The fragility of an incomplete monetary union: equations, figures, and Mathematica

1. Expectations are set before policy is chosen

The order of decisions is essential. Households and firms first make nominal commitments, such as wage agreements, using expected inflation. An unexpected shock may then occur. The authorities subsequently choose actual inflation, taking those expectations and commitments as given.

A policy can therefore have two different evaluations. Before commitments are made, low inflation may be desirable. After commitments are fixed, unexpected inflation can temporarily reduce unemployment. Private agents understand this incentive. Their expectations respond to the policy they anticipate, and those expectations alter the choices available to the authorities.

This is the time-consistency problem emphasized by Kydland and Prescott (1977). It does not require policymakers to be confused or private agents to be systematically mistaken. Each can act coherently at its own decision stage, yet the resulting outcome can be worse than an enforceable commitment.

In the derivation, we therefore hold expected inflation fixed when differentiating the loss function. We impose rational expectations only after finding the authorities’ conditional best response. Setting actual and expected inflation equal at the beginning would remove the very surprise channel that the policymaker considers.

2. The Phillips curve and the authorities’ objective

De Grauwe starts with the unemployment version of the Phillips curve:

If actual inflation exceeds expected inflation, \(\pi^e-\pi\) is negative, and unemployment falls below the level it would otherwise attain. If inflation is exactly anticipated and the shock is zero, unemployment equals \(U_N\). The model offers a temporary effect of an inflation surprise; it does not offer a permanent unemployment reduction from fully anticipated inflation.

The notes also provide an alternative output representation:

Here \(Y\) is output, \(Y_n\) is its natural level, and \(\theta\) measures the output response to unexpected inflation. Notice the shock convention: a positive \(\varepsilon\) increases output in this representation, whereas it increases unemployment in the preceding equation. We retain the two source equations as written and use the unemployment equation throughout the calculations below. The same signed disturbance should not be substituted interchangeably between the two formulations.

The authorities minimize a quadratic loss:

The first term penalizes inflation. The second penalizes unemployment above or below the target. Squaring a deviation makes its contribution nonnegative and makes a large deviation disproportionately costly. The parameter \(\beta\) describes preferences: a larger value means that the unemployment objective receives more weight.

The target lies below the natural unemployment rate:

The following abbreviation will make the later loss comparisons easier to read:

For \(U_N>0\) and \(\lambda<1\), the target gap \(k\) is positive. This is a gap between two reference unemployment rates. It is distinct from the realized cyclical deviation \(U-U_N\) and from the shock \(\varepsilon\). The ambition to eliminate \(k\) through monetary policy is what generates the incentive to create surprise inflation.

Reading Figure 1. Each curve joins combinations of inflation and unemployment that give the same loss. The preferred point is \((\bar U,0)\). Moving to an outer curve increases the loss. These curves describe preferences; they do not tell us which combinations can be achieved. The Phillips curve supplies that constraint. The horizontal origin shown at \(\bar U\) is a normalization of the diagram, not a claim that target unemployment must literally be zero.

Substituting the Phillips curve and the target into the loss function gives De Grauwe’s expression:

The entire expression inside square brackets is \(U-\bar U\). Keeping that interpretation in view is more useful than treating the formula as an arbitrary polynomial.

Reading Figure 2. Rearranging the no-shock Phillips curve gives \(\pi=\pi^e+(U_N-U)/a\). With unemployment on the horizontal axis and inflation on the vertical axis, its slope is \(-1/a\). Moving along a curve changes actual inflation while expectations remain fixed. Moving to the higher curve changes expected inflation. On the vertical line at \(U_N\), actual inflation equals expected inflation.

