Firstly, markets are perfectly competitive. The real wage is given by w and the rental rate on capital is given by r. Furthermore, households feature borrowing and lending opportunities. They can take unsecured loans of different sizes. Sim- ilarly to Chatterjee et al. (2007) we treat these loans as distinct financial assets.
Households candefault on their loans at any point of time, while default as in our previous chapter is strategic. In this sense, it arises as a result of an optimal household choice.
The default riskvaries with the size of the loan. Furthermore, itvaries with the labour earnings idiosyncratic state and the individual employment or unemploy- ment state. In this manner, we extend previous work in the consumer bankruptcy literature and we differentiate from models asLi and Sarte(2006) and Chatterjee et al. (2007). The model of Li and Sarte (2006) does not allow price to depend on the size of the loan, while the model of Chatterjee et al.(2007) does not allow price to depend on the idiosyncratic earnings state or unemployment state.
Households can take borrowing or saving decisions every period. However,
unlike in our previous chapter trade does not take place amongst households directly, due to the presence of a financial intermediary. Households can purchase a one-period non-contingent discount bond with a face value in the finite set
L ⊂R.
We turn now to the description of thecontract types. A purchase of a discount bond with positive face value`0 implies that the individual household has entered into a contract under which it will receive with certainty `0 units of consump- tion good next period. In this sense deposits are assumed to be risk-free and guaranteed.
We next discuss the risky borrowing contract. A purchase of a discount bond with negative face value `0, individual labour earnings y and employment- unemployment state s indicates that the individual household has entered into contract under which, it will receive q(`0, y, s)(−`0) units of consumption good to- day, promising to deliver, conditional on not defaulting −`0 units of consumption good tomorrow.
It is important to note that the total financial assets available in the market are NL·NY ·NS where N denotes the cardinality of the associated sets. As we
mentioned before trade does not take place amongst households, thus households can only purchase these bonds from financial intermediaries. We further assume that households are indifferent between investing in physical capital, or investing in bonds with positive face value. The underlying assumption of our statement is
that unlike in Chatterjee et al. (2007)’s economy, financial intermediaries in our model do not own capital and consequently do not rent capital to firms.
This structure implies that loan contracts are unenforceable, that is, any bor- rower can default on his loan at any time period. Furthermore, default induces a punishment that similarly to our previous chapter is described as temporary exclusion from the credit market. A defaulted household will regain full access to the credit market, with probability 1−θ ∈(0,1) and remain in a constrained state with probability θ.
Furthermore, any household that has agreed on a loan contract with a finan- cial intermediary decides whether to default or not based on the following value functions:
Vo(`, y, s) = maxVR(`, y, s), VD(0, y, s) , (2.11)
where VR(`, y, s) is the value associated with repayment, VD(0, y, s) is the value associated with defaulting, while Vo(`, y, s) depicts the value of the option which is always the maximum.
It is important to mention that households are still allowed to hold deposits after the default event, or in other words they are allowed to save in a risk-free asset. However, following the default event a proportion of a household’s labour income will be seized. This could be simply modelled as a wage garnishment. This default type resembles Chapter 13 bankruptcy in United States.
Under Chapter 13 bankruptcy law, a repayment plan is developed for around five years. Monthly instalments are scheduled and being supported by a wage garnishment. This garnishment simply acts as wage tax. In reality courts make an estimation of five-year income based on current earnings. For simplicity we assume that the amount of garnishment will be captured by a parameter γ ∈ (0,1). In line with theFederal Wage Garnishment Law and theConsumer Credit Protection Act III, we assume that the amount of earnings that could be garnished cannot exceed 25%.
the household’s problem. The reader for further details can refer to section 1.4.7 of our previous chapter.
2.4.2
Household’s Problem
The individual household’s problem is essentially to maximize expected discounted lifetime utility derived from consumption subject to the associated budget and borrowing constraints.
When households solve their optimisation problem, they take prices as given, that is, the loan price, the wage and the deposit rate. Note that we only study
stationary recursive competitive equilibria. In this sense the distribution across different states should be invariant, thus policy function will be time invariant.
We will first describe the problem of the unconstrained households and the problem of constrained households3 will follow.
2.4.2.1 The Unconstrained Households
The states for an unconstrained household can be described by the level of loans or deposits denoted by`∈ L ⊂R, the labour earnings draw denoted byy∈ Y ⊂ R+
and employment-unemployment draw denoted by s∈ S ∈[0,1].
We denote by VR(`, y, s) the value function of a household that repays his loan in the current period. We denote by VD(0, y, s) the current value associ- ated with defaulting. Finally,we denote Vo(`, y, s) the value of the option to be unconstrained.
Letcbe consumption in the current period and `0 denote the amount of loans or deposits that a households optimally chooses. Furthermore,wis the household wage, q(`0, y, s) is the price for a loan or a deposit, while 1s=u is an indicator function which takes the value 1 when household is in the unemployed state4 and 0 otherwise. Following these definitions, the unconstrained household’s problem could be presented as follows:
3The terms unconstrained and constrained households are described in detail in our previous
chapter. More details can be found in the studies ofLi and Sarte(2006) andAthreya(2002)
4In this case labour earnings are equal to the unemployment benefits, which are defined as
VR(`, y, s) = max {c≥0,`0}{u(c) +β X (y0,s0)∈(Y,S) π(s0|s)π(y0|y)Vo(`0, y0, s0)} (2.12) subject to c+q(`0, y, s)`0 =wy[1−(1−ρ)1s=u] +` `0 ≥ −B (2.13)
Let now VC(`, y, s) denote the value function of a household that defaulted on its loan and is currently borrowing constrained. In this case household is only allowed to hold deposits while also facing the wage garnishment. In other words a
γ proportion of its labour earnings is garnished, thus the household only receives 1−γ. Following these definitions, the problem associated with the value of default could be presented as follows:
VD(0, y, s) = max {c≥0,`0≥0}{u(c) +β X (y0,s0)∈(Y,S) π(s0|s)π(y0|y)VC(`0, y0, s0)} (2.14) subject to c+q(`0, y, s)`0 = [1−γ]wy[1−(1−ρ)1s=u] `0 ≥0 (2.15)
2.4.2.2 The Constrained Households
We now turn to the problem of borrowing constrained households, that is, house- holds that have defaulted in some previous period. Recall that we denote by θ
the exogenous probability related to the household remaining in the borrowing constrained state5. Following these definitions the problem of constrained house-
holds could be presented as follows:
VC(`, y, s) = max {c≥0,`0≥0}{u(c) +β X (y0,s0)∈(Y,S) π(s0|s)π(y0|y)θVC(`0, y0, s0) +(1−θ)Vo(`0, y0, s0)} (2.16) subject to c+q(`0, y, s)`0 = [1−γ]wy[1−(1−ρ)1s=u] +` `0 ≥0 (2.17)
wherecdenotes consumption in the current period and`0 denotes the amount of deposits that a households optimally chooses. Furthermore, wis the household wage, θ is the probability that a household will remain in the constrained state and 1s=u is an indicator function which takes the value 1 when household is in the unemployed state and 0 otherwise.
5The term borrowing constrained in this framework is related to no access to credit markets,