Exercise 2.17

Leontief Preferences for Consumption and Leisure

Problem

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

\[ \sum_{t=0}^{\infty} \beta^t \ln(x_t) \]

with

\[ x_t = \min\{c_t, 1 - h_t\}, \]

where $c_t$ denotes consumption in period $t$, and $h_t$ denotes labor effort in period $t$ and is restricted to reside in the interval $[0, 1)$. Households produce goods with the technology

\[ y_t = h_t, \]

where $y_t$ denotes output. They can also borrow or lend in one-period bonds that pay the constant interest rate $r > 0$. Let $d_t$ denote the debt acquired in $t$ and maturing in $t + 1$. Assume that households start period 0 with no debts or assets inherited from the past. Borrowing is limited by the no-Ponzi-game constraint

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

Finally, assume that the subjective and market discount rates are equal to each other

\[ \beta = \frac{1}{1 + r}. \]

Write down the household’s optimization problem.
Derive the first-order conditions associated with the household’s problem.
Calculate the equilibrium levels of consumption and the trade balance. These should be 2 numbers.
Now consider an environment in which households are relatively impatient, in the sense that the subjective discount factor is larger than the market discount factor. Specifically assume that the interest rate is 10 percent ($r = 0.1$) and that $\beta = 1/1.2$. Calculate the equilibrium levels of consumption and the trade balance in period 0 and characterize their evolution over time.

Answer

1.

The household’s optimization problem is

\[ \max \sum_{t=0}^{\infty} \beta^t \ln(x_t), \]

subject to

\[ x_t = \min\{c_t, 1 - h_t\}, \]

\[ y_t = h_t, \]

\[ d_t = (1 + r) d_{t-1} + c_t - y_t, \]

and the no-Ponzi-game constraint.

2.

It is optimal to set $c_t = 1 - h_t$. So the household’s problem becomes

\[ \max \sum_{t=0}^{\infty} \beta^t \ln(c_t), \]

\[ d_t = (1 + r) d_{t-1} + 2c_t - 1. \]

The first-order conditions associated with this problem are the sequential budget constraint, the no-Ponzi-game holding with equality, and

\[ c_{t+1} = c_t \quad \text{for all } t \geq 0. \]

3.

Because consumption is constant over time, so is debt ($d_t = d_{-1} = 0$ for $t \geq 0$). Then, the sequential budget constraint implies that

\[ c_t = h_t = \frac{1}{2}. \]

And the trade balance is

\[ tb_t = y - c = h_t - c_t = 0. \]

4.

Now the first-order condition is

\[ c_{t+1} = \beta (1 + r) c_t, \]

which is the same as before but with $\beta(1 + r) \neq 1$. The intertemporal resource constraint and the fact that $d_{-1} = 0$ implies that

\[ 0 = \sum_{t=0}^{\infty} (1 + r)^{-t}(2c_t - 1). \]

Combining the above two expressions, we get

\[ \frac{1 + r}{r} = 2 c_0 \cdot \frac{1}{1 - \beta}, \]

\[ c_0 = \frac{1}{2} \cdot \frac{1 + r}{r} (1 - \beta). \]

Using the given parameter values ($r = 0.1$, $\beta = 1/1.2$), we get:

\[ c_0 = 0.917. \]

The trade balance in period 0 is

\[ tb_0 = y_0 - c_0 = h_0 - c_0 = 1 - 2c_0 = -0.833. \]

From period 0 onward, consumption falls monotonically at the constant gross rate $\beta(1 + r)$, which happens to be 0.917. Consumption converges monotonically to zero. Since the trade balance is given by $1 - 2c_t$, we have that the trade balance improves monotonically and converges to 1.

--- title: "Exercise 2.17" subtitle: "Leontief Preferences for Consumption and Leisure" --- <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 $$ \sum_{t=0}^{\infty} \beta^t \ln(x_t) $$ with $$ x_t = \min\{c_t, 1 - h_t\}, $$ where $c_t$ denotes consumption in period $t$, and $h_t$ denotes labor effort in period $t$ and is restricted to reside in the interval $[0, 1)$. Households produce goods with the technology $$ y_t = h_t, $$ where $y_t$ denotes output. They can also borrow or lend in one-period bonds that pay the constant interest rate $r > 0$. Let $d_t$ denote the debt acquired in $t$ and maturing in $t + 1$. Assume that households start period 0 with no debts or assets inherited from the past. Borrowing is limited by the no-Ponzi-game constraint $$ \lim_{t \to \infty} (1 + r)^{-t} d_t \leq 0. $$ Finally, assume that the subjective and market discount rates are equal to each other $$ \beta = \frac{1}{1 + r}. $$ 1. Write down the household’s optimization problem. 2. Derive the first-order conditions associated with the household’s problem. 3. Calculate the equilibrium levels of consumption and the trade balance. These should be 2 numbers. 4. Now consider an environment in which households are relatively impatient, in the sense that the subjective discount factor is larger than the market discount factor. Specifically assume that the interest rate is 10 percent ($r = 0.1$) and that $\beta = 1/1.2$. Calculate the equilibrium levels of consumption and the trade balance in period 0 and characterize their evolution over time. ## Answer #### 1. The household’s optimization problem is $$ \max \sum_{t=0}^{\infty} \beta^t \ln(x_t), $$ subject to $$ x_t = \min\{c_t, 1 - h_t\}, $$ $$ y_t = h_t, $$ $$ d_t = (1 + r) d_{t-1} + c_t - y_t, $$ and the no-Ponzi-game constraint. #### 2. It is optimal to set $c_t = 1 - h_t$. So the household’s problem becomes $$ \max \sum_{t=0}^{\infty} \beta^t \ln(c_t), $$ $$ d_t = (1 + r) d_{t-1} + 2c_t - 1. $$ The first-order conditions associated with this problem are the sequential budget constraint, the no-Ponzi-game holding with equality, and $$ c_{t+1} = c_t \quad \text{for all } t \geq 0. $$ #### 3. Because consumption is constant over time, so is debt ($d_t = d_{-1} = 0$ for $t \geq 0$). Then, the sequential budget constraint implies that $$ c_t = h_t = \frac{1}{2}. $$ And the trade balance is $$ tb_t = y - c = h_t - c_t = 0. $$ #### 4. Now the first-order condition is $$ c_{t+1} = \beta (1 + r) c_t, $$ which is the same as before but with $\beta(1 + r) \neq 1$. The intertemporal resource constraint and the fact that $d_{-1} = 0$ implies that $$ 0 = \sum_{t=0}^{\infty} (1 + r)^{-t}(2c_t - 1). $$ Combining the above two expressions, we get $$ \frac{1 + r}{r} = 2 c_0 \cdot \frac{1}{1 - \beta}, $$ or $$ c_0 = \frac{1}{2} \cdot \frac{1 + r}{r} (1 - \beta). $$ Using the given parameter values ($r = 0.1$, $\beta = 1/1.2$), we get: $$ c_0 = 0.917. $$ The trade balance in period 0 is $$ tb_0 = y_0 - c_0 = h_0 - c_0 = 1 - 2c_0 = -0.833. $$ From period 0 onward, consumption falls monotonically at the constant gross rate $\beta(1 + r)$, which happens to be 0.917. Consumption converges monotonically to zero. Since the trade balance is given by $1 - 2c_t$, we have that the trade balance improves monotonically and converges to 1.