and little amplification. On the other hand, large negative shocks are amplified and can possibly lead to depression. At last, the system spends most of the time near the stochastic steady state, but it eventually happens that it descends to the depressed region with, as a consequence, misallocation of capital and low economic growth. The economy can get trapped in this region for a while. Mathematically, this is substan-tiated by the stationary distribution of the state variable η having high densities near the boundaries of its domain.

BrunSan’s paper is mathematically grounded in the theory of stochastic integration as, among others, put forth by Spreij (2014). I give the mathematical conditions under which the eco-nomic variables, which are mostly defined by stochastic integrals, are well defined. Although I give some definitions, for example of a filtration, some knowledge of stochastic processes and stochastic integration is required to understand these conditions and the derivations I make throughout this thesis. I also check some of BrunSan’s claims. I check their claim that η∗_{, the end point of the state space, is a reflecting boundary of the process η}

t using a

deﬁnition given by Chung and Williams (1983). In the process, I present an explicit formula for the experts’ consumption ζt.

Emerging economies are vulnerable to Sudden Stops. Mendoza (2010) identifies three main empirical regularities that define Sudden Stops: a reversal of international capital flows, a decline in production, and a correction in asset prices. I reinterpret the ‘experts’ and ‘households’ in BrunSan’s model as agents inside an emerging economy (A-agents) and for-eign investors respectively, such that the model can account for a Sudden Stop. I modelled the Sudden Stop as a sudden increase in the interest rate that foreign investors charge to A-agents. The impact of the Sudden Stop can be calculated by comparing the accounting equation of all A-agents at the time just before and the time right after the Sudden Stop with each other.

Mendoza’s three main empirical regularities are observed in this model. A Sudden Stop causes the capital price, and the A-agents’ net worth to decrease. Economic growth declines.

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Furthermore, the results show interesting state dependency. Proportionally, the A-agents’ net worth falls harder if they are undercapitalized at the moment before a Sudden Stop happens. If A-agents are undercapitalized, a Sudden Stop causes misallocation of capital. Less productive foreign investors have to manage a proportion of the total capital stock. This entails a reversal of capital ﬂows. If A-agents are well capitalized, the occurrence of a Sudden Stop does not bring about misallocation of capital. All things considered, a Sudden Stop is more painful if it hits when A-agents are less capitalized.

Finally, I have made a model of agents that anticipate the possibility of a Sudden Stop. Clymo (2015) studies the ex-ante effects of the fear of future financial crises. I do the same for the case that agents fear a Sudden Stop. A truncated Poisson process tells when the Sudden Stop happens. I give four definitions of the Poisson process, one of them is the famous martingale definition. According to Shreve (2004), “the Poisson process serves as the starting point for jump processes.” Notice that it is a stochastic process, so the agents cannot anticipate the exact time of a Sudden Stop. The derivations of the equilibrium of the model are inspired by Clymo’s derivations. I reproduce Shreve’s treatise on Itô’s formula for jump processes and apply it to ascertain that the math behind the equations is in order.

I give a quick overview of the results. The higher the agents assess the risk of a Sudden Stop, the lower the capital price. The endogenous end point of the state space η∗ _{moves to}

the right: if agents judge the probability of a Sudden Stop to be high, then A-agents build more net worth before they start to consume to cushion the expected blow of a Sudden Stop. Moreover, a lower capital price brings about a lower investment rate. This implies a lower growth rate. Thus, the fear of a Sudden Stop deteriorates economic growth.

In sum, my main contributions are the following. First, I grounded BrunSan’s model in the theory of stochastic integration and I checked some of their claims. Second, I derived the mathematical equations for the value of economic variables right after a sudden increase of the interest rate. These form a model for an unanticipated Sudden Stop. Third, I found a way to solve these equations numerically. Finally, I used a certain simpliﬁcation to numeri-cally calculate the economic variables in the adjusted model in which agents anticipate the probability of a Sudden Stop.

My thesis has a clear structure. Section 2 is concerned with BrunSan’s model which is the baseline model of this thesis. Sections 2.1-2.3 set up the model. I have tried to explain everything on a level that can be understood by mathematicians. These are followed by section 2.4 which states the main proposition, proposition 3, that exactly tells how a lot of the economic variables can be found as a function of η. In section 2.5 and 2.6 I prove that η∗ _{is a reﬂecting boundary of the process η}

t. Section 2.7 presents an algorithm based

on proposition 3, section 2.8 discusses the stationary distribution, and in section 2.9 I plot some sample paths of η with help of the Euler method.

The model of an unanticipated Sudden Stop is presented in section 3. Section 3.1 discusses the phenomenon of a Sudden Stop as deﬁned by Mendoza, and section 3.2 reinterprets the baseline model in terms of A-agents and foreign investors. Sections 3.3 and 3.4 build the intuition behind the model, and section 3.5 sets up the main equation. The results are presented in section 3.6.

Section 4 discusses the model in which agents anticipate the probability of a Sudden Stop before the Sudden Stop takes place. In section 4.1, I give four deﬁnitions of the Poisson

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process, and section 4.2 presents Itô’s formula for jump processes. Section 4.3 is devoted to the derivation of proposition 10 in subsection 4.3.7 which is similar to proposition 3 in section 2.4. It tells us that the economic variables are, from a practical, mathematical perspective, the result of diﬀerential equations. In section 4.4 I discuss the original algorithm I constructed. It turned out to be slow, so I constructed an alternative algorithm that is based on a simpliﬁcation. This algorithm is given in the next section. The results are presented in section 4.6. Section 5 concludes.

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2 The Baseline Model

2.1 Experts, households and an introduction of some economic

vari-ables

The input objects of BrunSan’s model are the parameters

r, ρ, a, a, δ, δ, σ _{∈ R}++,

the concave function Φ : Dom(Φ) → R, which satisﬁes Dom(Φ) ⊆ R, Φ(0) = 0, Φ0_{(0) =}

1, Φ0(.) > 0, Φ00(.) < 0, and a Brownian motion Z. The Brownian Motion plays a signif-icant role in the model. It generates exogenous aggregate shocks to the economy which instantaneously improve or aggravate the state of the economy.

The economic variables in the model are all stochastic processes. Stochastic processes model time- and chance-dependency at the same time, they can, for instance, be used to forecast the weather or to model future values of stock or option prices. BrunSan use real-valued continuous-time stochastic processes. A formal definition of such processes is given below. The mathematical definitions in sections 2.1 until 2.3 are taken from Spreij (2014). Notice that a Brownian motion is an example of a real-valued continuous-time stochastic process. Definition 1. Given is a probability space (Ω, F, P), here Ω is a set, F a sigma-algebra on this set and P is a probability measure defined on (Ω, F). A function X : Ω → R is called a random variable if it is F-measurable. A real-valued continuous-time stochastic

process X (hereafter referred to as a ‘stochastic process’) is a collection {Xt, t ∈ R+} of

random variables. Such a stochastic process is called continuous if for every ω ∈ Ω the path t 7→ Xt(ω) is continuous.

BrunSan divide the population in the economy into two classes: experts and households. “Both agents can own capital, but experts are able to manage it more productively” (page 6). The set of experts is I = [0, 1], which is indexed by i ∈ I, and the set households is J = (1, 2], which is indexed by j ∈ J.1 Physical capital managed by an individual expert is denoted by kt, where t ∈ R+ is the time measured in years. It produces output at rate

yt = akt (1)

Capital held by an expert evolves according to kt= k0+ Z t 0 (Φ(ιs) − δ)ksds + Z t 0 σksdZs. (2)

1_{Notice that the cardinalities of the sets I and J are inﬁnite, even uncountably inﬁnite. Why do BrunSan}

make this assumption? It seems to be a strange assumption, because in the real world the number of experts and households within an economy is finite. In this model, it does not matter what the cardinalities of these index sets are: infinite, finite or one. Experts and households may each be embodied by a single representative. However, in another paper by Brunnermeier and Sannikov (2011), agents are, in addition to aggregate shocks, exposed to agent-specific shocks. The agent-specific shocks cancel out in aggregate because the sets of agents are infinite. This is a consequence of the Law of large numbers. Thus, in our current baseline model, BrunSan make the assumption that |I| and |J| are infinite to ensure cohesion with their other papers.

