Exercise 4.9

Complete Markets and The Countercyclicality of the Trade Balance

Problem

Consider a small open economy with access to a complete array of internationally traded state-contingent claims. There is a single good, which is freely traded internationally. Let $q_{t,t+1}$ denote the period-$t$ price of a contingent claim that pays one good in a particular state of the world in period $t+1$ divided by the probability of occurrence of that state. The small open economy takes the process for $q_{t,t+1}$ as exogenously given.

Households have preferences over consumption $c_t$ and hours $h_t$ given by:

\[ E_0 \sum_{t=0}^{\infty} \beta^t \left[ \frac{\left(c_t - \frac{h_t^{\omega}}{\omega}\right)^{1-\sigma} -1}{1-\sigma} \right]; \quad \sigma, \omega > 1, \]

where $E_0$ denotes the expectations operator conditional on information available in period 0. Households produce goods according to the following production technology:

\[ A_t k_t^{\alpha} h_t^{1 - \alpha}, \]

where $A_t$ denotes an exogenous productivity factor, $k_t$ denotes the capital stock in period $t$, and $\alpha \in (0,1)$ denotes the elasticity of output with respect to capital.

Capital evolves according to:

\[ k_{t+1} = (1 - \delta)k_t + i_t, \]

where $i_t$ is investment in physical capital in period $t$, and $\delta \in (0,1)$ is the depreciation rate. In period 0, households are endowed with $k_0$ units of capital and hold contingent claims (acquired in period $-1$) that pay $d_0$ goods in period 0.

State the household’s period-by-period budget constraint.
Specify a borrowing limit that prevents households from engaging in Ponzi schemes.
State the household’s utility maximization problem. Indicate which variables/processes the household chooses and which it takes as given.
Derive the complete set of competitive equilibrium conditions.
Assume that in the nonstochastic steady state, $q_{0,t} = \beta^t$ and $A_t = 1$. Show that in response to a positive innovation in $A_t$, i.e., $\widehat{A}_t > 0$, the trade balance responds countercyclically only if investment rises. Then derive the minimum percentage increase in investment needed for the trade balance to fall. Show your answer is independent of $E_t A_{t+1}$.
Compare your findings in question 5 to those from Chapter 3’s model with no depreciation, no labor choice, and incomplete markets. Focus on the role of persistence in $A_t$. Provide intuition.
Find the threshold for $E_t \widehat{A}_{t+1}$ relative to $\widehat{A}_t$ that guarantees a deterioration of the trade balance. Express the condition as $\widehat{A}_t < M E_t \widehat{A}_{t+1}$ and compute $M$ using $\alpha = 1/3$, $\delta = 0.08$, $\beta^{-1} = 1.02$, and $\omega = 1.5$.
Discuss how your findings support or contradict Principle I derived in Chapter 3.
How would your answers to questions 5 and 7 change if utility were separable in $c_t$ and $h_t$?

Answer

1.

The household’s period-by-period budget constraint is:

\[ A_t k_t^{\alpha} h_t^{1 - \alpha} + d_t = c_t + k_{t+1} - (1 - \delta)k_t + E_t q_{t,t+1} d_{t+1} \]

2.

A borrowing limit that prevents Ponzi schemes is:

\[ \lim_{t \to \infty} E_0 \left[ q_{0,t} d_t \right] \ge 0 \]

where

\[ q_{0,t} \equiv q_{0,1} q_{1,2} \cdots q_{t-1,t} \]

3.

The household chooses $\{c_t, h_t, k_{t+1}, d_{t+1}\}$ to maximize:

\[ E_0 \sum_{t=0}^{\infty} \beta^t \left[ \frac{\left(c_t - \frac{h_t^{\omega}}{\omega}\right)^{1 - \sigma} - 1}{1 - \sigma} \right] \]

subject to:

\[ A_t k_t^{\alpha} h_t^{1 - \alpha} + d_t = c_t + k_{t+1} - (1 - \delta)k_t + E_t q_{t,t+1} d_{t+1} \]

\[ \lim_{t \to \infty} E_0 q_{0,t} d_t \ge 0 \]

taking as given the processes $\{A_t\}$, $\{q_{t,t+1}\}$, and initial conditions $k_0$, $d_0$.

4.

A competitive equilibrium is a set of plans $\{c_t, h_t, k_{t+1}\}$ and a constant $\xi > 0$ satisfying:

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

\[ h_t^{\omega - 1} = (1 - \alpha) A_t k_t^{\alpha} h_t^{-\alpha} \]

\[ 1 = E_t \left[ q_{t,t+1} \left( \alpha A_{t+1} k_{t+1}^{\alpha - 1} h_{t+1}^{1 - \alpha} + 1 - \delta \right) \right] \]

\[ E_0 \sum_{t = 0}^{\infty} q_{0,t} \left[ A_t k_t^{\alpha} h_t^{1 - \alpha} + (1 - \delta)k_t - c_t - k_{t+1} \right] + d_0 = 0 \]

5.

