A game theoretic model of economic crises

Applied Mathematics and Computation 266 (2015) 738–762

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A game theoretic model of economic crises Jonathan W. Welburn a, Kjell Hausken b,∗ a Department of Industrial Engineering, University of Wisconsin-Madison, 1550 Engineering Drive, Room 3237, Madison, Wisconsin 53706, USA b Faculty of Social Sciences, University of Stavanger, 4036 Stavanger, Norway

a r t i c l e

i n f o

Keywords: Game theory Sensitivity analysis Economic risk Default Contagion

a b s t r a c t Global financial crises have revealed the systemic risk posed by economic contagion as the increasing interconnectedness of the global economy has allowed adverse events to spread across countries more easily. These adverse economic events can be attributed to contagion through either credit or trade channels, or to common macroeconomic conditions that cause adverse events in multiple countries even without contagion. We model this system as a game between five types of players: countries; central banks; banks; firms; and households. In this framework, we model strategic choices, conduct sensitivity analysis, and analyze the impacts of random shocks in two examples. Our results demonstrate that each of the three causes discussed above (contagion through credit channels, contagion through trade channels, or common macroeconomic conditions with no contagion) can lead to crises even if all agents in the model behave rationally. © 2015 Elsevier Inc. All rights reserved.

1. Introduction We seek to add to the understanding of global economic risks that contribute to economic crises, at both an academic and a policy level. Global markets are imperfect and experience shocks from time to time. Shocks are adverse economic events that diminish the credit standing of countries, banks, as well as other borrowers. Such shocks can occur locally and subsequently spread globally. The propagation of shocks is referred to as contagion, and its potential is a key source of systemic risk. We distinguish between market risk, which is systematic, and idiosyncratic risk, which is specific to a single player and unsystematic. This risk is exacerbated by the problems of adverse selection1 and moral hazard2 that lead to country risk. We address these concerns through a proposed game-theoretic model with key themes of debt, capital, and trade. The 2007–2008 financial crises revealed the global economy’s vulnerability to economic risks marking a key departure from previous notions of country risk. Financial globalization and innovation have led to substantial growth in the derivatives market. This in turn led to an abundance of global credit, with resulting global risk exposures. Consequently, country risks were exacerbated. Increased derivative securities usage along with increased diversification led to increased information asymmetries and amplified contagion and common cause effects, by allowing adverse economic events to spread across countries. Additionally, abundant credit established an implicit promise of foreign lending, creating moral hazard. This structural shift in the global economy increased both country risk exposure and impact. The 2009–2012 Eurozone crisis underscored global economic risks’

∗

1 2

Corresponding author. Tel.: +47 51831632; fax: +47 51831550. E-mail addresses: [email protected] (J.W. Welburn), [email protected] (K. Hausken). For example, the poor choice in borrower that leads to less than full repayment of debts. For example, fiscal irresponsibility driven by an implicit promise of future credit or aid.

http://dx.doi.org/10.1016/j.amc.2015.05.093 0096-3003/© 2015 Elsevier Inc. All rights reserved.

J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

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importance as a systemic risk while exemplifying the debate between contagion and common cause failures as it is unclear whether the crises can be attributed to contagion or common cause failure. Various approaches have been applied to understand the crises and their spread. In their seminal work, Eaton and Gersovitz [1] discuss crises through a model of sovereign debt, default, and repudiation. The spread of crises defined as contagion is discussed by many in the literature; Allen and Gale [2] and Kaminsky and Reinhart [3] discuss contagion through financial sectors and common creditors; Hernández and Valdés [4] and Kaminsky et al. [5] highlight the fact that contagion can come from trade or credit channels; Forbes and Rigobon [6,7] discuss the plausibility that the perceived problem of contagion is only a problem of interdependence. Recent advances in game theory and applications demonstrate its importance in tackling large social problems. On the topic of evolutionary dynamics, Nowak [8] stresses the value of interpreting natural phenomena through the tools of evolutionary dynamics. Nowak [8] presents a strong case for the use of dynamic models and game theory in solving real world problems. Cressman and Tao [9] demonstrate the stability of Nash equilibria for dynamics and evolutionary game theory, extending them to symmetric, asymmetric, multiplayer, and population games. The importance of both network structure and strategy on optimal solutions to social problems is explored by Wang et al. [10– 13], Wang and Perc [14], and Wang et al. [15]. Discussing coevolution of network structure and strategy, Wang et al. [10] conclude that using interdependence in networks can lead to cooperative advantages in solving social problems. In what Wang et al. [10] describe as two classes of players resulting within the network structure, Wang et al. [11] find that in interconnected networks, “distinguished players” emerge with strong influence on cooperation. Furthermore, Huang et al. [16] discuss the importance of heterogeneity in players finding that under simple assumptions heterogeneity of investments has a positive effect on cooperation. Kokubo et al. [17] extend theory in the direction of realism from the discrete modeling choices of cooperative vs defective strategy. As solutions to social dilemma games Kokubo et al. [17] offer discrete, continuous, and mixed strategies. Gao et al. [18] show how the payoffs from mutualistic cooperation are differently impacted by faster evolving and slower evolving species. The literature exploring the use of game theory in economic crises is, however, small. Morris and Shin [19,20] model contagion in a coordination game of investors to currency crises, bank runs, and debt pricing to assess adverse selection. Arellano and Bai [21] include Nash bargaining for debt renegotiation in a model of linkages in sovereign debt markets. Hausken and Plumper [22] propose a contagion game for how a crisis spreads and can be contained through intervention by the IMF and collective action. Most similarly to our objectives, Baral [23] proposes a network formulation game between U.S. borrower banks, lender banks, and the Federal Reserve to model contagion. Similarly, Acemoglu et al. [24] compute Nash equilibria in a network formulation of the interbank market to describe contagion and counterparty risk. However, most of the literature applying game-theoretic approaches to economic contagion is limited in either its use of game-theoretic and decision-theoretic tools, or its ability to address the multiple-channel problems discussed by Hernández and Valdés [4] and Kaminsky et al. [5]. We develop a game-theoretic model consisting of five types of players; countries, central banks, banks, firms, and households. Countries produce, consume, trade, invest, borrow, and lend. Central banks lend and borrow. Banks, firms and households borrow and lend capital. A country’s debts and assets are endogenously determined. By determining strategic behavior and the outcome of adverse shocks, we endogenize financial frictions. We model contagion dynamically in a way that explains how the propagation of adverse economic events occurs. Additionally, we find equilibrium strategies which implicitly suggest how policy can be adapted to stem the spread of contagion. At the heart of contagion is lending by one player to another, exposing the lender to the risks of the borrower. Both borrowers and lenders have specified strategy sets. Moral hazard and adverse selection arise in such games when information asymmetries keep borrowers and lenders from knowing each other’s true characteristics (e.g., when the lender cannot easily or accurately assess the credit-worthiness of a borrower). The players’ strategy sets are quite large enabling extensive endogenization. Large strategy sets means that there are many strategic variables. The model can either be used with all strategic variables as strategic variables. Alternatively, the model can be used interpreting some of the strategic variables as exogenous parameters. In Section 2 we present a model with five types of players; households, firms, banks, countries, and central banks. We define player strategy sets and player interactions through goods and debt including endogenous default. We conclude by defining the Nash Equilibrium. In Section 3 we present a two country numerical example over ten periods. In Section 3.1 we conduct a numerical analysis of and solution to a game with one country, a set of households, and a set of firms. We explain how shocks change behavior and how under certain conditions, this can lead to default. In Section 3.2 we conduct a numerical analysis of and solution to a game with two countries, each with a set of households, and a set of firms. We explain how a shock in one country impacts the behavior of the other. In Section 4 we conclude and describe the model implications to global economic stability.

2. The model 2.1. Players, variables, and parameters We consider a complete information, non-cooperative, T period game where T ∈ [1, ∞], with five types of fully rational players which can make simultaneous or sequential moves. The players are interlinked as in Fig. 1 and defined in Table 1. Black dashed arrows represent the debt market and red dash-dotted arrows represent the market for goods and services. All parameters are common knowledge for all players. The strategic choice variables of the five players are shown in Table 2, the parameters in Table 3, and the dependent variables in Table 4.

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Fig. 1. International markets for goods and services and debt across five types of players. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.) Table 1 Player description. Name Countries, c

Central banks, e

Banks, b Firms, f

Households, h

Players p = {c, e, b, f, h}

Definition There are nc countries defined by their respective federal governments and their citizens. In this framing, countries exclude the role of the domestic/regional central bank. Countries can borrow B, lend (negative borrowing, B), set interest rates r, default D, penalize default P, consume C, invest I, lump sum transfers N. There are ne domestic and regional central banks being lenders of last resort. Examples include the Federal Reserve as a domestic central bank and the European Central Bank as a regional central bank. Central banks can borrow, lend, set interest rates, default, penalize default, set inflation target π ∗ . There are nb banking institutions who can borrow, lend, set interest rates, default, penalize default. There are n f firms who produce. To produce, they can demand labor L, invest, import M, export (determined by the imports of other firms), borrow B, lend, default. There are nh households who labor. They can consume, invest, borrow, lend, default.

c = {ce1 , . . . , cene } : ce j = {

Strategy set, {Si }i⊆p e e c1 j , . . . , cn j

}

Sc = {B, r, D, P, C, I, N }

e = {e1 , . . . , ene }

Se = {B, r, D, P, π ∗ }

c

c

b = {bc1 , . . . , bcnc } : bc j = {b1j , . . . , bnj } f = { f c1 , . . . , f cnc } : f c j = {

c c f1 j , . . . , fn j

c

Sb = {B, r, D, P }

}

c

h = {hc1 , . . . , hcnc } : hc j = {h1j , . . . , hnj }

S f = {L, I, M, B, D}

Sh = {C, I, B, D}

The set of players (c, e ) comprising countries, and central banks, is mutually exclusive, and collectively exhaustive in the sense of comprising all players. Each country has a central bank, and several countries may share a central bank. Thus we set e e c = {ce1 , . . . , cene } : ce j = {c1j , . . . , cnj }∀ j ∈ (1, . . . , ne ) where each of the nc countries is indexed by its respective central bank e j . The set (b, f, h ) comprising banks, firms, and households is mutually exclusive, and members of a given country, as shown in Fig. 1. For example, the aggregate production and consumption of a country also consist of the production and consumption c c c c of households, firms, and banks. We thus set b = {bc1 , . . . , bcnc } : bc j = {b1j , . . . , bnj }, f = { f c1 , . . . , f cnc } : bc j = { f1 j , . . . , fn j }, h = c

c

{hc1 , . . . , hcnc } : hc j = {h1j , . . . , hnj }, where c j = cej k ∀ j ∈ (1, . . . , nc ), k ∈ (1, . . . , ne ). A country’s strategic choices, thus, depend on the choices of its banks, firms, and households, but generally each country makes many additional choices. Assumption 1. Single currency.