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Similarly, capital held by an individual household evolves according to k_t= k₀+ Z t 0 (Φ(ι_s) − δ)k_sds + Z t 0 σk_sdZs (3)

and it produces output at rate

y

t= a kt. (4)

Here ιt (ιt) is the investment rate per unit of capital and δ (δ) is the depreciation rate for

experts (households). Notice that experts are more productive because a ≥ a and δ ≤ δ. The concavity of ι 7→ Φ(ι) “represents the technological illiquidity, i.e. adjustment costs of converting output to new capital and vice versa,” (BrunSan, page 7) and σ is the exogenous volatility.

kt and ktare deﬁned as Itô processes. Let X = R . 0µ

X_{dt +}R.

0σ

X_dZ _{be an Itô process. Under}

the conditions that

µX is a measurable process, (5) σX is a progressive process, (6) ∀t ∈ [0, ∞) : _E Z t 0 |µX_s | ds < ∞, (7) and ∀t ∈ [0, ∞) : _(E Z t 0 (σ_sX)2dhZis)1/2= (E Z t 0 (σX_s )2ds)1/2< ∞ (8) the process X is well-deﬁned. Notice that hZi denotes the quadratic variation of Z. All processes in BrunSan are adapted and continuous, hence conditions (5) and (6) are automat-ically satisﬁed. We assume that all Itô processes that appear in this thesis satisfy conditions (7) and (8).2

Next, consider the market price of capital, which is denoted by qt. Suppose, for a moment,

that the investment rates per unit of capital ι, ι ∈ Dom(Φ) ⊆ R are given. Given an expert’s capital holdings kt, her yearly dividend is akt − ιkt. Thus, the experts’ expected yearly

dividend per unit of capital is D = a − ι. By equation (2), the yearly growth rate, in dividends, is g = Φ(ι) − δ. According to the Gordon growth formula, the current value of all future dividends per unit of capital for experts is

D

r − g =

a − ι r − (Φ(ι) − δ),

where r < ρ is the interest rate for which households provide debt funding to experts. Besides, ρ is the experts’ discount rate. This parameter plays a role in the experts utility maximization problem. Similarly, the current value of all future dividends per unit of capital for households is

a − ι r − (Φ(ι) − δ).

The price of capital lies between these two values. If experts would manage all capital, the current value of all future dividends per unit of capital would be equal to the price of capital:

q = max

ι∈Dom(Φ)

a − ι r − (Φ(ι) − δ).

2_{The assumptions (7) and (8) are stricter than necessary to deﬁne the stochastic integrals. Both assumptions}

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The current value of all future dividends per unit of capital is maximized over ι, because the agents are to choose the investment rate themselves. This is called the ﬁrst-best price. If households would manage all capital, the price would equal the current value of all future dividends per unit of capital as well:

q = max

ι∈Dom(Φ)

a − ι r − (Φ(ι) − δ).

This is the liquidation value. From the foregoing, it follows that the capital price qtis bounded

between the liquidation value and the ﬁrst-best price. BrunSan postulate the law of motion of qtto take the following form.

qt= q0+ Z t 0 µq_sqsds + Z t 0 σ_sqqsdZs.

The market price of capital is determined endogenously in equilibrium.

Now suppose an expert holds kt units of capital at price qt. By the integration by parts rule3

ktqt= k0q0 + Z t 0 qsdks+ Z t 0 ksdqs+ hk, qit = k0q0 + Z t 0 qsdks+ Z t 0 ksdqs+ h Z t 0 σksdZs, Z t 0 σ_sqqsdZsi = k0q0 + Z t 0 qsdks+ Z t 0 ksdqs+ Z t 0 σσ_sqksqsdhZ, Zis = k0q0 + Z t 0 qsdks+ Z t 0 ksdqs+ Z t 0 σσ_sqksqsds,

where hk, qi denotes the quadratic covariance process of the processes k and q. It follows that

d(ktqt) = qtdkt+ ktdqt+ σσ q tktqtdt

= (Φ(ιt) − δ)qtktdt + σktqtdZt+ µqtktqtdt + σqtktqtdZt+ σσqtktqtdt.

Hence, the value of capital evolves according to d(ktqt) ktqt = (Φ(ιt) − δ + µqt + σσ q t) dt + (σ + σ q t) dZt.

This is the capital gains rate. Besides, “capital also generates a dividend yield of (a − ιt)/qt

from output remaining after internal investment.” Therefore, “the total return that experts earn from capital (per unit of wealth invested) is” (BrunSan, page 9)

drk_t = a − ιt qt dt | {z } dividend yield + (Φ(ιt) − δ + µqt + σσ q t) dt + (σ + σ q t) dZt | {z }

capital gains rate

. (9)

3_{The integration by parts rule is a special case of Itô’s formula. Itô’s formula only holds with probability}

one. Hence, if we us it to show that two processes are equal, we mean to say that they are modiﬁcations of each other. Processes X and Y are modiﬁcations of each other if P(Xt= Yt) = 1for all t. This is enough for

our purpose, because P(Xt= Yt) = 1implies that the random variables Xtand Ythave the same probability

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Assets Liabilities

qtkt dt

nt

Table 1: Balance sheet of an expert. Similarly, households earn a return of

d rk_t = a − ιt qt dt | {z } dividend yield + (Φ(ιt) − δ + µqt + σσ q t) dt + (σ + σ q t) dZt | {z }

capital gains rate

.

Consider the balance sheet of an expert, it is presented in Table 1. Her only assets are the capital holdings, worth qtkt. A liability is her indebtedness dt to households. What is left

on the liability side is the expert’s net worth nt. We want to ﬁnd an expression for the net

worth. Therefore, think of what happens between time t and time t + dt. The value of capital increases by qtktdrkt, and the value of debt decreases by dtr dt. Furthermore, experts

consume dct. The stochastic process ct is the cumulative consumption, it keeps track of the

consumption from time 0 until time t. This entails that ntevolves according to

dnt= qtktdrtk− dtr dt − dct. (10)

The leverage ratio is the value of capital managed by an expert divided by its net worth, xt = qtkt/nt. We can rewrite equation (10) in terms of leverage ratio and net worth. Rewrite

qtkt= ntxt and dt= qtkt− nt= (xt− 1)nt, hence nt = n0+ Z t 0 xsnsdrsk+ Z t 0 ns(1 − xs)r ds − ct. (11)

BrunSan give this equation in diﬀerential form, dnt nt = xtdrkt + (1 − xt)r dt − dct nt . The net worth of households evolves similarly,

n_t = n₀+ Z t 0 x_sn_sdrk_s + Z t 0 n_s(1 − x_s)r ds − c_t. (12)

In economics, theorems concerning equilibria often include one or more utility optimization problems. This is also the case in BrunSan (page 10). We ﬁrst present the expert’s problem and subsequently the household’s problem.

Deﬁnition 2 (Expert’s problem). Each experts solves max xt,ct,ιtE Z ∞ 0 e−ρtdct

subject to the solvency constraint ∀t : nt ≥ 0, the dynamic budget constraint (11) and the

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condition, as dct ≥ 0. This is common notation in the economics literature. Notice that

consumption cannot be negative. Als notice that e−ρ _{is the discount factor. The consumption}

of one unit of capital today gives utility 1, while the consumption of one unit of capital a year later gives utility e−ρ _{< 1}_.

It is important to understand that the expected utility is maximized over the stochastic pro-cesses xt, ct and ιt. This can be interpreted as follows: for every realization {Zt(ω), t ≥ 0},

where ω ∈ Ω, the expert chooses an optimal strategy concerning leverage {xt(ω), t ≥ 0},

consumption {ct(ω), t ≥ 0} and investment rate {ιt(ω), t ≥ 0}.

