Exercise 4.2

Variation of the PAC Model

Problem

This exercise aims to establish whether formulating portfolio adjustment costs as a function of the deviation of the household’s debt position from an exogenous reference point, $d_t - \bar{d}$, or as a function of the change in its debt position, $d_t - d_{t-1}$, has consequences for the stationarity of the model.

Consider a small open economy populated by a large number of infinitely-lived households with preferences described by the utility function

\[ E_0 \sum_{t=0}^\infty \beta^t \ln c_t, \]

where $\beta \in (0,1)$ denotes the subjective discount factor, and $c_t$ denotes consumption in period $t$. Each period, households receive an exogenous and stochastic endowment, $y_t$, and can borrow from (or lend to) international financial markets at the gross interest rate $1 + r$. Let $d_t$ denote the stock of foreign debt held by households at the end of period $t$. Households are subject to a portfolio adjustment cost of the form $\frac{\phi}{2} (d_t - d_{t-1})^2$, where $\phi$ is a positive constant. Assume that $\beta(1 + r) = 1$.

State the household’s period-by-period budget constraint.
State the household’s utility maximization problem.
Write the Lagrangian of the household’s problem.
Define a competitive equilibrium of this economy.
Suppose the endowment is nonstochastic and constant, $y_t = y$, for all $t$. Characterize the deterministic steady state. Does it exist? Is it unique? Hint: Consult appendix 4.14.2.
Consider now a temporary endowment shock. Suppose $y_0 > y$ and $y_t = y$ for all $t > 0$ deterministically. Suppose that prior to period 0 the economy was in a deterministic steady state with $d_{-1} = d^*$. Is the economy stationary, that is, is $d_t$ expected to return to $d^*$? Provide intuition.

Answer

1.

\[ y_t + (1 + r) b_{t-1} = c_t + b_t + \frac{\phi}{2}(b_t - b_{t-1})^2 \]

2.

\[ \max_{c_t, b_t} E_0 \sum_{t=0}^\infty \ln c_t \]

subject to

\[ y_t + (1 + r)b_{t-1} = c_t + b_t + \frac{\phi}{2}(b_t - b_{t-1})^2 \]

and some no-Ponzi game constraint taking as given the initial condition $b_{-1}$, the exogenous process $\{y_t\}$, the interest rate $r$.

3.

\[ \mathcal{L} = E_0 \sum_{t=0}^\infty \beta^t \left\{ \ln c_t + \lambda_t \left[ y_t + (1 + r)b_{t-1} - c_t - b_t - \frac{\phi}{2}(b_t - b_{t-1})^2 \right] \right\} \]

where $\lambda_t$ denote the Lagrange multiplier on the household’s period-by-period budget constraint.

4.

A competitive equilibrium is a set of contingent plans $\{c_t, b_t, \lambda_t\}$ satisfying:

\[ \frac{1}{c_t} = \lambda_t \]

\[ \lambda_t (1 + \phi(b_t - b_{t-1})) = \beta E_t \lambda_{t+1}(1 + r + \phi(b_{t+1} - b_t)) \]

\[ y_t + (1 + r)b_{t-1} = c_t + b_t + \frac{\phi}{2}(b_t - b_{t-1})^2 \]

and some no Ponzi game constraint, for given exogenous processes, $y_t$, and given initial conditions $b_{-1}$.

5.

In the deterministic steady state all endogenous variables are by definition forever constant. Thus a deterministic steady state is a triple $\{c, \lambda, b\}$ that solves:

\[ \frac{1}{c} = \lambda \]

\[ \lambda = \lambda \beta(1 + r) \]

\[ y + rb = c \]

Notice that the Euler equation does not impose a restriction, it just says $1 = 1$. Thus there exist a continuum of steady state equilibria. In fact any $b$ such that $c = y + rb > 0$ is a deterministic steady state. We do not need to check the no Ponzi restriction. It is satisfied because $b$ is constant and because $r > 0$.