We normalize all values to the same currency. That is, we assume all players produce, consume, invest, and trade in the same currency. Countries have many strategic choice variables, whereas the other players have relatively few. The fundamental strategic choice is borrowing, which all players have. We define borrowing as both positive and negative to indicate the two possible

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Table 2 Strategic choice variables. n ∈ {1, . . . , ni } m ∈ {1, . . . , n j }

Variable

Definition

Player domain

Bin jm t Din jm t

Amount player in borrows from player jm at time t Binary indicator for whether player in defaults on positive debt to player jm at time t (1=yes, 0=no) Interest rate at which player in borrows Bin jm t from player jm at time t. Amount player in penalizes player jm for defaulting on debt at time t Fraction of player jm ’s utility penalized for defaulting on debt owed to player in at time t Public consumption for country cn at time t Public capital investment for country cn at time t Target inflation rate of central bank en at time t Net lump sum transfer payment (taxes, subsidies, etc.) from player in to the country cm at time t Investment of firm fn at time t Labor demanded from household hm by firm fn at time t Investment of household hn at time t Imports of firm fn in country ci from firm fm in country c j at time t Consumption of household hn at time t

( i n , jm ) ∈ p ( i n , jm ) ∈ p

Bin jm t ∈ R2 Din jm t ∈ {0, 1}

( i n , jm ) ∈ p ( i n , jm ) ∈ p ( i n , jm ) ∈ p

rin jm t ∈ R2+ Pin jm t ∈ (0, ∞ )2 ψin jm t ∈ [0, 1]

cn ∈ c cn ∈ c en ∈ e in ∈ (b, h, f ), cm ∈ c

CcPn t ∈ R+ IcPn t ∈ R+ πe∗n t ∈ [0, 1] Nin cm t ∈ R

fn ∈ f hm ∈ h, fn ∈ f hn ∈ h fn ∈ f ci , fm ∈ f c j (ci , c j ) ∈ c hn ∈ h

I f n t ∈ R+ Lh m f n t ∈ R + Ihn t ∈ R+ M f n f m t ∈ R+ Chn t ∈ R+

rin jm t Pin jm t

ψin jm t CcPn t IcPn t

πe∗n t

Nin cm t I fn t Lh m f n t Ihn t M fn fm t Chn t

Variable domain

Table 3 Parameters. Parameter

Definition

Player domain

Parameter domain

Ain t Y¯in t

Utility discount factor for player in at time t GDP potential of player in at time t Inflation rate of central bank en ʼs monetary union cnu at time t Assumed equilibrium real interest rate of central bank en at time t Market risk (systematic) at time t Productivity (technology) factor for production in country cn at time t Elasticity parameter for country cn ’s utility Household hn ’s elasticity of intertemporal substitution Endogenous growth parameter (capital-output ratio) for production in country cn Elasticity parameter for country cn ’s utility Austerity parameter of country cn where increasing αcn increases the level of austerity Taylor rule weight on inflation Taylor rule weight on GDP Relative weight of central bank en ʼs monetary union’s utilities against profit and loss from banking activities Wage for household hn at time t as determined exclusively by the households exogenous ability Fraction of firm fm owned by household hn Capital depreciation of player in at time t (negative depreciation means appreciation) GDP-to-debt parameter Number of time periods Given time period

in ∈ p in ∈ (cn , en ) en ∈ e en ∈ e cn ∈ c cn ∈ c hn ∈ h cn ∈ c

βin t ∈ (0, 1 )  Y¯cn t ∈ R+ , Y¯en t = cmen ∈cen Y¯cn t πen t ∈ R re∗n t ∈ [0, 1] μt ∈ R Acn t ∈ R+ αcn ∈ [0, 1] ρhn ∈ [0, 1] ςcn ∈ [0, 1]

cn ∈ c cn ∈ c

αcn ∈ [0, 1] αcn ∈ R+

en ∈ e

aπ > 0 aY > 0 ωen ∈ [0, 1]

hn ∈ h

whn ∈ R+

hn ∈ h, fm ∈ f in ∈ p

ξhn fm ∈ [0, 1] δ in t ∈ R

T ∈ [1, ∞ ) t∈T

κ ∈ (0, ∞ ) T ∈ [1, ∞ ) t∈T

πen t

re∗n t

μt

Acn t

αcn ρhn ςcn αcn αcn

aπ aY

ωen

wh n t

ξhn fm δ in t κ T t

directions of cash flows. A player who borrows positively is taking on debt while a player who borrows negatively is taking on credit. In short, borrowing positively means receiving a loan, and borrowing negatively means giving a loan. Thus mathematically, lending is not a strategic choice variable since it is determined by borrowing. Henceforth, we use the following two terms: positive borrowing to refer to traditional borrowing from one player to another, and negative borrowing to refer to traditional lending from one player another.  +  Bi j t n m , where B+ is the amount We define borrowing as a 2 × 1 vector of positives and negatives such that Bin jm t def − in jm t −Bi j t player in borrows positively from player jm at time t, and B− i j

n m



n mt

t. Furthermore, borrowing is bounded by a minimum Bmin i j t n m

and Bimax,− ∈ R+ . j t n m

def

is the amount player in borrows negatively from player jm at time

Borrowing costs are defined similarly; we define rin jm t def [ri+n jm t

the interest rate on

B− . in jm t



0

−Bimin,− n jm t



def and maximum Bmax i j t n m

ri− j

n mt

], where ri+ j

n mt

Bimax,+ j t n m

0



, such that Bmin,+ ∈ R+ i j t n m

is the interest rate on B+ , and ri− j i j t n m

n mt

Furthermore, to simplify the mathematical formulation we assume that debt accumulates from

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Table 4 Dependent variables (depend on strategic choice variables and parameters). Variable

Definition

Player domain

Variable domain

di n t

Debt of player in to all players at time t

in ∈ p

di n t =

φin jm t

The percentage loss (haircut) on debt of player in to player jm given default at time t

( i n , jm ) ∈ p

ψ jm in t

Percentage of utility penalized for default imposed by player jm on player in at time t

( i n , jm ) ∈ p

in jm t

Debt of player in to player jm at time t conditional on default

( i n , jm ) ∈ p

in t

Debt of player in to all players at time t conditional on default

in ∈ p

Bin t

Net borrowing of player in in period t

in ∈ p

λ jm t

Idiosyncratic risk (unsystematic) on negative borrowing to player jm at time t

jm ∈ p

φin jm t = 1 − ψin trU+in t (xdin+t ,x−in t ) jm t in jm t  + in  ψ jm in t ψ jm in t def −ψ −  +jm in t  

in jm t (1 − φin jm t Din jm t )di+n jm t

in jm t = = − −

in jm t (1 − φ jm in t D jm in t )din jm t 

in t = jm ∈p in jm t  Bin t = jm ∈p [ 1(Bin jm t − in jm t ) − Pin jm t Din jm t + Pjm in t D jm in t ] λ jm t =φ jm in t Pr(D jm in t = 1 )

xin t

Strategy profile of player in at time t

in ∈ p

xin t ∈ Sin

x−in t

Strategy profile of all players not including player in at time t

−in ∈ p

x−in t ∈ Sin

U (xin t , x−in t )

Utility of player in at time t

in ∈ p

U (xin t , x−in t ) ∈ R

in t

Profit of player in at time t

in ∈ p

in t (xin t , x−in t ) ∈ R

CcHn t

Aggregate household consumption in country cn at time t

cn ∈ c

CcHn t =

IcHn t

Aggregate household investment in country cn at time t

cn ∈ c

IcHn t

IcFn t

Aggregate firm investment in country cn at time t

cn ∈ c

IcFn t =

Ccn t

Aggregate consumption of country cn at time t

cn ∈ c

Ccn t = CcPn t + CcHn t

Icn t

Aggregate investment of country cn at time t

cn ∈ c, hcmn ∈ h, flcn ∈ f

M fn t

Firm fn ’s imports at time t

fn ∈ f

Mcn t

Country cn ’s imports from all other countries at time t

(cn , cm ) ∈ c, cn ∈ c

Icn t = IcPn t + IcHn t + IcFn t  M fn t = c j ∈c, fm ∈ f c j sc j fn t M fn fm t  Mcn t = fi ∈ f c j M fi t

X fn fm t

Firm fn ’s exports to fm at time t

fn ∈ f ci , fm ∈ f c j , (ci , c j ) ∈ c, i = j

X fn t

Exports from firm fn to all other countries at time t

fn ∈ f

Xcn t

Exports from country cn to all other countries at time t

( c n , cm ) ∈ c

TBcn t

Trade balance for country cn at time t

cn ∈ c

CAcn t

Current account for country cn at time t

cn ∈ c

Yin t

GDP/production of player in at time t

in ∈ ( f, c, e )

Kin t

Total capital for player in at time t

in ∈ ( f, c )

Lin t

Labor supplied by player in at time t

in ∈ (h, c )

jm ∈p



 

din jm t

c

hmn ∈hcn

Chcmn t

c hmn ∈hcn Ihmn t c

c

fmn ∈ f cn

I fmcn t

X fn fm t = M fm fn t  X fn t = fn ∈ f X fn fm t  Xcn t = fn ∈ f cn X fn t TBcn t = Ycn t − Ccn t − Icn t = (Xcn t − Mcn t )  CAcn t = TBcn t − jm ∈p (rcn jm t × cn jm t )  ςcm Ycm t = Acm t Kcm t Lcm t 1−ςcm = f j ∈ f cm Y f j t ,  Yen t = cmen ∈cen Ycmen t  Kcn t = fl ∈cn K fl t ,  Kin t = tτ =1 (1 − δ )τ Iin τ   Lcn t = hi ∈hcn Lhi t , Lhi t = f j ∈ f cn Lhi f j t

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743

borrowing and is technically repaid each period. That is, a player in who positively borrows B+ i j

n mt

must repay B+ (1 + ri+ j i j t

from player jm at time t,

) in period t + 1. This assumption generalizes the debt structure where debt in each period can n m n m t+1 be rolled over to the next period (B+ = B+ (1 + ri+n jm t+1 )), partly rolled over, that is repaid over time (0 < B+ < in jm t+1 in jm t in jm t+1 + + + + + Bi j t (1 + ri j t+1 )), repaid at once in its entirety (B+ = 0), or expanded (B > B ( 1 + r ) ). This assumption in jm t+1 in jm t+1 in jm t in jm t+1 n m n m on debt, therefore, accommodates various term  structures  with mathematical simplicity. Similarly to borrowing, debt is defined + def din jm t as a vector of positives and negatives: din jm t . That is, debt is defined as −di− j t



din j m t

def

di+n jm t





=

−di−n jm t



B+ 1 + ri+n jm t in jm t−1



n m



−B− 1 + ri−n jm t in jm t−1



(1)

Assumption 2. No Ponzi schemes. Players cannot carry debt for all time periods. That is, there can be no Ponzi schemes, which is a common assumption [25]. Thus, at time T , i.e. in the last time period t = T , each player must carry zero debt:

 

din j m T =

0 0

∀ (in , jm ) ∈ p

(2)

Any player in can default on positive debt owed to any player jm , di+ j t , at time t. Default is defined as Din jm t , a binary indicator n m where Din jm t = 1 indicates that in has default and Din jm t = 0 indicates that in has not defaulted. Although default relieves player in from its debt obligations, default comes with repudiation, a concept discussed in the seminal work of Eaton and Gersovitz [1]. Player in is charged penalty Pin jm t for defaulting on debt di+ j t . n m The default penalty Pin jm t is presented as a strategic choice of player jm , a modeling assumption which is generically valid. In certain scenarios (e.g. households borrowing from banks and banks borrowing from central banks) it is clear that the lender has the capability of choosing appropriate repudiation for default within the legal framework of the country. However for sovereign debt (where countries borrow from households, firms, banks, central banks, and even other countries) the ability of the lender to adequately choose any repudiation measure is more opaque. As discussed by Eaton and Gersovitz [1], repudiation for sovereign default can come in many forms; limited access to international credit markets for an indefinite number of periods, elevated debt costs, stigma, and even the potential for direct costs. Consequently, we propose an alternative to penalties as a strategic choice through an endogenous default penalty. Assumption 3. Endogenous default penalty. The default penalty Pin jm t is either a strategic choice or an endogenous cost levied at the period of default. We determine this penalty as a cost equivalent to a fraction of utility lost by default. That is, the penalty is defined as follows



Pin jm t def ψin jm t Uin t xin t , x−in t



(3)

where ψin jm t ∈ (0, 1 ) is a parameter mapping utility to a fractional cost. The value of ψin jm t varies significantly by scenarios; in the scenario of households borrowing from banks, ψhn bm t should be quite high (e.g. the value of a repossessed home) while in the scenario of countries borrowing from households, ψcn hm t should be quite small (near zero). Assumption 4. A default rule. Default is a strategic choice. Alternatively, default can be endogenously determined using strategic choice variables as follows.



