Borrowing Constraints Financial Economics II

Yang-Ho Park

1. (Equilibrium with Exogenous Borrowing Constraints) Consider an economy with a continuum [0,1] of consumers of symmetric types who live forever. Consumers have utility

∑_{t=0}^{∞} β^{t} log(c_{t}^{i})

Consumers of type 1 have an endowment steam of the single good in each period (w_{0}^{1}, w_{1}^{1}, w_{2}^{1}, …) = (8, 1, 8, 1, …) while consumers of type 2 have (ω_{0}^{2}, ω_{1}^{2}, ω_{2}^{2}, …) = (1, 8, 1, 8, …). In addition there is one unit of trees that produses d = 1 units of good at each period. Each consumer of type i owns s̄_{0}^{i} of such a trees in period t = 0, s̄_{0}^{i} > 0, s̄_{0}^{1} + s̄_{0}^{2} = 1. Trees do not grow or decay over time.

(a)

Define an Arrow-Debreu equilibrium

Definition The following two equations are called the good and financial market clear conditions

c_{t}^{1} + c_{t}^{2} = ω^{g} + ω^{b} + d = ω ∀t  (1)
s_{t}^{1} + s_{t}^{2} ≤ 1 ∀t  (2)

Definition A household problem is defined by each agent’s utility optimization problem:

max ∑_{t=0}^{∞} β^{t} u(c_{t}^{i})  (3)

Subject to the following budget constraint

c_{t}^{i} + q_{t}s_{t+1}^{i} ≤ ω_{t}^{i} + (q_{t} + d) s_{t}^{i} ∀t  (4)

Definition An Arrow-Debreu equilibrium is an allocation {{c^{i}, s̄_{t+1}^{i}}_{t=0}^{∞}}_{i=1}^{2} and a sequence of prices {q_{t}}_{t=0}^{∞} such that the allocation solves each household problem and satisfies the market clear conditons.

Decide optimal consumptions From symmetric allocation, we can find optimal consumptions:

c^{∗} = ĉ_{t}^{1} = ĉ_{t}^{2} = \frac{ω^{g} + ω^{b} + d}{2} = \frac{8 + 1 + 1}{2} = 5 ∀t  (5)

Compute the equilibrium prices

Using the Lagrange Multiplier,

L = max ∑_{t=0}^{∞} β^{t} (u(c_{t}^{i}) + λ_{t}(ω_{t}^{i} + (q_{t} + d) s_{t}^{i} − c_{t}^{i} − q_{t}s_{t+1}^{i})

The following first-order conditions are obtained.

u′(c_{t}) + λ_{t} = 0
βλ_{t+1}(q_{t+1} + d) − λ_{t}q_{t} = 0

Using the above equations, we have reached the Euler Equation.

\frac{u′(c_{t})}{β u′(c_{t+1})} = \frac{q_{t+1} + d}{q_{t}}

Since u′(c^{g}) = u′(c^{b}) and prices are independent of time,

\frac{u′(c^{g})}{β u′(c^{b})} = \frac{q + d}{q}

\frac{1}{β} = \frac{q + d}{q}

Finally the equilibrium prices are obtained using d = 0.

q = \frac{β}{1 − β} d = \frac{β}{1 − β}  (6)

Find the intial asset holdings

From the budget constrains,

(q + d)s^{b} − qs^{g} = \frac{ω}{2} − ω^{g}

(q + d)s^{g} − qs^{b} = \frac{ω}{2} − ω^{b}

At the intial time, the agent 1 is in good shock and the agent 2 is in bad shock,

s̄_{0}^{1} = s^{g}
s̄_{0}^{2} = s^{b}

Next, the budget constraints can be written in a matrix form.

[ \begin{matrix}q + d & −q \\ −q & q + d \end{matrix} ] [ \begin{matrix} & s̄_{0}^{2} \\  & s̄_{0}^{1} \end{matrix} ] = [ \begin{matrix}\frac{ω}{2} − ω^{g} \\ \frac{ω}{2} − ω^{b} \end{matrix} ]  (7)

This is a system of linear equations. Given conditions,d = 1, ω^{g} = 8 and ω^{b} = 1, we can solve this linear system.