3. Deriving optimal inflation, one step at a time

The authorities choose \(\pi\), taking \(\pi^e\) and the observed shock as given. Differentiating requires the chain rule: inflation changes the direct inflation loss and also changes unemployment. Since \(dU/d\pi=-a\), the derivative of the unemployment component contains the factor \(-a\):

Dividing by two makes the economic balance visible:

The marginal inflation cost equals the marginal benefit of bringing unemployment closer to its target. A larger target gap strengthens the incentive to accommodate. The second derivative is \(2(1+\beta a^2)>0\), so the first-order condition identifies a unique minimum for given expectations.

To solve explicitly, expand the square-bracket term in the first-order condition. The term containing \(-a\pi\) generates an additional \(\beta a^2\pi\). Gather this term together with \(\pi\):

Move the remaining terms to the other side and divide by \(1+\beta a^2\). De Grauwe’s optimal inflation rule is:

Its three components have distinct interpretations. The first reflects the ambition to reduce unemployment below its natural rate. The second is the response to inflation already expected by the private sector. The third accommodates the newly observed unemployment shock. The common denominator arises from the optimization problem; it is not an additional parameter.

This rule is a conditional best response. It tells us what the authority chooses after expectations have been set. We still need to find the expectations consistent with the policy private agents should anticipate.

4. Rational expectations and the inflation bias

Set the shock to zero for now. If expectations are correct, \(\pi=\pi^e\). Substituting this restriction into the optimal policy rule gives:

Bring the term containing expected inflation on the right to the left and factor out \(\pi^e\):

The subtraction inside the brackets is especially simple. Expressing one over the common denominator leaves a numerator equal to one:

The common denominator cancels from the two sides. Thus:

Equivalently, equilibrium inflation is \(\beta a k\). Substituting \(\pi=\pi^e\) back into the Phillips curve gives \(U=U_N\). The surprise disappears, so the employment benefit disappears; positive inflation remains.

This is the inflation bias associated with discretion in the tradition of Barro and Gordon (1983). Expectations are correct in equilibrium. The difficulty is that the authorities would like to exploit a surprise after nominal commitments are fixed, and private agents anticipate that incentive.

Reading Figure 3. Each tangency between a loss curve and a Phillips curve is an optimal choice for one expectation. The dashed upward-sloping line collects these tangencies: it represents \(\pi=\beta a(U-\bar U)\). Rational expectations add another requirement, \(U=U_N\) in the absence of a shock. Their intersection gives the discretionary equilibrium.

The graph is a comparison of policy choices and expectation configurations. It does not specify how the economy moves through time between tangencies. Similarly, the formula identifies the equilibrium inflation bias without specifying a process by which beliefs converge to it.

5. Stabilization after an unexpected shock

The absence of a permanent unemployment gain from anticipated inflation does not make stabilization ineffective. Suppose the shock has mean zero and is observed after expectations have been set. Expected inflation remains equal to the anticipated component \(\beta a k\), while realized inflation responds to the new shock:

To recover unemployment, substitute this inflation rule into the Phillips curve. The anticipated inflation component cancels against expected inflation, leaving only the surprise:

Policy offsets part of the shock. Collecting the shock terms over a common denominator gives:

The factor \(1/(1+\beta a^2)\) is the unemployment response to one unit of the shock. For a positive stabilization weight, it lies below one. The greater willingness to accept inflation absorbs part of the unemployment disturbance.

The variance calculation follows from a general identity: multiplying a random variable by a constant multiplies its variance by the square of that constant. Natural unemployment is fixed in this exercise, so it contributes no variance:

For example, halving the unemployment response reduces its variance to one quarter of the shock variance. This is why the denominator is squared. Stronger stabilization can lower unemployment volatility while increasing average inflation bias when \(k>0\). These are different consequences of the same preference parameter.

6. Different preferences and asymmetric shocks

The next two diagrams separate two reasons countries may want different monetary policies. Figure 4 varies preferences. Figure 5 introduces an asymmetric shock while keeping preferences the same. This distinction helps explain both the attraction of monetary commitment and its potential stabilization cost, as discussed by De Grauwe (2020).

