Exercise 2.6

Anticipated Interest Rate Decline

Problem

Consider a small open endowment economy enjoying free capital mobility. Preferences are described by the utility function

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

with $\beta \in (0, 1)$. Agents have access to an internationally traded bond paying the interest rate $r_t$ when held from period $t$ to $t+1$. The representative household starts period zero with an asset position $b_{-1}$. Each period $t \geq 0$, the household receives an endowment $y_t$. Households know the time paths of $\{r_t\}$ and $\{y_t\}$ with certainty. The sequential budget constraint of the household is given by:

\[ c_t + \frac{b_t}{1 + r_t} = y_t + b_{t-1}. \]

And the household’s borrowing limit is:

\[ \lim_{j \to \infty} \frac{b_{t+j}}{\prod_{s=0}^{j}(1 + r_{t+s})} \geq 0. \]

Derive the household’s present value budget constraint.
Derive the equilibrium paths of consumption and assets in terms of $y_t$, $r_t$ and $b_{-1}$.

Assume now that in period 0 it is learned that in period $t^* \geq 0$ the interest rate will decline temporarily. Specifically, the new path of the interest rate is

\[ r'_t = \begin{cases} r_t & \text{for all } t \geq 0 \text{ and } t \neq t^* \\ r'_{t^*} < r_{t^*} & \text{for } t = t^* \end{cases} \]

Find the impact effect of this anticipated interest rate cut on consumption, that is, find $\ln (c'_0 / c_0)$, where $c'_t$ denotes the equilibrium path of consumption under the new interest rate path and $c_t$ denotes the equilibrium path of consumption under the old interest rate path. Distinguish two cases. First consider a storage economy with $y_t = 0$ for all $t$ and $b_{-1} > 0$. Discuss whether the anticipated future rate cut stimulates demand at the time it is announced. Provide intuition. Then consider an endowment economy with $b_{-1} = 0$ and $y_t = y > 0$ for all $t$. Analyze whether the response of consumption in period 0 is equal in size to the anticipated rate cut and whether it depends on the anticipation horizon $t^*$. In particular, do anticipated interest rate cuts have a smaller stimulating effect on current consumption the further in the future they will take place, that is, the larger $t^*$ is? Provide intuition for your findings.
Relate the insights obtained in this exercise to the debate on Forward Guidance as a monetary policy strategy. In particular, interpret the present real economy as a monetary economy with rigid nominal prices and a central bank that deploys the necessary monetary policy to fully control the real interest rate $r_t$. Address in particular the question of whether forward guidance is an effective tool to stimulate aggregate demand.

Answer

1.

The PVBC is

\[ \sum_{j=0}^{\infty} \left( \frac{c_{t+j}}{\prod_{s=0}^{j-1}(1 + r_{t+s})} \right) = b_{t-1} + \sum_{j=0}^{\infty} \left( \frac{y_{t+j}}{\prod_{s=0}^{j-1}(1 + r_{t+s})} \right), \tag{1} \]

which says that the PDV of consumption is equal to the sum of the initial asset position and the PDV of the endowment stream. Note that $\prod_{s=0}^{-1} (1 + r_{t+s})$ represents the empty product because the lower bound $s = 0$ is greater than the upper bound $s = -1$. By convention, an empty product (a product over an empty index set) is defined to be 1.

2.

To derive the equilibrium paths we first write the Lagrangian for the consumer

\[ \mathcal{L} = \sum_{t=0}^{\infty} \beta^t \left\{ \ln c_t + \lambda_t \left[ y_t + b_{t-1} - c_t - \frac{b_t}{1 + r_t} \right] \right\}. \]

This Lagrangian has first order conditions with respect to $c_t$ and $b_{t+1}$ respectively

\[ \frac{1}{c_t} = \lambda_t \]

\[ \frac{\lambda_t}{1 + r_t} = \beta \lambda_{t+1}. \]