Din jm t = 1 ri+n jm t di+n jm t > Pin jm t



(4)

where Pin jm t is given by (3) and a player defaults if the interest payments on debt ri+ j t di+ j t exceeds the default penalty Pin jm t . n m n m In the style of Eaton and Gersovitz [1], players are allowed to default even when they can afford to pay debt. The default rule in Assumption 4 specifies the maximum interest on debt that player in is willing to pay. This maximum equals the penalty imposed at default. Player in is indifferent to default when ri+ j t di+ j t = Pin jm t . Furthermore, given an interest rate on positive n m

n m

debt, ri+ j t , player in will be unwilling to pay interest on debt exceeding a threshold value. We define this value as player in ’s n m default threshold towards player jm at time t. The default threshold is found as the penalty divided by his interest rate, i.e. def Pin jm t d˜+ in j m t ri+n jm t

(5)

Assumption 5. Partial default. We define player in ’s fraction of default to player jm at time t as

φi n j m t 1 − def

d˜i+n jm t

di+n jm t

=1−

  ψin t Uin t xin t , x−in t ri+n jm t di+n jm t

(6)

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which expresses that player in at default pays player jm his default threshold of debt divided by his debt.3 That is, players pay what they can at default. Thus, a player repays debt in each period conditional on default and borrowing in the past period in the amount

   

+in jm t = 1 − φin jm t Din jm t di+n jm t , −in jm t = 1 − φ jm in t D jm in t di−n jm t

where

− in jm t

=

(7)

+jm in t .

Lemma 1. An increased interest rate ri+ j t with a constant penalty Pin jm t , and a constant interest rate with decreased penalty n m will increase the default likelihood. Proof. The default probability for player in on debt to player jm at time t is











Pr Din jm t = 1 = Pr ri+n jm t di+n jm t > Pin jm t = Pr ri+n jm t di+n jm t − Pin jm t > 0 +



(8)

+

Given an interest rate ri j t such that ri j t > ri+ j t and a constant penalty, the monotone increasing property of the cumulan m n m n m tive distribution function implies that









Pr rin+jm t di+n jm t − Pin jm t > 0 ≥ Pr ri+n jm t di+n jm t − Pin jm t > 0



(9)

Therefore, an increase in the borrowing rate leads to an increase in the probability of default.  Lemma 1 could explain a default crisis or contagion event. A vicious cycle may occur as one country’s default can lead a lender to raise interest rates for other countries. As shown by Lemma 1, this leads to increased default risk which may cause more defaults and a default crisis. In Table 2 strategic choice variables are presented as both vectors and scalars. The strategic choice variables for borrowing, interest rates, and default penalty are vectors to account for positive and negative transactions. In contrast, the strategic choice variables for default, government spending, inflation targeting, lump-sum transfers, investment, imports, and consumption are scalars since these are one-way transactions. Borrowing, Bin jm t , can be written as a scalar as player in ’s net borrowing, Bin t , in a given period t. Net borrowing, which is the and negative borrowing B− minus the difference of positive debt + and negative difference of positive borrowing B+ i j t i j t i j t n m

n m

n m

debt − minus penalties paid Pin jm t multiplied by the binary indicator Din jm t plus penalties received Pjm in t multiplied by the in jm t binary indicator D jm in t , summed over all jm ∈ p, is defined as follows

B in t =

    1 Bi j t − i j t − Pi j t Di j t + Pj i t D j i t n m n m n m n m m n m n

(10)

jm ∈p

Assumption 6. Firms are the sole importers and exporters. Neither households, banks, nor countries import or export. Households, banks, and countries consume foreign goods after trade from a domestic firm which imports. We define the interest rate r as a strategic choice variable chosen to maximize utility. We assume that if player in negatively borrows B− to player jm at time t, then in sets the cost of debt ri− j t to maximize its own utility. Moreover, with the terms of i j t n m

debt set by player in , player jm decides the optimal amount, B+j

n m

m in t

to borrow from player in at time t.

Assumption 7. Domestic capital ownership and investment. Households and firms are restricted to owning capital and making capital (through investment) in their respective country of residence. For example, BMW Norway owns capital and invests in Norway, while BMW U.S. owns capital and invests in the U.S. Assumption 8. Household ownership of firms. Firms are owned by households where ξhn fm is the fraction of firm fm owned by household hn . As a result, ξhn fm  fm t is household hn ’s dividend payment from firm fm ’s profit in period t. Assumption 9. Domestic financial investment.

ξhn fm = 0 ∀ hn ∈ h∈ci , fm ∈ f c j , i = j, ∀ fm ∈ f c j ,

hn ∈hci

ξhn fm = 1.

Assumption 9 states that households are restricted to owning shares of domestic firms only, and that each firm fm is owned exclusively by domestic households. Assumption 10. No negative investment.

ξhn fm ∈ [0, 1]. Assumption 10 states that households are not allowed to “short-sell” any firms. Households may only own a positive amount of each firm, designated by a positive fraction. 3

φin jm t is often referred to as the haircut on debt. For a relevant discussion of default risk and haircuts, see [26].

J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

745

2.2. Equations and utility functions for the players In this section we present a set of equations that guide the actions of each player. These include interest rate setting equations as well as utility functions for each player. 2.2.1. Households Households can borrow, consume, labor, invest, pay taxes, and default. They consume nondurable goods and services and invest in a single depreciable durable good (e.g. land). Household hn maximizes the isoelastic utility function with Greenwood et al. [27] preferences, i.e.

1−ρhn



Chn t + CcPn t /|hcn |

Uhn t =

(11)

1 − ρh n



U xhn t , x−hn t



=

T

τ =t

βhτn−t Uhn τ

(12)

which satisfies the Arrow–Pratt measure of absolute risk aversion, where 1/ρhn is the intertemporal elasticity of substitution. Eq. (12) discounts to period t, from period τ = t to period τ = T , all future utilities Uhn τ from (11). We raise to the τ − t to ensure that the current period t gets no discount. Household consumption includes individual consumption, Chn t , as well as public consumption spread equally across households in country cn , i.e. CcPn t /|hcn |. Household hn maximizes its utility subject to a budget constraint

Ihn t + Chn t + Nhn cm t = whn t Lhn t +



  ξhn fm  fm t + 1 − δhn t Ihn t−1 + Bhn t

(13)

fm ∈ f

Lemma 2. For nh equivalent households with zero borrowing and zero investment, household hn ’s consumption is



Chn t =

n−1 h





whn t Lhn t − Nhn c j t +

hn ∈h





 fm t

(14)

fm ∈ f

Proof. Inserting Ihn t = B− h

n jm t

= B+ h

n jm t

Chn t + Nhn c j t = whn t Lhn t +



= 0 into (13) yields

ξh n f m  f m t

(15)

fm ∈ f

Summing each term in (15) over nh households gives (14).  The designation of nh equivalent households in Lemma 2, each with zero borrowing and zero investment, leaves households without strategic choice. That is, in the absence of borrowing and investment, (14) demonstrates functionally that in each period household hn consumes what it earns from labor less lump sum transfers, plus its share in firm profits. 2.2.2. Firms Firms are profit seeking; they produce goods and services and receive income. Firms borrow, invest, produce, import, export, pay taxes, pay wages, and are able to default. Given its profit seeking nature, firm fn ’s utility is



U x f n t , x− f n t



=

T

τ =t

β τfn−t  fn τ

(16)

discounting to period t, from period τ = t to period τ = T , all future profits  fn τ . Firms make profits from producing Y fn t , with a generic production function



Y fn t = f K fn t ,





Lh i f n t

(17)

hi ∈h

with factors of production, capital, and labor. Firms invest I fn t in a durable good to produce capital according to the firm capital law of motion,





K fn t = 1 − δ fn t K fn t−1 + I fn t , Firm fn ʼs profit,  fn t is thus

 fn t = Y fn t − I fn t −



hi ∈h

(18)





w h i t Lh i f n t + X f n t − M f n t − N f n c m t + B f n t

(19)

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is the firm’s profit where

M fi t =



M fi c j t , X fi t =

c j ∈c



X fi c j t

(20)

c j ∈c

are import costs and export revenues. In (16) all future utilities (from period τ = t to τ = T ) are discounted to period t. Lemma 3. For n f equivalent firms with zero borrowing and zero investment firm fn ’s profit is

 fn t =

n−1 f

Ycm t + (Xcm − Mcm t ) −



w h j t Lh j t −

h j ∈h

Proof. Follows from inserting I fn t = B−f

n jm t

 N f n cm t

(21)

fn ∈ f

= B+f

ming each term over fn ∈ f , where  fn t = n−1 f



n jm t



fn ∈ f

= 0 into (19), where X fn t = n−1 Xcm t , M fn t = n−1 Mcm t , Y fn t = n−1 Ycm t , and sumf f f

 fn t . 

The designation of n f equivalent firms in Lemma 3, each with zero borrowing and zero investment, leaves firms without strategic choice. That is, in the absence of borrowing and investment, (21) demonstrates that firm fn earns profits as an equal share of domestic production and net exports less costs in the form of an equal share of labor income and lump sum transfers. 2.2.3. Banks Banks borrow and set interest rates to make profits. Bank bn sets the interest rate on negative borrowing to each other bank, bm ∈ b, based on central bank ek ’s lending rate (i.e. the rate at which bank bn can borrow from central bank ek ) and a market risk premium

rb−n bm t = rb+n e t + μt k

∀ (bn , bm ) ∈ b, ek ∈ e

We assume that rb− b

n mt

(22)

= rb− b t ∀ (bn , bm , bi , b j ) ∈ b. i j

The rate at which banks borrow from other banks is rbn bm t . Although we define the interbank rate endogenously in (22), it can be measured empirically through LIBOR. Banks then use this as the base for setting interest rates on borrowing to firms and households as follows:

rh−m bn t = rb+n b t + λhm t k

∀ (bn , bk ) ∈ b, hm ∈ h

(23)

In effect, changing interest rates is a way of changing borrowing and lending levels. Bank bn ’s utility is a function of interest payments on positive and negative debt, and penalties from default, i.e.





U xbn t , x−bn t =

T

βbτn−t bn τ

τ =t

(24)

discounting all future profits bn τ (from period τ = t to τ = T ) to period t, where

bn t = Bbn t − Nbn cm t

(25)

2.2.4. Countries A country’s utility is commonly defined as the aggregate utility of the households [28]. We introduce additional realism and generality. Each country maximizes two objectives assigned different weights. The first is aggregate household utility which expresses social welfare. The second is the GDP-to-debt ratio where we consider a max function where Ycn t /dc+n t applies for countries with large positive debt and 1/κ applies for countries with low debt. The realism of this is that countries with no debt otherwise get infinite utility, and countries with low debt are concerned about the other objective. We model country cn ’s utility as a Cobb–Douglas function with two inputs, i.e.