6.

Consider first the economy from period $t \geq 1$. In that case there is no uncertainty and one equilibrium is to set:

\[ b_t = b_0 \quad \forall t \geq 1 \]

\[ c_t = y + rb_0 \]

\[ \lambda = \frac{1}{y + rb_0} \]

Now consider the economy in period $t = 0$:

\[ y_0 + (1 + r)b_{-1} = c_0 + b_0 + \frac{\phi}{2}(b_0 - b_{-1})^2 \]

and use the Euler equation:

\[ \frac{1}{c_0}(1 + \phi(b_0 - b_{-1})) = \frac{1}{c_1}(1 + r + \phi(b_1 - b_0)) \]

We already know that $b_1 = b_0$ so this expression becomes:

\[ \frac{1}{c_0}(1 + \phi(b_0 - b_{-1})) = \frac{1}{c_1} \]

where we used $\beta(1 + r) = 1$ now use $c_1 = y + rb_0$ and rearrange:

\[ c_0 = (1 + \phi(b_0 - b_{-1}))(y + rb_0) \]

Now use the sequential budget constraint for $t = 0$:

\[ c_0 + b_0 + \frac{\phi}{2}(b_0 - b_{-1})^2 = y_0 + (1 + r)b_{-1} \]

Now for $b_t \to b_{-1}$, notice that $\lim_{t \to \infty} b_t = b_0$. Thus let’s try $b_0 = b_{-1}$. Then the above expression becomes:

\[ c_0 + b_{-1} = y_0 + (1 + r)b_{-1} \]

and replace $c_0$ with the expression from the Euler equation

\[ c_0 = (1 + \phi(b_0 - b_{-1}))(y + rb_0) = y + rb_{-1} \]

to get:

\[ y + rb_{-1} + b_{-1} = y_0 + (1 + r)b_{-1} \]

This holds only if $y_0 = y$, which can never be the case. Thus the model is not stationary.