An equilibrium is paths $\{c_t, b_t\}_{t=0}^{\infty}$ such that for all $t \geq 0$ we have the combined first order conditions, sequential budget constraint, and the no-Ponzi game condition

\[ c_{t+1} = \beta(1 + r_t)c_t \tag{2} \]

\[ c_t + \frac{b_t}{1 + r_t} = y_t + b_{t-1} \tag{3} \]

\[ \lim_{j \to \infty} \frac{b_{t+j}}{\prod_{s=0}^j (1 + r_{t+s})} = 0 \tag{4} \]

given $\{r_t\}_{t=0}^{\infty}$, $\{y_t\}_{t=0}^{\infty}$, and $b_{-1}$.

Equivalently, equilibrium is a $c_0$ such that

\[ c_0 = (1 - \beta)b_{-1} + (1 - \beta) \sum_{t=0}^{\infty} \left( \frac{y_t}{\prod_{s=0}^{t-1}(1 + r_s)} \right). \tag{5} \]

To see this, iterate (2) backwards to obtain

\[ c_t = \beta^t \left( \prod_{s=0}^{t-1} (1 + r_s) \right) c_0. \tag{6} \]

Rearranging yields

\[ \frac{c_t}{\prod_{s=0}^{t-1}(1 + r_s)} = \beta^t c_0. \]

Now sum both sides from $t=0$ to $t=T$

\[ \sum_{t=0}^{T} \frac{c_t}{\prod_{s=0}^{t-1}(1 + r_s)} = \sum_{t=0}^{T} \beta^t c_0 \]

Take limit for $T \rightarrow \infty$

\[ \sum_{t=0}^{\infty} \frac{c_t}{\prod_{s=0}^{t-1}(1 + r_s)} = \sum_{t=0}^{\infty} \beta^t c_0 = \frac{1}{1 - \beta}c_0 \]

Next use this expression in equation (1) evaluated for $t = 0$ to obtain

\[ \frac{1}{1 - \beta}c_0 = b_{-1} + \sum_{t=0}^{\infty} \left( \frac{y_t}{\prod_{s=0}^{t-1}(1 + r_s)} \right) \]

Finally, multiply by $(1− \beta)$, which yields the equation (5).

How to find the associated path of $b_t$ and $c_t$? Use (3) evaluated at $t = 0$. This gives $b_0$. Then use (5) evaluated at $t = 0$. This gives $c_1$. With $b_0$ and $c_1$ in hand evaluate (3) for $t = 1$ to obtain $b_1$. Continue in this way to construct the sequences $\{c_t\}_{t=0}^{\infty}$ and $\{b_t\}_{t=0}^{\infty}$.

3.

Storage economy, $y = 0$ and $b_{-1} > 0$

\[ \frac{c_t'}{c_t} = \begin{cases} 1 & \text{for all } t \leq t^* \\ \left( \frac{1 + r_{t^*}'}{1 + r_{t^*}} \right) < 1 & \text{for all } t > t^* \end{cases} \]

This shows that the anticipated future rate cut does not stimulate demand at the time it is announced. Importantly, there is no effect on current consumption regardless of the anticipation horizon $t^*$. But the rate cut will permanently lower future consumption starting from the period it takes place. Intuition: The rate cut has a SE whereby current consumption rises and future consumption falls. But it also has an IE. Here the IE is negative because the positive stock of bonds will pay lower interest in the future. With log utility, the SE exactly cancels the IE effect and the future rate cut fails to stimulate the current economy.

Endowment Economy: $y_t = y > 0$ for all $t$ and $b_{-1} = 0$.