U (xcn t , x−cn t ) =

T

τ =t

βcτn−t

hm ∈cn

Uhm τ

ζcn

αcn





Ycn τ

max + cn τ , κ Ycn τ



(26)

summed from period τ = t to period τ = T , where βcn is the time discount factor and 0 ≤ ζcn , αcn ≤1 are parameters. ζcn + αcn = 1 implies constant returns to scale, ζcn + αcn < 1 implies decreasing returns to scale, and ζcn + αcn > 1 implies increasing returns to scale. In (26), all future utilities (from period τ = t to τ = T ) are discounted to period t. Production in country cn at time t is a product of the productivity factor (productivity for short) Acn t , capital Kcn t , and labor Lcn t such that: ς

Ycn t = Acn t Kcnctn Lcn t 1−ςcn

(27)

where the productivity Acn t grows as the growth rate gcn increases, i.e.

Acn t+1 = (1 + gcn )Acn t

(28)

J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

747

Moreover, Ycn t is the level of production supplied to the market while the level of production demanded is a sum of consumption Ccn t , investment Icn t , and net exports (Xcn t − Mcn t ):

Ccn t + Icn t + (Xcn t − Mcn t )     = CcPn t + IcPn t + Chn t + Ihn t + C fn t + I fn t + (Xcn t − Mcn t ) hn ∈hcn

=

CcPn t

+

IcPn t

+ CcHn t

+

fn ∈ f cn

IcHn t

+

IcFn t

+ (Xcn t − Mcn t )

(29)

Furthermore, we add the following market clearing condition for goods,



Ycn t =

cn ∈c



(Ccn t + Icn t + (Xcn t − Mcn t ))

(30)

cn ∈c

enforcing the fact that all goods produced must be consumed. That is, global supply equals global demand. Additionally, aggregate capital, Kcn t , grows according to the capital law of motion:

Kcn t = (1 − δcn )Kcn t−1 + Icn t−1

(31)

The domestic production (27) is often criticized for its lack of micro-foundations. We endogenously determine all production through the players’ strategic choices. The budget constraint is



Nhm cn t +

hm ∈hcn



fl

N fl cn t + Bcn t = CcPn t + IcPn t

(32)

∈ f cn

where Nhm cn t and N fl cn t reflect the cumulative sum of lump sum transfers (tax payments) made to the government by households and firms and transfers made from the government to households and firms. Lemma 4. Given a single country, c1 , with nh equivalent households (Lemma 2), n f equivalent firms (Lemma 3), and no other max U (xc1 1 , x−c1 1 ) subject to the domestic production function in (27), players, country c1 s best response follows from CcP t , IcP t ,Bc t 1 1 1

the market clearing condition for goods in (30), the capital law of motion in (31), the country budget constraint in (32), and the equations in Lemmas 2 and 3. Proof. Household hn ’s consumption follows from inserting Lemma 3 into Lemma 2, i.e.



Chn t = n−1 Yc1 t + (Xc1 t − Mc1 t ) − n f N fm c1 t − nh Nhn c1 t h



(33)

Inserting (33) into (11) gives household hn ’s utility

1−ρhn



Uhn t =

Yc1 t + (Xc1 t − Mc1 t ) − n f N fm c1 t − nh Nhn c1 t + CcP1 t /|hcn | 1 − ρh n

(34)

 Thus household hn ’s utility depends on country variables and the strategic behavior of country c1 through public consumption CcP t . Furthermore, country c1 ’s utility in (26) depends only on its strategic behavior and not the behavior of any household or 1 firm or other player. Thus, country c1 ’s optimal behavior follows from maximizing its own utility. 2.2.5. Central banks We define the central bank, en , as a monetary authority that is the lender of last resort with the exclusive power of setting interest rates. We define a monetary union, cnu , as a group of countries with a common central bank. The fundamental action of the central bank en is setting the lending rate. This interest rate refers to the rate at which central banks negatively borrow to banks; re− b t . Actions of central bank en can be described using the Taylor [29] rule as follows: n m

re−n bm t = max



      Yu ,0 πen t + re∗n t + aπ πen t − πe∗n t + aY log cn t Y cnu t

(35)

where the target inflation rate πe∗n t is the strategic choice variable for the central bank en , and aπ > 0, aY > 0. The right hand side of (35) is the maximum of the Taylor rule and zero in order to accommodate for the “zero lower bound” (ZLB) on interest rates (a topic discussed by Eggertsson [30] and others). In addition to objectives captured in the Taylor rule, central bank en has a variety of objectives such as maintaining full employment, ensuring liquidity and financial stability, meeting inflation targets, possibly handling of monetary policy at the ZLB and the potential for negative real rates, and various other objectives. Although we assume labor markets clear, the objective of full employment can be achieved through maximizing household utility. Furthermore, maximizing household utility can serve as a proxy for financial market stability; a stable credit system with sufficient liquidity will provide households access to credit. Consequently, we assume that central bank en ’s utility Uen t is a Cobb–Douglas function with two inputs assigned different weights. The first, assigned exponential weight αen , is the aggregate household utility defined in (11), i.e. across all households within these households’ monetary union. The second, assigned exponential weight χen , is an unspecified input en t , such as those

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J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

mentioned above or alternatives. Discounting these two inputs to period t, from period τ = t to period τ = T, gives central bank en ’s utility

U (xen t , x−en t ) =

T

τ =t





β τ −t en

αen

Uhm τ

eχneτn

(36)

u hm ∈hcn

0 ≤ αen , χen ≤1 are parameters. αen + χen = 1 implies constant returns to scale, αen + χen < 1 implies decreasing returns to scale, and αen + χen > 1 implies increasing returns to scale. Central bank en ’s budget constraint is simply that its assets equal its liabilities:

Ben t = 0

(37)

2.3. Conditions for Nash equilibrium We determine a pure-strategy Nash equilibrium where all players maximize their utility functions. We define an equilibrium as a set of strategies (x∗cn t , x∗en t , x∗b t , x∗h t ) such that

U



x∗in t , x∗−in t





n

≥ U xin t , x−in t



n

∀ in ∈ p, xin t ∈ Si , xin t = x∗in t

(38)

Lemma 5. Given a continuum of countries (c1 , . . . , cm ), each with nh equivalent households (Lemma 2), n f equivalent firms (Lemma 3), and no other players, country cn ’s optimal strategy, x∗cn t , is the one that solves4





U x∗cn t , x∗−cn t ≥ U (xcn t , x−cn t ) ∀ cn ∈ (c1 , . . . , cm ), xcn t ∈ Sc , xcn t = x∗cn t

(39)

Proof. Follows from (38) which involves more than n players, whereas (39) involves a game of n players.  3. Numerical analysis of examples In Section 3 we present two examples: a one country example in Section 3.1 and a two country example in Section 3.2. In Section 3.1 we evaluate behavior for one country populated by households and firms. We demonstrate how an adverse shock in this country changes its behavior with respect to consumption, investment, borrowing, and default. In Section 3.2 we evaluate behavior for two countries, each populated by households and firms. We demonstrate how an adverse shock in one country not only changes its behavior, but the behavior of the other country. In both subsections we model default strictly as a strategic choice and the default penalty endogenously as shown in (3). In Section 3.1, we conduct sensitivity analysis on strategic choice variables varying key input parameters. Furthermore, in Section 3.1 we test the impact of an exogenously imposed shock to productivity Acn t , where a negative shock of X% imposed on productivity Acn t results in post-shock productivity Acn t = (1 − X )Acn t . Given the production function in (27), the shock to productivity directly drives a shock to production, Ycn t . Henceforth, we refer to negative shocks to productivity as shocks. We calculate the equilibrium allocation of strategic choice variables numerically in a ten period model with and without a shock. Section 3.1 elucidates information on the ability of countries to respond to negative shocks. In Section 3.2 we conduct sensitivity analysis around strategic choice variables in a two country, borrower-lender game. We calculate the equilibrium outcome of this game in a two period model. Thereafter, we determine the changes in the equilibrium outcome resulting from exogenously imposed shocks of varying magnitude. Section 3.2 elucidates information on the ability of adverse events to impact borrower–lender decisions offering some insight into the problem of contagion. 3.1. One country Consider a world with one country, n f equivalent firms, and nh equivalent households such that Lemmas 2–4 hold, where no player borrows or invests except country c1 . The large number of strategic choices of the framework in Section 2 quickly complicates. Consequently, we parameterize certain values for these players at time t in Table 5 based on 2012 values for the United States. We set αc1 = 1 − ζc1 in (26) to ensure constant returns to scale. Table 5 gives the parameter values for country c1 , and parameterizes the strategic choice variables; lump sum transfer for   households and firms, hm ∈hc1 Nhm c1 t and f ∈hc1 N fl c1 t , imports Mc1 t , and interest rates on borrowing rc+ t . For the remainder of 1 l Section 3.1, the strategy set for country c1 is Sc1 = {B, D, C, I}, i.e. borrow, default, consume, and invest. Using the parameters of Table 5, Fig. 2 plots country c1 ’s utility U (xc1 1 , x−c1 1 ) in (24) and aggregate household utility, UcH 1 ≡ 1  P hn ∈hc1 U (xhn t , x−hn t ), as a function of country c1 ’s public consumption Cc 1 . In a one-period model, households benefit from 1

public consumption. Thus, aggregate household utility UcH 1 increases with slight concavity as CcP 1 increases without bound. 1

1

Country c1 ’s utility is high for consumption levels until CcP∗1 = $25,058,583 billion and thereafter decreases asymptotically toward 1

4 For a single country, the optimization procedure in Lemma 4 resembles small open economy models in the macroeconomics literature. However, for multiple countries, the optimization procedure in Lemma 5 has added advantages in allowing, each country to act not only to achieve the objectives of Lemma 4, but also to account for the best response of all other countries.

J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

749

Table 5 Baseline parameter values for one country. Parameter

Value

Parameter

Value

 ($ billions) c N hm ∈h 1 hm c1 t fl ∈hc1 N fl c1 t ($ billions) IcHn t = IcFn t c Kc1 1 ($ billions) Lc1 t (thousands) Ac1 1 Xc1 t ($ billions) Mc1 t ($ billions) rc+ jm t = rc+1 t 1 D jm c1 t = Dc1 jm t dc1 0 ($ billions) nf d

1075.2 304.2 0 48,724 133,737 1700 2195.9 2743.1 2.5% {0, 1} 14,764.2 17, 671

βa δc1 b αc1 ζc1

0.90 0 0.5 1 − αc1 =0.5 0.5 1 − απ = 0.5 0.3 0.5

aπ aY

ςc1 ρhn κc1

0.2 2.5% 115,226,802

gc1 nh e

Source: Federal Reserve Economic Data (FRED), Federal Reserve Bank of St. Louis, NIPA, BEA, United States Office of Management and Budget, and the IMF, each based on 2012 values for the United States. For the complete explanation of the estimation of the capital-output ratio, ςc1, and productivity, Ac1 1, see Appendix A and Appendix B respectively. a The intertemporal discount factor, β , is set to 0.90, a common number for borrower countries in the sovereign default literature where the discount factor is often between 0.88 and 0.95 [32] J. Park, Contagion of Sovereign Default Risk: the Role of Two Financial Frictions, in: Working paper, National University of Singapore, 2012. b The depreciation rate, δc1, is set to 10%, the common value in the macroeconomic literature. c We assume zero investment from households and firms (where low levels of investment are realistic during shocks) in order to apply Lemmas 2–5. That is, we allow investment decisions to be made solely by the central planner, country c1 . d Total U.S. firms with more than 500 employees [33] Census, United States Census Bureau: Statistics of U.S. Businesses, U.S. Department of Commerce, 2014. e Number of households as given by US Census data [34] Census, United States Census Bureau: State & Country QuickFacts, U.S. Department of Commerce, 2014.

