Exercise 2.15

An Economy With Labor and Technological Progress

Problem

Consider a small open economy inhabited by a large number of identical households with preferences described by the utility function

\[ \sum_{t=0}^{\infty} \beta^t \ln \left( C_t - \frac{X_t}{2} h_t^2 \right), \]

where $C_t$ denotes consumption in period $t$, $h_t$ denotes hours worked in period $t$, $\beta \in (0, 1)$ is a subjective discount factor, and $X_t$ is an exogenous and deterministic factor governing household and market technological progress, which evolves according to the expression

\[ X_t = \mu X_{t-1}, \]

for all $t \geq 0$, where $\mu > 1$ is a parameter, and $X_{-1} = 1$. Households face the sequential budget constraint

\[ C_t + D_{t-1} (1 + r_{t-1}) = Y_t + D_t, \]

where $D_t$ denotes the amount of one-period debt assumed in period $t$, $r_t$ denotes the interest rate in period $t$, and $Y_t$ denotes output in period $t$, which is produced via the technology

\[ Y_t = X_t h_t. \]

Households are subject to the no-Ponzi-game constraint $\lim_{j \to \infty} \frac{D_{t+j}}{\prod_{s=0}^{j-1} (1 + r_{t+s})} \leq 0, \quad \text{for all } t.$

The country interest rate, $r_t$, is the sum of a constant world interest rate, $r^*$, and a country spread, denoted $\rho \left( \hat{D}_t / X_t \right)$, where $\hat{D}_t$ denotes the cross-sectional average of $D_t$.
Assume the functional form $\rho(x) = \gamma \left[ \exp(x - \bar{d}) - 1 \right]$, where $\gamma, \bar{d} > 0$ are parameters.

Finally, assume that

\[ 1 + r^* = \frac{\mu}{\beta} \quad \text{and that} \quad (1 + r_{t-1}) D_{-1} / X_{-1} = (1 + r^*) \bar{d}. \]

State the household’s maximization problem.
Derive the optimality conditions associated with the household’s problem.
Derive the complete set of equilibrium conditions in stationary form by appropriately scaling variables that display perpetual growth in equilibrium.
Derive equilibrium paths of consumption, hours, the trade balance, the current account, and external debt. Provide intuition.

Answer

1.

The household’s problem is

\[ \max_{\{C_t, h_t\}} \sum_{t=0}^{\infty} \beta^t \ln\left( C_t - \frac{X_t}{2} h_t^2 \right), \]

subject to

\[ C_t + D_{t-1}(1 + r_{t-1}) = X_t h_t + D_t, \]

\[ \lim_{j \to \infty} \frac{D_{t+j}}{\prod_{s=0}^{j}(1 + r_{t+s})} \leq 0 \]

2.

The associated first-order conditions are

\[ h_t = 1 \]

\[ \left( C_{t+1} - \frac{X_{t+1}}{2} \right) = \beta (1 + r_t) \left( C_t - \frac{X_t}{2} \right) \]

3.

The complete set of equilibrium conditions in stationary form are

\[ c_{t+1} - 1/2 = \left[ 1 + \gamma \frac{e^{d_t - \bar{d}} - 1}{1 + r^*} \right] (c_t - 1/2) \]

\[ c_t + d_{t-1} \left( \frac{1 + r_{t-1}}{\mu} \right) = 1 + d_t \]

4.

Since $d_{-1}(1 + r_{-1}) = \bar{d}(1 + r^*)$, the equilibrium (i.e., a solution to the above system) is

\[ d_t = \bar{d} \]

\[ c_t = 1 + \bar{d}(1 - 1/\beta) < 1 \]

\[ r_t = r^* \]

\[ tb_t = 1 - c_t > 0 \]

\[ ca_t = \bar{d}(1/\mu - 1) < 0 \]

for all $t \geq 0$. So $C_t$, $D_t$, $TB_t$, and $CA_t$ are proportional to $X_t$, which means that they all grow at the gross rate $\mu$ which is less than $1 + r^*$. The country is a debtor, runs a perpetual trade balance surplus and a perpetual current account deficit (tb surplus not enough to pay for all interest on the debt but enough to prevent debt growing at rate $r^*$). Absent growth, debtor country would not be able to run a perpetual current account deficit.

