Exercise 8.5

An Economy With a Leontief Aggregator

Problem

Consider a small open economy inhabited by identical consumers with preferences described by the utility function

\[ \sum_{t=0}^{\infty} \beta^t[ \ln c_t - \gamma h_t], \]

where $c_t$ denotes consumption, $h_t$ denotes hours worked, and $\beta\in (0,1)$ and $\gamma>0$ are parameters. The consumption good is a composite made of tradable and nontradable consumption goods via a Leontief aggregator. Formally,

\[ c_t = \min\{c^T_t,c^N_t\}, \]

where $c^T_t$ and $c^N_t$ denote, respectively, domestic absorption of tradables and nontradables in period $t$. To produce his nontraded consumption, each consumer operates a linear technology that uses labor as the sole input:

\[ c^N_t = A h_t, \]

where $A>0$ is a parameter. In addition, households can borrow or lend in the international financial market at the rate $r>0$. Their sequential budget constraint is given by

\[ d_t = (1+r) d_{t-1} + c^T_t - y^T, \]

where $d_t$ denotes the level of net external debt assumed in period $t$ and maturing in period $t+1$, and $y^T>0$ denotes a constant endowment of tradable goods. In period 0, households start with outstanding debt equal to $d_{-1}>0$. Finally, households are subject to a no-Ponzi-game constraint of the form

\[ \lim_{t\rightarrow \infty} (1+r)^{-t}d_t \le 0. \]

Characterize the equilibrium levels of consumption, consumption of nontradables, and hours worked.
Suppose that in period 0 and unexpectedly, foreign lenders decide to forgive an amount $\Delta^d>0$ of the debt. Assuming that $\Delta^d$ is relatively small, characterize the effect of this debt forgiveness shock on consumption, consumption of nontradables, and hours worked.
Now suppose that $\Delta^d=0$. Instead, assume that in period 0 the nontraded sector experiences a permanent increase in productivity. Specifically the productivity factor $A$ increases by $\Delta^A>0$. Characterize the effect of this positive productivity shock on consumption, consumption of nontradables, and hours worked.

Answer

1.

Note first that in equilibrium $c_t=c^N_t$. To see this, assume, on the contrary, that in some period $t\ge 0$ $c_t=c^T_t<c^N_t$. This level of nontradable consumption is welfare dominated by $c^N_t=c^T_t$ because it requires less labor to be produced and utility is strictly decreasing in effort. Therefore, the equilibrium can be characterized as the pair of sequences $\{h_t,c^T_t\}$ that solves

\[ \max \sum_{t=0}^{\infty} \beta^t [\ln A + \ln h_t - \gamma h_t] \]

subject to

\[ c^T_t \ge Ah_t \tag{*} \]

\[ d_t = (1+r) d_{t-1} +c^T_t -y^T \tag{**} \]

\[ \lim_{t\rightarrow \infty} (1+r)^{-t}d_t \le 0 \tag{***} \]

Consider the less constrained problem consisting in chosing the sequence $\{h_t\}$ so as to

\[ \max \sum_{t=0}^{\infty} \beta^t [\ln A + \ln h_t - \gamma h_t] \]

The solution to this problem is

\[ h_t = \frac 1 {\gamma} \]

Clearly, if this solution satisfies ()-(**), then we have found the equilibrium. This will be the case if

\[ y^T - rd_{-1}\ge \frac A {\gamma} \]

Thus, the equilibrium is characterized as follows:

If $y^T-rd_{-1}\ge A/\gamma$, then $h_t = 1/\gamma$, $c^N_t=A/\gamma$, $c_t=A/\gamma$, if $y^T-rd_{-1}\ge A/\gamma$. In this case, $c^T_t \ge A/\gamma$ and is bounded by (**) and (***), but is otherwise indetermined.
If $y^T-rd_{-1}<A/\gamma$, then $c^T_t=y^T-rd_{-1}$, $c^N_t = y^T-rd_{-1}$, $h_t = (y^T-rd_{-1})/A$,

2.

If $y^T-rd_{-1}>A/\gamma$ the debt forgiveness has no effect on consumption, consumption of nontradables, or hours. Otherwise, consumption of tradables increases by $r\Delta^d$ and so does consumption of nontradables and totoal consumption. In this case, hours increase by $r\Delta^d/A$.

3.

If $y^T-rd_{-1}>A/\gamma$, then consumption of nontradables, and total consumption increase by $\Delta^A/\gamma$ and hours are unchanged. Otherwise, neither consumption (of any type) nor hours change.