,

5E+8

4E+4

4E+8

Country utility

3E+4 3E+8 2E+4 2E+8 1E+4

1E+8 0E+0 $0

$100,000,000

$200,000,000

Aggregate household utility

5E+4

6E+8

0E+0 $300,000,000

Country public consumption Aggregate household utility Fig. 2. Country c1 ’s utility U (xc1 1 , x−c1 1 ) and aggregate household period.

utility UcH1 1

Country utility

as functions of country c1 ’s public consumption CcP1 1 assuming one country and one

a constant. The reason for this high optimal value is that in a single period model debt does not need to be repaid. The U.S. 2012 public consumption is CcP 1 = $2549.7 billion.5 That is, no public consumption hurts a country. Increasing CcP 1 benefits country c1 1

1

up to a certain point, and thereafter country c1 ’s utility U (xc1 1 , x−c1 1 ) decreases as CcP 1 increases. The one period model greatly 1 overpredicts consumption levels for reasons that we will explore in detail later in this section through a ten period model.   Spending over and above country c1 s revenue from lump sum transfers, hm ∈hc1 Nhm c1 t + f ∈hc1 N fl c1 t = $1379.4 billion, greatly l diminishes country c1 ’s utility as it forces the country to take on debt. A one period model illustrates the preferences of the country and its households. To obtain more complete insight into player strategies as well as their responses to shocks, we henceforth assume a two period model. Although simplistic, this time frame generalizes to initial endowment before the game starts, beginning of life as period 1, and end of life as period 2, thus having implications on longer time frames.

5

2012 value of U.S. general government final consumption expenditure (current billion US$) from the World Bank.

Country utility

6E+8

3.0E+6

5E+8

2.5E+6

4E+8

2.0E+6

3E+8

1.5E+6

2E+8

1.0E+6

1E+8

5.0E+5

0E+0 $0

$50,000,000

$100,000,000

Aggregate household utility

J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

,

750

0.0E+0 $150,000,000

Country public consumption Aggregagte household utility

Country utility

Fig. 3. Country c1 ’s utility U (xc1 1 , x−c1 1 ) and aggregate household utility UcH1 1 as functions of country c1 ’s public consumption CcP1 1 assuming one country and two periods.

Fig. 3 plots for a two period model country c1 ’s utility U (xc1 1 , x−c1 1 ) in (24), and aggregate household utility UcH 1 in (12) as 1

functions of country c1 ’s public consumption CcP 1 . Both these two utilities are the sum of the period 1 utility and discounted 1  period 2 utility. In period 1, country c1 uses borrowing, B+ , to augment income from lump sum transfers, hm ∈hc1 Nhm c1 1 + c1 1  P∗ f ∈hc1 N fl c1 1 = $1379.4 billion, in order to choose public consumption, Cc 1 = $1382 billion. In period 2, the government pays l

1

debt on previous borrowing and uses the remaining income from lump sum transfers to finance public consumption CcP∗2 = 1

$1377 billion. As a consequence of costs to borrowing through accrued debt caused by the interest rate, borrowing in period 1 stifles future consumption adversely affecting consumption smoothing households. Thus, the country finds borrowing B+ c1 1   optimal in the first period, lifting public consumption CcP 1 above the sum of lump sum transfers hm ∈hc1 Nhm c1 1 + f ∈hc1 N fl c1 1 = l

1

$1379.4 billion in period 1 and dropping it below the sum of lump sum transfers in period 2. The two period model, however, underpredicts the actual value of CcP 1 = 2549.7. To some extent, this is driven by the presence of exogenous shocks. 1 Consider a temporary shock to productivity, Ac1 1 , in period 1, where productivity returns to normal in the second period. Fig. 4 shows the impact of four different realizations of negative shocks (no shock, a −30% shock, a −50% shock, and a −70% shock). Shown in Fig. 4a, country c1 ’s utility U (xc1 t , x−c1 t ), expectedly, decreases with negative shocks. Temporary negative productivity , from the country. As shocks increase in magnitude, households increasingly benefit from country shocks induce borrowing, B+ c 1 1

deficit spending on public consumption, CcP 1 , shown through household utility in Fig. 4b. Large negative shocks demonstrate 1 substantial marginal utility of borrowing. Despite concerns for debt levels, the country finds it optimal to borrow further from , as does spending CcP∗1 , the second period to smooth consumption during the shock in the first period. Optimal borrowing B+∗ c 1 1

1

increases as shocks increase in magnitude. Now consider the impact of austerity on country c1 with the assumptions in Table 5 except preference toward austerity, αc1 . Consider two options; a high austerity country placing high significance on constraining debt such that αc1 = 0.9, and a low austerity country placing high significance on social welfare such that αc1 = 0.1. Fig. 5 displays the difference in behavior towards consumption and borrowing, expressed within country c1 s utility function. Maximum austerity, αc1 = 1.0, makes the country indifferent to the utility of households while minimum austerity, αc1 = 0, makes the country maximize social welfare. This difference in preference is further demonstrated as countries face a negative productivity shock. A country with austerity, shown in Fig. 6a, responds to negative productivity with strong fiscal tightening reducing spending on public consumption substantially with strong diminishing utility to borrowing. Fig. 6a displays six curves for utility with increasing magnitudes of shocks. With no shock, utility decreases slightly with increasing consumption, CcP∗1 , until consumption crosses over a point, and 1 then decreases more rapidly thereafter. As the magnitude of shocks increase, the curves shift downward and the initial decrease flattens out and for large shocks causes an increase in utility with increasing consumption, CcP∗1 . A country without austerity, 1 shown in Fig. 6b, however, responds to negative shocks by increasing borrowing to finance a temporary increase in spending on public consumption smoothing household consumption over periods one and two. Fig. 6b also displays six curves. For each curve, utility is high until consumption, CcP∗1 , crosses over a point, and thereafter decreases. As the magnitude of shocks increase, 1

the points at which consumption, CcP∗1 , decreases, decrease. 1 In Fig. 7, country c1 shows investment in period 1 while consumption is assumed to be held constant at the value of the sum   of lump sum transfers, hm ∈hc1 Nhm c1 1 + f ∈hc1 N fl c1 1 = $1379.4 billion. Both aggregate household utility, UcH 1 , and country l

1

utility, U (xc1 t , x−c1 t ), increase nearly linearly as investment, IcP∗1 , increases. Investment in period 2 is 0 following the fact that the 1

marginal utility of investment in period 2 is 0. Thus, second period consumption, CcP∗2 , is equal to the value of lump sum 1

J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

751

a) Country utility

Country utility

,

2.6E+6 2.4E+6 2.2E+6 2.0E+6 1.8E+6 1.6E+6 1.4E+6 1.2E+6 1.0E+6

0% -30% -50% -70% $0

$50,000,000

$100,000,000

$150,000,000

Country public consumption b) Household utility Aggregate household utility

6.0E+8 5.5E+8 5.0E+8

0%

4.5E+8

-30% -50%

4.0E+8

-70% 3.5E+8 $0

$50,000,000

$100,000,000

$150,000,000

Country public consumption Fig. 4. Country c1 ’s utility U (xc1 1 , x−c1 1 ) and aggregate household utility UcH1 1 as functions of country c1 ’s public consumption CcP1 1 assuming one country, two periods and negative productivity shocks, where the percentages express shocks relative to productivity Ac1 1 .

,

,

70

1E+8

60

8E+7

40

4E+7

30 20

2E+7

10

0E+0 $0

$50,000,000

0 $150,000,000

$100,000,000

Country utility , = 0.9

Country utility , = 0.1

50 6E+7

Country public consumption = 0.1

= 0.9

Fig. 5. Country c1 ’s utility U (xc1 1 , x−c1 1 ) as a function of country c1 ’s public consumption CcP1 1 assuming one country, two periods, and two austerity levels αc1 = 0.1 and αc1 = 0.9.

transfers

less

debt.

That

(1 + rc+1 t )Ic1 1 .

is,

CcP 2 = 2



hm ∈hc1

Nhm c1 1 +



fl ∈hc1

N fl c1 1 − (1 + rc+ t )B+ = c 1 1

1



hm ∈hc1

Nhm c1 1 +



fl ∈hc1

N f l c1 1 −

The fact that the marginal utility of period 1 investment is positive over the feasible range makes it simple to see that country c1 ’s utility U (xc1 1 , x−c1 1 ) is maximized at the corner point solution with no consumption in period 2. That point     is 0 = hm ∈hc1 Nhm c1 1 + f ∈hc1 N fl c1 1 − (1 + rc+ t )Ic1 1 ⇒ Ic1 1 = (1 + rc+ t )−1 ( hm ∈hc1 Nhm c1 1 + f ∈hc1 N fl c1 1 ) = 1379.4/1.025 = l

1

1

l

$1345.76 billion. We consider a ten period model for country c1 with nh equivalent households and n f equivalent firms. Country c1 invests Ic1 t , bounded above by a 15% absorptive capacity rate [31]. A mixed-integer nonlinear program (MINLP) is developed in GAMS (gams.com) to solve Lemma 4, using household utility in (11) and (12), production, Yc1 t , in (27), the goods market clearing condition in (30), the capital law of motion for Kc1 t in (31), the country budget constraint in (32), and household consumption, Chn t , in (33) to find country c1 ’s optimal strategy to maximize utility. Furthermore, in solving Lemma 4, we assume that country c1 has the sole objective of maximizing social welfare (i.e. country c1 has utility in (26) with ζc1 = 1). Fig. 8 plots the results of this

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J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

a) Utility with austerity

b) Utility without austerity

Fig. 6. Country c1 ’s utility U (xc1 1 , x−c1 1 ) as a function of country c1 ’s public consumption CcP1 1 assuming one country, two periods, two austerity levels αc1 = 0.1 and αc1 = 0.9, and negative productivity shocks.

2.594E+6 2.593E+6

,

5.450E+8

2.592E+6

Country utility

5.445E+8 2.591E+6 5.440E+8

2.590E+6

5.435E+8

2.589E+6 0

200

400

600

800

1000

Aggregate household utility

5.455E+8

1200

Public investment Country utility

Aggregagte household utility

Fig. 7. Country c1 ’s utility U (xc1 1 , x−c1 1 ) and aggregate household utility UcH1 1 as functions of country c1 ’s public investment Ic1 1 assuming one country and two periods.

optimization problem over ten periods with three strategic choice variables for country c1 ; CcP∗t , IcP∗t , and B+∗ c t , and three en1

1

1

dogenously determined variables; Yc∗ t , Kc∗ t , and CcH∗t . For Fig. 8, we assume no borrowing (B+ c1 t = 0), an assumption that is later 1 1 1 relaxed. The top three panels in Fig. 8a–c assume no shocks occur over the ten periods. Fig. 8a plots optimal allocations of capital, Kc∗ t , 1 and production, Yc∗ t , for country c1 at time t = 1, . . . , 10. Capital increases by 25% from Kc∗ 1 in period 1, and production increases 1

ςc1

by 50%, from Yc1 1 = Ac1 1 Kc

11

1

Lc1 1 1−ςc1 in period 1, both nearly linearly, over the 10 periods.

