Exercise 12.5

A Two-Period Economy with a Flow Collateral Constraint

Problem

Consider a two-period small open endowment economy populated by identical households with preferences given by

\[ \ln c^T_1 + \ln c^N_1 + \beta \ln c^T_2, \]

where $c^T_1$ and $c^T_2$ denote consumption of traded goods in periods 1 and 2, respectively; $c^N_1$ denotes consumption of nontraded goods in period 1; and $\beta\in(0,1)$ is the subjective discount factor. The household’s budget constraints in periods 1 and 2 are, respectively,

\[ c^T_1 + p_1 c^N_1 = y^T_1 + p_1 y^N_1 + \frac{d_1}{1+r} \]

and

\[ c^T_2 + d_1 = y^T_2 , \]

where $p_1$ denotes the relative price of nontradable in period 1; $y^T_i$ denotes the endowment of tradables in period $i=1,2$; $y^N_1$ denotes the endowment of nontradables in period 1; $d_1$ denotes debt due in period 2; and $r$ denotes the interest rate at which the country can borrow internationally. Assume that $\beta(1+r)=1$. Implicit in the period-2 constraint is the requirement that the debt position at the end of period 2 be nil. Debt accumulation in period 1 is subject to the following flow collateral constraint:

\[ \frac{d_1}{1+r} \le \kappa [y^T_1 + p_1 y^N_1] \]

where $0<\kappa$ is a parameter.

Derive the first-order conditions associated with the household’s optimization problem.
Derive the complete set of equilibrium conditions.
Find conditions on the structural parameters of the model, ($\beta$, $r,$ $\kappa$, $y^T_1$, $y^T_2,$ and $y^N_1$), such that an equilibrium exists in which the collateral constraint is slack.
Under those conditions, does there also exist an equilibrium in which the collateral constraint is binding, that is, an equilibrium with a self-fulfilling financial crisis. If so, discuss whether there is underborrowing in the financial crisis and whether in the crisis the value of collateral suffers a Fisherian deflation.

Answer

1.

\[ \begin{eqnarray*}\mathcal{L} & =& \ln c^T_1 + \ln c^N_1 + \beta \ln c^T_2\\ & +& \lambda_1 \left[ y^T_1 + p_1 y^N_1 + \frac{d_1}{1+r} - c^T_1 - p_1 c^N_1 \right]\\ & +& \lambda_2 \left[ y^T_2 - c^T_2 - d_1\right]\\ & +& \mu\left[ \kappa [y^T_1 + p_1 y^N_1] - \frac{d_1}{1+r} \right] \end{eqnarray*} \]

FOCs:

\[ \begin{eqnarray*} \frac{1}{c^T_1} = \lambda_1 \\ \frac{1}{c^N_1} = \lambda_1 p_1\\ \beta \frac{1}{c^T_2} = \lambda_2\\ \lambda_1 \frac{1}{1+r} =\lambda_2 + \mu \frac{1}{1+r}\\ \mu\ge0\\ \mu\left[ \kappa [y^T_1 + p_1 y^N_1] - \frac{d_1}{1+r} \right]=0\\ \kappa [y^T_1 + p_1 y^N_1] \ge \frac{d_1}{1+r} \\ c^T_1 + p_1 c^N_1=y^T_1 + p_1 y^N_1 + \frac{d_1}{1+r} \\ c^T_2 + d_1 = y^T_2 \end{eqnarray*} \]

2.

Short answer HH optimality conditions plus $c^N_1 = y^N_1$. Long answer combines some of the conditions to eliminate some variables. A competitive equilibrium are values $\{c^T_1, c^T_2, d_1, p_1, \mu\}$ satisfying $c^T_1, c^T_2 >0$, and

\[ \begin{eqnarray} \frac{c^T_1}{y^N_1} = p_1 \tag{1} \\ \frac{1}{c^T_1} =\frac{1}{c^T_2} + \mu \tag{2}\\ \mu\ge0 \tag{3} \\ \mu\left[ \kappa [y^T_1 + p_1 y^N_1] - \frac{d_1}{1+r} \right]=0 \tag{4} \\ \kappa [y^T_1 + p_1 y^N_1] \ge \frac{d_1}{1+r} \tag{5} \\ c^T_1 =y^T_1 + \frac{d_1}{1+r} \tag{6} \\ c^T_2 + d_1 = y^T_2 \tag{7} \end{eqnarray} \]

3.