--- title: "Exercise 8.5" subtitle: "An Economy With a Leontief Aggregator" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider a small open economy inhabited by identical consumers with preferences described by the utility function $$ \sum_{t=0}^{\infty} \beta^t[ \ln c_t - \gamma h_t], $$ where $c_t$ denotes consumption, $h_t$ denotes hours worked, and $\beta\in (0,1)$ and $\gamma>0$ are parameters. The consumption good is a composite made of tradable and nontradable consumption goods via a Leontief aggregator. Formally, $$ c_t = \min\{c^T_t,c^N_t\}, $$ where $c^T_t$ and $c^N_t$ denote, respectively, domestic absorption of tradables and nontradables in period $t$. To produce his nontraded consumption, each consumer operates a linear technology that uses labor as the sole input: $$ c^N_t = A h_t, $$ where $A>0$ is a parameter. In addition, households can borrow or lend in the international financial market at the rate $r>0$. Their sequential budget constraint is given by $$ d_t = (1+r) d_{t-1} + c^T_t - y^T, $$ where $d_t$ denotes the level of net external debt assumed in period $t$ and maturing in period $t+1$, and $y^T>0$ denotes a constant endowment of tradable goods. In period 0, households start with outstanding debt equal to $d_{-1}>0$. Finally, households are subject to a no-Ponzi-game constraint of the form $$ \lim_{t\rightarrow \infty} (1+r)^{-t}d_t \le 0. $$ 1. Characterize the equilibrium levels of consumption, consumption of nontradables, and hours worked. 2. Suppose that in period 0 and unexpectedly, foreign lenders decide to forgive an amount $\Delta^d>0$ of the debt. Assuming that $\Delta^d$ is relatively small, characterize the effect of this debt forgiveness shock on consumption, consumption of nontradables, and hours worked. 3. Now suppose that $\Delta^d=0$. Instead, assume that in period 0 the nontraded sector experiences a permanent increase in productivity. Specifically the productivity factor $A$ increases by $\Delta^A>0$. Characterize the effect of this positive productivity shock on consumption, consumption of nontradables, and hours worked. ## Answer #### 1. Note first that in equilibrium $c_t=c^N_t$. To see this, assume, on the contrary, that in some period $t\ge 0$ $c_t=c^T_t<c^N_t$. This level of nontradable consumption is welfare dominated by $c^N_t=c^T_t$ because it requires less labor to be produced and utility is strictly decreasing in effort. Therefore, the equilibrium can be characterized as the pair of sequences $\{h_t,c^T_t\}$ that solves $$ \max \sum_{t=0}^{\infty} \beta^t [\ln A + \ln h_t - \gamma h_t] $$ subject to $$ c^T_t \ge Ah_t \tag{*} $$ $$ d_t = (1+r) d_{t-1} +c^T_t -y^T \tag{**} $$ $$ \lim_{t\rightarrow \infty} (1+r)^{-t}d_t \le 0 \tag{***} $$ Consider the less constrained problem consisting in chosing the sequence $\{h_t\}$ so as to $$ \max \sum_{t=0}^{\infty} \beta^t [\ln A + \ln h_t - \gamma h_t] $$ The solution to this problem is $$ h_t = \frac 1 {\gamma} $$ Clearly, if this solution satisfies (*)-(***), then we have found the equilibrium. This will be the case if $$ y^T - rd_{-1}\ge \frac A {\gamma} $$ Thus, the equilibrium is characterized as follows: (I) If $y^T-rd_{-1}\ge A/\gamma$, then $h_t = 1/\gamma$, $c^N_t=A/\gamma$, $c_t=A/\gamma$, if $y^T-rd_{-1}\ge A/\gamma$. In this case, $c^T_t \ge A/\gamma$ and is bounded by $\*\*$ and $\*\*\*$, but is otherwise indetermined. (II) If $y^T-rd_{-1}<A/\gamma$, then $c^T_t=y^T-rd_{-1}$, $c^N_t = y^T-rd_{-1}$, $h_t = (y^T-rd_{-1})/A$, #### 2. If $y^T-rd_{-1}>A/\gamma$ the debt forgiveness has no effect on consumption, consumption of nontradables, or hours. Otherwise, consumption of tradables increases by $r\Delta^d$ and so does consumption of nontradables and totoal consumption. In this case, hours increase by $r\Delta^d/A$. #### 3. If $y^T-rd_{-1}>A/\gamma$, then consumption of nontradables, and total consumption increase by $\Delta^A/\gamma$ and hours are unchanged. Otherwise, neither consumption (of any type) nor hours change.