Reading Figure 4. In the illustration, Italy puts more weight on the unemployment objective than Germany. Holding the other parameters fixed, a higher \(\beta\) produces a higher discretionary inflation rate, since \(\pi=\beta a k\). Both countries nevertheless end up at their natural unemployment rates when inflation is correctly anticipated.

The dashed horizontal line, labeled \(\pi^*\), marks Germany’s reference inflation. In the four-loss comparison below, foreign inflation is normalized to zero.

For Italy, a credible commitment to Germany’s lower inflation can remove part of the extra inflation bias without a permanent unemployment cost in this no-shock comparison. The word “credible” matters. If actual inflation is forced down while people still expect the earlier, higher rate, unemployment rises. The result with expectations already adjusted should not be confused with the cost of surprising private agents with disinflation.

Reading Figure 5. The countries now have the same policy preferences, but Italy experiences an adverse unemployment shock. Its Phillips curve shifts upward in inflation-unemployment space. Italy would prefer some inflation accommodation; Germany, which has not experienced the same disturbance, has no corresponding reason to adjust.

Algebraically, the shock raises Italy’s desired inflation by \(\beta a\varepsilon/(1+\beta a^2)\) when prior expectations are fixed. Keeping inflation unchanged leaves more of the unemployment shock unabsorbed. The disagreement here comes from different economic disturbances, even with identical preferences. The figure helps identify the source of the policy tension; it does not itself solve a two-country central bank’s joint optimization problem.

7. Connecting inflation to a fixed exchange rate

The notes use a purchasing-power-parity relation to connect domestic inflation to the exchange rate:

Here \(\dot S\) denotes the depreciation rate, interpreted as the change in a log exchange rate, and \(\pi^*\) denotes foreign inflation. Under this simplifying relation, a fixed exchange rate requires domestic inflation to equal foreign inflation. Normalizing foreign inflation to zero means that maintaining the peg corresponds to \(\pi=0\).

The axes of Figures 6 and 7 use \(\dot e\) for the exchange-rate change. I retain this source convention. The dot on the exchange-rate symbol denotes a change; the superscript \(e\) in \(\pi^e\) denotes an expectation. These two uses of the letter should be read differently.

From this point, set \(\varepsilon=0\). We want to isolate how expectations affect the incentive to defend the peg, with the other fundamentals held fixed. Let \(C\) be the additional cost of abandoning the exchange-rate commitment, measured in the same loss units as the policy objective. The following four losses exclude \(C\). We add it once, when comparing a devaluation decision with continued defense.

8. Four situations and four policy losses

There are two expectation configurations and two possible decisions. Private agents may expect the peg to survive or expect devaluation. The authorities may then maintain the peg or devalue. Writing out all four combinations prevents us from comparing policies evaluated under different expectations.

8.1 Expected inflation is delivered: discretion

Start with the discretionary equilibrium already derived:

Because inflation is expected and delivered, unemployment equals its natural rate. The target gap remains \((1-\lambda)U_N\). Substituting inflation and this gap into the loss function gives:

Both terms contain the squared target gap. Factoring it out yields:

The single factor \(1+\beta a^2\) combines the unemployment loss with the additional inflation loss. The notation \(L_{\mathrm{DIS}}\) is the macroeconomic policy loss. If this outcome is reached by abandoning an existing peg, total loss at the regime decision is \(L_{\mathrm{DIS}}+C\).

8.2 Zero inflation is expected and delivered: the credible peg

There is neither surprise inflation nor an unemployment shock. Unemployment is therefore at its natural rate, while the inflation component of loss is zero:

This is smaller than the discretionary loss when \(a,\beta,k>0\). Commitment avoids inflation without changing natural unemployment. However, this comparison alone does not establish credibility. We must check whether the authority wants to depart from zero inflation after private agents have formed zero-inflation expectations.