Fig. 8b plots aggregate household consumption, CcH∗t , and public consumption, CcP∗t . Aggregate household consumption in1 1 creases by 36%, nearly linearly, over the ten periods. Public consumption remains flat at zero for periods 1–9 and rises to the sum P∗ of lump sum transfers from households and firms in period 10, Cc 10 = n f N fm c1 10 + nh Nhm c1 10 = $1379 billion. 1

J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

753

Fig. 8. Behavior of households, firms, and country c1 shown over ten periods. The top three panels (a)–(c) show the strategic allocations of capital Kc∗1 t , production , public consumption CcP∗ , public investment IcP∗1 t , and country borrowing B+∗ Yc∗1 t , aggregate household consumption CcH∗ c1 t , in a world with no shocks. The bottom nt 1t three (d)–(f) panels show the allocations and the strategic response to a negative 30% shock to productivity, Ac1 3 , in period 3.

Fig. 8c plots public investment, IcP∗t , and borrowing, B+∗ c t . Borrowing is zero for all time periods t by assumption. Investment is 1

1

equal to the sum of lump sum transfers from households and firms remaining flat for periods 1–9, IcP∗t = n f N fm c1 t + nh Nhm c1 t = 1 $1379 billion and falling to zero in period 10 as there is no incentive for investment in the final period. The bottom three panels in Fig. 8d–f assume a negative productivity shock. The negative shock to productivity, Ac1 t , is presented as a temporary negative 30% shock to productivity in period 3 such that Ac1 3 = (1 + gc1 )Ac1 2 (1 − 0.3 ) applying (28), multiplying (28) for t = 2 with (1 − 0.3) to reflect the shock. Productivity returns in period 4 such that Ac1 4 = (1 + gc1 )2 Ac1 2 , applying (28) twice. Nonanticipativity is enforced in solving the MINLP by applying the no-shock solution for the first two periods. Consequently, in the bottom three panels, the first two periods are equivalent to the top three panels for each series due to nonanticipativity. Fig. 8d plots the optimal allocations of capital, Kc∗ t , and production, Yc∗ t , assuming a negative shock in period 3. In the third 1 1 period, production decreases sharply due to the productivity shock. By changing incentives away from investment, third period capital growth is flat. Both production and capital resume to near linear growth after the shock. However, the shock damages both; capital increases by 23% in the presence of a shock as opposed to 25% without, and production increases by 47% with a shock as opposed to 50% without. Fig. 8e plots aggregate household and public consumption with a shock in period 3. Household consumption follows production with a large dip in period 3. As a response to the drop in household consumption, country c1 spends its entire period 3 income from lump sum transfers on public consumption to smooth household consumption. This is visible through a spike in period 3 public consumption, CcP∗3 . Long-term household consumption growth is impacted. It increases 28% over the ten periods 1 with the shock as opposed to 36% without. Public consumption subsequently returns to zero remaining flat for periods 4–9 until returning to the sum of lump sum transfers in period 10. Fig. 8f plots public investment and borrowing with a shock in period 3. Borrowing is still zero for all periods by assumption. The large decrease in household consumption from the shock in period 3 increases country c1 ’s utility from spending on public

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J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

Fig. 9. Behavior of households, firms, and country c1 shown over ten periods with first period borrowing. Panels (a)–(c) show the strategic allocations of capital , public consumption CcP∗ , public investment IcP∗1 t , and country borrowing B+∗ Kc∗1 t , production Yc∗1 t , household consumption CcH∗ c1 t , without shocks. Panels (d)–(f) nt 1t show the outcomes in response to a negative 30% shock to productivity, Ac1 3 , in period 3, while panels (d)–(f) show the outcomes when a negative productivity shock in period 3 results in default.

consumption above investment. Investment in period 3 falls to zero as a result. Investment returns to the sum of lump sum transfers remaining flat for periods 4–9. In the final period, 10, investment falls to zero. Next, consider the same 10 period scenario as in Fig. 8 with the same assumptions, except that country c1 can borrow. That is, in Fig. 8, the country is hit by a shock with no prior debt. Relaxing this assumption means that a country may carry debt when a shock occurs, greatly impacting its ability to respond. Fig. 9a plots the optimal allocations of capital, Kc∗ t , and production, Yc∗ t , over ten periods without shocks. Capital expands 1 1 by 12% after enduring significant growth from periods 1–4 before tapering off in period 5 to remain flat through period 10. Production expands by 37% with rapid growth in periods 1–4 following the large capital expansion, and with decelerated growth in periods 5–10.

J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

755

Fig. 9b plots aggregate household consumption, CcH∗t , and public consumption, CcP∗t . Aggregate household consumption in1 1 creases by 31% following the growth in production over the ten periods. Public consumption is high in period 1 which acts to “smooth” the total amount consumed by households by supporting households in the beginning when production is low. As household consumption grows in subsequent periods, country c1 spends less on public consumption in period 2 until public consumption falls to zero in period 3 and remains flat at zero through period 9. In period 10, public consumption rises to the sum of lump sum transfers from households and firms in period 10, CcP∗10 = n f N fm c1 10 + nh Nhm c1 10 = $1379 billion. 1

Fig. 9c plots public investment, IcP∗t , and borrowing, B+∗ c t . Investment rises gradually from period 1 to period 3 before tapering 1

1

off in period 4 and falling to zero in period 5 where it remains for all subsequent periods. Borrowing, B+∗ c1 t , expands for the first three periods to finance large spending on consumption and investment. Borrowing rounds out to a peak in period 3 and subsequently falls as debt is repaid from period 4 to period 9. The middle three panels in Fig. 9d–f assume a negative productivity shock. The negative shock to productivity, Ac1 t , is presented as a temporary negative 30% shock to productivity in period 3 such that Ac1 3 = (1 + gc1 )Ac1 2 (1 − 0.3 ) where productivity returns in period 4 such that Ac1 4 = (1 + gc1 )2 Ac1 2 . Nonanticipativity is enforced in solving the MINLP by applying the no-shock solution for the first two periods. Consequently, for the first two periods all curves in the middle three panels are equivalent to curves in the top three panels due to nonanticipativity. Fig. 9d plots the optimal allocations of capital, Kc∗ t , and production, Yc∗ t , when there is a negative shock in period 3. Capital 1 1 grows by a modest 8% and production by 33% over the 10 periods as opposed to 12% and 37% respectively when there is no shock (in the top three panels). Production drops sharply in period 3, and although production growth resumes in period 4, the result is permanently lower production for the remaining ten periods. Capital growth is impeded by the third period shock diminishing investment. As previously, capital, Kc∗ t , remains flat after period 4. 1

Fig. 9e plots aggregate household consumption, CcH∗t , and public consumption, CcP∗t , with a shock in period 3. With a large dip 1 1 in period 3, aggregate household consumption increases by 23%, lower than 31% growth experienced without shocks. High debt , restrict the ability of country c1 to respond fully to the third period shock. levels incurred from significant initial borrowing, B+∗ c 1 1

Consequently, country c1 responds with a modest public consumption increase in period 3. , for country c1 with a shock in period 3. The shock leads to reduced Fig. 9f plots public investment, IcP∗t , and borrowing, B+∗ c 1 1

1

investment IcP∗3 in period 3. Investment subsequently falls to zero in period 4 where it remains for all future periods. Borrow1 ing expands sharply to a peak in period 3 out of the need to finance increased spending on consumption while keeping some investment. Debt, dc+ 3 , is repaid from period 4 to period 9. 1

In Figs. 8 and 9 the penalty, Pc1 jm t , for default, Dc1 jm t , is set high enough (Pc1 jm t = 0.01), to ensure that no defaults will occur. As Lemma 1 and the budget constraint in (32) imply, decreasing the penalty on default increases country c1 ’s willingness to default. As a result, in the numerical analysis driving Fig. 9 a critical penalty level exists for which a default exists. That is, when the default penalty is lowered to Pc1 jm t = 0.0004, default occurs. This default occurs in period 4 (Dc1 jm 4 = 1) after substantial , by country c1 in period 3. In fact, the foresight of a default in Fig. 4 changes the optimal behavior in period 3. borrowing, B+∗ c 3 1

at the maximum level tolKnowledge that debt levels, dc+ 3 , are high enough to trigger a default drives country c1 to borrow B+∗ c 3 1

1

erated the period before default, i.e. period 3. Significant debt increases in period 3 allow for larger consumption, CcH∗t , smoothing 1

responses in period 3 with increased public consumption, CcP∗t , and increased investment, IcP∗t . Following the period 4 default, 1 1 borrowing resumes financing investment and public consumption. Investment in periods 5 and 6, financed by borrowing, restore growth in capital, Kc∗ t , leading to rapid growth in production, Kc∗ t , immediately following default. The impact of default 1 1 results in violent swings in debt and investment. The results at the critical penalty level are showed in the bottom three panels in Fig. 9g–i. Fig. 9g plots the optimal allocations of capital, Kc∗ t , and production, Yc∗ t , when a negative shock in period 3 drives a default 1 1 in period 4. The result leads to dramatic swings in production and capital growth. Production drops harshly in period 3 due to the shock. Production growth resumes in period 4 while capital flattens driven by reduced investment, IcP∗t , during default. 1 Investment, however returns in period 5 driving a large expansion in capital and elevated production growth following the default episode. Capital remains flat from periods 7–10 leading to slower production growth. Fig. 9h plots aggregate household consumption, CcH∗t , and public consumption, CcP∗t , when a negative shock in period 3 drives 1 1 a default in period 4. Aggregate household consumption decreases sharply in period 3 in a similar fashion as the middle center panel. However, aggregate consumption increases at a quicker pace after the period 4 default due to rapid production growth. The default in period 4 allows country c1 to deficit spend at the limit in period 3. Consequently, country c1 responds to the third period shock with high public consumption. Furthermore, country c1 enters period 5 without debts and thus resumes borrowing and spending on public consumption. Fig. 9i plots public investment, IcP∗t , and borrowing, B+∗ c1 t , when a negative shock in period 3 drives a default in period 4. 1 Borrowing and investment swing violently as a result of default. The period 3 shock leaves country c1 in a situation of inevitable default in period 4. Consequently, the best response of country c1 is to borrow B+∗ at its debt ceiling in period 3 viewed through c 3 1

a harsh peak in period 3 borrowing. This props up third period investment IcP∗3 . The penalty from default in period 4 tightens the 1 countries budget constraint eliminating all spending possibilities. Following the default episode in period 4, country c1 borrows in period 5 in order to increase investment and fuel capital growth. Borrowing and investment taper off following period 6, investment falls to zero in period 7 and borrowing in 9, both remain flat at zero thereafter.

