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I'm studying the Dornbusch overshooting model of the exchange rate. Specifically, I'm studying the model presented in a textbook by Copeland (2014).

The economy is represented by the following equations:

$y^d=h(e-p)$

$m_s-p=k\bar{y}-lr$

$\Delta p =\pi(y^d-\bar{y})$

$r=r^*+\Delta e^e$

$\Delta e^e=\theta(\bar{e}-e), \theta>0$

where $e$ is the log nominal exchange rate, $\bar{e}$ is the long-run value of the log nominal exchange rate, 𝑝(π‘βˆ—) is the log domestic (foreign) price level, $m_s$ stands for log nominal money supply, $\bar{y}$ is potential log output, π‘Ÿ(π‘Ÿβˆ—) is the log domestic (foreign) nominal interest rate, $\Delta e^e$ is the expected change in the log nominal exchange rate, and βˆ†π‘ is the rate of change for log prices.

You can simplify these equations into 2:

$p=m_s-ky+lr^*-l\theta(e-\bar{e})$

$\Delta p = \pi(h(e-p)-\bar{y})$

You then can find the long-run values of e, p, and q:

$\bar{q}=\frac{\bar{y}}{h}$

$\bar{p}=\bar{m}-k\bar{y}+lr^*$

$\bar{e}=(\frac{1}{h}-k)\bar{y}-k\bar{y}+lr^*$

What I'm confused about is the functioning of fiscal policy. If there is a permanent increase in government spending, $y^d=h(e-p)+g$, correct? From what I can gather, real shocks in the model won't lead to any changes in long-run prices. Indeed, if we plug this equation into the model, only $\bar{q}$ and $\bar{e}$ change.

I have seen an exam question which asks how fiscal policy might be used to bring $q, e, p$ to their equilibrium values after an increase in $r^*$. But surely this would necessitate a decrease in $\bar{p}$ (since $\bar{p}$ has increased with an increase in $r^*$), from which I gather isn't possible in the Dornbusch model?

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There are some mistakes in your presentation of the Dornbusch model.

$𝑝(π‘βˆ—)$ is the domestic (foreign) price level, $m_s$ stands for nominal money supply, $\bar{y}$ is potential output,

These actually stand for the log of price level, log of money supply and log of potential output. In international macro an unwritten rule is that lowercase letters represent logs of uppercase letters. For example, money supply is $M$ and $m = \ln M$. To be consistent you cannot add to the Dorrnbush model just government spending but you should add log of government spending. So $g$ instead of $G$:

$$y^d=h(eβˆ’p)+g$$

From what I can gather, real shocks in the model won't lead to any changes in long-run prices. Indeed, if we plug this equation into the model, only $\bar{q}$ and $\bar{e}$ change.

The permanent government spending can in principle affect the price level through $\bar{y}$, but it depends on the background assumptions of the model. In Copeland, exchange rates and international finance (6th ed), you can see on pp207 that long run equilibrium values are derived based on arguing that in long run equilibrium $y^d =\bar{y}$. So in principle it could be affected by $g$ through $\bar{y}$ unless they are offset by changes in the other variables.

However, Copeland is not really clear on all background assumptions he makes in the model that he presents (at least not in the edition I have). For example, if we assume flexible exchange rate perfect capital mobility and the shock was completely unanticipated the $\bar{y}$ wont change in response to fiscal expansion as any change in $g$ will be just offset by changes to the exchange rate (due to capital inflows) and fiscal policy will only be able to expand output of its recipients at the expense of tradable sector in the economy. If some of the assumptions above are violated then the price level can actually change in the Dornbusch model.

In case you are interested in this for your own sake and you want to understand how price level responds to fiscal policy in general I recommend the following paper Devereux & Purvis (1990). Fiscal policy and the real exchange rate. They actually show that in the Dornbusch model fiscal policy can affect the price level and also discuss in greater detail how the sign of the effect depends on model parameters and assumptions of the model. If you are studying for an exam I would recommend clarifying this with your supervisor because Copeland does not provide detailed list of assumptions (at least not in the chapter 7.3 where he gives formal exposition of the model).

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    $\begingroup$ Thanks very much for the reply, and apologies for the imperfect exposition of the model, you're correct. I believe in the original 1976 paper, Dornbusch assumes perfect capital mobility (viz. $r=r^*$ after $e$ adjusts) as well as flexible exchange rates. As you explain, it makes sense that $\bar{p}$ won't be affected: since $r=r^*$, any increase in government spending will be offset by a fall in the current account because of appreciation. Thank you for the reference, I will check that out. It is certainly tricky understanding what assumptions are made without expert knowledge of the topic. $\endgroup$ – Chaerephon Jun 2 at 13:15

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