Deﬁnition 3 (Household’s problem). Each household solves max x_t≥0,c_t,ι_tE Z ∞ 0 e−rtdc_t

subject to nt ≥ 0 and the dynamic budget constraint (12). Unlike the experts, “households

consumption dct can be both positive and negative.” Hence, ct is allowed to be decreasing.

BrunSan (page 11) summarize the three diﬀerences between experts and households as follows. 1. “Experts are more productive than households” because a ≥ a and δ < δ.

2. “Experts are less patient than households,” since ρ > r.

3. “Experts’ consumption has to be positive while the consumption of households is allowed to be negative.”

2.2 Deﬁnition of the equilibrium

BrunSan give a definition of an equilibrium. We make some adjustments in order to ame-liorate the connection with the theory of stochastic integration. First we give two definitions that are needed to understand the definition of the equilibrium.

Deﬁnition 4. Given is a probability space (Ω, F, P). A ﬁltration F = {Ft, t ≥ 0} is a

collection of sub-σ-algebras of F such that

for all s ≤ t : Fs⊂ Ft.

Let X be a stochastic process, on probability space (Ω, F, P). FX

t is the smallest

sub-σ-algebra of F such that

for all s ≤ t : Xs is FtX-measurable.

The ﬁltration generated by X is the ﬁltration FX _{= {F}X

t , t ≥ 0}.

Deﬁnition 5. A stochastic process X is adapted to a ﬁltration F if for all t ≥ 0 the random variable Xt is Ft-measurable. In particular, note that X is adapted to FX.

Secondly, we discuss the total demand for and supply of consumption goods. An expert i’s cumulative demand for consumption goods at time t is ci

t, hence the experts’ total cumulative

demand for consumption goods is

C_te = Z

I

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The households’ total cumulative demand for consumption goods at time t is C_th =

Z

J

cj_tdj.

At time t expert i produces yi

t= akti and it reinvests ιitkti to form new capital. What is left is

its supply of consumption goods (a − ιi

t)kit. The experts’ total supply of consumption goods

is

S_te= Z

I

(a − ιi_t)ki_tdi.

Similarly, the households’ total supply of consumption goods is S_th =

Z

J

(a− ιj_t)kj_tdj.

It follows that the market clearing condition for consumption goods can be expressed as C_te+ C_th = Z t 0 S_seds + Z t 0 S_shds. (13)

Now we are ready to give the deﬁnition of an equilibrium in our model. The deﬁnition is the same as in BrunSan, but the presentation is slightly altered.

Deﬁnition 6 (Deﬁnition of an equilibrium). Given is a probability space. For any initial endowments of capital {ki 0, k j 0 : i ∈ I, j ∈ J} such that Z I k₀idi + Z J kj₀dj = K0,

an equilibrium is described by stochastic processes that are adapted to the ﬁltration generated by the Brownian Motion {Zt, t ≥ 0}: the price of capital {qt}, net worths {nit, n

j t ≥ 0}, capital holdings {ki t, k j t ≥ 0}, investment decisions {ιit, ι j

t}, and consumption choices {cit

(non-decreasing) , cj

t}of individual agents i ∈ I, j ∈ J; such that

(i) initial net worths satisfy ni

0 = ki0q0 and nj0 = k j 0q0.

(ii) each expert i ∈ I and each household j ∈ J solve their problems given prices.

(iii) market for consumption goods clears, i.e. equation (13) holds, and the capital market clears, i.e. Z I ki_tdi + Z J kj_tdj = Kt, where Kt= K0+ Z t 0 Z I (Φ(ιs_t) − δ)ki_sdi + Z J (Φ(ιj_s) − δ)kj_sdj ds + Z t 0 σKsdZs.

Walras’ Law implies that if these the two markets clear, then the remaining market for risk-free lending and borrowing at rate r clears automatically.

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2.3 More economic variables and optimality conditions

We have given the deﬁnition of an equilibrium in the BrunSan model. This gives us the opportunity to derive the optimal internal investment rate ιt, which is the path ιt follows in

equilibrium. BrunSan claim the following.

Claim 1. The optimal internal investment rate ιt = ιt is given by the ﬁrst-order condition

Φ0(ιt) = 1/qt. Hence4,

ιt= (Φ0)−1(

1 qt

). (14)

Proof. Let t ∈ [0, ∞) arbitrary. The optimal ιt maximizes the total return on capital drtk.

Hence the optimal ιtis the solution to the problem

max ιt Φ(ιt) − ιt qt .

Deﬁne f : R+_{→ R, by f(y) = Φ(y) −} y

qt. By the concavity of Φ, ∀y ∈ R+ : f00(y) = Φ00(y) < 0.

Therefore, the optimal ιt satisﬁes

Φ0(ιt) −

1 qt

= f0(ιt) = 0.

The proof for the optimal internal investment rate for households, ι, is similar.

From now on, if we write ιt we mean the optimal internal investment rate at time t. Next,

BrunSan introduce three new economic variables. ψt:= 1 Kt Z I k_tidt

is the fraction of physical capital held by experts. It follows from the deﬁnition that for all t, ψt∈ [0, 1]. ζt:= Z t 0 1 ns dcs

is the cumulative consumption rate (of the experts). This process is called the cumulative consumption rate since dζt = dc_n_tt. BrunSan deﬁne a strategy as a pair of processes {xt, ζt}

that speciﬁes leverage and the (cumulative) consumption rate. Note that E Z ∞ t e−ρ(s−t)dcs|FtZ ,

gives the expert’s future expected payoﬀ under this strategy. “The following proposition pro-vides necessary and suﬃcient conditions for the strategy to be optimal.”

4_{The function Φ}0 _:_{Dom(Φ) → Image(Φ}0₎_{is bijective because Φ(.) is concave, therefore the inverse function}

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Proposition 1 (BrunSan II.2). Suppose θtis an Itô process of the form θt= θ0+ Z t 0 µθ_sθsds + Z t 0 σ_sθθsdZs.

Then ntθt represents the maximal future expected payoﬀ that an expert with net worth nt can

attain, i.e. θtnt:= E Z ∞ t e−ρ(s−t)dcs|FtZ , (15)

and {xt, ζt}is an optimal strategy if and only if the following four conditions hold true.

(i) θt ≥ 1at all times, and ζt increases only when θt = 1.

(ii) µθ

t = ρ − r.

(iii) Either xt> 0 and

a − ιt qt + Φ(ιt) − δ + µqt + σσ q t − r | {z }

expected excess return on capital

= −σθ_t(σ + σq_t), | {z } risk premium (16) or xt= 0 and a − ιt qt + Φ(ιt) − δ + µqt + σσ q t − r ≤ −σ θ t(σ + σ q t).

(iv) The transversality condition,

lim

t→∞E[e −ρt

θtnt] = 0,

holds under the strategy {xt, ζt}.

If (i) to (iv) hold, then θtis the experts’ marginal utility of wealth, which is given by formula

(15). Subsequently, BrunSan present two other economic variables. Let Nt :=

Z

I

ni_tdi

denote the total wealth of experts and

ηt :=

Nt

qtKt

the experts’ wealth share. By a balance sheet argument, Nt ≤ qtKt and therefore ηt∈ [0, 1].

The latter is the key economic variable. In the next section we will explain why. Lemma 2 gives the equilibrium law of motion of ηt.

Lemma 2 (BrunSan II.3). The equilibrium law of motion of ηtis5

dηt ηt = ψt− ηt ηt (dr_tk− r dt − (σ + σ_tq)2dt) +a − ι(qt) qt dt + (1 − ψt)(δ − δ) dt − dζt,

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where ζt = Rt 0 1 Nt dC e

s is the aggregate expert consumption rate. Moreover, if ψt > 0, then (16)

implies that we can write

ηt= η0+ Z t 0 µη_sηsds + Z t 0 σ_sηηsdZs− ζt, (17) where ση t = ψt− ηt ηt (σ + σ_tq) and µη_t = −ση(σ + σ_tq+ σ_tθ) +a − ι(qt) qt + (1 − ψt)(δ + δ).