[ \begin{matrix}q + d & −q \\ −q & q + d \end{matrix} ] [ \begin{matrix} & s̄_{0}^{2} \\  & s̄_{0}^{1} \end{matrix} ] = [ \begin{matrix}5 − 8 \\ 5 − 1 \end{matrix} ]

[ \begin{matrix}\frac{1}{1 − β} & −\frac{β}{1 − β} \\ −\frac{β}{1 − β} & \frac{1}{1 − β} \end{matrix} ] [ \begin{matrix} & s̄_{0}^{2} \\  & s̄_{0}^{1} \end{matrix} ] = [ \begin{matrix}−3 \\ 4 \end{matrix} ]

We solved for s̄_{0}^{1} and s̄_{0}^{2}

[ \begin{matrix} & s̄_{0}^{2} \\  & s̄_{0}^{1} \end{matrix} ] = [ \begin{matrix}\frac{1}{1 − β} & −\frac{β}{1 − β} \\ −\frac{β}{1 − β} & \frac{1}{1 − β} \end{matrix} ]^{−1} [ \begin{matrix}−3 \\ 4 \end{matrix} ]

(8)

[ \begin{matrix} & s̄_{0}^{2} \\  & s̄_{0}^{1} \end{matrix} ] = [ \begin{matrix}\frac{1}{1 + β} & \frac{β}{1 + β} \\ \frac{β}{1 + β} & \frac{1}{1 − β} \end{matrix} ] [ \begin{matrix}−3 \\ 4 \end{matrix} ]

[ \begin{matrix} & s̄_{0}^{2} \\  & s̄_{0}^{1} \end{matrix} ] = [ \frac{4β − 3}{1 + β} ; \frac{4 − 3β}{1 + β} ]  (9)

I have obtained the initial asset holdings:

s̄_{0}^{2} = \frac{4β − 3}{1 + β}

s̄_{0}^{1} = \frac{4 − 3β}{1 + β}

(b) Suppose that consumers cannot borrow or lend. Define an equilibrium for this liquidity constraint economy.

The crucial feature of the liquidity model is that physical capital cannot be held in negative amounts, and that there is no other securities that can be traded besides physical capital. Therefore, I want to add nonnegativity of physical capital holdings to the Arrow-Debreu market.

Definition The following two equations are called the good and financial market clear conditions

c_{t}^{1} + c_{t}^{2} = ω^{g} + ω^{b} + d = ω ∀t  (10)
s_{t}^{1} + s_{t}^{2} ≤ 1 ∀t  (11)

Definition A household problem is defined by each agent’s utility optimization problem:

max ∑_{t=0}^{∞} β^{t} u(c_{t}^{i})  (12)

Subject to the following budget constraint

c_{t}^{i} + q_{t}s_{t+1}^{i} ≤ ω_{t}^{i} + (q_{t} + d) s_{t}^{i} ∀t  (13)
s̄_{t}^{i} ≥ 0

Definition An Arrow-Debreu equilibrium is an allocation {{c^{i}, s̄_{t+1}^{i}}_{t=0}^{∞}}_{i=1}^{2} and a sequence of prices {q_{t}}_{t=0}^{∞} such that the allocation solves each household problem and satisfies the market clear conditons.

(c) Compute the stochastic discount factor for this economy and compare it with the complete markets solution

Before solving this question, I want to make it clear that the stochastic discount factor is the price of pure discount bond so that it has a negative relationship with corresponding interest rate. We can switch from one to the other using the following equation:

m = \frac{1}{1 + r}

where m and r mean stochastic discount factor and interest rate respectively.

The case of complete market

The stochastic discount factor is determined by the Euler equation.

m^{c} = \frac{βu′(c_{t+1})}{u′(c_{t})}

where the superscript c indicates the complete market.

In the case of complete market, we can get perfect risk sharing, meaning that consumptions are constant across periods.

c_{t} = c_{t+1} = c_{t+2} = .......

Hence, the stochastic discount factor in the complete market is derived as:

m^{c} = \frac{βu′(c_{t+1})}{u′(c_{t})} = β

The case of liquidity constrained market In the liquidity constrained market, the stochastic discount factor

is determined by the agents with good shock.

m^{l} = \frac{βu′(c^{b})}{u′(c^{g})}

where the superscript l indicates the complete market.

In the liquidity constrained market, the consumption with good shock is greater than or equal to the consumption with bad shock.

c^{g} ≥ c^{b}

Since the marginal utility is decreasing,

u′(c^{g}) ≤ u′(c^{b})

We arrived at the stochastic discount factor in the liquidity constrained market.

m^{l} = \frac{βu′(c^{b})}{u′(c^{g})} ≥ β

m^{l} ≥ m^{c}

The conclusion is that the stochastic discount factor is bigger in the liquidity constrained market than in the complete market. More exactly, if the liqudity constraint does not bind, two stochastic discount factors will be the same. Otherwise, they could be different. I would like to mention that the interest rate is smaller in the liquidity constrained market than in the complete market because interest rate is negatively related to the stochastic discount factor.

