Exercise 12.2

Self-Fulfilling Financial Crises In A Two-Period Economy

Problem

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

\[ \ln c_1 + \ln c_2, \]

where $c_1$ and $c_2$ denote consumption in periods 1 and 2, respectively. The household’s budget constraints in periods 1 and 2 are, respectively,

\[ d_2 = d_1 + c_1 + q (k_1-k) - F(k) \]

and

\[ c_2 = F(k_1) -d_2, \]

where $d_1$ denotes debt due in period 1 and $d_2$ the debt assumed in period 1 and due in period 2, $k$ denotes an exogenous initial stock of capital, $k_1$ denotes capital purchased in period 1, $q$ denotes the relative price of capital in terms of consumption, and $F(\cdot)$ is an increasing and concave production function. 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 collateral constraint:

\[ d_2 \le \kappa q k_1 \]

where $0<\kappa<1$ is a parameter.

Derive the first-order conditions associated with the household’s optimization problem.
Assume that the aggregate stock of capital is fixed. Derive the complete set of equilibrium conditions.
Characterize the range of initial debt positions, $d_1$, for which the collateral constraint does not bind in equilibrium.
Find sufficient conditions on the initial level of debt, $d_1$, for which the economy possesses multiple equilibria (in particular, at least one equilibrium in which the collateral constraint binds and one equilibrium in which it does not).

Answer

1.

The first-order conditions are the 2 sequential budget constraints and

\[ c_2 (1-\mu) = c_1 \]

\[ c_2 q (1-\kappa\mu) = c_1 F'(k_1) \]

\[ \mu\ge 0; \mu (\kappa q k_1-d_2) =0 \]

given initial conditions $k$ and $d_1$ and also taking as given the equilibrium price of capital $q$.

2.

In equilibrium, $k_1=k$. So we have

\[ \begin{eqnarray} d_2 = d_1 + c_1 - F(k) \tag{1}\\ c_1 + c_2 = 2F(k)-d_1 \tag{2} \\ c_2 (1-\mu) = c_1 \tag{3}\\ c_2 q (1-\kappa\mu) = c_1 F'(k) \tag{4}\\ \mu\ge 0 \tag{5} \\ d_2 \le \kappa q k \tag{6}\\ \mu (\kappa q k-d_2) =0 \tag{7} \end{eqnarray} \]

3.

Set $\mu=0$. Then by equation (3),

\[ c_1 = c_2. \]

and by (2)

\[ c_1 = F(k) - \frac12 d_1 \]

and by (1)

\[ d_2 = \frac12 d_1 \]

A requirement of equilibrium is that $c_1,c_2>0$, thus an unconstrained equilibrium exists only if

\[ \fbox{$F(k) - \frac12 d_1 >0$} \]

From (4)

\[ q= F'(k) \ge 0 \]

Finally, check that the collateral constraint (6) is satisfied, that is, does $\frac12 d_1 \le \kappa F'(k) k$ hold. But this is not true in general thus, we need to impose:

\[ \fbox{$ \frac12 d_1 \le \kappa F'(k) k $} \]

This restriction is the tighter one of the two, because $\kappa <1$ and because $F(k)> F'(k) k$.

Thus, the answer is that an unconstrained equilibrium exists for any $d_1$ satisfying $\frac12 d_1 \le \kappa F'(k) k$.

4.

We want to show that there exists an equilibrium in which $\mu >0$.

Combining the 2 Euler equations, (3) and (4) yields

\[ q = \frac{1-\mu}{1-\kappa \mu} F'(k) < F'(k) \]

The last inequality follows from the facts that $\kappa<1$. Thus, we have already shown that should an equilibrium exist with $\mu>0$, then $q$ is lower. Now solve this equation for $1-\mu$ to obtain

\[ 1- \mu = \frac{q -\kappa q}{F'(k) - \kappa q} \tag{8} \]

At $q=F'(k)$ (which is the max value that $q$ can take), we have $1-\mu =1$. At $q=0$ (which is the min value that $q$ can take), we have $1-\mu = 0$. What is the slope for values of $q$ between 0 and $F'(k)$?

\[ \frac{\partial (1-\mu)}{\partial q} = \frac{F'(k) (1-\kappa)}{(F'(k) - \kappa q)^2}>0 \]

The last inequality follows from $\kappa<1$.

Now plot (8) in the space $(q, 1-\mu)$. It is an upward sloping function that is zero at $q=0$ and is 1 at $q=F'(k)$.