8.3 The peg is expected, but the authority devalues

Keep \(\pi^e=0\) and use the conditional best response. The authority chooses some surprise inflation, reducing unemployment toward the target:

The corresponding loss contains both the inflation cost and the remaining unemployment-target gap:

The crucial simplification is the target gap. In the shorter \(k\) notation, it is:

Substituting this gap produces the next line in De Grauwe’s derivation:

Factor out the common squared fraction, then cancel one factor of \(1+\beta a^2\):

The subscript “cheat” identifies a departure from the announced peg. This outcome is useful for checking the temptation to deviate; it is not a rational-expectations equilibrium in which agents are permanently surprised. Whether deviation occurs also depends on the additional cost \(C\).

Reading Figure 6. With zero inflation expected, compare the zero-inflation peg with the point where the Phillips curve is tangent to an inner loss curve. The latter delivers a smaller macroeconomic loss. The gap between the relevant loss levels is the temptation to devalue. The marked distances in the drawing illustrate the comparison; their lengths are not numerical welfare units.

8.4 Devaluation is expected, but the authority defends the peg

Now change the expectation configuration. Private agents expect the discretionary inflation rate, but the authority keeps actual inflation at zero:

Inflation falls below expectations. The Phillips curve therefore implies unemployment above its natural rate:

The authority avoids inflation but bears the full unemployment cost:

Replacing the target by \(\lambda U_N\) and gathering the unemployment terms gives:

The square on \(1+\beta a^2\) has a direct economic origin. The unemployment-target gap is now \(k(1+\beta a^2)\), and the loss function squares that entire gap. Compared with the credible peg, expectations of devaluation have made defending the same exchange-rate commitment more painful.

Reading Figure 7. At the maintained peg, zero inflation is below the rate private agents expect, so unemployment is high. If the authority instead validates the expected inflation, unemployment returns to \(U_N\) and the macroeconomic loss becomes \(L_{\mathrm{DIS}}\). The relevant comparison is between these two outcomes at the same expected inflation. Again, the marked horizontal interval is schematic.

8.5 Compare decisions within an expectation row

Read across a row. Expectations are already set when the authority chooses its action. Comparing the credible-peg loss in the first row with discretion in the second row tells us about outcomes under different expectations. It does not tell us whether the authority wants to deviate from an announcement at the moment of decision.

9. Temptation and the cost of defending the peg

De Grauwe summarizes the incentives using two differences between policy losses. When no devaluation is expected, the gross gain from abandoning the peg is:

Factoring out \(\beta k^2\) gives:

This is the temptation to devalue. The authority accepts the separate cost \(C\) only if the saving in macroeconomic loss is large enough to compensate.

When devaluation is expected, the saving from abandoning defense is:

Factor out \(\beta k^2(1+\beta a^2)\). The remaining difference is \((1+\beta a^2)-1=\beta a^2\), giving:

This is the expression labeled “cost of defence” in the notes. More precisely, it is the extra macroeconomic loss from defending rather than allowing the already-expected devaluation. It is distinct from \(C\), which is the cost of abandoning the peg.

A useful additional observation follows by dividing the two expressions, for positive \(a\), \(\beta\), and \(k\):

Pessimistic expectations magnify the incentive to abandon the commitment because they make defense more costly. This is the channel through which beliefs can help produce the outcome they anticipate.

10. Three regions and the possibility of two equilibria

For positive \(a\), \(\beta\), and \(k\), the defense loss difference exceeds temptation. The location of \(C\) relative to these two quantities determines whether the authority’s decision depends on expectations.

10.1 The cost of devaluation is high

Even when devaluation is expected, the saving from abandoning defense is smaller than \(C\). The authority maintains the peg in both expectation configurations. Expectations of devaluation would not be fulfilled, so they cannot sustain the adverse equilibrium in this comparison. The peg is credible because the incentives support it.

10.2 The cost of devaluation is low

Devaluation is attractive even when private agents expect the peg to survive. Optimistic expectations therefore cannot sustain the peg. Agents anticipate devaluation, and the authority validates that expectation. The fundamentals place the economy in a region with a unique devaluation equilibrium.