(d) Consider the function

F^{L}(c^{g}) = u′(c^{g})(c^{g} − 8) + β u′(10 − c^{g})(9 − c^{g})  (14)

The case of β = 0.9

Show that when β = 0.9, F^{L}(c^{g}) > 0 and that ĉ_{t}^{i} = 5 satisifies all of the equilibrium conditions for the liquidity constraint economy for the right choice of s̄_{0}^{i}.

F^{L}(c^{g}) = u′(c^{g})(c^{g} − 8) + β u′(10 − c^{g})(9 − c^{g})
= \frac{c^{g} − 8}{c^{g}} + \frac{β (9 − c^{g})}{10 − c^{g}}
= \frac{5 − 8}{5} + \frac{(.9)(9 − 5)}{10 − 5}
= .12  (15)

I have proved that F^{L}(c^{g}) = .12 > 0 when ĉ_{t}^{i} = 5 and β = .9. Next let me show that those consumptions are in equilibrium. To prove this, we need to find price system and securities holdings and check whether or not they satisfy the budget constraint, the Euler equation, and the market clearing conditions.

From the budget constraints,

s^{b} = \frac{4β − 3}{1 + β} = \frac{4 × 0.9 − 3}{1 + 0.9} = \frac{0.6}{1.9} = 0.316

s^{g} = \frac{4 − 3β}{1 + β} = \frac{−3 × 0.9 + 4}{1 + 0.9} = \frac{1.3}{1.9} = 0.684

From the pricing equations,

q = \frac{β}{1 − β} = \frac{0.9}{1 − 0.9} = 9  (16)

Equilibrium solution

q = 9
s^{b} = 0.316
s^{g} = 0.684
c^{b} = c^{g} = 0.5

This equilibrium solution is calculated such that it satisfies all of the equilibrium conditions: the budget constraint, the Euler equation, and the market clearing conditions.

The case of β = 0.2 Show that when β = 0.2, the solution of F^{L}(c^{g}) = 0 and c^{g} ∈ [5, 8] is such that

ĉ_{t}^{i} = \{ \begin{matrix}c^{g} & if ω_{t}^{i} = 8 \\ 10 − c^{g} & if ω_{t}^{i} = 1 \end{matrix}

is an equilibrium allocation for the right choice of s̄_{0}^{i}.

Since u(c^{i}) = log(c_{t}^{i}) and β = 0.2,

F^{L}(c^{g}) = \frac{c^{g} − 8}{c^{g}} + \frac{β (9 − c^{g})}{10 − c^{g}}  (17)

Setting F^{L}(c^{g}) = 0

F^{L}(c^{g}) = \frac{c^{g} − 8}{c^{g}} + \frac{β (9 − c^{g})}{10 − c^{g}} = 0

\frac{c^{g} − 8}{c^{g}} = \frac{(c^{g} − 9)}{5(10 − c^{g})}

5(c^{g} − 8)(10 − c^{g}) − c^{g}(c^{g} − 9) = 0

6(c^{g})^{2} − 99c^{g} + 400 = 0

By the quadratic formular, we solve for c^{g}.

c^{g} = \frac{99 ± \sqrt{(99)^{2} − 4(6)(400)}}{12}

The following solutions have been obtained.

c^{g} = 7.069 or 9.43  (18)

The choice of the first solution c^{g} = 7.069 gives us F^{L}(c^{g}) = 0 and c^{g} ∈ [5, 8]

Like the previous case, we need to find price system and securities holdings and check whether or not they satisfy the budget constraint, the Euler equation, and the market clearing conditions. Since the liquidity constraint binds in this case, the securities holdings are determined as:

s^{b} = 0
s^{g} = 1

From the Euler equation of agent with good shock,

\frac{βu′(c^{b})}{u′(c^{g})} = \frac{q}{q + d}
\frac{βc^{g})}{u′(c^{b})} = \frac{q}{q + d}
qc^{b} = 0.2c^{g}(q + d)
q = \frac{(0.2)c^{g}}{c^{b} − 0.2c^{g}} = 0.9318

Equilibrium solution

q = 0.9318
s^{b} = 0
s^{g} = 1
c^{b} = 2.931
c^{g} = 7.069

This equilibrium solution is calculated such that it satisfies all of the equilibrium conditions: the budget constraint, the liquidity constraint, and the market clearing conditions.