Next we combine the remaining equilibrium conditions to obtain a second relationship between $q $ and $1-\mu$.

Because $\mu>0$, the collateral constraint must bind:

\[ d_2 = \kappa q k \]

Using the period 1 and period 2 budget constraints, we then have

\[ c_1 = F(k) +\kappa q k - d_1 \]

and

\[ c_2 = F(k) - \kappa q k \]

The Euler for debt, (3), is

\[ c_1 = (1-\mu) c_2 \]

Using the solution for $c_1$ and $c_2$ in this expression yields

\[ (1-\mu ) = \frac{c_1}{c_2} = \frac{F(k) +\kappa q k - d_1}{ F(k) - \kappa q k} \tag{9} \]

Notice that if $\mu>0$, then it must be the case that $c_1<c_2$. Also $\mu$ must be less than 1. Draw this relation in the space $(q, 1-\mu)$ for values of $q$ between $q=0$ and $q=F'(k)$, the highest value it can take, namely the one associated with the unconstrained equilibrium.

Consider $q=0$, then $1-\mu = (F(k) - d_1)/ F(k)$. Suppose that

\[ F(k) < d_1 \]

Then the intercept is negative.

Consider $q=F'(k)$, then $1-\mu = \frac{F(k) +\kappa F'(k) k - d_1}{ F(k) - \kappa F'(k) k}$ this will be greater than 1 as long as $\kappa F'(k) k > \frac12 d_1$, which holds as long as an unconstrained equilibrium exists in which the collateral constraint is slack. What is the slope?

\[ \frac{\partial (1-\mu)}{\partial q} = \frac{\kappa k (c_1+c_2)}{c_2^2}>0 \]

Therefore this locus, ie (9), must intersect the locus given in equation (8) at least once.

In summary an equilibrium in which collateral constraint binds exists if:

\[ 2 \kappa F'(k) k > d_1 > F(k) \]

Suppose $F(k) = k^{\alpha}$, then this condition is:

\[ 2 \kappa \alpha F(k) > d_1 > F(k) \]

That is we need $2 \kappa \alpha >1$. This is possible even though both $\kappa$ and $\alpha$ are less than 1.

