<p>This will be long, but the subject is worth it.</p>
<p>The gist of the &quot;exchange rate overshooting&quot; model in <a href="https://www.jstor.org/stable/1831272" rel="nofollow noreferrer">Dornbusch, R. (1976). Expectations and exchange rate dynamics. The Journal of Political Economy, 1161-1176.</a> can be given using two equations:</p>
<p><strong>Uncovered Interest Rate Parity (UIRP)</strong>
<span class="math-container">$$i  = i^*+ \dot e\tag{1}$$</span></p>
<p>where <span class="math-container">$i$</span> is local interest rate, <span class="math-container">$i^*$</span> is international interest rate, and <span class="math-container">$e$</span> is the logarithm of the <em>foreign</em> exchange rate, i.e. local currency units per unit of international currency basket. So an <em>increase</em> in <span class="math-container">$e$</span> reflects <em>depreciation</em> of the local currency. A <em>decrease</em> in <span class="math-container">$e$</span> reflects <em>appreciation</em> of the local currency (I would prefer to use the local exchange rate, where conceptually we link &quot;rise&quot; with &quot;rise&quot;, but it is unfriendly to the phase diagram that it will follow). Then <span class="math-container">$\dot e$</span> is the expected percentage change on the exchange rate.</p>
<p><strong>Money-market Equilibrium (in logarithms)</strong></p>
<p><span class="math-container">$$m-p = -\lambda i +\phi y \tag{2}$$</span></p>
<p>where <span class="math-container">$m$</span> is money supply, <span class="math-container">$p$</span> is the local price level, and <span class="math-container">$y$</span> is local output.</p>
<p>The assumptions are: international interest rate is fixed. There is perfect capital mobility, so the local interest rate is a &quot;jump&quot; variable, being able to adjust instantaneously (and so eq. <span class="math-container">$(2)$</span> holds always). Local output is at full-employment level. Local prices are &quot;sticky&quot;: this means that the local price-level is not a &quot;jump&quot; variable anymore, but its evolution is governed by a difference/differential equation that reflects gradual adjustment. Money supply is long-term neutral.</p>
<p>The famous &quot;overshooting&quot; effect from an <em>unanticipated</em> permanent expansion of <span class="math-container">$m$</span> can then be derived: an increase in <span class="math-container">$m$</span> raises the left-hand side of <span class="math-container">$(2)$</span>. Since prices are sticky (don't jump) and output is fixed, the only way that equilibrium can continue to hold in the money market is by a <em>decrease</em> in <span class="math-container">$i$</span> (due to the minus sign), so that that eq. <span class="math-container">$(2)$</span> continues to hold.<br />
Given the fall in <span class="math-container">$i$</span>, and the assumption of constant <span class="math-container">$i^*$</span>, the only way now that <span class="math-container">$(1)$</span> can hold with a lower <span class="math-container">$i$</span> is for <span class="math-container">$\dot e$</span> to be <em>negative</em>, i.e. if the local currency is expected to <em>appreciate</em>. But raising the long-term neutral local money supply can only create expectations for local currency <em>depreciation</em>, not appreciation.<br />
<strong>&quot;Overshooting&quot;</strong> cuts the Gordian Knot: the <em>immediate</em> response of the exchange rate <span class="math-container">$e$</span> is to <em>depreciate by more</em> (overshooting) than the eventual depreciation that is consistent with the increase in the money supply, and then it starts to <em>appreciate</em> to reach its new level (and so we can have <span class="math-container">$\dot e&lt;0$</span>).</p>
<p>The question asks about &quot;<strong>fiscal expansion</strong>&quot;. How is this to be represented in the model? The only possible way is to assume that output can rise due to the fiscal expansion.</p>
<p>A fiscal expansion increases <span class="math-container">$y$</span> and raises the right-hand-side of <span class="math-container">$(2)$</span>. Assuming that money supply is unchanged (and prices of course sticky), to equilibrate the money market, the local interest rate must now <em>rise</em> (to reduce demand for money balances that tend to rise to facilitate transactions on the higher <span class="math-container">$y$</span>). Turning to <span class="math-container">$(1)$</span>, with a higher local interest rate, we need expectations for currency <em>depreciation</em> (<span class="math-container">$\dot e &gt;0$</span>) in order for UIRP to hold. But a fiscal expansion, be it temporary or permanent, given a full-employment output or not, is usually associated with local currency <em>appreciation</em> (lower <span class="math-container">$e$</span>).<br />
<strong>So it appears that here too the overshooting effect obtains</strong>: the local currency jumps and appreciates by more than it is consistent with the fiscal expansion, and then it starts to depreciate, and we obtain the needed <span class="math-container">$\dot e &gt;0$</span>.</p>
<p><strong>Let's put this in a (semi) phase diagram</strong>. We need a bit more specification, how the exchange rate and the price level change. Denoting <span class="math-container">$\bar e$</span> and <span class="math-container">$\bar p$</span> their long-run equilibrium values, we specify</p>
<p><span class="math-container">$$\dot e = \theta(\bar e -e)\; \;\;\;\text{eq.}(3a)\\ \dot p = -v(p-\bar p)\; \;\;\;\text{eq.}(3b)$$</span></p>
<p><strong>Notes:</strong> Both parameters are positive. The above look like ad-hoc &quot;adaptive&quot; schemes, but: the <span class="math-container">$\theta$</span> parameter can be determined so as to be compatible with rational expectations, the <span class="math-container">$v$</span> parameter is essentially determined endogenously, while the sluggishness of the price level is an  a priori assumption. This means that while both are governed optimally by a differential equation, the exchange rate <em>can jump</em>, while the price level <em>cannot</em> -it is &quot;slaved&quot; to its differential equation, while the exchange rate is only <em>optimally</em> characterized by its own equation, not chained to it.</p>