Let

\[ q_{0,t} \equiv \prod_{s=0}^{t-1} (1 + r_s); \quad \text{and } q_{0,t}' = \prod_{s=0}^{t-1} (1 + r_s') \]

Use (5) to obtain

\[ \begin{aligned} c_0 &= (1 - \beta) \sum_{t=0}^{\infty} \left( \frac{y_t}{q_{0,t}} \right) \\ &= (1 - \beta) \sum_{t=0}^{t^*} \left( \frac{y_t}{q_{0,t}} \right) + (1 - \beta) \sum_{t = t^* + 1}^{\infty} \left( \frac{y_t}{q_{0,t}} \right) \end{aligned} \]

\[ \begin{aligned} c_0' &= (1 - \beta) \sum_{t = 0}^{\infty} \left( \frac{y_t}{q_{0,t}'} \right) \\ &= (1 - \beta) \sum_{t = 0}^{t^*} \left( \frac{y_t}{q_{0,t}} \right) + (1 - \beta) \cdot \frac{1 + r_{t^*}}{1 + r_{t^*}'} \sum_{t = t^* + 1}^{\infty} \left( \frac{y_t}{q_{0,t}} \right) \end{aligned} \]

so that

\[ c_0' - c_0 = (1 - \beta) \sum_{t = t^* + 1}^{\infty} \left( \frac{y_t}{q_{0,t}} \right) \left[ \frac{1 + r_{t^*}}{1 + r_{t^*}'} - 1 \right] > 0 \]

What is different now? Why does consumption go up on impact in response to the future rate cut? There is a positive IE associated with the expected future rate cut. Why? Because of the rate cut in $t^*$, the PDV of endowment stream is higher. Thus, the SE and the IE reinforce each other for $t \leq t^*$ and pull in opposite directions for $t > t^*$.

Size of increase in consumption for $t \leq t^*$ now depends on $t^*$. The further in the future is the rate cut, the smaller is the positive IE and hence the smaller is increase in demand.

4.

By contrast in the NK model, increase in consumption depends only on size of rate cut, but not on anticipation horizon, i.e., a rate cut in 100 years has same power as a rate cut in 1 year from now and hence there is no Forward Guidance puzzle. In the current environment Forward Guidance is a less effective tool to stimulate the economy than in the NK model.

