Exercise 2.2

An Economy with Endogenous Labor Supply

Problem

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

\[ E_0 \sum_{t=0}^{\infty} \beta^t U(c_t, h_t), \]

where $U$ is a period utility function given by

\[ U(c, h) = -\frac{1}{2} \left[ (c - \bar{c})^2 + h^2 \right], \]

where $\bar{c} > 0$ is a satiation point. The household’s budget constraint is given by

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

where $d_t$ denotes real debt acquired in period $t$ and due in period $t+1$, and $r > 0$ denotes the world interest rate. To avoid inessential dynamics, we impose

\[ \beta(1 + r) = 1. \]

The variable $y_t$ denotes output, which is assumed to be produced by the linear technology

\[ y_t = A h_t. \]

Households are also subject to the no-Ponzi-game constraint

\[ \lim_{j \to \infty} E_t \left[ \frac{d_{t+j}}{(1 + r)^j} \right] \leq 0. \]

Compute the equilibrium laws of motion of consumption, debt, the trade balance, and the current account.
Assume that in period 0, unexpectedly, the productivity parameter $A$ increases permanently to $A' > A$. Establish the effect of this shock on output, consumption, the trade balance, the current account, and the stock of debt.

Answer

1.

Household solves:

\[ \max E_0 \sum_{t=0}^{\infty} \beta^t \left[ -\frac{1}{2} \left( (\bar{c} - c_t)^2 + h_t^2 \right) \right] \]

subject to:

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

\[ y_t = A h_t \]

\[ \lim_{j \to \infty} E_t \left[ \frac{d_{t+j}}{(1 + r)^j} \right] \leq 0 \]

Lagrangian of this problem can be written as:

\[ \mathcal{L} = E_0 \sum_{t=0}^{\infty} \beta^t \left[ -\frac{1}{2} \left( (\bar{c} - c_t)^2 + h_t^2 \right) - \lambda_t (c_t + (1 + r)d_{t-1} - A h_t - d_t) \right] \]

First order conditions:

\[ (\bar{c} - c_t) - \lambda_t = 0 \]

\[ - h_t + \lambda_t A = 0 \]

\[ \lambda_t = \beta (1 + r) E_t [\lambda_{t+1}] \]

This yields Euler Equation and optimal labor supply condition:

\[ \bar{c} - c_t = \beta(1 + r) E_t [\bar{c} - c_{t+1}] \]

\[ h_t = A (c_t - \bar{c}) \]

Recall that $(1 + r)\beta = 1$, then Euler Equation becomes:

\[ c_t = E_t [c_{t+1}] \]

\[ h_t = A (\bar{c} - c_t) \]

Intertemporal budget constraint is given by:

\[ (1 + r) d_{t-1} = \sum_{j=0}^{\infty} \frac{E_t (y_{t+j} - c_{t+j})}{(1 + r)^j} \]

Note that:

\[ E_t(y_{t+j} - c_{t+j}) = A^2 \bar{c} - E_t(A^2 c_{t+j}) - E_t(c_{t+j}) = A^2 \bar{c} - (A^2 + 1) c_t \]

Then the intertemporal budget constraint becomes:

\[ (1 + r) d_{t-1} = \frac{1 + r}{r} \left[ A^2 \bar{c} - (A^2 + 1) c_t \right] \]

Solve for $c_t$:

\[ c_t = \frac{1}{A^2 + 1} \left[ A^2 \bar{c} - r d_{t-1} \right] \]

From the optimality condition for labor and the budget constraint:

\[ h_t = \frac{A}{A^2 + 1} \left[ \bar{c} + r d_{t-1} \right] \]

\[ tb_t = y_t - c_t = r d_{t-1} \]

\[ ca_t = tb_t - r d_{t-1} = 0 \]

2.

Recall that:

\[ c_t = \frac{1}{A^2 + 1} (A^2 \bar{c} - r d_{t-1}) \]

\[ y_t = A h_t = \frac{A^2}{A^2 + 1} (\bar{c} + r d_{t-1}) \]

\[ tb_t = y_t - c_t = r d_{t-1} \]

\[ ca_t = tb_t - r d_{t-1} = 0 \]

Therefore, consumption will increase once and for all at period $t = 0$, output will also increase, but trade balance and current account will not change.