From the FOC:

\[ \widehat{c}_t = \frac{h^\omega}{c} \widehat{h}_t \]

Labor condition:

\[ (\omega - 1)\widehat{h}_t = \widehat{A}_t - \alpha \widehat{h}_t \Rightarrow \widehat{h}_t = \frac{1}{\omega - (1 - \alpha)} \widehat{A}_t \]

Then:

\[ \widehat{c}_t = \frac{h^\omega}{c} \cdot \frac{1}{\omega - (1 - \alpha)} \widehat{A}_t \]

Output:

\[ \widehat{y}_t = \widehat{A}_t + (1 - \alpha) \widehat{h}_t = \frac{\omega}{\omega - (1 - \alpha)} \widehat{A}_t \]

Using:

\[ y_t = c_t + i_t + tb_t \]

Linearized:

\[ \widehat{y}_t = \frac{c}{y} \widehat{c}_t + \frac{\delta k}{y} \widehat{i}_t + \frac{1}{y}(tb_t - tb) \]

So:

\[ (tb_t - tb) = y \widehat{A}_t - \delta k \widehat{i}_t \]

The trade balance deteriorates if:

\[ \widehat{i}_t > \frac{y}{\delta k} \widehat{A}_t \]

6.

In the current model, productivity shocks create income effects, leading to consumption booms and trade deficits.
In this complete markets model, there’s no income effect — only substitution between consumption and leisure.
Intertemporal prices $q_{t,t+1}$ are fixed and not influenced by $E_t A_{t+1}$, unlike in incomplete markets.
Trade balance dynamics now depend more heavily on expectations of future productivity and capital accumulation.

7.

We begin with the expression for the trade balance response: \[ \frac{tb_t - tb}{y} = \widehat{A}_t - \frac{\delta k}{y} \widehat{i}_t \]

Eliminate $\widehat{i}_t$ using the law of motion for capital: \[ \widehat{i}_t = \frac{1}{\delta} \widehat{k}_{t+1} \]

Substitute the expression for $\widehat{k}_{t+1}$ derived earlier, and replace $k/y$ using the steady-state Euler equation: \[ \frac{k}{y} = \frac{\alpha}{\beta^{-1} - 1 + \delta} \]

Combining terms yields: \[ \frac{tb_t - tb}{y} = \widehat{A}_t - \left( \frac{\omega}{\omega - 1} \cdot \frac{1}{1 - \alpha} \cdot \frac{\alpha}{\beta^{-1} - 1 + \delta} \right) E_t \widehat{A}_{t+1} \]

Hence, the trade balance deteriorates if: \[ \widehat{A}_t < M E_t \widehat{A}_{t+1} \] where \[ M = \left( \frac{\omega}{\omega - 1} \right) \left( \frac{1}{1 - \alpha} \right) \left( \frac{\alpha}{\beta^{-1} - 1 + \delta} \right) \]

Plug in:

$\omega = 1.5$
$\alpha = 1/3$
$\delta = 0.08$
$\beta^{-1} = 1.02$

Then:

\[ M = \frac{3}{1} \cdot \frac{3}{2} \cdot \frac{1}{3} \cdot 10 = 15 \]

So the condition becomes:

\[ E_t \widehat{A}_{t+1} > \frac{1}{15} \widehat{A}_t \]

8.

Principle I states that the more persistent a productivity shock is, the more likely it is that the trade balance deteriorates on impact. Our findings support this principle: the greater the ratio $E_t \widehat{A}_{t+1} / \widehat{A}_t$, the more likely it is that the condition $\widehat{A}_t < M E_t \widehat{A}_{t+1}$ holds, leading to a trade deficit. Moreover, the size of the trade deficit increases with the persistence of the shock.

9.

If the period utility function is separable in consumption and hours, the answers to both Questions 5 and 7 would change. In this case, the marginal utility of consumption would depend only on consumption, so when the price of contingent claims remains constant (i.e., $q_{0,t} = \beta^t$), consumption does not respond to a temporary productivity shock. There would be neither a substitution effect nor an income effect on consumption, so consumption would remain constant. Labor supply, and hence output, would still increase following the shock, but the resulting increase in output will have to be absorbed entirely by either higher investment or a trade surplus (given the lack of an increase in consumption). Whether the trade balance deteriorates on impact will still depend on the strength of the investment response. However, since consumption no longer increases, a larger increase in investment is required to generate a trade deficit. This implies that a more persistent productivity process (i.e., a higher $E_t \widehat{A}_{t+1} / \widehat{A}_t$) is needed for the economy to run a trade deficit.