2.4 Markov equilibrium

By deﬁnition of a Markov equilibrium, all economic variables are a function of the state. In BrunSan the state is represented by the two variables K, total amount of capital in the economy, and η, the experts’ wealth share. Hence, the state at time t is the the pair (Kt, ηt).

Most of the economic variables are only a function of ηt, for example

qt= q(ηt), ιt= ι(ηt), θt= θ(ηt), ψt = ψ(ηt) and xt = x(ηt).

In BrunSan’s proposition II.4 it is claimed that

Claim 2. For all t ∈ R+ : ηt∈ [0, η∗], where η∗ ≤ 1is a reﬂecting boundary.

The point η∗ _{is deﬁned as follows.}

Deﬁnition 7. η∗ _{is deﬁned as the point where experts consume, expert optimization implies}

that θ(η∗_{) = 1}_.

We consider this as a sufficient definition. In section 2.6 we will show that η∗ _{is a reflecting}

boundary of the process η. Proposition II.4 is important because it shows how to calculate ψ, q and θ as a function of η. This proposition also gives formulas for the volatility and drift terms. Just like the economic variables mentioned earlier, the volatility and drift terms

ση_t = ση(ηt), µ η t = µ η (ηt), σ q t = σ q (ηt), µ q t = µ q (ηt), σtθ = σ θ (ηt), µθt = µ θ (ηt),

are functions of the state variable ηt. The remainder of Proposition II.4 is given below.

Proposition 3 (BrunSan II.4). The function η 7→ q(η) is strictly increasing, η 7→ θ(η) is strictly decreasing and the boundary conditions are

q(0) = q, θ(η∗) = 1, q0(η∗) = 0, θ0(η∗) = 0, lim

η↓0θ(η) = ∞.

The following procedure can be used to compute ψ(η), q00_(η)_{and θ}00_(η)_{from (η, q(η), q}0_{(η), θ(η), θ}0_(η)_.

1. Find ψ ∈ (η, η + q(η)/q0_(η))_{such that}

a − a q(η) + δ − δ + (σ + σ q (η))σθ(η) = 0, where ση_{(η)η =} (ψ − η)σ 1 − (ψ − η)q0_(η)/q(η), (18) σq(η) = q 0_(η) q(η)σ η_(η)η _{and σ}θ_{(η) =} θ 0_(η) θ(η)σ η_(η)η. ₍₁₉₎

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If ψ > 1, set ψ = 1 and recalculate (18) and (19). 2. Compute µη(η) = −ση(η)(σ + σq(η) + σθ(η)) + a − ι(q(η)) q(η) + (1 − ψ)(δ − δ), (20) µq(η) = r − a − ι(q(η)) q(η) − Φ(q(η)) + δ − σσ q_{(η) − σ}θ_{(η)(σ + σ}q_(η)), _µθ _{= ρ − r,} ₍₂₁₎ q00(η) = 2(µ q_{(η)q(η) − q}0_(η)µη_(η)η) (ση_(η))2_η2 and θ 00 (η) = 2(µ θ_{θ(η) − θ}0_(η)µη_(η)η) (ση_(η))2_η2 . (22)

2.5 Mathematical background: reﬂection at the boundary

In the next section we will show that η∗ _{is a reﬂecting boundary of the process η. First, we}

describe what it means for the function α to be reﬂected in some point r ∈ R. Consider a continuous function α : R+→ R. Suppose that the function exceeds an under bound rL ∈ R

and we want the function to stay above this under bound, i.e. we want that ∀t ∈ R+ : γ(t) ≥ rL,

where γ = α + β and β : R+ → R is some function. However, we want that β is constant

for all t ∈ R+ such that γ(t) > rL. Can we ﬁnd such functions γ, β? If this the case, (γ, β) is

the solution to the problem of reﬂection of α. A similar description can be given for the case that we do not want the function α to exceed an upper bound rH ∈ R. Chung and Williams

(1983) present the following formulation of the problem of reﬂection.

Definition 8 (Problem of Reflection.). Let C denote the class of continuous functions from R+ to R. Given α ∈ C, a pair (γ, β) is called a solution of the problem of reflection for α,

denoted by PR(α ≥ 0), if γ ∈ C, β ∈ C, and the following three conditions are satisﬁed: (i) γ = α + β

(ii) γ ≥ 0

(iii) β(0) = 0, β is non-decreasing on R+, and R ∞

0 γ(t)dβ(t) = 0.

The zero integral

Z ∞

0

γ(t)dβ(t) = 0

expresses that T ∈ R+can only be a point of increase of β if γ(T ) = 0. Chung and Williams

also give the solution to the problem of reﬂection.

Proposition 4. Let α ∈ C with α(0) ≥ 0. Then PR(α ≥ 0) has a unique solution given by (γ, β)where

γ = α + β; β(t) = max

0≤s≤tα −

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We give an example.

Example. Given is the function α : R+ → R, α(t) = sin(t). The solution to the reﬂection

problem is not trivial because α(t) < 0 for t in the collection of intervals S_k∈N(2kπ−π, 2kπ). According to proposition 4, the solution is given by the pair (γ, β), where

β(t) =    0 if t ∈ [0, π) − sin(t) if t ∈ [π,3 2π) 1 if t ≥ 3₂π. and γ(t) =    sin(t) if t ∈ [0, π) 0 if t ∈ [π,3₂π) sin(t) + 1 if t ≥ 3₂π. α, β and γ are plotted in ﬁgure 1.

t 0 5 10 15 20 , (t), -(t), . (t) -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 , -.

Figure 1: α, β and γ out of the example are presented. α is the original function, a sine wave in our example. γ is the solution to the problem of reﬂection PR(α ≥ 0). Notice that

γ = α + β, where β is a non-decreasing function that only increases when γ is zero.

2.6 η

∗

is a reﬂecting boundary of the proces η

We show that η∗ _{is a reﬂecting boundary of η. Consider the following.}

Proposition 5. η∗_{is a reﬂecting boundary of the process η if and only if ζ}

t = max0≤s≤t(˜ηs−η∗)+, where ˜ ηt= η0+ Z . 0 µη_tηtdt + Z . 0 σ_tηηtdZt.

Proof. “⇐.” Suppose that ζt = max0≤s≤t(˜ηs− η∗)+. Deﬁne α := −˜η + η∗. Notice that

ζt = max0≤s≤tα−s. Hence PR(α ≥ 0) has solution (γ, ζ), where γ = α + ζ = −˜η + η ∗ _{+ ζ}_.

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By deﬁnition 8, η = ˜η − ζ = η∗− γ ≤ η∗_, ₍₂₃₎ ζ(0) = 0, ζ is increasing (24) and _Z _∞ 0 (ηt− η∗) dζt= Z ∞ 0 (˜ηt− ζt− η∗) dζt= − Z ∞ 0 γtdζt = 0. (25)

Equations (23), (24) and (25) imply that η∗ _{is a reﬂecting boundary of the process η.}

“⇒.” By proposition 4 combined with a translation-and-mirror argument, PR(˜η ≤ η∗₎_{has a}

unique solution γ = ˜η− β, where βt = max0≤s≤t(˜η − η∗)+. This solution is unique, therefore

η = γ, hence ζ = β.

Proposition 5 is in line with BrunSan. If ζt= max

0≤s≤t(˜ηs− η ∗

)+,

then dζt> 0 only if ηt= η∗. By deﬁnition, dct= ntdζt, hence

dct> 0 only if ηt = η∗.

From (9), we deduce that the experts’ expected return from capital is Et[drkt] dt = a − ιt qt + Φ(ιt) − δ + µqt+ σσ q t.