2. (Equilibrium with Endogenous Borrowing Constraints) Consider the economy from the previous question. However, we assume that consumers face borrowinf constraints of the form

∑_{j=0}^{∞} β^{i} log c_{t+j}^{i} ≥ ∑_{j=0}^{∞} β^{i} log ω_{t+j}^{i} ∀t, i

but otherwise the definition of equilibrium is the same as in part (a).

(a) Provide a motivation for this environment and explain the economic initution of the constraint.

At any point in time, households have an incentive to renege on their claims and walk away from the credit market. If they choose to default, they are excluded from all further participation in intertemporal trade. However, they still have claim on endowment from human capital as we consider the human capital as inalienable. This model requires some central authority to keep track of who has gone bankrupt and to assure that their physical capital is seized. If they decide to default, households consume like autarky economy because they cannot lend nor borrow any more from financial markets. Therefore, the households face the following rationality constraint:

(1 − β) ∑_{τ=t}^{∞} β^{τ−t}u(c_{τ}^{i}) ≥ (1 − β) ∑_{τ=t}^{∞} β^{τ−t}u(ω_{τ}^{i})

This says that in every period, the value of continuing to participate in the economy is no less than the value of dropping out. The credit agency will never lend so much to consumers that they will choose bankruptcy.

(b) Define an equilibrium for this debt constaint economy.

Definition The following equation is called the financial market clear condition

c_{t}^{1} + c_{t}^{2} = ω^{g} + ω^{b} + d = ω ∀t  (19)

Definition A household problem is defined by each agent’s utility optimization problem:

max ∑_{t=0}^{∞} β^{t} u(c_{t}^{i})  (20)

Subject to the following budget constraint and rationality constraint

∑_{t=0}^{∞} p_{t} c_{t}^{i} = ∑_{t=0}^{∞} p_{t}(ω_{t}^{i} + s_{0}^{i}d)  (21)

(1 − β) ∑_{τ=t}^{∞} β^{τ−t}u(c_{τ}^{i}) ≥ (1 − β) ∑_{τ=t}^{∞} β^{τ−t}u(ω_{τ}^{i}) ∀ t

Definition An equilibrium in the debt constraint economy is an allocation {{c^{i}}_{t=0}^{∞}}_{i=1}^{2} and a sequence of prices {p_{t}}_{t=0}^{∞} such that the allocation solves each household problem and satisfies the market clear conditon.

(c) Consider the function

F^{D}(c^{g}) = u(c^{g}) − u(8) + β[u(10 − c^{g}) − u(1)]  (22)

The case of β = 0.9

Show that when β = 0.9, F^{D}(c^{g}) > 0 and that ĉ_{t}^{i} = 5 satisifies all of the equilibrium conditions for the debt constraint economy.

F^{D}(c^{g}) = u(c^{g}) − u(8) + β (u(10 − c^{g}) − u(1))
= u(5) − u(8) + (0.9)(u(10 − 5) − u(1))
= log(5) − log(8) + (0.9)(log(5) − log(1))
= 0.9785  (23)

I have proved that F^{D}(c^{g}) = 0.9785 > 0 when ĉ_{t}^{i} = 5 and β = .9. Next let me show that those consumptions are in equilibrium. To prove this, we need to find price system and check whether or not they satisfy the budget constraint, the Euler equation, and the market clearing conditions.

From the Euler equation of agent with good shock,

\frac{βu′(c^{b})}{u′(c^{g})} = \frac{q}{q + d}
beta = \frac{q}{q + d}
q = (0.9)(q + d)
(0.1)q = (0.9)
q = 9

Equilibrium solution

q = 9
c^{b} = 5
c^{g} = 5

This equilibrium solution is calculated such that it satisfies all of the equilibrium conditions: the budget constraint, the Euler equation, and the market clearing conditions.

The case of β = 0.2 Show that when β = 0.2, the solution of F^{D}(c^{g}) = 0 and c^{g} ∈ [5, 8] is such that

ĉ_{t}^{i} = \{ \begin{matrix}c^{g} & if ω_{t}^{i} = 8 \\ 10 − c^{g} & if ω_{t}^{i} = 1 \end{matrix}

is an equilibrium allocation for the right choice of s̄_{0}^{i}. If β = .2, the solution to F^{D}(c^{g}) = 0 and c^{g} ∈ [5, 8] is approximately 6.09076.