Collateral constraint slack means $\mu=0$.

By (2), $c^T_1=c^T_2$. By (6) and (7) $c^T_1 (1 + 1/(1+r)) = y^T_1 + y^T_2/(1+r)$, solving yields

\[ c^T_1 = \frac{1+r}{2+r}\left[y^T_1 + y^T_2/(1+r)\right] \]

This is always positive as long as at least 1 of the 2 endowments is strictly positive. The associated value of $d_1$ can be read off from (7)

\[ \frac{d_1}{1+r} = c^T_1 - y^T_1 = \frac{1}{2+r}\left(y^T_2-y^T_1\right) \]

This is intuitive. Debt is positive, ie, you borrow against future income, when future income exceeds current income.

When does that value of $d_1$ satisfy the collateral constraint (5), That is, we need that

\[ \begin{eqnarray*} \frac{d_1}{1+r} &\le& \kappa [y^T_1 + p_1 y^N_1]\\ \frac{1}{2+r}\left(y^T_2-y^T_1\right)& \le& \kappa [y^T_1 + c^T_1]\\ \frac{1}{2+r}\left(y^T_2-y^T_1\right)& \le& \kappa [y^T_1 + \frac{1+r}{2+r}\left[y^T_1 + \frac{y^T_2}{1+r}\right]]\\ \left(y^T_2-y^T_1\right)& \le& \kappa [(2+r)y^T_1 + \left[(1+r)y^T_1 + y^T_2\right]]\\ \left(y^T_2-y^T_1\right)& \le& \kappa [(3+2r)y^T_1 + y^T_2] \end{eqnarray*} \]

Thus an equilibrium with a nonbinding constraint exists if

\[ \fbox{$(1-\kappa)y^T_2 \le (1+ \kappa (3+2r))y^T_1 \tag{8} $} \]

Suppose $\kappa\ge1$, then this condition is always satisfied. Suppose $y^T_1=0$, then this condition can only be satisfied by a $\kappa>1$. As we saw before debt is only positive in the unconstrained equilibrium if $y^T_2>y^T_1$.

4.

If the collateral constraint binds,we have by (5),

\[ \frac{d_1}{1+r} =\kappa [y^T_1 + p_1 y^N_1] \]

Recall that $p_1 y^N_1 = c^T_1$ and that by (6) $c^T_1 = y^T_1 + d_1/(1+r)$. Thus we can write: $c^T_1 - y^T_1 =\kappa [y^T_1 + c^T_1]$ or

\[ \fbox{$ c^T_1 = \frac{1+\kappa}{1-\kappa} y^T_1 $} \]

Thus a binding constraint with a positive $c^T_1$ exists only if $0<\kappa<1$. In this case $c^T_1>y^T_1$, that is, $d_1>0$.

Start with (7)

\[ \begin{eqnarray*} c^T_2 & = & y^T_2 - d_1\\ &=&y^T_2 - (1+r) \kappa (y^T_1 + c^T_1)\\ &=&y^T_2 - (1+r) \kappa (y^T_1 + \frac{1+\kappa}{1-\kappa} y^T_1 )\\ &=&y^T_2 - (1+r) \kappa ( \frac{2}{1-\kappa} y^T_1 )\\ &=&\frac1{1-\kappa}\left((1-\kappa)y^T_2 - (1+r)2 \kappa y^T_1 \right)\\ &\le &\frac1{1-\kappa}\left(1+\kappa(3+2r) - (1+r)2 \kappa \right)y^T_1\\ &= &\frac1{1-\kappa}\left(1+\kappa \right)y^T_1 \\ &= &c^T_1 \end{eqnarray*} \]

The inequality follows from (8). Thus we have that if the collateral constraint binds, then $c^T_1\ge c^T_2$. This implies by (2) that $\mu\le0$. It follows that no equilibrium with $\mu>0$ exists. We have shown that if an unconstrained equilibrium exists, then there does not exist a second equilibrium in which the constraint binds and $\mu>0$. That is, no second equilibrium with a financial crisis exists.