In Appendix A we give the mathematical deﬁnition of the operator Et[d . ]

dt that works on

stochastic processes X and we show how to calculate Et[dXt]

dt if X is an Itô process such as

rk. At η = η∗ the experts’ expected returns has decreased to the risk-free rate r.6 Therefore, experts are, at this moment, indiﬀerent between consuming and holding capital. Proposition 5 is in accordance with this observation: experts consume their excess net worth.

2.7 How to determine the economic variables and η

∗

_{in practice?}

Proposition 3 allows us to determine ψ, q, θ and the drift and volatility terms as a function of η. ψ(η) can be found by solving step 1 of proposition 3. The volatilities are given by equations (18) and (19), and the drift terms are given by equations (20) and (21). The following recipe, set up by BrunSan, yields η 7→ q(η), η 7→ θ(η) and η∗ _{∈ (0, 1]}_.

1. Set q(0) = q, θ(0) = 1 and θ0_{(0) = −10}10_.

2. Set qL= 0 and qH = 1015.

3. Guess that q0_{(0) = (q}

L+ qH)/2. Solve the following two diﬀerential equations

q00(η) = 2(µ q tq(η) − q0(η)µ η tη) (ση_t)2_η2 and θ 00 (η) = 2(µ θ tθ(η) − θ0(η)µ η tη) (σ_tη)2_η2

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This gives the functions q(η) and θ(η). Stop if either a. q(η) reaches q;

b. θ0_(η)_{reaches 0;}

c. q0_(η)_{reaches 0.}

If event c. happens ﬁrst, then increase the guess of q0₍₀₎_{by setting q}

L= q0(0). Otherwise, let

qH = q0(0). Repeat until convergence (BrunSan repeat the procedure 50 times). Notice that

event a. stops the evaluation of the diﬀerential equations if the guess of q0₍₀₎ _{is way too high.}

4. If qH was chosen large enough in step 2 and if the number of repetitions is large enough,

then θ0_(η)_{and q}0_(η)_{will reach 0 at the same point η}∗_.

5. Divide the function η 7→ θ(η) by θ(η∗₎_{to match the boundary condition θ(η}∗_{) = 1}_.

This algorithm can be used to calculate most of the relevant economic variables as a function of η. The only input objects are the parameters given in equation (2.1) and the concave function Φ : Dom(Φ) → R. BrunSan use the function Φ(ι) = 1

κ(

√

1 + 2κι − 1), with κ = 2.7 Notice that the boundary conditions

q0(η∗) = 0, θ0(η∗) = 0 (26)

together with the upper limit q for q(η) are used to approximate q0₍₀₎_{. After 50 iterations}

q0(0) is approximated with an error, qH − qL 2number of iterations = 1015 250 = 515 235 ≈ 0.8882.

This may seam inaccurate. However, with BrunSan’s parameter values which are given by equation (27), the approximated value of q0_{(0) ≈ 1.8579 · 10}5 _{is large enough for the error to}

be suﬃciently small, i.e. 0.00048 %. If q0₍₀₎ _{is determined, then the functions η 7→ q(η) and}

η 7→ θ(η) are pinned down by the diﬀerential equations (22) and η∗ can be determined by the boundary conditions (26).

2.8 The stationary distribution

In BrunSan the parameter values are set at

a = 11%, a = 7%, ρ = 6%, r = 5%, σ = 10%, δ = δ = 5%. (27)

Under these assumptions q = 0.8 and q = 1.2. In ﬁgure 2 their plot of the stationary density of η is presented. We are going to plot the stationary distribution ourselves.

Consider an Itô process Y satisfying the stochastic diﬀerential equation dYt= µY(Yt) dt + σY(Yt) dZt.

7_{On BrunSan page 18 it is stated that the authors set κ = 2. However, in the provided Matlab code the}

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Figure 2: Stationary distribution of η plotted by Brunnermeier and Sannikov.

Ghosh (2010) points out that the coeﬃcients µY _{and σ}Y _{have to meet some conditions. They}

have to be Lipschitz continuous, i.e. there exists K > 0 such that

∀y, z ∈ R : |µY(y) − µY(z)| + |σY(y) − σY(z)| ≤ K|y − z|. (28)

The stationary distribution is an application of the Kolmogorov forward equation. The forward equation is only well deﬁned if the derivatives

∂µY

∂y : S → R and

∂2(σY)2

∂y2 : S → R (29)

exist. Here S is the state space of process Y .

BrunSan (page 55 and 56) explain how to plot the stationary distribution of such an Itô process Xt which evolves over the state space S = [yL, yR]. The stationary density g(y) can

be found by g(y) = G(y)

σX_(y)2 where G satisﬁes the ordinary diﬀerential equation G0(y) = 2µ

Y_(y)

σY_(y)2G(y). (30)

Note that the stationary distribution distribution is determined up to multiplication by a constant. If G(y) is a solution to (30) and g(y) is the corresponding density then, for c ∈ R, cG(y) is also a solution to (30) and cg(y) is a valid stationary distribution as well. If we have found a solution, we normalize the density such that

Z yR

yL

g(y)dy = 1.

Next, we try to determine the stationary distribution of the experts’ wealth share η. In section 2.3 and 2.4 we showed that the process ηt evolves on [0, η∗]and is given by

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In section 2.6 we proved that dζt = 0if η ∈ [0, η∗). Hence,

dηt= µη(ηt)ηtdt + ση(ηt)ηtdZt

if ηt ∈ [0, η∗). For the calculations of the stationary density we take the state space to be

S = [0, η∗).

We have that the conditions (28) and (29) are satisﬁed in the case of the process ηt. First, we

consider condition (29). Notice that BrunSan ﬁgure 2 (top left panel) shows that η 7→ µη_(η)η

is suﬃciently smooth on [0, η∗_{) \ {η}φ_}_{. BrunSan (page 18) deﬁne η}ψ _{as the point such that}

for η ≥ ηψ _{experts hold all capital, ψ(η) = 1. Also notice that η 7→ σ}η_(η)η _{(top right}

panel) seems to be suﬃciently smooth for ∂2_(ση_η)2

∂η2 to exist on [0, η

∗_{) \ {η}ψ_}_{. We suppose that}

for other parameter values the ﬁgures demonstrate the same smoothness.8 _{The observed}

smoothness substantiates our assumption that the derivatives ∂(µη_η) ∂η : [0, η ∗ ) → R and ∂ 2_(ση_η)2 ∂η2 : [0, η ∗ ) → R exist, where ∂(µηη) ∂η (η ψ ) := 0 and ∂ 2_(ση_η)2 ∂η2 (η ψ ) := 0.

To check if condition (28) is met, we again give a visual argument. From BrunSan ﬁgure 2 it can be observed that

∀η ∈ [0, η∗) : ∂(µ η_η) ∂η (η) ≤ ∂(µη_η) ∂η (0) and ∂(ση_η) ∂η (η) ≤ ∂(ση_η) ∂η (0), (31) hence ∀η1, η2 ∈ [0, η∗) : |µη(η1) − µη(η2)| + |ση(η1) − ση(η2)| ≤ K|η1− η2|, where K = ∂(µη_η) ∂η (0) + ∂(ση_η)

∂η (0). We assume that (31) also holds for other parameter values.

Now that we have checked the conditions, we can use the previously presented theory to determine the stationary density g of η. By equation (30), g(η) = G(η)

η2_ση_(η)2, where G satisﬁes the ordinary diﬀerential equation

G0(η) = 2 µ

η_(η)

η ση_(η)2G(η). (32)

We have established that the function η 7→ g(η) is well-deﬁned. However, it may not be integrable. BrunSan give the following condition for g to be integrable.

Proposition 6 (BrunSan III.2). Denote by Λ = (a − a)/q + δ − δ the risk premium that experts earn from capital at η = 0. As long as

2(ρ − r)σ2 < Λ2

the stationary density η 7→ g(η) exists and satisﬁes g0_(η∗_{) > 0} _{and g(η) → ∞ as η → 0. If}

2(ρ − r)σ2 _{> Λ}2 _{then the stationary density does not exist and in the long run η}

t ends up in an

arbitrarily small neighbourhood of 0 with probability close to 1.