<p>By setting <span class="math-container">$\dot e=0$</span> we can obtain the expression for the long-run price level:</p>
<p><span class="math-container">$$\bar p = m+\lambda i^* - \phi y \Rightarrow m = \bar p-\lambda i^* + \phi y\tag{4}$$</span>
Inserting equation <span class="math-container">$(3a)$</span> into <span class="math-container">$(1)$</span>, then <span class="math-container">$(1)$</span> and <span class="math-container">$(4)$</span> into <span class="math-container">$(2)$</span>, we have</p>
<p><span class="math-container">$$\bar p-\lambda i^* + \phi y -p = -\lambda[i^*+ \theta(\bar e -e)] +\phi y$$</span></p>
<p>Simplifying and re-arranging we get</p>
<p><span class="math-container">$$ e = \bar e -\frac {1}{\lambda \theta}(p-\bar p) \equiv QQ\tag{5}$$</span></p>
<p>Equation <span class="math-container">$(5)$</span> is the <em>saddle-path</em> of the economy in the <span class="math-container">$e-p$</span> space (although in Dornbusch's paper time it was rarely called that). And it alone can tell us about overshooting. Note that the current exchange rate level depends positively on both its long-run value but also on the long-run price level. Following Dornbusch, we normalize the initial relative price of local to foreign output to unity, and so we have that <span class="math-container">$\bar e = \bar p$</span>. This means that in the <span class="math-container">$e-p$</span> space the long-run equilibrium will be on the <span class="math-container">$45^{\text{o}}$</span> line. Putting <span class="math-container">$p$</span> on the vertical axis, the <span class="math-container">$QQ$</span> schedule will be downward sloping. Then the phase-diagram (without fixed-point loci, we don't need them since we have the saddle-path) is, for the case of a fiscal expansion,</p>
<p><img src="https://i.sstatic.net/hm7Y4.png" alt="enter image description here" /></p>
<p>We assume that we start at the long-run equilibrium point <span class="math-container">$A$</span>, (and we will discuss this point later). Then, fiscal expansion happens. If it is permanent, it is associated with long-run currency appreciation and a lower long-run price level. This means that the saddle-path of the economy moves permanently to the left and becomes the <span class="math-container">$Q'Q'$</span> schedule. How can the economy find itself on the new saddle-path? It must jump. But it cannot jump vertically because prices are sticky. It can only jump <em>horizontally</em>: so it goes from point <span class="math-container">$A$</span> to point <span class="math-container">$B$</span> which is one the new saddle-path, and then starts to travel towards the new equilibrium, that is point <span class="math-container">$C$</span>. But at <span class="math-container">$B$</span> corresponds exchange rate <span class="math-container">$e_B&lt;\bar e_2$</span>, the latter being the new long-run equilibrium exchange rate. And lower <span class="math-container">$e$</span> means <em>more</em> appreciated currency. So the exchange rate &quot;overshoots&quot; the appreciation level, and then starts to depreciate as the economy moves towards <span class="math-container">$C$</span>.</p>
<p>Assume that the fiscal expansion is considered &quot;temporary&quot; (for decades, this was a good laugh in real-world macroeconomics -not so in the last couple of years). Still, even though the long-run equilibrium point remains <span class="math-container">$A$</span>, there is short-term and mid-term horizons that must adjust. Here the point <span class="math-container">$C$</span> is some sort of &quot;temporary&quot; equilibrium point, which subsequently will start again to approach <span class="math-container">$A$</span>. But the initial adjustment will happen exactly in the same way, and we will have overshooting.</p>
<p>Imagining a monetary expansion, amounts to shift the saddle-path outwards -and obtain overshooting related to depreciation.</p>
<p><strong>A final note:</strong> if one plays around with the diagram, he will realize that
<strong>if we do <em>not</em> start at the long run-equilibrium level, and if we are sufficiently far away from equilibrium on the <span class="math-container">$QQ$</span> schedule, we may obtain <em>undershooting</em> in either case of policy.</strong> For the case at hand (fiscal expansion), imagine that we are at point <span class="math-container">$D$</span> on <span class="math-container">$QQ$</span> that at the time of the expansion corresponds to a price level <em>below</em> the one that corresponds to new equilibrium point <span class="math-container">$C$</span>, and we are moving upwards towards <span class="math-container">$A$</span>. When the fiscal expansion happens and the new saddle-path is <span class="math-container">$Q'Q'$</span> we will jump horizontally alright -but we will jump on the <em>upward moving</em> part of the new schedule (draw an horizontal line from <span class="math-container">$D$</span> to see that), i.e. we will <em>undershoot</em> the needed appreciation, and then we will gradually continue to appreciate the currency.
But this case is not unique to the fiscal expansion scenario -it can also happen in the monetary expansion scenario.</p>
<p>But if in principle the model permits everything to happen, why everybody went bonkers over Dornbusch paper back then, and it continues to be considered one of the high peaks of economic theory? For the historical framework <a href="http://www.imf.org/external/np/speeches/2001/112901.htm#P41_10723" rel="nofollow noreferrer">one can read this</a>. The fact that the model offered a clear and <em>optimizing</em> reasoning to account for the continuous ups-and-downs of the exchange rates (a new &quot;bizarre&quot; phenomenon at the time) I guess was reason enough (and also perhaps because open economies sufficiently far away from the equilibrium point of their nominal variables are not considered very likely -prices do adjust, after all).</p>