756

J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762 Table 6 Baseline parameters values for two countries. Country c1 Parameter  N ($ billions) hm ∈c1 hm c1 t fl ∈c1 N fl c1 t ($ billions) Ac1 t Kc1 t ($ billions) Lc1 t (thousands) IcF1 t + IcH1 t ($ billions) Xc1 t ($ billions) Mc1 t ($ billions) dc1 0 ($ billions) rc+1 c2 t = rc+1 t Dc1 c2 t nh nf

βc1 δc1 αc1 = 1 − ζc1

aπ = aY

ςc1 ρhn κc1

Country c2 Value 1075.2 304.2 1700 48,724 133,737 0 2195.9 2743.1 500 0.18%a {0, 1} 115, 226, 802 17, 671 0.90 0.10 0.5 0.5 1/3 0.5 0.2

Parameter

Value

 N ($ billions) hm ∈c2 hm c2 t fl ∈c1 N fl c2 t ($ billions) Ac2 t Kc2 t ($ billions) Lc2 t (thousands) IcF2 t + IcH2 t ($ billions) Xc2 t ($ billions) Mc2 t ($ billions) dc2 0 ($ billions) rc−2 c1 t = rc−2 t Dc1 c2 t = Dc2 jm t nh nf

βc2 b δc2 αc2 = 1 − ζc2 aπ = aY

ςc2 ρhn κc2

5376 1521 1700 243,620 133,737 0 Mc1 t Xc1 t 0 rc+1 t {0, 1} 115, 226, 802 17, 671 0.98 0.10 0.5 0.5 1/3 0.5 0.2

a In this table, the actual 2012 U.S. crisis value 0.18% for the short term interest rate for 12 months, rc+1 t , is 14 times smaller than the more common pre crisis value 2.5% in Table 5. (The latter value would have caused substantial borrowing and other distortions in this table.) b In order to induce borrowing, we set the lender country discount factor such that βc1 < βc2 < 1/(1 + rc+1 t ).

3.2. Two countries This section considers two countries, equivalent with the exception of initial capital and tax revenue, with baseline parameter values shown in Table 6.6 Country c1 is a borrower and country c2 is a lender. Both countries consist of a set of households and firms with the same characteristics as in Section 3.1, except that the exports of country c2 are the imports of country c1 and the imports of c2 are the exports of c1 . Table 6 gives the parameter values for country cn ∈ (c1 , c2 ), and parameterizes the strategic choice variables; lump sum trans  fer for households and firms, hm ∈hcn Nhm cn t and f ∈hcn N fl cn t , interest imports Mcn t , rates on borrowing rc+n t . For the remainder l of Section 3.2 the strategy set for country cn ∈ (c1 , c2 ) is Scn = {B, r, D, C, I} for, borrow, default, consume, and invest. Country c2 is the sole creditor to country c1 making the total borrowing of country c1 in each period B+ c c t . Similarly, the 1 2

budget constraint of country c2 , in (32), is tied to the borrowing behavior of country c1 as its lending position is B− c c

2 1t

= B+ c c t. 1 2

Now, consider a two period world as in Section 3.1. Both countries trade in both periods while country c1 may borrow, B+ c c t, 1 2

from country c1 in period 1 to finance investment, IcP t , or public consumption, CcP t , it must also repay these debts, + c c t , in 1

1

1 2

P period 2 providing income, − c c t , which country c2 may use to finance public consumption, Cc t . This is illustrated in Fig. 10 2 1

2

through almost overlapping concave decreasing utilities for country c1 and c2 in period 1 as public consumption, CcP 1 , in period 1 1 increases. Furthermore, we consider the impact of a negative productivity shock (i.e. temporary decrease in Ac1 t ) in country c1 . The negative shock to productivity is presented as a temporary negative 30% shock to productivity in period 1, Ac1 1 , where productivity in period 2, Ac1 1 , returns to normal. The productivity shock in country c1 is assumed to proportionally impact demand for goods produced in country c1 and country c2 thus diminishing the demand for imports, Mc1 c2 t , thereby decreasing the exports of country c2 , Xc2 c1 t = Mc1 c2 t . This has a heavy impact on the trade balance increasing net exports (Xc1 c2 t − Mc1 c2 t ) in country c1 and decreasing net exports in country c2 (Xc2 c1 t − Mc2 c1 t ). Furthermore, the fact that country c1 is still borrowing from country c2 makes c2 less able to act on decreased net exports. The result leads to a disproportionate impact on the utility of country c2 in the sense that country c2 ’s utility thus decreases more rapidly than country c1 ’s utility when public consumption CcP t increases, 1 as shown in Fig. 11 which assumes a negative 30% shock to productivity, Ac1 1 , in country c1 in period 1. Next, we evaluate the solution to a two period game between countries c1 and c2 with sequential moves. In the first period, country c2 (the lender) moves first, and c1 (the borrower) moves second. Moreover, country c1 borrows from country c2 in

6



Country c2 is parameterized equivalently to c1 for the U.S. except that country c2 is wealthier to enable lending, i.e. with five times larger lump sum transfers    Nhm c2 t = 5 hm ∈hc1 Nhm c1 t , fl ∈c1 N fl c2 t = 5 fl ∈c1 N fl c1 t , and five times more capital, i.e. Kc2 t = 5Kc1 t .

hm ∈c2

J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

757

5.6E+08 5.4E+08

,

5.2E+08 5.0E+08

Country utility

4.8E+08 4.6E+08 4.4E+08 4.2E+08 4.0E+08 $0

$50,000,000

Country

$100,000,000

$150,000,000

public consumption Country

Country

Fig. 10. Utility U (xcn 1 , x−cn 1 ), n = 1, 2, of countries c1 and c2 in period 1 as functions of country c1 ’s public consumption CcP1 1 assuming two countries and two periods.

5.2E+08

,

5.0E+08 4.8E+08

Country utility

4.6E+08 4.4E+08 4.2E+08 4.0E+08 $0

$50,000,000

Country

$100,000,000

$150,000,000

public consumption Country

Country

Fig. 11. Utility U (xcn 1 , x−cn 1 ), n = 1, 2, of countries c1 and c2 in period 1 as functions of country c1 ’s public consumption CcP1 1 assuming two countries, two periods, and a negative 30% shock to productivity, Ac1 1 , in country c1 in period 1.

, carried from period 0 and borrows, B+ , in addiperiod 1. Furthermore, in period 1 country c1 pays debts, (1 + rc+ c 0 )B+ c c 0 c c 1 1 2

1 2

1 2

tion to spending on consumption, CcP 1 , and investment, IcP 1 . Both countries receive income in periods 1 and 2 from production 1 1 and lump sum transfers. In the second, and final, period country c1 pays down all debts and spends the remaining income on consumption (recall the suboptimalitlty of investment in the final period following zero marginal utility from investment). For country c2 , the debts paid by country c1 provide additional income while country c1 ’s period 1 borrowing creates additional expenses, the balance is spent on consumption and investment. In the second, and final, period country c2 augments income with debt payments and spends the remaining income on consumption. A key assumption is made in arranging the sequence of this game; we assume that country c2 is the first mover. The assumption is justified by the notion that the lender has control in lending, B− c c t ; the lender may choose whether or not to lend and, 2 1

of course the interest rate, rc+ c t , on borrowing, B+ c1 c2 t . The borrower may then decide whether or not to borrow, how much to 1 2 borrow and whether or not to default. The decisions of each country are constrained by the domestic production function in (27), the market clearing condition for goods in (30), the capital law of motion in (31), and the country budget constraint in (32). Given the definition of countries c1 and c2 , each with a set of equivalent households and a set of equivalent firms, the solution to the sequential move game is the one that solves Lemma 5. The eight optimal values B+∗ = B−∗ , r+∗ = rc−∗c 1 , CcP∗ , IP∗ for the four strategic choice variables for the two countries, c c 1 c c 1 c c 1 n t cn t 1 2

2 1

1 2

2 1

for n ∈ (1, 2 ), are determined using the following enumerative optimization procedure developed in GAMS (gams.com) to solve Lemma 5 using a nonlinear program and a loop through all possible strategic choice values for each strategy for each country.7

7 This algorithm is a heuristic that will find an approximate solution. Existence of an equilibrium solution is guaranteed, however this algorithm does not guarantee uniqueness.

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J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

b) Period 2 public consumption, C cP*2

a) Period 1 public consumption, C cP*1

n

n

7E+3

8E+3 6E+3 4E+3 2E+3 0E+0

5E+3 3E+3 1E+3 0%

-10%

-20%

-30%

-40%

-50%

0%

-10%

-20%

-30%

Shock to productivity

Shock to productivity

Country

Country

Country

n

8E+6

, r+* )

1c21

1 2

9E+6

-50%

Country

d) Debt market, (B+c *c 1 , c

c) Utility, U* (xc* 1 , x−c * 1) n

-40%

1E+3

60%

1E+3

40%

9E+2

20%

7E+6 6E+6 5E+6

7E+2 0%

-10%

-20%

-30%

-40%

-50%

0%

Shock to productivity Country

-10%

-20%

-30%

0% -50%

-40%

Shock to productivity Borrowing, (left axis)

Country

e) Period 1 household consumption, CcHn 2*

*

Interest rate, (right axis)

*

f) Period 2 household consumption, C H* cn 2

3E+4

3E+4

2E+4

2E+4

2E+4

2E+4

8E+3

8E+3 0%

-10%

-20%

-30%

-40%

-50%

0%

Shock to productivity Country

-10%

-20%

-30%

-40%

-50%

Shock to productivity

Country

Country

Country

Fig. 12. Sensitivity of equilibrium solution to the magnitude of a shock to productivity Ac1 1 , in period 1, assuming two countries and two periods. Equilibrium , utility U ∗ (x∗cn 1 , x∗−cn 1 ), debt expressed with borrowing, B+∗ , interest rate rc+∗ , and aggregate household consumption CcHn t are values of public consumption CcP∗ c1 c2 1 nt 1 c2 1 plotted for each country in each period.

In this procedure, r and C are increments on interest rates and consumption respectively. In the calculation leading to Fig. 12, we set r = 1% and C = 50. Smaller increments lead to increased accuracy but also substantially increased run time.8 Initialize Ucmax =0, CcP,max =nh Nhi c2 1 + n f N f j c2 1 + (1 + rc− c 0 )B+ c c 1 2 For any rc− c

2 11

2

= 0, r , . . . , 1 − r , 1

2 1

1 20

For any  = 0, 1, 2, . . . , CcP,max / C 1 Set CcP 1 =  C 2 Solve maxCP , IP c1 t

+ c1 t ,Bc1 c2 1

2

U (xc1 1 , x−c1 1 ) subject to (27), (30)–(32), and Lemmas 2 and 3.

Set IcP 1 = nh Nhi c2 1 + n f N f j c2 1 + (1 + rc− c 0 )B+ − B+ c1 c2 0 c1 c2 1 2 2 1 − + P Set Cc 2 = nh Nhi c2 2 + n f N f j c2 2 + (1 + rc c 1 )Bc c 1 2 2 1 1 2

− CcP 1 2

Calculate U (xc2 1 , x−c2 1 ) If(Ucmax < U (xc2 1 , x−c2 1 ), Ucmax = U (xc2 1 , x−c2 1 ), Set x∗cn t = xcn t ) 2 2 Calculate U (x∗cn t , x∗−cn t ) As demonstrated in Section 3.1, the behavior of countries will adjust in the presence of a negative shock to productivity Acn t . Consequently, we use the procedure to find the equilibrium outcome of the two country sequential game in the presence of an exogenously imposed negative shock to period 1 productivity Ac1 1 , in country c1 . Production is assumed to return to normal in period 2. Fig. 12, plots the impact of a sensitivity analysis in six panels. The exogenously imposed shock to country c1 is , for both countries varied from 0% to −50% in −10% intervals. Fig. 12a plots the sensitivity of first period public consumption, CcP∗ n1 n = 1, 2 as a function of the magnitude of a negative shock to productivity Ac1 1 . The equilibrium allocation of public consumption

8

Run time is proportional to | r × C | where time complexity is O(n2 ).