--- title: "Exercise 2.6" subtitle: "Anticipated Interest Rate Decline" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Consider a small open endowment economy enjoying free capital mobility. Preferences are described by the utility function $$ \sum_{t=0}^{\infty} \beta^t \ln c_t, $$ with $\beta \in (0, 1)$. Agents have access to an internationally traded bond paying the interest rate $r_t$ when held from period $t$ to $t+1$. The representative household starts period zero with an asset position $b_{-1}$. Each period $t \geq 0$, the household receives an endowment $y_t$. Households know the time paths of $\{r_t\}$ and $\{y_t\}$ with certainty. The sequential budget constraint of the household is given by: $$ c_t + \frac{b_t}{1 + r_t} = y_t + b_{t-1}. $$ And the household’s borrowing limit is: $$ \lim_{j \to \infty} \frac{b_{t+j}}{\prod_{s=0}^{j}(1 + r_{t+s})} \geq 0. $$ 1. Derive the household’s present value budget constraint. 2. Derive the equilibrium paths of consumption and assets in terms of $y_t$, $r_t$ and $b_{-1}$. Assume now that in period 0 it is learned that in period $t^* \geq 0$ the interest rate will decline temporarily. Specifically, the new path of the interest rate is $$ r'_t = \begin{cases} r_t & \text{for all } t \geq 0 \text{ and } t \neq t^* \\ r'_{t^*} < r_{t^*} & \text{for } t = t^* \end{cases} $$ 3. Find the impact effect of this anticipated interest rate cut on consumption, that is, find $\ln (c'_0 / c_0)$, where $c'_t$ denotes the equilibrium path of consumption under the new interest rate path and $c_t$ denotes the equilibrium path of consumption under the old interest rate path. Distinguish two cases. First consider a storage economy with $y_t = 0$ for all $t$ and $b_{-1} > 0$. Discuss whether the anticipated future rate cut stimulates demand at the time it is announced. Provide intuition. Then consider an endowment economy with $b_{-1} = 0$ and $y_t = y > 0$ for all $t$. Analyze whether the response of consumption in period 0 is equal in size to the anticipated rate cut and whether it depends on the anticipation horizon $t^*$. In particular, do anticipated interest rate cuts have a smaller stimulating effect on current consumption the further in the future they will take place, that is, the larger $t^*$ is? Provide intuition for your findings. 4. Relate the insights obtained in this exercise to the debate on Forward Guidance as a monetary policy strategy. In particular, interpret the present real economy as a monetary economy with rigid nominal prices and a central bank that deploys the necessary monetary policy to fully control the real interest rate $r_t$. Address in particular the question of whether forward guidance is an effective tool to stimulate aggregate demand. ## Answer #### 1. The PVBC is $$ \sum_{j=0}^{\infty} \left( \frac{c_{t+j}}{\prod_{s=0}^{j-1}(1 + r_{t+s})} \right) = b_{t-1} + \sum_{j=0}^{\infty} \left( \frac{y_{t+j}}{\prod_{s=0}^{j-1}(1 + r_{t+s})} \right), \tag{1} $$ which says that the PDV of consumption is equal to the sum of the initial asset position and the PDV of the endowment stream. Note that $\prod_{s=0}^{-1} (1 + r_{t+s})$ represents the empty product because the lower bound $s = 0$ is greater than the upper bound $s = -1$. By convention, an empty product (a product over an empty index set) is defined to be 1. #### 2. To derive the equilibrium paths we first write the Lagrangian for the consumer $$ \mathcal{L} = \sum_{t=0}^{\infty} \beta^t \left\{ \ln c_t + \lambda_t \left[ y_t + b_{t-1} - c_t - \frac{b_t}{1 + r_t} \right] \right\}. $$ This Lagrangian has first order conditions with respect to $c_t$ and $b_{t+1}$ respectively $$ \frac{1}{c_t} = \lambda_t $$ $$ \frac{\lambda_t}{1 + r_t} = \beta \lambda_{t+1}. $$ An equilibrium is paths $\{c_t, b_t\}_{t=0}^{\infty}$ such that for all $t \geq 0$ we have the combined first order conditions, sequential budget constraint, and the no-Ponzi game condition $$ c_{t+1} = \beta(1 + r_t)c_t \tag{2} $$ $$ c_t + \frac{b_t}{1 + r_t} = y_t + b_{t-1} \tag{3} $$ $$ \lim_{j \to \infty} \frac{b_{t+j}}{\prod_{s=0}^j (1 + r_{t+s})} = 0 \tag{4} $$ given $\{r_t\}_{t=0}^{\infty}$, $\{y_t\}_{t=0}^{\infty}$, and $b_{-1}$. Equivalently, equilibrium is a $c_0$ such that $$ c_0 = (1 - \beta)b_{-1} + (1 - \beta) \sum_{t=0}^{\infty} \left( \frac{y_t}{\prod_{s=0}^{t-1}(1 + r_s)} \right). \tag{5} $$ To see this, iterate (2) backwards to obtain $$ c_t = \beta^t \left( \prod_{s=0}^{t-1} (1 + r_s) \right) c_0. \tag{6} $$ Rearranging yields $$ \frac{c_t}{\prod_{s=0}^{t-1}(1 + r_s)} = \beta^t c_0. $$ Now sum both sides from $t=0$ to $t=T$ $$ \sum_{t=0}^{T} \frac{c_t}{\prod_{s=0}^{t-1}(1 + r_s)} = \sum_{t=0}^{T} \beta^t c_0 $$ Take limit for $T \rightarrow \infty$ $$ \sum_{t=0}^{\infty} \frac{c_t}{\prod_{s=0}^{t-1}(1 + r_s)} = \sum_{t=0}^{\infty} \beta^t c_0 = \frac{1}{1 - \beta}c_0 $$ Next use this expression in equation (1) evaluated for $t = 0$ to obtain $$ \frac{1}{1 - \beta}c_0 = b_{-1} + \sum_{t=0}^{\infty} \left( \frac{y_t}{\prod_{s=0}^{t-1}(1 + r_s)} \right) $$ Finally, multiply by $(1− \beta)$, which yields the equation (5). How to find the associated path of $b_t$ and $c_t$? Use (3) evaluated at $t = 0$. This gives $b_0$. Then use (5) evaluated at $t = 0$. This gives $c_1$. With $b_0$ and $c_1$ in hand evaluate (3) for $t = 1$ to obtain $b_1$. Continue in this way to construct the sequences $\{c_t\}_{t=0}^{\infty}$ and $\{b_t\}_{t=0}^{\infty}$. #### 3. Storage economy, $y = 0$ and $b_{-1} > 0$ $$ \frac{c_t'}{c_t} = \begin{cases} 1 & \text{for all } t \leq t^* \\ \left( \frac{1 + r_{t^*}'}{1 + r_{t^*}} \right) < 1 & \text{for all } t > t^* \end{cases} $$ This shows that the anticipated future rate cut does not stimulate demand at the time it is announced. Importantly, there is no effect on current consumption regardless of the anticipation horizon $t^*$. But the rate cut will permanently lower future consumption starting from the period it takes place. Intuition: The rate cut has a SE whereby current consumption rises and future consumption falls. But it also has an IE. Here the IE is negative because the positive stock of bonds will pay lower interest in the future. With log utility, the SE exactly cancels the IE effect and the future rate cut fails to stimulate the current economy. Endowment Economy: $y_t = y > 0$ for all $t$ and $b_{-1} = 0$. Let $$ q_{0,t} \equiv \prod_{s=0}^{t-1} (1 + r_s); \quad \text{and } q_{0,t}' = \prod_{s=0}^{t-1} (1 + r_s') $$ Use (5) to obtain $$ \begin{aligned} c_0 &= (1 - \beta) \sum_{t=0}^{\infty} \left( \frac{y_t}{q_{0,t}} \right) \\ &= (1 - \beta) \sum_{t=0}^{t^*} \left( \frac{y_t}{q_{0,t}} \right) + (1 - \beta) \sum_{t = t^* + 1}^{\infty} \left( \frac{y_t}{q_{0,t}} \right) \end{aligned} $$ $$ \begin{aligned} c_0' &= (1 - \beta) \sum_{t = 0}^{\infty} \left( \frac{y_t}{q_{0,t}'} \right) \\ &= (1 - \beta) \sum_{t = 0}^{t^*} \left( \frac{y_t}{q_{0,t}} \right) + (1 - \beta) \cdot \frac{1 + r_{t^*}}{1 + r_{t^*}'} \sum_{t = t^* + 1}^{\infty} \left( \frac{y_t}{q_{0,t}} \right) \end{aligned} $$ so that $$ c_0' - c_0 = (1 - \beta) \sum_{t = t^* + 1}^{\infty} \left( \frac{y_t}{q_{0,t}} \right) \left[ \frac{1 + r_{t^*}}{1 + r_{t^*}'} - 1 \right] > 0 $$ What is different now? Why does consumption go up on impact in response to the future rate cut? There is a positive IE associated with the expected future rate cut. Why? Because of the rate cut in $t^*$, the PDV of endowment stream is higher. Thus, the SE and the IE reinforce each other for $t \leq t^*$ and pull in opposite directions for $t > t^*$. Size of increase in consumption for $t \leq t^*$ now depends on $t^*$. The further in the future is the rate cut, the smaller is the positive IE and hence the smaller is increase in demand. #### 4. By contrast in the NK model, increase in consumption depends only on size of rate cut, but not on anticipation horizon, i.e., a rate cut in 100 years has same power as a rate cut in 1 year from now and hence there is no Forward Guidance puzzle. In the current environment Forward Guidance is a less effective tool to stimulate the economy than in the NK model.