CC BY-SA 4.0
2020-06-02T08:27:29.430Z
I'm a little confused by this answer. In the case there was a permanent increase in government spending, wouldn't this imply a real exchange rate appreciation? I'm not sure equilibrium should be on the $\bar{e}=\bar{p}$ line: rather it should be on the locus of points such that $0=\Delta p$ (viz. equilibrium in the goods market).
I had thought that, in general, in the Dornbusch model, changes in real variables (i.e. aggregate demand and long run output) do not change prices, since all adjustment takes place via changes in the exchange rate (no overshooting).
Chaerephon
CC BY-SA 4.0
2014-12-14T18:12:09.103Z
2022-03-20T20:04:47.107Z
2022-03-20T20:04:47.107Z
Alecos Papadopoulos
<p>Is it possible to reproduce the nominal exchange rate undershooting within the framework of the Dornbusch model?</p>
<p>If the government uses fiscal expansion, does undershooting happen?<img src="https://i.sstatic.net/R7ST1.png" alt="enter image description here" /></p>

CC BY-SA 3.0
2014-12-13T15:13:08.657Z
Please provide a reference that contains the Dornbusch model, paper or textbook.
Alecos Papadopoulos
CC BY-SA 3.0
2014-12-13T23:43:16.280Z
Since it's on my teacher's slide, I'm not sure what is the appropriate reference. But the figure I attached on this question is the adapted version of what is on the slides.(My teacher only explained "overshooting" model of Dornbusch model)
user51966
CC BY-SA 3.0
2014-12-14T00:06:09.500Z
In the version of Dornbusch model you examined, was output treated as being fixed at full-employment level, or as being variable?
Alecos Papadopoulos
CC BY-SA 3.0
2014-12-14T02:47:14.653Z
I realized I am not sure about what you mean by the word "undershooting". "Overshooting" is the effect of a initial higher depreciation of the currency compared to its eventual depreciation (so it "overshoots" and then starts to appreciate). What "undershoot" represents? Simply that the currency will appreciate if we have fiscal expansion? (if this the intended meaning, the term "undershooting" is inappropriate). Or what?
Alecos Papadopoulos
CC BY-SA 3.0
2014-12-14T10:06:34.780Z
The lecture says exchange rate undershooting is initial appreciation (or depreciation) but it is not so large that the gradual appreciation (depreciation) follows until exchange rate reaches at certain level.