J.W. Welburn, K. Hausken / Applied Mathematics and Computation 266 (2015) 738–762

759

CcP∗1 in country c1 adjusts up and down with changes in borrowing B+∗ . Public consumption increases as the shock magnitude c c 1 1

1 2

strengthens 10%, i.e. from 0 to −10%, since the −0% shock induces deficit spending. As the shock magnitude intensifies beyond 10% (i.e., becomes lower than −10%), debt is curtailed, and public consumption wanes. Changes in country c2 ’s equilibrium allocation of public consumption CcP∗1 in period 1 are equal and opposite. Country c1 2 uses the balance of debts paid by country c1 and country c1 ’s period 1 borrowing to spend on public consumption. Consequently, country c2 ’s public consumption decreases as the shock in country c1 goes to −10% and then increases in magnitude as the shock , for both countries n = 1, 2 as a intensifies thereafter. Fig. 12b plots the sensitivity of second period public consumption, CcP∗ n2 function of the magnitude of the negative shock to Ac1 1 . Again, the impact of the shock corresponds to changing borrowing levels . Without a shock, country c1 has little to spend on public consumption in period 2 after and the resulting debt, (1 + rc+∗c 1 )B+∗ c c 1 1 2

1 2

paying its debt. The increased debt levels with higher debt costs result in near zero public consumption, CcP∗2 , when there is a 1 −10% shock. As shocks increase above 20% in magnitude (i.e. below −20%), second period public consumption falls to zero in country c1 . It is the opposite for country c2 , where second period public consumption, CcP∗2 , is elevated from increased revenues 2 from debt payments. Furthermore, the equilibrium allocation of public investment, IcP∗ , is zero for all values for both countries n = 1, 2, and is, n1 therefore, not included in Fig. 12. The implication of this outcome is that the marginal utility of investment is less than the marginal utility of consumption in period 1 for both countries. The lack of investment by both countries is driven by the interest rate rc+ c 1 , the discount rate βc1 < βc2 , and the return on investment, IcPn t . For country c2 , the equilibrium interest rate, 1 2

rc+∗c 1 , exceeds the return on investment making lending the preferred method of intertemporal substitution. For country c1 , a 1 2

discount rate, βc1 , not only incentivizes borrowing but also incentivizes more consumption in period 1. That is, for country c2 the benefit from using investment as an intertemporal transfer of wealth, is outweighed by the marginal utility from immediate consumption. Fig. 12c plots the sensitivity of utility U ∗ (x∗cn t , x∗−cn t ) for both countries n = 1, 2 as a function of the magnitude of a negative shock to Ac1 1 . The utility of country c1 decreases nearly linearly as the magnitude of the negative shock increases. The utility of country c2 is, however, mostly unchanged as the shock increases, a fact that is mostly explained by its behavior in the upper right panel in Fig. 12. Fig. 12d plots the behavior of both countries with respect to the debt market; the period 1 borrowing, B+∗ , of country c1 and the interest rate, rc+∗c 1 , charged on debt, dc+∗c 1 , by country c2 are plotted as functions of the shock. c c 1 1 2

1 2

1 2

Unsurprisingly, a negative shock leads to increased debt. Contagion effects are not observed by changes in country c2 ’s utility but rather the changes in its best response following a shock to country c1 . Country c2 responds with a dramatic increase in interest rates. As shocks intensify, interest rates pick up. In fact, as shocks rise above 10%, the rise in rates hurts the borrower’s ability to borrow leading to decrease borrowing. Fig. 12e, the lower left panel, plots the sensitivity of first period aggregate household consumption, CcHn 1 , for both countries n = 1, 2 as a function of the magnitude of a negative shock to Ac1 1 . The changes

reflect the changes in production Ycn 1 . Therefore, aggregate household consumption, CcH 1 , decreases linearly as negative shocks 1 to production increase. On the other hand, production, Yc2 1 , in country c2 is unchanged as is its aggregate household production, CcH 2 . Second period aggregate household consumption,CcHn 2 , plotted in Fig. 12f is flat for both countries n = 1, 2. This follows from 2 the assumption that the shock is temporary and second period production, Ycn 2 , is therefore unaffected. In this example we do not consider default because the outcome of default is straightforward. Any default from country c1 would significantly harm country c2 ; country c2 would diminish expected income. Reductions in consumption during the default period would harm household utility and, thus, the utility of country c2 . Summing up, Fig. 12 shows a number of ways in which country c2 reacts to a shock in country c1 . First, its public consumption, CcP∗1 , in period 1 decreases initially and then increases in response to the shock. Second, in period 2, country c2 ’s public consump2

tion, CcP∗2 , increases overall in response to the shock. Third, that increase is due to increasing interest rates, rc+∗c 1 , with high 2

1 2

, by country c1 as the shock increases. Fourth, aggregate household consumption in country c2 , CcH∗1 , remains borrowing, B+∗ c c 1 1 2

2

relatively constant in both periods. This causes relatively constant utility for country c2 . Furthermore, Fig. 12 gives insight into contagion. That is, in order for contagion to be plausible, country c2 would have to either endure changes in public consumption, investment, or utility, regardless of changes interest rates. Changes in country c2 ’s public consumption in period 1, thus, demonstrate contagion. This means that contagion can result in changes to consumption without changing utility or investment. Fig. 12 does not discuss problems in trade as the balance of trade is fixed. It is, however, conceivable that large shock magnitudes in country c1 could impact its demand for imported goods. A decrease in the relative demand for imported goods would impact the balance of trade, driving down net exports (Xc1 t − Mc1 t ) for country c1 . The decrease in net exports would effectively act as a negative productivity shock for country c2 . Thus, Fig. 12 does not rule out the possibility of contagion from a sharp change in the balance of trade. It is, therefore, apparent that contagion is plausible without default or changes in trade in a world with perfect information. 4. Conclusion It is increasingly rare for financial crises to be isolated to a single country or economic entity. The growth of the global financial market coupled with increased financial innovation and economic trade has led to financial crises that are increasingly global

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in nature. The 2007–2008 global financial crisis provides a strong example of the possibilities produced by global economic risks. The game-theoretic model of the global financial system we introduce seeks to increase our understanding of these global economic risks. To model the interactions between countries, central banks, banks, households, and firms that result in global economic risks, we formulate, find Nash equilibrium conditions for, and illustrate how the players interact. Through numerical analysis, we show how an adverse shock in a single country can change the behavior of countries with respect to consumption, investment, borrowing, default, and costs of debt. We explain how, in response to a shock, a country changes allocations of consumption, investment, and borrowing. Furthermore, we demonstrate that under certain conditions, a shock can result in default. Additionally, the numerical analysis presented demonstrates how an exogenously imposed negative shock in a borrower country can lead to changes in behavior in a lender country. In this two country example, spill-over effects from a shock in one country are moderate. The lender adjusts behavior e.g. by changing consumption and increasing interest rates with increasingly negative shocks. Consequently, in this framework, spill-over effects are possible with and without default. In fact, this is a major part of the contagion problem. This happened during the 2009 Eurozone crisis, the 1997 Asian financial crisis, the 1998 Russian crisis, etc. Furthermore, we illustrate how a negative shock in one country, which impacts households and firms in that country impacts the countries behavior and utility. The numerical analysis in Section 3 demonstrates the behavior of countries before and after adverse shocks. Section 3.1 demonstrates that countries invest when without shocks. However, following a negative shock in one country, this country’s best response is to increase borrowing which is spent on public consumption, smoothing out aggregate household consumption during the period of the shock. Furthermore, as demonstrated in Section 3.2, the best response of a lender country following a shock in a borrower country is to increase interest rates, thereby minimizing the damage of a crisis by preventing excessive debt and stopping its spread. The literature on economic crises is largely not game-theoretic. Recently, some research has used game theory to model some players in the financial system. We present a model with five types of players capable of introducing additional realism. The result is a framework that constitutes a tool which provides insight into decision making in general, and especially decision making associated with adverse events. The developed model is large and can be illustrated further in future research. In this paper we demonstrate how sub-games between countries, households, and firms can contribute to our understanding and how mitigation strategies can be implemented. Possible mitigation strategies explained in this paper include borrowing to ease shocks and increasing interest rates to minimize their spread. Additionally, Section 2 explains how other players are incentivized to change actions in response to a shock. The central bank’s utility depends on their member countries, where their strategy set includes changing the interest rates that, through banks, govern all interest rates in this model. That is, in Section 3.1 we illustrate how high levels of sovereign debt without a shock can lead to default when a shock occurs. Section 2 explains that central banks, thus, have the ability to, and the incentive to, increase interests rates when borrowing levels are too high. That is, when debt levels are at levels that risk default in the event of a future shock, central banks, banks, and lenders alike have the incentive to mitigate future default by raising interest rates. The two country world presented in Section 3.2 illustrates this behavior as the lending country vigorously increases interest rates in the presence of shocks. However, this model also demonstrates a need for future work. Future work can build on this model, to compute increasingly realistic interactions between players where debt, trade, and common cause failures are explored in increasingly complex analyses. Acknowledgments We thank Vicki M. Bier, Robin Keller, Robin Dillon-Merrill, Jay Simon, Jun Zhuang, and participants of INFORMS 2013 and 2014, ISERC 2014, CAARMS20, and the University of Wisconsin-Madison Economics brownbag lunch seminar for their comments. Paper presented at the conference “Games and Decisions in Reliability and Risk,” Dublin, July 8–10, 2013. We thank two anonymous referees of this journal for useful comments. This work was supported by jointly by the National Science Foundation and the Research Council of Norway, grant no. DGE-0718123. Additionally, this material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under grant no. DGE-1256259. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Appendix A. Estimation of capital-output ratio The capital-output ratio of a given country, ςcn , can be estimated directly from the data as ςcn = Kcn t /Ycn t . The U.S. capitaloutput ratio is estimated from 1960–2012 in Fig. 13. The result finds a near constant ratio of 0.3 throughout this time period. Thus, 0.3 is assumed the capital-output ratio for the U.S. Appendix B. Estimation of productivity Using the capital-output ratio estimated above with time series data for production YUSt , capital KUSt , and labor, LUSt , proY ductivity, AUSt can be estimated using (27). That is, productivity can be found by solving AUSt = ςUS USt1−ς in each period t. KUSt LUSt

US

Empirical assessment results in the series for AUSt shown in Fig. 14. The productivity, AUSt , increases 403% from 338 in 1961 to

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U.S. capital-ouput ratio 1960 -2012 0.4 0.3 0.2 0.1 0.0 1960

1970

1980

1990

2000

2010

Time t Fig. 13. Annual U.S. capital-output (KUSt /YUSt ) ratio 1960–2012.

U.S. productivity parameters 1961-2012 1800 1600 1400 1200 1000 800 600 400 200 0 1961

9% 8% 7% 6% 5% 4% 3% 2% 1% 0% 1971

1981

1991

2001

2011

Time t Productivity

Year-over-year growth

Productivity growth trend

Fig. 14. Annual U.S. productivity, AUSt , its year-over-year growth, and the linear productivity growth trend line 1961–2012.

1700 in 2012. Furthermore, productivity grows over time. The year-over-year growth rate is not constant, and is best described through a random process. It fluctuates between 1% and 8%. In our numerical analysis we simplify and use a fixed year-overyear growth for productivity, determined by inspection to 2.5%, combined with specific negative shocks to productivity AUSt . The linear productivity growth trend line is determined by linearly interpolating the year-over-year growth. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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