Since u(c^{i}) = log(c_{t}^{i}) and β = 0.2,

F^{D}(c^{g}) = \frac{c^{g} − 8}{c^{g}} + \frac{β (9 − c^{g})}{10 − c^{g}}  (24)

Setting F^{D}(c^{g}) = 0

F^{D}(c^{g}) = \frac{c^{g} − 8}{c^{g}} + \frac{β (9 − c^{g})}{10 − c^{g}} = 0
log(c^{g}) − log(8) + (0.2)(log(10 − c^{g}) − log(1)) = 0
log(c^{g}) − log(8) + (0.2) log(10 − c^{g}) = 0
log(c^{g})^{5}(10 − c^{g}) = log(8^{5})
(c^{g})^{5}(10 − c^{g}) = (8^{5})
(c^{g})^{6} − 10(c^{g})^{5} + 8^{5} = 0

Let us see if c^{g} = 6.09076 is a solution to the above equation. Since (6.09076)^{6} − 10(6.09076)^{5} + 8^{5} = 0, c^{g} = 6.09076 is the solution to F^{D}(c^{g}) = 0.

Like the previous case, we need to find price system and check whether or not they satisfy the budget constraint, the Euler equation, and the market clearing condition. From the Euler equation of agent with good shock,

\frac{βu′(c^{b})}{u′(c^{g})} = \frac{q}{q + d}
\frac{βc^{g})}{u′(c^{b})} = \frac{q}{q + d}
qc^{b} = 0.2c^{g}(q + d)
q = \frac{(0.2)c^{g}}{c^{b} − 0.2c^{g}} = 0.452

Equilibrium solution

q = 0.452
c^{b} = 3.90924
c^{g} = 6.09076

This equilibrium solution is calculated such that it satisfies all of the equilibrium conditions: the budget constraint, the debt rationality constraint, and the market clearing conditions.

(d) Compute the stochastic discount factor for this economy and compare it with the liquidty constraint model. Do you think these two economies can predict the same pattern for asset price movements?

Let me start by saying the final conclusion. The stochastic discount factor in the debt constraint economy is smaller than that in the liquidity constraint economy. Regarding interest rates, the interest rate in the debt constraint economy is larger than that in the liquidity constraint economy because the interest rates are negatively related to the stochastic discount factor.

The reason is that the liquidity constraint has less consumption smoothing and the debt constraint exhibits a large degree of consumption smoothing as seen in the previous questions 1(d) and 2(c). The stochastic discount factor is derived by the following equation:

m = \frac{βu′(c^{b})}{u′(c^{g})}

As more consumption smoothing is obtained in the debt constraint economy, c^{b} and c^{g} are closer to each other and so the stochastic discount factor gets closer to β. However, in the liquidity constraint economy,c^{b} and c^{g} keep away from each other and so the stochastic discount factor (m) can be much bigger than β. Therefore,the stochastic discount factor in the debt constraint economy is smaller than that in the liquidity constraint economy.

(e) Unfortunatly, the value of s̄_{0}^{i} that you calculated in the previous section when β = 0.2 is negative. Can you think of another way to make the proposed steady state an equilibrium? Explain. In these borrowing constraint economy, the consumer who first has good productivity has a permanent advantage over the other. The easiest way to compensate for this advantage and arrive at the steady state is to impose a transfer payment from one consumer type to the other. In the liquidity model, we need the budget constraint for the consumer who first has high productivity in the first period.

c^{g} + q ≤ ω^{g} + s_{0}^{1}(q + d) − τ

In the debt constraint economy, we need to transfer enough income so that the present discounted value of lifetime incomes are equal.

∑_{t=0}^{∞} p_{t}(ω_{t}^{1} + θ_{0}^{1}d) − τ = ∑_{t=0}^{∞} p_{t}(ω_{t}^{2} + θ_{0}^{2}d) + τ

Alternatively, in the debt constraint economy, we could introduce uncertainty before the first period, giving both consumer types equal chances of having the high productivity first, and allowing them to write contingent contracts against this initial uncertainty.

(f) Write down the endogenous borrowing constraint if the agent is allowed to save in an storage technology with a return R > 0, following a default period.

(g) Bankruptcy law only preclude individuals from trading just for a finite number of periods. How would you omodify this constraint to accomodate this institutional feature.