In contrast, in overshooting, initial appreciation (depreciation) is so large that it then depreciates (appreciates) and reach at certain level.
user51966
CC BY-SA 3.0
2014-12-14T10:09:15.523Z
And to your first comment, I think the only change is government's fiscal expansion, so figure captures only the short-term (price hasn't changed yet).
user51966
<exchange-rates><ode>
Dornbusch model for exchange rate undershooting
user51966
# Dornbusch model for exchange rate undershooting

(Asked by: username_0 on 2014-12-13)

Is it possible to reproduce the nominal exchange rate undershooting within the framework of the Dornbusch model?

If the government uses fiscal expansion, does undershooting happen?![enter image description here](https://i.sstatic.net/R7ST1.png)

**Tags:** exchange-rates, ode
> **Comments:**
> **username_1**: Please provide a reference that contains the Dornbusch model, paper or textbook.  
> **username_0**: Since it's on my teacher's slide, I'm not sure what is the appropriate reference. But the figure I attached on this question is the adapted version of what is on the slides.(My teacher only explained "overshooting" model of Dornbusch model)  
> **username_1**: In the version of Dornbusch model you examined, was output treated as being fixed at full-employment level, or as being variable?  
> **username_1**: I realized I am not sure about what you mean by the word "undershooting". "Overshooting" is the effect of a initial higher depreciation of the currency compared to its eventual depreciation (so it "overshoots" and then starts to appreciate). What "undershoot" represents? Simply that the currency will appreciate if we have fiscal expansion? (if this the intended meaning, the term "undershooting" is inappropriate). Or what?  
> **username_0**: The lecture says exchange rate undershooting is initial appreciation (or depreciation) but it is not so large that the gradual appreciation (depreciation) follows until exchange rate reaches at certain level.  In contrast, in overshooting, initial appreciation (depreciation) is so large that it then depreciates (appreciates) and reach at certain level.  
> **username_0**: And to your first comment, I think the only change is government's fiscal expansion, so figure captures only the short-term (price hasn't changed yet).  

---

## 1 answers

### Answer by username_1 on 2014-12-14. Score: 2 | Accepted answer

This will be long, but the subject is worth it.

The gist of the "exchange rate overshooting" model in [Dornbusch, R. (1976). Expectations and exchange rate dynamics. The Journal of Political Economy, 1161-1176.](https://www.jstor.org/stable/1831272) can be given using two equations:

**Uncovered Interest Rate Parity (UIRP)**
$$i = i^\*+ \dot e\tag{1}$$

where $i$ is local interest rate, $i^\*$ is international interest rate, and $e$ is the logarithm of the *foreign* exchange rate, i.e. local currency units per unit of international currency basket. So an *increase* in $e$ reflects *depreciation* of the local currency. A *decrease* in $e$ reflects *appreciation* of the local currency (I would prefer to use the local exchange rate, where conceptually we link "rise" with "rise", but it is unfriendly to the phase diagram that it will follow). Then $\dot e$ is the expected percentage change on the exchange rate.