8_{One may prove the existence of} ∂(µη_η)

∂η and

∂2_(ση_η)2

∂η2 on [0, η

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In our case, Λ = 0.11−0.07

0.8 + 0.05 − 0.05 = 0.05and

2(ρ − r)σ2 = 2(0.06 − 0.05) · 0.12 = 0.0002 < 0.00025 = Λ2.

Thus, the density exists; g is integrable.

In order to calculate the density, we solved ordinary differential equation (32). In figure 3, we plotted the stationary distribution of η using the same parameter values as BrunSan, see equation (27). We observe that this graph is the same as the plot by Brunnermeier and Sannikov presented in figure 2.

2 0 0.1 0.2 0.3 0.4 0.5 st at io n ar y d en si ty g( 2 ) 0 2 4 6 8 10 12 14 16 18 20 22 2*

Figure 3: The stationary distribution of the experts’ wealth share η. It measures the average amount of time that the variable η spends at diﬀerent parts of its domain. The stationary distribution has high densities near the boundaries of the state space. BrunSan’s model is designed such that the economy is subject to systemic risk. If η is far below its attracting point η∗ _{it is in the volatile middle region. It stays there for a short time, but there is a}

chance that a few bad shocks take the system to depressed states near η = 0, where it can get trapped for a while (BrunSan, page 24).

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2.9 Sample paths

Now that we know what the stationary distribution looks like, we may want to know what the paths t 7→ ηtlook like in reality. These paths are called sample paths. We choose a time

step ∆t = 1

N, with N ∈ N, and we approximate the paths with a diﬀerence equation

ηi+1= ηi+ ∆ηi. (33)

We derive the diﬀerence term from stochastic diﬀerential equation (17). The consumption rate dζt= 0 if ηt< η∗ and dηt= µη(ηt)ηtdt + ση(ηt)ηtdZt Therefore, ∆ηi = µη(ηi)ηi · ∆t + ση(ηi)ηi· φ, where φ ∼ N(0, ∆t). Hence, ∆ηi = µη(ηi)ηi · ∆t + ση(ηi)ηi· √ ∆t · ϕ,

where ϕ ∼ N(0, 1). Equation (33) holds only if ηi + ∆ηi ∈ [0, η∗]. The iteration that gives

the sample paths is

ηi+1=    0, if ηi+ ∆ηi < 0 ηi+ ∆ηi, if ηi+ ∆ηi ∈ [0, η∗] η∗, else. (34) This method to ﬁnd approximate sample paths of a stochastic diﬀerential equation is called the Euler method. For our plots we set N = 100 and η0 = 1₂η∗ ≈ 1₂ · 0.4678 ≈ 0.2339.

A plot with a time window of three years is presented in figure 4 and figure 5 has a time window of fifteen years. In both figures we observe that ηt evolves jerkily. This is caused

by the Brownian shock term dZt in the law of motion of η. We also observe that ηt has the

tendency to increase, this is caused by the drift term µη_tηtdt.

The volatility term ση

tηtdZt works in both ways, negative and positive. At every moment in

time the chance a negative shock happens is the same as the probability of a positive shock. On the other hand, the drift term pushes ηt in the positive direction because ∀η ∈ (0, η∗] :

µη_{(η)η > 0}. In ﬁgure 5 we clearly perceive that η

thas an upper bound. This is caused by the

consumption (rate) term dζt. If η has the tendency to exceed the upper bound η∗, then the

experts start to consume the excess net worth such that the experts’ wealth share, ηt, stays at

η∗.

Finally, in both ﬁgures we clearly observe that the system’s reaction to shocks is nonlinear and asymmetric. If ηt is the volatile middle region, the shocks are strongly ampliﬁed, while

the system quickly recovers from most shocks near η∗_{. Near the steady state, positive shocks}

lead to consumption and ηt does not change. However, large negative shocks can bring the

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t 0 0.5 1 1.5 2 2.5 3 2 t 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Figure 4: Four sample paths of η, the expert net worth as a fraction of the total stock value. The experts’ wealth share has the tendency to increase over time.

t 0 5 10 15 2 t 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 2* !

Figure 5: Four sample paths of η over a longer period of time. We clearly perceive that η is bounded by the endogenous reﬂecting boundary η∗_{. This is the point that experts have built}

an optimal portfolio, and start to consume. At this point, the households’ savings in bonds are worth 53 % of the capital stock. However, all capital is in hands of the experts who

manage it more productively. Notice that at η = η∗ _{the expected economic growth is}

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3 Unanticipated Sudden Stops: including a jump to the

baseline model

3.1 What is a Sudden Stop?

Since the 1980s many emerging economies were hit by Sudden Stops, often in the aftermath of a financial crash. These Sudden Stops all show similar characteristics. Mendoza (2010) therefore argues that “they provide a unique laboratory to study the linkages between finan-cial collapse and macroeconomic crisis.” He points out that Sudden Stops are defined by three main empirical regularities:

(i) correction in asset prices, (ii) decline in production,

(iii) reversal of international capital ﬂows.

While Mendoza uses a business cycle model with a collateral constraint to model Sudden Stops of emerging economies, we use an adapted version of BrunSan’s model. This has the advantage that we do not have make a log-linearised approximation of the equations. In our model, the cause of the sudden stop will be an abrupt increase in the interest rate that foreign investors charge to agents in the emerging economy. First, we need to reinterpret BrunSan’s model.

3.2 Reinterpretation of the model

Consider a small, open, emerging economy A such as the Czech Republic, Mexico or Taiwan. In order to turn BrunSan’s model into a model that accounts for Sudden Stops, we need to create a dichotomy between the agents in economy A and foreign investors. Foreign investors often return part of the accumulated capital to their country of origin, or to stakeholders out-side economy A. Hence, in terms of productivity that beneﬁts economy A, agents inout-side the emerging economy manage capital more productively than foreign investors. Furthermore, foreign investors are often focused on the long or medium term while agents in the emerging economy are focused on the medium or short term. Thus, agents inside the emerging econ-omy are less patient. Therefore, it is intelligible to reinterpret BrunSan’s model such that the ‘experts’ become the agents in economy A (A-agents) and the ‘households’ become foreign investors.9 _{As a consequence of this reinterpretation, an A-agent’s consumption must be}

positive while a foreign investor’s consumption is allowed to be negative. In the next sections we will show what happens if the interest rate r is increased.

3.3 Two models and the price of capital

Consider the same parameter values as in section 2.8. However, we look at two models now. In Model 0 the risk-free rate is r = r0 and in Model 1 it is r1 with r0 < r1. This is the

interest rate foreign investors demand from A-agents. Let η 7→ q0_(η)_{and η 7→ q}1_(η) _denote

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the capital price functions in Model 0 and Model 1 respectively. We calculate the average capital prices E[q0_] _{and E[q}1_]_{in the case that}

r0 = 4.5 % and r1 = 5.5 %. (35)

We can calculate the average capital price by taking the expectation with respect to the stationary density g:

E[q] = Z η∗

0

q(η)g(η) dη. The obtained results are

E[q0] ≈ 1.0339 and E[q1] ≈ 1.0077. Note that E[q0_{] > E[q}1_]_{, and}

E[q0] − E[q1] ≈ 0.0262.

We also plotted the functions η 7→ q0_(η) _{and η 7→ q}1_(η)_{, they are presented in ﬁgure 6.}

Notice that the capital price in Model 0 is higher than the capital price in Model 1. The economic interpretation is as follows. If the interest rate that A-agents have to pay is higher, they have a smaller incentive to lend money from foreign investors. Therefore, their demand for capital is smaller and the price of capital is lower. In ﬁgure 6 one can also observe that η∗ in Model 1 is not the same as η∗ in Model 0. From a mathematical perspective, this is due to the fact that in the two models the expressions for µq_(η) _{and µ}θ_(η)_{, in proposition}

3, are different. This entails that the differential equations (22) differ. Hence, the boundary conditions (26) give unequal values for the reflecting boundary. We can explain from an economic perspective why

η1∗> η0∗.

