Exercise 3.5

Unbalanced Growth

Problem

Consider the same economy as in exercise 3.4, except that the interest rate and the subjective discount factor now satisfy

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

and $r > 0$.

Characterize the equilibrium dynamics of output, consumption, investment, the trade balance, the current account, and external debt.
Does a balanced growth path exist?
Considering jointly the results obtained in this exercise and in exercise 3.4, what general conclusion can you derive?

Answer

1.

The equilibrium conditions are the same, that is, (3.10)–(3.13).

By (3.11), $\Lambda_t = \Lambda_0$ for all $t \geq 0$.
By (3.10), $C_t = C_0$ for all $t \geq 0$.
By (3.12)

\[ k_{t+1} = k_1; \quad \text{where } k_1 \text{ solves } r = \alpha k_1^{\alpha - 1} \]

and

\[ K_{t+1} = A_{t+1} k_1 \]

With the path of $K_t$ in hand, we find the path for

\[ Y_t = A_t^{1 - \alpha} K_t^{\alpha} = A_t k_1^{\alpha} \]

For $t > 0$, the path of investment is

\[ I_t = K_{t+1} - K_t = A_t (\gamma - 1) k_1 \]

For $t = 0$,

\[ I_0 = A_1 k_1 - K_0 / A_0 \]

To find $C_0$ use the intertemporal budget constraint we derived above

\[ \sum_{t=0}^{\infty} \frac{A_t^{1 - \alpha} K_t^{\alpha} + K_t - K_{t+1} - C_t}{(1 + r)^t} = (1 + r) D_{-1} \]

\[ A_0 (K_0 / A_0)^{\alpha} + K_0 - A_1 k_1 + \sum_{t=1}^{\infty} \frac{A_t [k_1^{\alpha} + k_1 (1 - \gamma)]}{(1 + r)^t} - \frac{1 + r}{r} C_0 = (1 + r) D_{-1} \]

\[ A_0 (K_0 / A_0)^{\alpha} + K_0 - A_1 k_1 + A_0 \cdot \frac{\gamma}{1 + r - \gamma} [k_1^{\alpha} + k_1 (1 - \gamma)] - (1 + r) D_{-1} = \frac{1 + r}{r} C_0 \]

The trade balance:

\[ TB_t = Y_t - I_t - C_0 \]

How to find the path to $D_t$? We know the path of $Y_t$, $K_t$, $I_t$, and $C_t$. Use the sequential budget constraint (3.9) to construct the path of $D_t$ sequentially. With $D_t$ in hand, find the current account

\[ CA_t = -D_t + D_{t-1} \]

2.

Balanced growth for $Y_t$, $I_t$, $K_t$, that is, investment and the capital stock grow at the same rate as output.

No balanced growth for consumption, the trade balance, external debt, and the current account.

\[ TB_t = Y_t - I_t - C_0 \]

Consumption does not grow at all. This implies that the consumption-output ratio is not constant over time. In fact, it converges to 0.

The trade balance-to-output ratio is also not constant over time, it converges to

\[ 1 - \frac{(\gamma - 1)\alpha}{r} > 0 \]

And the debt-to-output ratio is not constant over time either.
Since debt is not growing at a constant rate, the current account won’t either.

Therefore, overall no balanced growth path exists.

3.

For a balanced growth path for consumption to exist and to be able to express the economy in terms of stationary variables it is necessary that

\[ \beta(1 + r) = \sqrt{\gamma} \]

or for a general CRRA utility of the form

\[ \frac{c^{1 - \sigma} - 1}{1 - \sigma} \]

that

\[ \beta(1 + r) = \gamma^{\sigma} \]

--- title: "Exercise 3.5" subtitle: "Unbalanced Growth" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider the same economy as in exercise 3.4, except that the interest rate and the subjective discount factor now satisfy $$ \beta(1 + r) = 1, $$ and $r > 0$. 1. Characterize the equilibrium dynamics of output, consumption, investment, the trade balance, the current account, and external debt. 2. Does a balanced growth path exist? 3. Considering jointly the results obtained in this exercise and in exercise 3.4, what general conclusion can you derive? ## Answer #### 1. The equilibrium conditions are the same, that is, (3.10)–(3.13). By (3.11), $\Lambda_t = \Lambda_0$ for all $t \geq 0$. By (3.10), $C_t = C_0$ for all $t \geq 0$. By (3.12) $$ k_{t+1} = k_1; \quad \text{where } k_1 \text{ solves } r = \alpha k_1^{\alpha - 1} $$ and $$ K_{t+1} = A_{t+1} k_1 $$ With the path of $K_t$ in hand, we find the path for $$ Y_t = A_t^{1 - \alpha} K_t^{\alpha} = A_t k_1^{\alpha} $$ For $t > 0$, the path of investment is $$ I_t = K_{t+1} - K_t = A_t (\gamma - 1) k_1 $$ For $t = 0$, $$ I_0 = A_1 k_1 - K_0 / A_0 $$ To find $C_0$ use the intertemporal budget constraint we derived above $$ \sum_{t=0}^{\infty} \frac{A_t^{1 - \alpha} K_t^{\alpha} + K_t - K_{t+1} - C_t}{(1 + r)^t} = (1 + r) D_{-1} $$ $$ A_0 (K_0 / A_0)^{\alpha} + K_0 - A_1 k_1 + \sum_{t=1}^{\infty} \frac{A_t [k_1^{\alpha} + k_1 (1 - \gamma)]}{(1 + r)^t} - \frac{1 + r}{r} C_0 = (1 + r) D_{-1} $$ $$ A_0 (K_0 / A_0)^{\alpha} + K_0 - A_1 k_1 + A_0 \cdot \frac{\gamma}{1 + r - \gamma} [k_1^{\alpha} + k_1 (1 - \gamma)] - (1 + r) D_{-1} = \frac{1 + r}{r} C_0 $$ The trade balance: $$ TB_t = Y_t - I_t - C_0 $$ How to find the path to $D_t$? We know the path of $Y_t$, $K_t$, $I_t$, and $C_t$. Use the sequential budget constraint (3.9) to construct the path of $D_t$ sequentially. With $D_t$ in hand, find the current account $$ CA_t = -D_t + D_{t-1} $$ #### 2. Balanced growth for $Y_t$, $I_t$, $K_t$, that is, investment and the capital stock grow at the same rate as output. No balanced growth for consumption, the trade balance, external debt, and the current account. $$ TB_t = Y_t - I_t - C_0 $$ Consumption does not grow at all. This implies that the consumption-output ratio is not constant over time. In fact, it converges to 0. The trade balance-to-output ratio is also not constant over time, it converges to $$ 1 - \frac{(\gamma - 1)\alpha}{r} > 0 $$ And the debt-to-output ratio is not constant over time either. Since debt is not growing at a constant rate, the current account won’t either. Therefore, overall no balanced growth path exists. #### 3. For a balanced growth path for consumption to exist and to be able to express the economy in terms of stationary variables it is necessary that $$ \beta(1 + r) = \sqrt{\gamma} $$ or for a general CRRA utility of the form $$ \frac{c^{1 - \sigma} - 1}{1 - \sigma} $$ that $$ \beta(1 + r) = \gamma^{\sigma} $$