**Money-market Equilibrium (in logarithms)**

$$m-p = -\lambda i +\phi y \tag{2}$$

where $m$ is money supply, $p$ is the local price level, and $y$ is local output.

The assumptions are: international interest rate is fixed. There is perfect capital mobility, so the local interest rate is a "jump" variable, being able to adjust instantaneously (and so eq. $(2)$ holds always). Local output is at full-employment level. Local prices are "sticky": this means that the local price-level is not a "jump" variable anymore, but its evolution is governed by a difference/differential equation that reflects gradual adjustment. Money supply is long-term neutral.

The famous "overshooting" effect from an *unanticipated* permanent expansion of $m$ can then be derived: an increase in $m$ raises the left-hand side of $(2)$. Since prices are sticky (don't jump) and output is fixed, the only way that equilibrium can continue to hold in the money market is by a *decrease* in $i$ (due to the minus sign), so that that eq. $(2)$ continues to hold.  
Given the fall in $i$, and the assumption of constant $i^\*$, the only way now that $(1)$ can hold with a lower $i$ is for $\dot e$ to be *negative*, i.e. if the local currency is expected to *appreciate*. But raising the long-term neutral local money supply can only create expectations for local currency *depreciation*, not appreciation.  
**"Overshooting"** cuts the Gordian Knot: the *immediate* response of the exchange rate $e$ is to *depreciate by more* (overshooting) than the eventual depreciation that is consistent with the increase in the money supply, and then it starts to *appreciate* to reach its new level (and so we can have $\dot e<0$).

The question asks about "**fiscal expansion**". How is this to be represented in the model? The only possible way is to assume that output can rise due to the fiscal expansion.

A fiscal expansion increases $y$ and raises the right-hand-side of $(2)$. Assuming that money supply is unchanged (and prices of course sticky), to equilibrate the money market, the local interest rate must now *rise* (to reduce demand for money balances that tend to rise to facilitate transactions on the higher $y$). Turning to $(1)$, with a higher local interest rate, we need expectations for currency *depreciation* ($\dot e >0$) in order for UIRP to hold. But a fiscal expansion, be it temporary or permanent, given a full-employment output or not, is usually associated with local currency *appreciation* (lower $e$).  
**So it appears that here too the overshooting effect obtains**: the local currency jumps and appreciates by more than it is consistent with the fiscal expansion, and then it starts to depreciate, and we obtain the needed $\dot e >0$.

**Let's put this in a (semi) phase diagram**. We need a bit more specification, how the exchange rate and the price level change. Denoting $\bar e$ and $\bar p$ their long-run equilibrium values, we specify

$$\dot e = \theta(\bar e -e)\; \;\;\;\text{eq.}(3a)\\ \dot p = -v(p-\bar p)\; \;\;\;\text{eq.}(3b)$$

**Notes:** Both parameters are positive. The above look like ad-hoc "adaptive" schemes, but: the $\theta$ parameter can be determined so as to be compatible with rational expectations, the $v$ parameter is essentially determined endogenously, while the sluggishness of the price level is an a priori assumption. This means that while both are governed optimally by a differential equation, the exchange rate *can jump*, while the price level *cannot* -it is "slaved" to its differential equation, while the exchange rate is only *optimally* characterized by its own equation, not chained to it.

