Exercise 4.12

GHH Preferences and No Capital

Problem

Consider a small open economy populated by an infinite number of identical households with preferences of the form

\[ (1-\sigma)^{-1}\sum_{t=0}^{\infty} \beta^t \left(c_t-\frac{h_t^{\omega}}{\omega} \right)^{1-\sigma}, \]

where $c_t$ denotes consumption of a perishable good in period $t$, $h_t$ denotes labor effort in period $t$, and $\beta\in(0,1)$, $\sigma>1$, and $\omega>1$ are parameters. Each household operates a technology that produces consumption goods according to the relationship

\[ y_t = h_t^{\alpha}, \]

where $y_t$ denotes output, and $\alpha\in(0,1)$ is a parameter. The household can borrow or lend in international financial markets at the interest rate $r_t = r^*+p(\tilde{d}_t)$, where $r^*$ denotes the world interest rate and satisfies $\beta(1+r^*)=1$. The function $p(\tilde{d}_t)$ is a country interest-rate premium in period $t$, satisfying $p(0)=0$, and $p(x)\neq0$ for $x\neq0$, where $\tilde{d}_t$ denotes the cross-sectional average debt holdings in period $t$ and is taken as given by the individual household. Let $d_t$ denote the household’s debt holdings in period $t$ maturing in $t+1$. Households cannot play Ponzi games.

Write down the household’s optimization problem.
Derive the first-order conditions associated with the household’s optimization problem.
Display the complete set of equilibrium conditions.
Derive the steady state of the economy. In particular, compute the steady-state values of consumption, hours, output, the trade balance, the current account, and external debt, denoted, respectively, $c$, $h$, $y$, $tb$, $ca$, and $d$.
Derive analytically a first-order approximation to the equilibrium conditions. Express it as a first-order difference equation in the vector $[\hat{d}_{t-1} \,\, \hat{c}_t]'$, where $\hat{d}_{t-1}\equiv d_{t-1}-d$, and $\hat{c}_t\equiv \ln(c_t/c)$.
Derive conditions under which the perfect-foresight equilibrium is locally unique.

Answer

The household chooses $\{c_t, h_t, d_t\}_{t=0}^\infty$ to maximize \[ \sum_{t=0}^{\infty} \beta^t \frac{\left(c_t - \frac{h_t^{\omega}}{\omega} \right)^{1 - \sigma}}{1 - \sigma} \] subject to the sequence of budget constraints: \[ c_t + d_t = h_t^{\alpha} + (1 + r_{t-1}) d_{t-1}, \] taking $r_t$ and $d_{-1}$ as given and satisfying the no-Ponzi condition.
The first-order conditions are:

With respect to $c_t$: \[ \left(c_t - \frac{h_t^{\omega}}{\omega}\right)^{-\sigma} = \lambda_t \]
With respect to $h_t$: \[ \left(c_t - \frac{h_t^{\omega}}{\omega}\right)^{-\sigma} h_t^{\omega - 1} = \lambda_t \alpha h_t^{\alpha - 1} \]
With respect to $d_t$: \[ \lambda_t = \beta (1 + r_t) \lambda_{t+1} \]

The complete set of equilibrium conditions includes:

Intra-temporal optimality: \[ \left(c_t - \frac{h_t^{\omega}}{\omega}\right)^{-\sigma} h_t^{\omega - 1} = \lambda_t \alpha h_t^{\alpha - 1} \]
Inter-temporal Euler equation: \[ \lambda_t = \beta (1 + r_t) \lambda_{t+1} \]
Budget constraint: \[ c_t + d_t = h_t^{\alpha} + (1 + r_{t-1}) d_{t-1} \]
Interest rate function: \[ r_t = r^* + p(d_t) \]
No-Ponzi condition: \[ \lim_{T \to \infty} \frac{d_T}{\prod_{s=0}^{T-1} (1 + r_s)} = 0 \]