--- title: "Exercise 12.2" subtitle: "Self-Fulfilling Financial Crises In A Two-Period Economy" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider a two-period small open economy populated by identical households with preferences given by $$ \ln c_1 + \ln c_2, $$ where $c_1$ and $c_2$ denote consumption in periods 1 and 2, respectively. The household's budget constraints in periods 1 and 2 are, respectively, $$ d_2 = d_1 + c_1 + q (k_1-k) - F(k) $$ and $$ c_2 = F(k_1) -d_2, $$ where $d_1$ denotes debt due in period 1 and $d_2$ the debt assumed in period 1 and due in period 2, $k$ denotes an exogenous initial stock of capital, $k_1$ denotes capital purchased in period 1, $q$ denotes the relative price of capital in terms of consumption, and $F(\cdot)$ is an increasing and concave production function. 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 collateral constraint: $$ d_2 \le \kappa q k_1 $$ where $0<\kappa<1$ is a parameter. 1. Derive the first-order conditions associated with the household's optimization problem. 2. Assume that the aggregate stock of capital is fixed. Derive the complete set of equilibrium conditions. 3. Characterize the range of initial debt positions, $d_1$, for which the collateral constraint does not bind in equilibrium. 4. Find sufficient conditions on the initial level of debt, $d_1$, for which the economy possesses multiple equilibria (in particular, at least one equilibrium in which the collateral constraint binds and one equilibrium in which it does not). ## Answer #### 1. The first-order conditions are the 2 sequential budget constraints and $$ c_2 (1-\mu) = c_1 $$ $$ c_2 q (1-\kappa\mu) = c_1 F'(k_1) $$ $$ \mu\ge 0; \mu (\kappa q k_1-d_2) =0 $$ given initial conditions $k$ and $d_1$ and also taking as given the equilibrium price of capital $q$. #### 2. In equilibrium, $k_1=k$. So we have $$ \begin{eqnarray} d_2 = d_1 + c_1 - F(k) \tag{1}\\ c_1 + c_2 = 2F(k)-d_1 \tag{2} \\ c_2 (1-\mu) = c_1 \tag{3}\\ c_2 q (1-\kappa\mu) = c_1 F'(k) \tag{4}\\ \mu\ge 0 \tag{5} \\ d_2 \le \kappa q k \tag{6}\\ \mu (\kappa q k-d_2) =0 \tag{7} \end{eqnarray} $$ #### 3. Set $\mu=0$. Then by equation (3), $$ c_1 = c_2. $$ and by (2) $$ c_1 = F(k) - \frac12 d_1 $$ and by (1) $$ d_2 = \frac12 d_1 $$ A requirement of equilibrium is that $c_1,c_2>0$, thus an unconstrained equilibrium exists only if $$ \fbox{$F(k) - \frac12 d_1 >0$} $$ From (4) $$ q= F'(k) \ge 0 $$ Finally, check that the collateral constraint (6) is satisfied, that is, does $\frac12 d_1 \le \kappa F'(k) k$ hold. But this is not true in general thus, we need to impose: $$ \fbox{$ \frac12 d_1 \le \kappa F'(k) k $} $$ This restriction is the tighter one of the two, because $\kappa <1$ and because $F(k)> F'(k) k$. Thus, the answer is that an unconstrained equilibrium exists for any $d_1$ satisfying $\frac12 d_1 \le \kappa F'(k) k$. #### 4. We want to show that there exists an equilibrium in which $\mu >0$. Combining the 2 Euler equations, (3) and (4) yields $$ q = \frac{1-\mu}{1-\kappa \mu} F'(k) < F'(k) $$ The last inequality follows from the facts that $\kappa<1$. Thus, we have already shown that should an equilibrium exist with $\mu>0$, then $q$ is lower. Now solve this equation for $1-\mu$ to obtain $$ 1- \mu = \frac{q -\kappa q}{F'(k) - \kappa q} \tag{8} $$ At $q=F'(k)$ (which is the max value that $q$ can take), we have $1-\mu =1$. At $q=0$ (which is the min value that $q$ can take), we have $1-\mu = 0$. What is the slope for values of $q$ between 0 and $F'(k)$? $$ \frac{\partial (1-\mu)}{\partial q} = \frac{F'(k) (1-\kappa)}{(F'(k) - \kappa q)^2}>0 $$ The last inequality follows from $\kappa<1$. Now plot (8) in the space $(q, 1-\mu)$. It is an upward sloping function that is zero at $q=0$ and is 1 at $q=F'(k)$. Next we combine the remaining equilibrium conditions to obtain a second relationship between $q $ and $1-\mu$. Because $\mu>0$, the collateral constraint must bind: $$ d_2 = \kappa q k $$ Using the period 1 and period 2 budget constraints, we then have $$ c_1 = F(k) +\kappa q k - d_1 $$ and $$ c_2 = F(k) - \kappa q k $$ The Euler for debt, (3), is $$ c_1 = (1-\mu) c_2 $$ Using the solution for $c_1$ and $c_2$ in this expression yields $$ (1-\mu ) = \frac{c_1}{c_2} = \frac{F(k) +\kappa q k - d_1}{ F(k) - \kappa q k} \tag{9} $$ Notice that if $\mu>0$, then it must be the case that $c_1<c_2$. Also $\mu$ must be less than 1. Draw this relation in the space $(q, 1-\mu)$ for values of $q$ between $q=0$ and $q=F'(k)$, the highest value it can take, namely the one associated with the unconstrained equilibrium. Consider $q=0$, then $1-\mu = (F(k) - d_1)/ F(k)$. Suppose that $$ F(k) < d_1 $$ Then the intercept is negative. Consider $q=F'(k)$, then $1-\mu = \frac{F(k) +\kappa F'(k) k - d_1}{ F(k) - \kappa F'(k) k}$ this will be greater than 1 as long as $\kappa F'(k) k > \frac12 d_1$, which holds as long as an unconstrained equilibrium exists in which the collateral constraint is slack. What is the slope? $$ \frac{\partial (1-\mu)}{\partial q} = \frac{\kappa k (c_1+c_2)}{c_2^2}>0 $$ Therefore this locus, ie (9), must intersect the locus given in equation (8) at least once. In summary an equilibrium in which collateral constraint binds exists if: $$ 2 \kappa F'(k) k > d_1 > F(k) $$ Suppose $F(k) = k^{\alpha}$, then this condition is: $$ 2 \kappa \alpha F(k) > d_1 > F(k) $$ That is we need $2 \kappa \alpha >1$. This is possible even though both $\kappa$ and $\alpha$ are less than 1.