--- title: "Exercise 2.15" subtitle: "An Economy With Labor and Technological Progress" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider a small open economy inhabited by a large number of identical households with preferences described by the utility function $$ \sum_{t=0}^{\infty} \beta^t \ln \left( C_t - \frac{X_t}{2} h_t^2 \right), $$ where $C_t$ denotes consumption in period $t$, $h_t$ denotes hours worked in period $t$, $\beta \in (0, 1)$ is a subjective discount factor, and $X_t$ is an exogenous and deterministic factor governing household and market technological progress, which evolves according to the expression $$ X_t = \mu X_{t-1}, $$ for all $t \geq 0$, where $\mu > 1$ is a parameter, and $X_{-1} = 1$. Households face the sequential budget constraint $$ C_t + D_{t-1} (1 + r_{t-1}) = Y_t + D_t, $$ where $D_t$ denotes the amount of one-period debt assumed in period $t$, $r_t$ denotes the interest rate in period $t$, and $Y_t$ denotes output in period $t$, which is produced via the technology $$ Y_t = X_t h_t. $$ Households are subject to the no-Ponzi-game constraint $\lim_{j \to \infty} \frac{D_{t+j}}{\prod_{s=0}^{j-1} (1 + r_{t+s})} \leq 0, \quad \text{for all } t.$ The country interest rate, $r_t$, is the sum of a constant world interest rate, $r^*$, and a country spread, denoted $\rho \left( \hat{D}_t / X_t \right)$, where $\hat{D}_t$ denotes the cross-sectional average of $D_t$. Assume the functional form $\rho(x) = \gamma \left[ \exp(x - \bar{d}) - 1 \right]$, where $\gamma, \bar{d} > 0$ are parameters. Finally, assume that $$ 1 + r^* = \frac{\mu}{\beta} \quad \text{and that} \quad (1 + r_{t-1}) D_{-1} / X_{-1} = (1 + r^*) \bar{d}. $$ 1. State the household’s maximization problem. 2. Derive the optimality conditions associated with the household’s problem. 3. Derive the complete set of equilibrium conditions in stationary form by appropriately scaling variables that display perpetual growth in equilibrium. 4. Derive equilibrium paths of consumption, hours, the trade balance, the current account, and external debt. Provide intuition. ## Answer #### 1. The household’s problem is $$ \max_{\{C_t, h_t\}} \sum_{t=0}^{\infty} \beta^t \ln\left( C_t - \frac{X_t}{2} h_t^2 \right), $$ subject to $$ C_t + D_{t-1}(1 + r_{t-1}) = X_t h_t + D_t, $$ $$ \lim_{j \to \infty} \frac{D_{t+j}}{\prod_{s=0}^{j}(1 + r_{t+s})} \leq 0 $$ #### 2. The associated first-order conditions are $$ h_t = 1 $$ $$ \left( C_{t+1} - \frac{X_{t+1}}{2} \right) = \beta (1 + r_t) \left( C_t - \frac{X_t}{2} \right) $$ #### 3. The complete set of equilibrium conditions in stationary form are $$ c_{t+1} - 1/2 = \left[ 1 + \gamma \frac{e^{d_t - \bar{d}} - 1}{1 + r^*} \right] (c_t - 1/2) $$ $$ c_t + d_{t-1} \left( \frac{1 + r_{t-1}}{\mu} \right) = 1 + d_t $$ #### 4. Since $d_{-1}(1 + r_{-1}) = \bar{d}(1 + r^*)$, the equilibrium (i.e., a solution to the above system) is $$ d_t = \bar{d} $$ $$ c_t = 1 + \bar{d}(1 - 1/\beta) < 1 $$ $$ r_t = r^* $$ $$ tb_t = 1 - c_t > 0 $$ $$ ca_t = \bar{d}(1/\mu - 1) < 0 $$ for all $t \geq 0$. So $C_t$, $D_t$, $TB_t$, and $CA_t$ are proportional to $X_t$, which means that they all grow at the gross rate $\mu$ which is less than $1 + r^*$. The country is a debtor, runs a perpetual trade balance surplus and a perpetual current account deficit (tb surplus not enough to pay for all interest on the debt but enough to prevent debt growing at rate $r^*$). Absent growth, debtor country would not be able to run a perpetual current account deficit.