--- title: "Exercise 4.2" subtitle: "Variation of the PAC Model" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem This exercise aims to establish whether formulating portfolio adjustment costs as a function of the deviation of the household’s debt position from an exogenous reference point, $d_t - \bar{d}$, or as a function of the change in its debt position, $d_t - d_{t-1}$, has consequences for the stationarity of the model. Consider a small open economy populated by a large number of infinitely-lived households with preferences described by the utility function $$ E_0 \sum_{t=0}^\infty \beta^t \ln c_t, $$ where $\beta \in (0,1)$ denotes the subjective discount factor, and $c_t$ denotes consumption in period $t$. Each period, households receive an exogenous and stochastic endowment, $y_t$, and can borrow from (or lend to) international financial markets at the gross interest rate $1 + r$. Let $d_t$ denote the stock of foreign debt held by households at the end of period $t$. Households are subject to a portfolio adjustment cost of the form $\frac{\phi}{2} (d_t - d_{t-1})^2$, where $\phi$ is a positive constant. Assume that $\beta(1 + r) = 1$. 1. State the household’s period-by-period budget constraint. 2. State the household’s utility maximization problem. 3. Write the Lagrangian of the household’s problem. 4. Define a competitive equilibrium of this economy. 5. Suppose the endowment is nonstochastic and constant, $y_t = y$, for all $t$. Characterize the deterministic steady state. Does it exist? Is it unique? Hint: Consult appendix 4.14.2. 6. Consider now a temporary endowment shock. Suppose $y_0 > y$ and $y_t = y$ for all $t > 0$ deterministically. Suppose that prior to period 0 the economy was in a deterministic steady state with $d_{-1} = d^*$. Is the economy stationary, that is, is $d_t$ expected to return to $d^*$? Provide intuition. ## Answer #### 1. $$ y_t + (1 + r) b_{t-1} = c_t + b_t + \frac{\phi}{2}(b_t - b_{t-1})^2 $$ #### 2. $$ \max_{c_t, b_t} E_0 \sum_{t=0}^\infty \ln c_t $$ subject to $$ y_t + (1 + r)b_{t-1} = c_t + b_t + \frac{\phi}{2}(b_t - b_{t-1})^2 $$ and some no-Ponzi game constraint taking as given the initial condition $b_{-1}$, the exogenous process $\{y_t\}$, the interest rate $r$. #### 3. $$ \mathcal{L} = E_0 \sum_{t=0}^\infty \beta^t \left\{ \ln c_t + \lambda_t \left[ y_t + (1 + r)b_{t-1} - c_t - b_t - \frac{\phi}{2}(b_t - b_{t-1})^2 \right] \right\} $$ where $\lambda_t$ denote the Lagrange multiplier on the household’s period-by-period budget constraint. #### 4. A competitive equilibrium is a set of contingent plans $\{c_t, b_t, \lambda_t\}$ satisfying: $$ \frac{1}{c_t} = \lambda_t $$ $$ \lambda_t (1 + \phi(b_t - b_{t-1})) = \beta E_t \lambda_{t+1}(1 + r + \phi(b_{t+1} - b_t)) $$ $$ y_t + (1 + r)b_{t-1} = c_t + b_t + \frac{\phi}{2}(b_t - b_{t-1})^2 $$ and some no Ponzi game constraint, for given exogenous processes, $y_t$, and given initial conditions $b_{-1}$. #### 5. In the deterministic steady state all endogenous variables are by definition forever constant. Thus a deterministic steady state is a triple $\{c, \lambda, b\}$ that solves: $$ \frac{1}{c} = \lambda $$ $$ \lambda = \lambda \beta(1 + r) $$ $$ y + rb = c $$ Notice that the Euler equation does not impose a restriction, it just says $1 = 1$. Thus there exist a continuum of steady state equilibria. In fact any $b$ such that $c = y + rb > 0$ is a deterministic steady state. We do not need to check the no Ponzi restriction. It is satisfied because $b$ is constant and because $r > 0$. #### 6. Consider first the economy from period $t \geq 1$. In that case there is no uncertainty and one equilibrium is to set: $$ b_t = b_0 \quad \forall t \geq 1 $$ $$ c_t = y + rb_0 $$ $$ \lambda = \frac{1}{y + rb_0} $$ Now consider the economy in period $t = 0$: $$ y_0 + (1 + r)b_{-1} = c_0 + b_0 + \frac{\phi}{2}(b_0 - b_{-1})^2 $$ and use the Euler equation: $$ \frac{1}{c_0}(1 + \phi(b_0 - b_{-1})) = \frac{1}{c_1}(1 + r + \phi(b_1 - b_0)) $$ We already know that $b_1 = b_0$ so this expression becomes: $$ \frac{1}{c_0}(1 + \phi(b_0 - b_{-1})) = \frac{1}{c_1} $$ where we used $\beta(1 + r) = 1$ now use $c_1 = y + rb_0$ and rearrange: $$ c_0 = (1 + \phi(b_0 - b_{-1}))(y + rb_0) $$ Now use the sequential budget constraint for $t = 0$: $$ c_0 + b_0 + \frac{\phi}{2}(b_0 - b_{-1})^2 = y_0 + (1 + r)b_{-1} $$ Now for $b_t \to b_{-1}$, notice that $\lim_{t \to \infty} b_t = b_0$. Thus let’s try $b_0 = b_{-1}$. Then the above expression becomes: $$ c_0 + b_{-1} = y_0 + (1 + r)b_{-1} $$ and replace $c_0$ with the expression from the Euler equation $$ c_0 = (1 + \phi(b_0 - b_{-1}))(y + rb_0) = y + rb_{-1} $$ to get: $$ y + rb_{-1} + b_{-1} = y_0 + (1 + r)b_{-1} $$ This holds only if $y_0 = y$, which can never be the case. Thus the model is not stationary.