--- title: "Exercise 12.5" subtitle: "A Two-Period Economy with a Flow Collateral Constraint" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider a two-period small open endowment economy populated by identical households with preferences given by $$ \ln c^T_1 + \ln c^N_1 + \beta \ln c^T_2, $$ where $c^T_1$ and $c^T_2$ denote consumption of traded goods in periods 1 and 2, respectively; $c^N_1$ denotes consumption of nontraded goods in period 1; and $\beta\in(0,1)$ is the subjective discount factor. The household's budget constraints in periods 1 and 2 are, respectively, $$ c^T_1 + p_1 c^N_1 = y^T_1 + p_1 y^N_1 + \frac{d_1}{1+r} $$ and $$ c^T_2 + d_1 = y^T_2 , $$ where $p_1$ denotes the relative price of nontradable in period 1; $y^T_i$ denotes the endowment of tradables in period $i=1,2$; $y^N_1$ denotes the endowment of nontradables in period 1; $d_1$ denotes debt due in period 2; and $r$ denotes the interest rate at which the country can borrow internationally. Assume that $\beta(1+r)=1$. Implicit in the period-2 constraint is the requirement that the debt position at the end of period 2 be nil. Debt accumulation in period 1 is subject to the following flow collateral constraint: $$ \frac{d_1}{1+r} \le \kappa [y^T_1 + p_1 y^N_1] $$ where $0<\kappa$ is a parameter. 1. Derive the first-order conditions associated with the household's optimization problem. 2. Derive the complete set of equilibrium conditions. 3. Find conditions on the structural parameters of the model, ($\beta$, $r,$ $\kappa$, $y^T_1$, $y^T_2,$ and $y^N_1$), such that an equilibrium exists in which the collateral constraint is slack. 4. Under those conditions, does there also exist an equilibrium in which the collateral constraint is binding, that is, an equilibrium with a self-fulfilling financial crisis. If so, discuss whether there is underborrowing in the financial crisis and whether in the crisis the value of collateral suffers a Fisherian deflation. ## Answer #### 1. $$ \begin{eqnarray*}\mathcal{L} & =& \ln c^T_1 + \ln c^N_1 + \beta \ln c^T_2\\ & +& \lambda_1 \left[ y^T_1 + p_1 y^N_1 + \frac{d_1}{1+r} - c^T_1 - p_1 c^N_1 \right]\\ & +& \lambda_2 \left[ y^T_2 - c^T_2 - d_1\right]\\ & +& \mu\left[ \kappa [y^T_1 + p_1 y^N_1] - \frac{d_1}{1+r} \right] \end{eqnarray*} $$ FOCs: $$ \begin{eqnarray*} \frac{1}{c^T_1} = \lambda_1 \\ \frac{1}{c^N_1} = \lambda_1 p_1\\ \beta \frac{1}{c^T_2} = \lambda_2\\ \lambda_1 \frac{1}{1+r} =\lambda_2 + \mu \frac{1}{1+r}\\ \mu\ge0\\ \mu\left[ \kappa [y^T_1 + p_1 y^N_1] - \frac{d_1}{1+r} \right]=0\\ \kappa [y^T_1 + p_1 y^N_1] \ge \frac{d_1}{1+r} \\ c^T_1 + p_1 c^N_1=y^T_1 + p_1 y^N_1 + \frac{d_1}{1+r} \\ c^T_2 + d_1 = y^T_2 \end{eqnarray*} $$ #### 2. Short answer HH optimality conditions plus $c^N_1 = y^N_1$. Long answer combines some of the conditions to eliminate some variables. A competitive equilibrium are values $\{c^T_1, c^T_2, d_1, p_1, \mu\}$ satisfying $c^T_1, c^T_2 >0$, and $$ \begin{eqnarray} \frac{c^T_1}{y^N_1} = p_1 \tag{1} \\ \frac{1}{c^T_1} =\frac{1}{c^T_2} + \mu \tag{2}\\ \mu\ge0 \tag{3} \\ \mu\left[ \kappa [y^T_1 + p_1 y^N_1] - \frac{d_1}{1+r} \right]=0 \tag{4} \\ \kappa [y^T_1 + p_1 y^N_1] \ge \frac{d_1}{1+r} \tag{5} \\ c^T_1 =y^T_1 + \frac{d_1}{1+r} \tag{6} \\ c^T_2 + d_1 = y^T_2 \tag{7} \end{eqnarray} $$ #### 3. Collateral constraint slack means $\mu=0$. By (2), $c^T_1=c^T_2$. By (6) and (7) $c^T_1 (1 + 1/(1+r)) = y^T_1 + y^T_2/(1+r)$, solving yields $$ c^T_1 = \frac{1+r}{2+r}\left[y^T_1 + y^T_2/(1+r)\right] $$ This is always positive as long as at least 1 of the 2 endowments is strictly positive. The associated value of $d_1$ can be read off from (7) $$ \frac{d_1}{1+r} = c^T_1 - y^T_1 = \frac{1}{2+r}\left(y^T_2-y^T_1\right) $$ This is intuitive. Debt is positive, ie, you borrow against future income, when future income exceeds current income. When does that value of $d_1$ satisfy the collateral constraint (5), That is, we need that $$ \begin{eqnarray*} \frac{d_1}{1+r} &\le& \kappa [y^T_1 + p_1 y^N_1]\\ \frac{1}{2+r}\left(y^T_2-y^T_1\right)& \le& \kappa [y^T_1 + c^T_1]\\ \frac{1}{2+r}\left(y^T_2-y^T_1\right)& \le& \kappa [y^T_1 + \frac{1+r}{2+r}\left[y^T_1 + \frac{y^T_2}{1+r}\right]]\\ \left(y^T_2-y^T_1\right)& \le& \kappa [(2+r)y^T_1 + \left[(1+r)y^T_1 + y^T_2\right]]\\ \left(y^T_2-y^T_1\right)& \le& \kappa [(3+2r)y^T_1 + y^T_2] \end{eqnarray*} $$ Thus an equilibrium with a nonbinding constraint exists if $$ \fbox{$(1-\kappa)y^T_2 \le (1+ \kappa (3+2r))y^T_1 \tag{8} $} $$ Suppose $\kappa\ge1$, then this condition is always satisfied. Suppose $y^T_1=0$, then this condition can only be satisfied by a $\kappa>1$. As we saw before debt is only positive in the unconstrained equilibrium if $y^T_2>y^T_1$. #### 4. If the collateral constraint binds,we have by (5), $$ \frac{d_1}{1+r} =\kappa [y^T_1 + p_1 y^N_1] $$ Recall that $p_1 y^N_1 = c^T_1$ and that by (6) $c^T_1 = y^T_1 + d_1/(1+r)$. Thus we can write: $c^T_1 - y^T_1 =\kappa [y^T_1 + c^T_1]$ or $$ \fbox{$ c^T_1 = \frac{1+\kappa}{1-\kappa} y^T_1 $} $$ Thus a binding constraint with a positive $c^T_1$ exists only if $0<\kappa<1$. In this case $c^T_1>y^T_1$, that is, $d_1>0$. Start with (7) $$ \begin{eqnarray*} c^T_2 & = & y^T_2 - d_1\\ &=&y^T_2 - (1+r) \kappa (y^T_1 + c^T_1)\\ &=&y^T_2 - (1+r) \kappa (y^T_1 + \frac{1+\kappa}{1-\kappa} y^T_1 )\\ &=&y^T_2 - (1+r) \kappa ( \frac{2}{1-\kappa} y^T_1 )\\ &=&\frac1{1-\kappa}\left((1-\kappa)y^T_2 - (1+r)2 \kappa y^T_1 \right)\\ &\le &\frac1{1-\kappa}\left(1+\kappa(3+2r) - (1+r)2 \kappa \right)y^T_1\\ &= &\frac1{1-\kappa}\left(1+\kappa \right)y^T_1 \\ &= &c^T_1 \end{eqnarray*} $$ The inequality follows from (8). Thus we have that if the collateral constraint binds, then $c^T_1\ge c^T_2$. This implies by (2) that $\mu\le0$. It follows that **no** equilibrium with $\mu>0$ exists. We have shown that if an unconstrained equilibrium exists, then there does not exist a second equilibrium in which the constraint binds and $\mu>0$. That is, no second equilibrium with a financial crisis exists.