In ﬁgure 7 the A-agents’ expected returns are presented. A-agents start to consume when they do not expect to make proﬁt from investing in capital. This is the case when the A-agents’ expected return equals the interest rate ri with i ∈ {0, 1}. We observe that, although

r1 > r0, the expected return hits ri for a higher value of η if we are in Model 1. Generally,

A-agents tend to build more net worth before they start to consume if the interest rate is higher.

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2 0 0.1 0.2 0.3 0.4 0.5 q 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 q0 q1

Figure 6: A higher interest rate r1 > r0 leads to a lower price of capital.

2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

A-agents` expected return

0 0.02 0.04 0.06 0.08 0.1 Model 0_{Model 1} r₁ r₀

Figure 7: The (expected) excess return on capital over the risk-free interest rate Et[drkt]

dt − r

hits zero for a larger value of η if the interest rate is high. Therefore, η1∗_{> η}0∗_{. In this case,}

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3.4 The stationary average η

0

and the associated leverage x

0

_(η

0

₎

Let η0 be the stationary average η in Model 0, it is given by

η0 _{= E[η] =}

Z η∗

0

ηg(η) dη ≈ 0.3329

The associated leverage is denoted by x0_(η0₎. The mapping η 7→ x0_(η) _{is given by}

x0(η) = ψ

0_(η)

η ,

where the mapping η 7→ ψ0_(η)_{can be found by executing the algorithm given in section 2.7.}

The value of x0_(η0₎is found by evaluating the mapping η 7→ x0_(η)_{at η}0. The resulting value

is x0_(η0_{) ≈ 3.0038}.

3.5 Where will η and q jump to after a Sudden Stop?

We are in the Model 0 world and suppose that at t = T it happens to be that ηT = η0. Next,

suppose that the interest rate jumps to r1 and everyone thinks it will stay at r1 forever. We

are now in a Model 1. We will determine the impact of the jump in two steps. Step 1. Determine where q and η jump to using the accounting equation. Step 2. Calculate where the other variables10 _α _{jump to using that α}1

T = α1(η1T).

In this section we focus on Step 1. Let KA

t denote the total amount of capital managed by

A-agents, in the notation of BrunSan: KA t =

R

Ik i

tdi. Consider the total balance sheet of all

experts. The accounting equation before the jump is N_T0 = q_T0K_TA,0 − D0

T. (36)

Here N denotes the total net worth of A-agents and D their total debt. After the jump, the net worth has to satisfy

N_T1 = q_T1K_TA,0 − D0

T. (37)

This equation incorporates the losses the A-agents make as a consequence of the capital price drop. We can rewrite equations (36) and (37) as

N_T1 = N_T0 − q0 TK A,0 T + q 1 TK A,0 T .

Recall that in BrunSan

ηt:=

Nt

qtKt

,

10_{We introduce new notation. If α is a variable in the model, then we distinguish between two values at}

the time T of the jump. α0

T is the value right before the jump calculated with the parameters from Model 0,

α1

T is the value right after the jump calculated with the parameters from Model 1. We only use the ambiguous

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where K denotes the total amount of capital in the economy. This gives us an equation11 _for η1 T: η1_T = N 1 T q1 TKT = N 0 T q1 TKT − q 0 TK A,0 T q1 TKT +q 1 TK A,0 T q1 TKT (38) Also recall that ψt =

KA t Kt , hence ψ 0 T = KA,0_T KT . Thus, equation (38) becomes

η_T1 = N 0 T q0 TKT q0 T q1 T − q 0 T q1 T ψ0_T + ψ0_T = η_T0q 0 T q1 T − q 0 T q1 T − 1 ψ_T0.

For the new capital price q1

T to be self-fulﬁlling, or sustainable in equilibrium, we need that

q_T1 = q1(η1_T).

Given η0

T, qT0, ψT0 and the function η1 7→ q1(η1) we can ﬁnd qT1 and η1T using the following

system of equations: ( q_T1 − q1_(η1 T) = 0 η_T1 − η0 T q0 T q1 T +qT0 q1 T − 1ψ0_T = 0. (39) The values of q0

T = q0(ηT0)and ψT0 = ψ0(ηT0) are obtained in the same way as we found the

value of x0_(η0

T)in section 3.4. We solved the system of equations numerically, the results are

presented in the table below.

moment right ‘before’ the jump moment right ‘after’ the jump

η 0.3329 0.2520

q 1.05064 0.9370

3.6 Results

In the previous section we calculated the values where η and q jump to, which are denoted by η1

T and qT1, after an instant interest rate jump from r0 to r1. We only considered the case

that the value of η right before the jump, denoted by η0

T, is equal to its stationary average η0

in Model 0. Next, we would like to know what the impact of the jump is for other values of η0

T.

We determined the value where the variables jump to as a function of η0

T in the same way as

we calculated the impact of the jump in the situation that η0

T = η0. The plots of ηT1, q1T, ψ1T

and θ1

T are presented in ﬁgures 8, 9, 10 and 11 respectively, the values of the variables right

before the jump are added as a solid, red line.

11_{Note that K remains the same at the jump: K}0

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We observe that, during the Sudden Stop, the A-agents’ wealth share η and the price of capital q jump down. In section 3.3 we explained that a higher interest rate brings about a decreased incentive to invest and therefore a lower capital price. By the accounting equation

N = q KA− D,

a lower capital price causes the A-agents’ net worth to fall. There is an interesting state dependency to be perceived. The absolute value of the drop in the A-agents’ wealth share is greatest in the middle region of the state space of η0_{. Notice that}

η1 T(η0T) η0 T ≈ 0.67 if η0 T ≤ η ψ0 , and η1_T(η_T0) η0 T is increasing in η0 T if η 0 T > η ψ , where ηψ0 is deﬁned as ψ0(η0_T) = 1 ⇐⇒ η_T0 ≥ ηψ0 . Proportionally, the A-agents’ net worth falls harder if η0

T is lower. A Sudden Stop is more

painful if it hits when A-agents are less capitalized.

We also observe that the fraction of physical capital held by A-agents ψ decreases during the Sudden Stop if η0

T is not near its optimum η0∗. BrunSan’s model is constructed in such a way

that the optimal leverage ratio x(η) entails that ψ(η) = x(η)η is increasing in η if ψ(η) < 1 and constant if ψ(η) = 1. Intuitively, it makes sense that ψ should be increasing in η. Let η0ψ1 be deﬁned by

ψ1(η1(η0_T)) = 1 ⇐⇒ η_T0 ≥ η0ψ1 . If a Sudden Stop occurs at time T , η0

T falls to η1T. Moreover, if ηT0 < η0ψ

1

, then η1

T ≤ η1ψ

1 where η1ψ1 is deﬁned similarly. In this case, an A-agent revises her portfolio such that ψ1

T =

ψ1(η_T1) < ψ_T0. Hence, if agents are undercapitalized, a Sudden Stop causes a misallocation; the foreign investors, who are less productive, have to manage a proportion of the total capital stock. If the A-agents are well capitalized, their net worth is relatively high, η0

T is near

η0∗in the state space, and there is no misallocation of capital if a Sudden Stop occurs. Finally, we observe that the A-agents’ marginal utility of wealth θ increases at a Sudden Stop. Outside a very small interval near zero, θ(η) is primarily dependent on η and not on the interest rate level. Hence, the increase in θ is mainly due to the decline in η.