In steady state, $r = r^*$, $d = 0$, and $h$ satisfies: \[ h = \bar{h} = \alpha^{\frac{1}{\omega - \alpha}} \] Then, \[ c = y = \bar{h}^{\alpha}, \quad tb = ca = d = 0 \]
Let $\hat{d}_{t-1} = d_{t-1}$ and $\hat{c}_t = \ln(c_t / c)$. Then the linearized system becomes: \[ \hat{d}_t = \beta^{-1} \hat{d}_{t-1} + c \hat{c}_t \] \[ \hat{c}_{t+1} = \frac{p'(0)}{\sigma } \cdot \frac{\omega - \alpha}{\omega}\hat{d}_{t-1} +\left(1+ c \cdot \frac{\beta p'(0)}{\sigma}\cdot\frac{\omega - \alpha}{\omega}\right) \hat{c}_t \] which can be written in matrix form: \[ \begin{bmatrix} \hat{d}_{t} \\ \hat{c}_{t+1} \end{bmatrix} = \begin{bmatrix} \beta^{-1} & c \\ \frac{p'(0)}{\sigma } \cdot \frac{\omega - \alpha}{\omega} & 1+ c\beta \cdot \frac{p'(0)}{\sigma} \cdot \frac{\omega - \alpha}{\omega} \end{bmatrix} \begin{bmatrix} \hat{d}_{t-1} \\ \hat{c}_t \end{bmatrix} \]
For local uniqueness, the system must have one eigenvalue inside and one outside the unit circle. This requires:

$p'(0) > 0$
$\det(M) = \beta^{-1} > 1$
$\operatorname{tr}(M) = \beta^{-1} + 1 + c\beta X > 2$ where $X = \frac{p'(0)}{\sigma} \cdot \frac{\omega - \alpha}{\omega} > 0$

This ensures a saddle-path stable equilibrium with a unique perfect foresight path.

