Exercise 2.3

An Open Economy with Habit Formation

Problem

Consider a two-period small open economy populated by a large number of identical households with preferences specified by the utility function

\[ \ln c_1 + \ln(c_2 - x) \]

where $c_1$ and $c_2$ denote, respectively, consumption in periods 1 and 2. Households are endowed with $y > 0$ units of goods each period and are born in period 1 with no assets or debts. In period 1, households can borrow or lend at a zero interest rate. Derive the equilibrium level of consumption and the trade balance under the following three formulations:

$x = 0$ (no habits).
$x = 0.5c_1$ (internal habit formation).
$x = 0.5\tilde{c}_1$, where $\tilde{c}_1$ denotes the economy’s per capita level of consumption in period 1 (external habit formation).

Compare economies (1) and (2) and provide intuition. Similarly, compare economies (2) and (3) and provide intuition.

Answer

1.

The intertemporal budget constraint is

\[ c_2 = 2y - c_1. \]

In the economy without habits, the optimality condition is

\[ \frac{1}{c_1} = \frac{1}{2y - c_1} \]

which yields

\[ c_1 = y \]

2.

With internal habits the household’s problem is to pick $c_1$ to maximize $\ln c_1 + \ln(2y - 1.5c_1)$. The optimality condition is

\[ \frac{1}{c_1} = \frac{1.5}{2y - 1.5c_1} \]

which yields

\[ c_1 = \frac{2}{3}y \]

3.

with external habits the household’s problem is to pick $c_1$ to maximize $\ln c_1 + \ln(2y - c_1 - 0.5\tilde{c}_t)$. The optimality condition is

\[ \frac{1}{c_1} = \frac{1}{2y - c_1 - 0.5\tilde{c}_1} \]

In equilibrium, $c_1 = \tilde{c}_1$. Using this expression to eliminate $\tilde{c}_1$, we obtain

\[ c_1 = \frac{4}{5}y \]

Comparison of no habits with internal habits:
Internal habits delivers less consumption in period 1, because households internalize that the more they consume in period 1, the less happy they are in period 2.

Comparison of internal and external habits:
Again, with internal habits, households internalize the fact that period-1 consumption makes them unhappy in period 2. This internalization is absent under external habits, so household consume more in period 1 under the latter formulation.

It is of interest to note that period-1 consumption is lower under external habits than under no habits. This is because under external habits, when $c_1 = c_2$, the marginal utility of consumption is higher in period 2 than in period 1, tilting consumption toward the future.

--- title: "Exercise 2.3" subtitle: "An Open Economy with Habit Formation" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider a two-period small open economy populated by a large number of identical households with preferences specified by the utility function $$ \ln c_1 + \ln(c_2 - x) $$ where $c_1$ and $c_2$ denote, respectively, consumption in periods 1 and 2. Households are endowed with $y > 0$ units of goods each period and are born in period 1 with no assets or debts. In period 1, households can borrow or lend at a zero interest rate. Derive the equilibrium level of consumption and the trade balance under the following three formulations: 1. $x = 0$ (no habits). 2. $x = 0.5c_1$ (internal habit formation). 3. $x = 0.5\tilde{c}_1$, where $\tilde{c}_1$ denotes the economy’s per capita level of consumption in period 1 (external habit formation). Compare economies (1) and (2) and provide intuition. Similarly, compare economies (2) and (3) and provide intuition. ## Answer ##### 1. The intertemporal budget constraint is $$ c_2 = 2y - c_1. $$ In the economy without habits, the optimality condition is $$ \frac{1}{c_1} = \frac{1}{2y - c_1} $$ which yields $$ c_1 = y $$ #### 2. With internal habits the household’s problem is to pick $c_1$ to maximize $\ln c_1 + \ln(2y - 1.5c_1)$. The optimality condition is $$ \frac{1}{c_1} = \frac{1.5}{2y - 1.5c_1} $$ which yields $$ c_1 = \frac{2}{3}y $$ #### 3. with external habits the household’s problem is to pick $c_1$ to maximize $\ln c_1 + \ln(2y - c_1 - 0.5\tilde{c}_t)$. The optimality condition is $$ \frac{1}{c_1} = \frac{1}{2y - c_1 - 0.5\tilde{c}_1} $$ In equilibrium, $c_1 = \tilde{c}_1$. Using this expression to eliminate $\tilde{c}_1$, we obtain $$ c_1 = \frac{4}{5}y $$ **Comparison of no habits with internal habits:** Internal habits delivers less consumption in period 1, because households internalize that the more they consume in period 1, the less happy they are in period 2. **Comparison of internal and external habits:** Again, with internal habits, households internalize the fact that period-1 consumption makes them unhappy in period 2. This internalization is absent under external habits, so household consume more in period 1 under the latter formulation. It is of interest to note that period-1 consumption is lower under external habits than under no habits. This is because under external habits, when $c_1 = c_2$, the marginal utility of consumption is higher in period 2 than in period 1, tilting consumption toward the future.