--- title: "Exercise 2.2" subtitle: An Economy with Endogenous Labor Supply --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider a small open economy populated by a large number of households with preferences described by the utility function $$ E_0 \sum_{t=0}^{\infty} \beta^t U(c_t, h_t), $$ where $U$ is a period utility function given by $$ U(c, h) = -\frac{1}{2} \left[ (c - \bar{c})^2 + h^2 \right], $$ where $\bar{c} > 0$ is a satiation point. The household’s budget constraint is given by $$ d_t = (1 + r) d_{t-1} + c_t - y_t, $$ where $d_t$ denotes real debt acquired in period $t$ and due in period $t+1$, and $r > 0$ denotes the world interest rate. To avoid inessential dynamics, we impose $$ \beta(1 + r) = 1. $$ The variable $y_t$ denotes output, which is assumed to be produced by the linear technology $$ y_t = A h_t. $$ Households are also subject to the no-Ponzi-game constraint $$ \lim_{j \to \infty} E_t \left[ \frac{d_{t+j}}{(1 + r)^j} \right] \leq 0. $$ 1. Compute the equilibrium laws of motion of consumption, debt, the trade balance, and the current account. 2. Assume that in period 0, unexpectedly, the productivity parameter $A$ increases permanently to $A' > A$. Establish the effect of this shock on output, consumption, the trade balance, the current account, and the stock of debt. ## Answer #### 1. Household solves: $$ \max E_0 \sum_{t=0}^{\infty} \beta^t \left[ -\frac{1}{2} \left( (\bar{c} - c_t)^2 + h_t^2 \right) \right] $$ subject to: $$ c_t + (1 + r) d_{t-1} = y_t + d_t $$ $$ y_t = A h_t $$ $$ \lim_{j \to \infty} E_t \left[ \frac{d_{t+j}}{(1 + r)^j} \right] \leq 0 $$ Lagrangian of this problem can be written as: $$ \mathcal{L} = E_0 \sum_{t=0}^{\infty} \beta^t \left[ -\frac{1}{2} \left( (\bar{c} - c_t)^2 + h_t^2 \right) - \lambda_t (c_t + (1 + r)d_{t-1} - A h_t - d_t) \right] $$ First order conditions: $$ (\bar{c} - c_t) - \lambda_t = 0 $$ $$ - h_t + \lambda_t A = 0 $$ $$ \lambda_t = \beta (1 + r) E_t [\lambda_{t+1}] $$ This yields Euler Equation and optimal labor supply condition: $$ \bar{c} - c_t = \beta(1 + r) E_t [\bar{c} - c_{t+1}] $$ $$ h_t = A (c_t - \bar{c}) $$ Recall that $(1 + r)\beta = 1$, then Euler Equation becomes: $$ c_t = E_t [c_{t+1}] $$ $$ h_t = A (\bar{c} - c_t) $$ Intertemporal budget constraint is given by: $$ (1 + r) d_{t-1} = \sum_{j=0}^{\infty} \frac{E_t (y_{t+j} - c_{t+j})}{(1 + r)^j} $$ Note that: $$ E_t(y_{t+j} - c_{t+j}) = A^2 \bar{c} - E_t(A^2 c_{t+j}) - E_t(c_{t+j}) = A^2 \bar{c} - (A^2 + 1) c_t $$ Then the intertemporal budget constraint becomes: $$ (1 + r) d_{t-1} = \frac{1 + r}{r} \left[ A^2 \bar{c} - (A^2 + 1) c_t \right] $$ Solve for $c_t$: $$ c_t = \frac{1}{A^2 + 1} \left[ A^2 \bar{c} - r d_{t-1} \right] $$ From the optimality condition for labor and the budget constraint: $$ h_t = \frac{A}{A^2 + 1} \left[ \bar{c} + r d_{t-1} \right] $$ $$ tb_t = y_t - c_t = r d_{t-1} $$ $$ ca_t = tb_t - r d_{t-1} = 0 $$ #### 2. Recall that: $$ c_t = \frac{1}{A^2 + 1} (A^2 \bar{c} - r d_{t-1}) $$ $$ y_t = A h_t = \frac{A^2}{A^2 + 1} (\bar{c} + r d_{t-1}) $$ $$ tb_t = y_t - c_t = r d_{t-1} $$ $$ ca_t = tb_t - r d_{t-1} = 0 $$ Therefore, consumption will increase once and for all at period $t = 0$, output will also increase, but trade balance and current account will not change.