--- title: "Exercise 4.9" subtitle: "Complete Markets and The Countercyclicality of the Trade Balance" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider a small open economy with access to a complete array of internationally traded state-contingent claims. There is a single good, which is freely traded internationally. Let $q_{t,t+1}$ denote the period-$t$ price of a contingent claim that pays one good in a particular state of the world in period $t+1$ divided by the probability of occurrence of that state. The small open economy takes the process for $q_{t,t+1}$ as exogenously given. Households have preferences over consumption $c_t$ and hours $h_t$ given by: $$ E_0 \sum_{t=0}^{\infty} \beta^t \left[ \frac{\left(c_t - \frac{h_t^{\omega}}{\omega}\right)^{1-\sigma} -1}{1-\sigma} \right]; \quad \sigma, \omega > 1, $$ where $E_0$ denotes the expectations operator conditional on information available in period 0. Households produce goods according to the following production technology: $$ A_t k_t^{\alpha} h_t^{1 - \alpha}, $$ where $A_t$ denotes an exogenous productivity factor, $k_t$ denotes the capital stock in period $t$, and $\alpha \in (0,1)$ denotes the elasticity of output with respect to capital. Capital evolves according to: $$ k_{t+1} = (1 - \delta)k_t + i_t, $$ where $i_t$ is investment in physical capital in period $t$, and $\delta \in (0,1)$ is the depreciation rate. In period 0, households are endowed with $k_0$ units of capital and hold contingent claims (acquired in period $-1$) that pay $d_0$ goods in period 0. 1. State the household's period-by-period budget constraint. 2. Specify a borrowing limit that prevents households from engaging in Ponzi schemes. 3. State the household's utility maximization problem. Indicate which variables/processes the household chooses and which it takes as given. 4. Derive the complete set of competitive equilibrium conditions. 5. Assume that in the nonstochastic steady state, $q_{0,t} = \beta^t$ and $A_t = 1$. Show that in response to a positive innovation in $A_t$, i.e., $\widehat{A}_t > 0$, the trade balance responds countercyclically only if investment rises. Then derive the minimum percentage increase in investment needed for the trade balance to fall. Show your answer is independent of $E_t A_{t+1}$. 6. Compare your findings in question 5 to those from Chapter 3's model with no depreciation, no labor choice, and incomplete markets. Focus on the role of persistence in $A_t$. Provide intuition. 7. Find the threshold for $E_t \widehat{A}_{t+1}$ relative to $\widehat{A}_t$ that guarantees a deterioration of the trade balance. Express the condition as $\widehat{A}_t < M E_t \widehat{A}_{t+1}$ and compute $M$ using $\alpha = 1/3$, $\delta = 0.08$, $\beta^{-1} = 1.02$, and $\omega = 1.5$. 8. Discuss how your findings support or contradict Principle I derived in Chapter 3. 9. How would your answers to questions 5 and 7 change if utility were separable in $c_t$ and $h_t$? ## Answer #### 1. The household’s period-by-period budget constraint is: $$ A_t k_t^{\alpha} h_t^{1 - \alpha} + d_t = c_t + k_{t+1} - (1 - \delta)k_t + E_t q_{t,t+1} d_{t+1} $$ #### 2. A borrowing limit that prevents Ponzi schemes is: $$ \lim_{t \to \infty} E_0 \left[ q_{0,t} d_t \right] \ge 0 $$ where $$ q_{0,t} \equiv q_{0,1} q_{1,2} \cdots q_{t-1,t} $$ #### 3. The household chooses $\{c_t, h_t, k_{t+1}, d_{t+1}\}$ to maximize: $$ E_0 \sum_{t=0}^{\infty} \beta^t \left[ \frac{\left(c_t - \frac{h_t^{\omega}}{\omega}\right)^{1 - \sigma} - 1}{1 - \sigma} \right] $$ subject to: $$ A_t k_t^{\alpha} h_t^{1 - \alpha} + d_t = c_t + k_{t+1} - (1 - \delta)k_t + E_t q_{t,t+1} d_{t+1} $$ $$ \lim_{t \to \infty} E_0 q_{0,t} d_t \ge 0 $$ taking as given the processes $\{A_t\}$, $\{q_{t,t+1}\}$, and initial conditions $k_0$, $d_0$. #### 4. A competitive equilibrium is a set of plans $\{c_t, h_t, k_{t+1}\}$ and a constant $\xi > 0$ satisfying: $$ \beta^t \left(c_t - \frac{h_t^{\omega}}{\omega} \right)^{-\sigma} = \xi q_{0,t} $$ $$ h_t^{\omega - 1} = (1 - \alpha) A_t k_t^{\alpha} h_t^{-\alpha} $$ $$ 1 = E_t \left[ q_{t,t+1} \left( \alpha A_{t+1} k_{t+1}^{\alpha - 1} h_{t+1}^{1 - \alpha} + 1 - \delta \right) \right] $$ $$ E_0 \sum_{t = 0}^{\infty} q_{0,t} \left[ A_t k_t^{\alpha} h_t^{1 - \alpha} + (1 - \delta)k_t - c_t - k_{t+1} \right] + d_0 = 0 $$ #### 5. From the FOC: $$ \widehat{c}_t = \frac{h^\omega}{c} \widehat{h}_t $$ Labor condition: $$ (\omega - 1)\widehat{h}_t = \widehat{A}_t - \alpha \widehat{h}_t \Rightarrow \widehat{h}_t = \frac{1}{\omega - (1 - \alpha)} \widehat{A}_t $$ Then: $$ \widehat{c}_t = \frac{h^\omega}{c} \cdot \frac{1}{\omega - (1 - \alpha)} \widehat{A}_t $$ Output: $$ \widehat{y}_t = \widehat{A}_t + (1 - \alpha) \widehat{h}_t = \frac{\omega}{\omega - (1 - \alpha)} \widehat{A}_t $$ Using: $$ y_t = c_t + i_t + tb_t $$ Linearized: $$ \widehat{y}_t = \frac{c}{y} \widehat{c}_t + \frac{\delta k}{y} \widehat{i}_t + \frac{1}{y}(tb_t - tb) $$ So: $$ (tb_t - tb) = y \widehat{A}_t - \delta k \widehat{i}_t $$ The trade balance deteriorates if: $$ \widehat{i}_t > \frac{y}{\delta k} \widehat{A}_t $$ #### 6. - In the current model, productivity shocks create **income effects**, leading to consumption booms and trade deficits. - In this complete markets model, there’s no income effect — only **substitution** between consumption and leisure. - Intertemporal prices $q_{t,t+1}$ are fixed and not influenced by $E_t A_{t+1}$, unlike in incomplete markets. - Trade balance dynamics now depend more heavily on **expectations of future productivity** and capital accumulation. #### 7. We begin with the expression for the trade balance response: $$ \frac{tb_t - tb}{y} = \widehat{A}_t - \frac{\delta k}{y} \widehat{i}_t $$ Eliminate $\widehat{i}_t$ using the law of motion for capital: $$ \widehat{i}_t = \frac{1}{\delta} \widehat{k}_{t+1} $$ Substitute the expression for $\widehat{k}_{t+1}$ derived earlier, and replace $k/y$ using the steady-state Euler equation: $$ \frac{k}{y} = \frac{\alpha}{\beta^{-1} - 1 + \delta} $$ Combining terms yields: $$ \frac{tb_t - tb}{y} = \widehat{A}_t - \left( \frac{\omega}{\omega - 1} \cdot \frac{1}{1 - \alpha} \cdot \frac{\alpha}{\beta^{-1} - 1 + \delta} \right) E_t \widehat{A}_{t+1} $$ Hence, the trade balance deteriorates if: $$ \widehat{A}_t < M E_t \widehat{A}_{t+1} $$ where $$ M = \left( \frac{\omega}{\omega - 1} \right) \left( \frac{1}{1 - \alpha} \right) \left( \frac{\alpha}{\beta^{-1} - 1 + \delta} \right) $$ Plug in: - $\omega = 1.5$ - $\alpha = 1/3$ - $\delta = 0.08$ - $\beta^{-1} = 1.02$ Then: $$ M = \frac{3}{1} \cdot \frac{3}{2} \cdot \frac{1}{3} \cdot 10 = 15 $$ So the condition becomes: $$ E_t \widehat{A}_{t+1} > \frac{1}{15} \widehat{A}_t $$ #### 8. Principle I states that the more persistent a productivity shock is, the more likely it is that the trade balance deteriorates on impact. Our findings support this principle: the greater the ratio $E_t \widehat{A}_{t+1} / \widehat{A}_t$, the more likely it is that the condition $\widehat{A}_t < M E_t \widehat{A}_{t+1}$ holds, leading to a trade deficit. Moreover, the size of the trade deficit increases with the persistence of the shock. #### 9. If the period utility function is separable in consumption and hours, the answers to both Questions 5 and 7 would change. In this case, the marginal utility of consumption would depend only on consumption, so when the price of contingent claims remains constant (i.e., $q_{0,t} = \beta^t$), consumption does not respond to a temporary productivity shock. There would be neither a substitution effect nor an income effect on consumption, so consumption would remain constant. Labor supply, and hence output, would still increase following the shock, but the resulting increase in output will have to be absorbed entirely by either higher investment or a trade surplus (given the lack of an increase in consumption). Whether the trade balance deteriorates on impact will still depend on the strength of the investment response. However, since consumption no longer increases, a larger increase in investment is required to generate a trade deficit. This implies that a more persistent productivity process (i.e., a higher $E_t \widehat{A}_{t+1} / \widehat{A}_t$) is needed for the economy to run a trade deficit.