By setting $\dot e=0$ we can obtain the expression for the long-run price level:

$$\bar p = m+\lambda i^\* - \phi y \Rightarrow m = \bar p-\lambda i^\* + \phi y\tag{4}$$
Inserting equation $(3a)$ into $(1)$, then $(1)$ and $(4)$ into $(2)$, we have

$$\bar p-\lambda i^\* + \phi y -p = -\lambda[i^\*+ \theta(\bar e -e)] +\phi y$$

Simplifying and re-arranging we get

$$ e = \bar e -\frac {1}{\lambda \theta}(p-\bar p) \equiv QQ\tag{5}$$

Equation $(5)$ is the *saddle-path* of the economy in the $e-p$ space (although in Dornbusch's paper time it was rarely called that). And it alone can tell us about overshooting. Note that the current exchange rate level depends positively on both its long-run value but also on the long-run price level. Following Dornbusch, we normalize the initial relative price of local to foreign output to unity, and so we have that $\bar e = \bar p$. This means that in the $e-p$ space the long-run equilibrium will be on the $45^{\text{o}}$ line. Putting $p$ on the vertical axis, the $QQ$ schedule will be downward sloping. Then the phase-diagram (without fixed-point loci, we don't need them since we have the saddle-path) is, for the case of a fiscal expansion,

![enter image description here](https://i.sstatic.net/hm7Y4.png)

We assume that we start at the long-run equilibrium point $A$, (and we will discuss this point later). Then, fiscal expansion happens. If it is permanent, it is associated with long-run currency appreciation and a lower long-run price level. This means that the saddle-path of the economy moves permanently to the left and becomes the $Q'Q'$ schedule. How can the economy find itself on the new saddle-path? It must jump. But it cannot jump vertically because prices are sticky. It can only jump *horizontally*: so it goes from point $A$ to point $B$ which is one the new saddle-path, and then starts to travel towards the new equilibrium, that is point $C$. But at $B$ corresponds exchange rate $e\_B<\bar e\_2$, the latter being the new long-run equilibrium exchange rate. And lower $e$ means *more* appreciated currency. So the exchange rate "overshoots" the appreciation level, and then starts to depreciate as the economy moves towards $C$.

Assume that the fiscal expansion is considered "temporary" (for decades, this was a good laugh in real-world macroeconomics -not so in the last couple of years). Still, even though the long-run equilibrium point remains $A$, there is short-term and mid-term horizons that must adjust. Here the point $C$ is some sort of "temporary" equilibrium point, which subsequently will start again to approach $A$. But the initial adjustment will happen exactly in the same way, and we will have overshooting.

Imagining a monetary expansion, amounts to shift the saddle-path outwards -and obtain overshooting related to depreciation.

**A final note:** if one plays around with the diagram, he will realize that
**if we do *not* start at the long run-equilibrium level, and if we are sufficiently far away from equilibrium on the $QQ$ schedule, we may obtain *undershooting* in either case of policy.** For the case at hand (fiscal expansion), imagine that we are at point $D$ on $QQ$ that at the time of the expansion corresponds to a price level *below* the one that corresponds to new equilibrium point $C$, and we are moving upwards towards $A$. When the fiscal expansion happens and the new saddle-path is $Q'Q'$ we will jump horizontally alright -but we will jump on the *upward moving* part of the new schedule (draw an horizontal line from $D$ to see that), i.e. we will *undershoot* the needed appreciation, and then we will gradually continue to appreciate the currency.
But this case is not unique to the fiscal expansion scenario -it can also happen in the monetary expansion scenario.

But if in principle the model permits everything to happen, why everybody went bonkers over Dornbusch paper back then, and it continues to be considered one of the high peaks of economic theory? For the historical framework [one can read this](http://www.imf.org/external/np/speeches/2001/112901.htm#P41_10723). The fact that the model offered a clear and *optimizing* reasoning to account for the continuous ups-and-downs of the exchange rates (a new "bizarre" phenomenon at the time) I guess was reason enough (and also perhaps because open economies sufficiently far away from the equilibrium point of their nominal variables are not considered very likely -prices do adjust, after all).

> **Comments:**
> **username_2**: I'm a little confused by this answer. In the case there was a permanent increase in government spending, wouldn't this imply a real exchange rate appreciation? I'm not sure equilibrium should be on the $\bar{e}=\bar{p}$ line: rather it should be on the locus of points such that $0=\Delta p$ (viz. equilibrium in the goods market). I had thought that, in general, in the Dornbusch model, changes in real variables (i.e. aggregate demand and long run output) do not change prices, since all adjustment takes place via changes in the exchange rate (no overshooting).  

---