--- title: "Exercise 4.12" subtitle: "GHH Preferences and No Capital" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider a small open economy populated by an infinite number of identical households with preferences of the form $$ (1-\sigma)^{-1}\sum_{t=0}^{\infty} \beta^t \left(c_t-\frac{h_t^{\omega}}{\omega} \right)^{1-\sigma}, $$ where $c_t$ denotes consumption of a perishable good in period $t$, $h_t$ denotes labor effort in period $t$, and $\beta\in(0,1)$, $\sigma>1$, and $\omega>1$ are parameters. Each household operates a technology that produces consumption goods according to the relationship $$ y_t = h_t^{\alpha}, $$ where $y_t$ denotes output, and $\alpha\in(0,1)$ is a parameter. The household can borrow or lend in international financial markets at the interest rate $r_t = r^*+p(\tilde{d}_t)$, where $r^*$ denotes the world interest rate and satisfies $\beta(1+r^*)=1$. The function $p(\tilde{d}_t)$ is a country interest-rate premium in period $t$, satisfying $p(0)=0$, and $p(x)\neq0$ for $x\neq0$, where $\tilde{d}_t$ denotes the cross-sectional average debt holdings in period $t$ and is taken as given by the individual household. Let $d_t$ denote the household's debt holdings in period $t$ maturing in $t+1$. Households cannot play Ponzi games. 1. Write down the household's optimization problem. 2. Derive the first-order conditions associated with the household's optimization problem. 3. Display the complete set of equilibrium conditions. 4. Derive the steady state of the economy. In particular, compute the steady-state values of consumption, hours, output, the trade balance, the current account, and external debt, denoted, respectively, $c$, $h$, $y$, $tb$, $ca$, and $d$. 5. Derive analytically a first-order approximation to the equilibrium conditions. Express it as a first-order difference equation in the vector $[\hat{d}_{t-1} \,\, \hat{c}_t]'$, where $\hat{d}_{t-1}\equiv d_{t-1}-d$, and $\hat{c}_t\equiv \ln(c_t/c)$. 6. Derive conditions under which the perfect-foresight equilibrium is locally unique. ## Answer 1. The household chooses $\{c_t, h_t, d_t\}_{t=0}^\infty$ to maximize $$ \sum_{t=0}^{\infty} \beta^t \frac{\left(c_t - \frac{h_t^{\omega}}{\omega} \right)^{1 - \sigma}}{1 - \sigma} $$ subject to the sequence of budget constraints: $$ c_t + d_t = h_t^{\alpha} + (1 + r_{t-1}) d_{t-1}, $$ taking $r_t$ and $d_{-1}$ as given and satisfying the no-Ponzi condition. 2. The first-order conditions are: - With respect to $c_t$: $$ \left(c_t - \frac{h_t^{\omega}}{\omega}\right)^{-\sigma} = \lambda_t $$ - With respect to $h_t$: $$ \left(c_t - \frac{h_t^{\omega}}{\omega}\right)^{-\sigma} h_t^{\omega - 1} = \lambda_t \alpha h_t^{\alpha - 1} $$ - With respect to $d_t$: $$ \lambda_t = \beta (1 + r_t) \lambda_{t+1} $$ 3. The complete set of equilibrium conditions includes: - Intra-temporal optimality: $$ \left(c_t - \frac{h_t^{\omega}}{\omega}\right)^{-\sigma} h_t^{\omega - 1} = \lambda_t \alpha h_t^{\alpha - 1} $$ - Inter-temporal Euler equation: $$ \lambda_t = \beta (1 + r_t) \lambda_{t+1} $$ - Budget constraint: $$ c_t + d_t = h_t^{\alpha} + (1 + r_{t-1}) d_{t-1} $$ - Interest rate function: $$ r_t = r^* + p(d_t) $$ - No-Ponzi condition: $$ \lim_{T \to \infty} \frac{d_T}{\prod_{s=0}^{T-1} (1 + r_s)} = 0 $$ 4. In steady state, $r = r^*$, $d = 0$, and $h$ satisfies: $$ h = \bar{h} = \alpha^{\frac{1}{\omega - \alpha}} $$ Then, $$ c = y = \bar{h}^{\alpha}, \quad tb = ca = d = 0 $$ 5. Let $\hat{d}_{t-1} = d_{t-1}$ and $\hat{c}_t = \ln(c_t / c)$. Then the linearized system becomes: $$ \hat{d}_t = \beta^{-1} \hat{d}_{t-1} + c \hat{c}_t $$ $$ \hat{c}_{t+1} = \frac{p'(0)}{\sigma } \cdot \frac{\omega - \alpha}{\omega}\hat{d}_{t-1} +\left(1+ c \cdot \frac{\beta p'(0)}{\sigma}\cdot\frac{\omega - \alpha}{\omega}\right) \hat{c}_t $$ which can be written in matrix form: $$ \begin{bmatrix} \hat{d}_{t} \\ \hat{c}_{t+1} \end{bmatrix} = \begin{bmatrix} \beta^{-1} & c \\ \frac{p'(0)}{\sigma } \cdot \frac{\omega - \alpha}{\omega} & 1+ c\beta \cdot \frac{p'(0)}{\sigma} \cdot \frac{\omega - \alpha}{\omega} \end{bmatrix} \begin{bmatrix} \hat{d}_{t-1} \\ \hat{c}_t \end{bmatrix} $$ 6. For local uniqueness, the system must have one eigenvalue inside and one outside the unit circle. This requires: - $p'(0) > 0$ - $\det(M) = \beta^{-1} > 1$ - $\operatorname{tr}(M) = \beta^{-1} + 1 + c\beta X > 2$ where $X = \frac{p'(0)}{\sigma} \cdot \frac{\omega - \alpha}{\omega} > 0$ This ensures a saddle-path stable equilibrium with a unique perfect foresight path.