Notice that there is a break point in the graphs of the values of the variables right after the jump. The break point is at η0

T = ηψ

0

This is in line with the theory. The value the economic variables jump to at the moment of the Sudden Stop are, in particular, a function of ψ0

T, and

the function η0

T 7→ ψ0(η0T) has a break point at η0T = ηψ

0

, so the functions η0

T 7→ ηT1(η0T),

η0

T 7→ q1(η0T), ηT0 7→ ψ1(η0T)and ηT0 7→ θ1(ηT0)are expected to have a break point at ηT0 = ηψ

0 as well.

It is important to notice that the three main empirical regularities of Sudden Stops, that we mentioned in section 3.1, are indeed observed.

(i). Correction in asset prices.

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η_T0 ∈ [0, η0∗_]_{, the price of capital after the Sudden Stop is lower than the price before the}

Sudden Stop, q1_(η0

T) < q0(η0T).

(ii). Decline in production.

The decline in production knows two sources. First, aggregating equations (1) and (4) gives the total output of the economy:

Yt= (aψt+ a(1 − ψt))Kt. (40)

If η0

T < η0ψ

1

, then ψ decreases at a Sudden Stop. Because a < a, this has an immediate negative eﬀect on output. Second, the law of motion of the total amount of capital, under the condition that δ = δ,

dKt = (Φ(ιt) − δ)Ktdt + σKtdZt (41)

is obtained by aggregating equations (2) and (3). With equation (85), we know that Φ(ι(qt)) =

qt− 1

κ .

Thus a Sudden Stop brings about a lower investment rate, therefore the expected production is decreased. This is a long term effect because the price of capital is structurally lower if the interest rate is high. We conclude that the decline in output is caused by a general long term effect due to a lower investment rate, and an immediate effect due to misallocation of capital in the case that A-agents are undercapitalized.

(iii). Reversal of international capital ﬂows.

Recall from section 2.9 that before the Sudden Stop, when we are still in Model 0, the A-agents’ wealth share has an upward trend. In this case we can derive from ﬁgure 10 that ψ has an upward trend as well. Before the Sudden Stop there is a trend that the fraction of physical capital held by A-agents is increasing. If, at the time T that the Sudden Stop occurs, η_T0 < ηψ1, then ψ falls from ψ_T0 to ψ_T1. Hence, there is a reversal of international capital ﬂows. If η0

T ≥ ηψ

1

, all capital remains in hands of the A-agents although their wealth share falls from η0

T to ηT1.

Figures 12, 13 and 14 plot a simulated sample path of the economy under the circumstance that a Sudden Stop takes place after five years. The figures present the values of η, q and the output Y respectively. To obtain the sample path of Y , we construct an Euler scheme based on stochastic differential equation (41) to simulate a path of the total amount of capital Kt.

Subsequently, equation (40) gives the corresponding values of Yt.

This example illustrates the potential harmfulness of a Sudden Stop. Before the Sudden Stop, ηtis in the volatile middle region of the state space. The A-agents are moderately capitalized,

so they are vulnerable to the threat of a Sudden Stop. After ﬁve years, the interest rate is raised by 1 percentage point, the Sudden Stops hits the A-agents and they lose 33% of their wealth share. In ﬁgure 13, we clearly perceive that the price of capital is a function of ηt,

its value follows approximately the same patters. At the time of the Sudden Stop, the price drops by 10%, so the A-agents’ net worth suﬀers a blow of 39%. Their net worth is eaten away because the value of their capital holdings decreases while their indebtedness to foreign investors remains the same. In our example, the Sudden Stop takes the state variable ηt to

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the depressed region of its domain. However, ηt is not decreasing to zero, so it may ﬁnd its

way back to a healthier state.

Economic growth is already negative before the Sudden Stop. The baseline economy with default parameters given by (27) and (35) is expected to be in a recession. Thus, although the state variable ηt is expected to ﬁnd its way back to a healthier state, the output is expected

to decline towards zero. In section 4.6, we discuss how the parameters can be changed to remedy this. Nevertheless, the immediate negative effect on output due to misallocation is indeed observed, because A-agents are undercapitalized right before the moment of the Sudden Stop. The long term effect due to a lower investment rate is not observed, since the Brownian motion delivers more positive shocks to the system in the five years after the Sudden Stop compared to the three years before the event.

2 T 0 0 0.1 0.2 0.3 0.4 0.5 2T 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 2_T0 2_T1

Figure 8: The dashed line represents the A-agents’ wealth share at the moment that a Sudden Stop occurs as a function of her wealth share right before the event. The solid, red

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2 T 0 0 0.1 0.2 0.3 0.4 0.5 qT 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 q0_T q1_T

Figure 9: The impact of a Sudden Stop on q applied to the entire state space [0, η0∗_]_.

2 T 0 0 0.1 0.2 0.3 0.4 0.5 A T 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A0_T A1_T

Figure 10: The impact of a Sudden Stop on the fraction of physical capital managed by A-agents. If the A-agents are undercapitalized, a Sudden Stop brings about a misallocation

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2 T 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 3 T 0 1 2 3 4 5 6 7 30_T 31_T

Figure 11: The impact of a Sudden Stop on the A-agents’ marginal utility of wealth. A Sudden Stop causes θ to increase.

t 0 2 4 6 8 10 2t 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

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t 0 1 2 3 4 5 6 7 8 9 10 qt 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

Figure 13: The capital price qt over time based on the same simulated sample path. The

Sudden Stop causes the price of capital to fall by 10%.

t 0 1 2 3 4 5 6 7 8 9 10 Y t 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Figure 14: Evolution of the economy’s output based on the same simulation. The Sudden Stop causes output to fall due to misallocation. This is an immediate eﬀect that only takes place if A-agents are undercapitalized. The long term negative eﬀect on output due to less

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4 Agents anticipate the probability of a Sudden Stop

4.1 What is a Poisson Process?

In order to make a model of the behaviour of agents that anticipate the possibility of a Sud-den Stop, we need to create a stochastic process that tells us when the interest rate jumps from r0 to r1. Clymo (2015) uses a Poisson process or counting process for this purpose.

A Poisson process is a special case of a counting process. First, we give the deﬁnition of a counting process which is taken from Björk (2011).

Deﬁnition 9. A counting process is a stochastic process {Yt, t ∈ R+} which satisﬁes the

following conditions.

(i). The trajectories of Y are, with probability one, right continuous and piecewise constant. (ii). Y0 = 0.

(iii). For every t ∈ R+: ∆Yt= 1 or ∆Yt= 0 with probability one, where ∆Yt = Yt− Yt−.

Yt counts the number of ‘arrivals’ in the time interval (0, t). A Poisson process is

charac-terised by the arrival intensity λ > 0. It is well-known that a Poisson process F with arrival intensity λ can be deﬁned in, at least, four ways.

Deﬁnition 10 (Poisson process I, informal). In an inﬁnitesimal time interval dt, Ft increases

by one with probability λ dt. Such an increase is called an arrival. The chance of an arrival in the time interval dt is independent of arrivals outside this interval.

Deﬁnition 11 (Poisson process II). The number of arrivals Ft in a ﬁnite time interval of

length t obeys the Poisson(λt) distribution, P(Ft= n) =

(λt)n

n! e

−λt

.

Moreover, the number of arrivals (Ft2 − Ft1) and (Ft4 − Ft3) in non-overlapping intervals, t1 ≤ t2 ≤ t3 ≤ t4, are independent.

Deﬁnition 12 (Poisson process III). The interarrival times are independent and obey the Exp(λ) distribution:

P(interarrival time > t) = e−λt.

Deﬁnition 13 (Poisson process IV). F is a counting process and the process Mt= Ft− λtis

an FF_{-martingale. In this case, M is called the compensated Poisson process.}

A proof that deﬁnitions 10-12 are equivalent can be found in Virtamo (2005). Notice that deﬁnition 12 implies that

the expected interarrival time = 1

λ. (42)

In Appendix C we proof that the martingale deﬁnition is equivalent to the other deﬁnitions of the Poisson process.

A Macroeconomic Model with Sudden Stops

Master thesis in Stochastics and Financial Mathematics