10.3 The cost lies between the two incentives

When the peg is expected to survive, temptation is smaller than \(C\), and the authority maintains it. When devaluation is expected, the cost of defense exceeds \(C\), and the authority devalues. Both expectations can be validated by the decisions they induce.

This is the central multiple-equilibrium result. The parameters \(a\), \(\beta\), \(k\), and \(C\) need not change. Expected inflation changes the unemployment cost of keeping actual inflation at zero, and that changes the preferred policy. This mechanism is closely related to the self-fulfilling currency-crisis logic in Obstfeld (1996).

The strict inequalities identify clear regions. At equality, at least one policy choice is tied. Also, the model does not determine the date at which expectations change or assign probabilities to the two equilibria. It identifies the conditions under which expectations can matter in this particular way.

11. Reading the threshold diagram

Figure 8 varies the unemployment weight \(\beta\), holding \(a\), \(k\), and the cost \(C=C_0\) fixed. The source writes the two curves as:

Read the horizontal axis as a preference parameter. It is neither time nor the size of a new unemployment shock. As \(\beta\) rises, both gains from devaluation rise. The defense curve lies above the temptation curve and crosses the horizontal cost line first.

The first intersection, \(\beta_1\), is where defense becomes just as costly as devaluation. The second, \(\beta_2\), is where devaluation becomes attractive even without pessimistic expectations. Thus the graph contains three regions:

The steep defense curve has no finite vertical asymptote: its formula is a polynomial in \(\beta\). The drawing illustrates how fast it rises over the chosen range. The plot’s numerical scale is illustrative; the threshold ordering comes from the model.

12. From an exchange-rate commitment to an incomplete monetary union

The exchange-rate model establishes an expectations mechanism. To understand the title of this article, we must now explain how that mechanism carries over to sovereign borrowing within a monetary union.

Under a national exchange-rate peg, the authority can eventually choose devaluation. A member government in a monetary union does not independently choose inflation in the common currency. In the sovereign-debt argument developed by De Grauwe (2020, Chapter 5, especially pp. 111–115), the relevant commitment is repayment, and the adverse decision is default. The economic link is that expectations change the conditions under which the government must honor its commitment.

Suppose investors expect repayment. They are willing to hold government bonds and refinance maturing debt at relatively favorable terms. Servicing the debt is then easier, helping validate their expectation. If investors become pessimistic, they may demand a higher return or refuse to roll over debt. Refinancing becomes more difficult, fiscal adjustment becomes more painful, and default becomes more attractive.

The pass-through is gradual for previously issued long-maturity fixed-rate debt: a higher market yield does not instantly change every existing coupon. It matters as debt is refinanced or new borrowing is required. De Grauwe explicitly notes the importance of maturity for the gap between expected and unexpected default incentives.

De Grauwe (2011) emphasizes the institutional vulnerability: national governments borrow in a common currency they cannot individually create to meet a liquidity shortfall. A deterioration in confidence can therefore affect their capacity to refinance and help generate the default risk investors fear. The analogy with the peg is that defending the commitment becomes more costly under pessimistic expectations.

An incomplete monetary union combines a common currency with important national fiscal responsibilities. In De Grauwe’s analysis, institutional arrangements governing liquidity support and fiscal risk sharing are therefore central to stability. A credible liquidity backstop can prevent a cash shortage alone from forcing default. It cannot eliminate every incentive to default when the underlying solvency problem is severe.

This is a theoretical distinction between liquidity and solvency, not a claim that every sovereign crisis is caused exclusively by market sentiment. Weak fundamentals may make default attractive under both expectation configurations, just as sufficiently low \(C\) makes devaluation attractive in both rows of the exchange-rate model.

De Grauwe and Ji (2013) investigate this mechanism empirically for the euro-area crisis. They find that changes in debt and fiscal-space indicators do not account for the entire increase in several peripheral countries’ sovereign spreads during 2010–2011, and interpret an additional component through self-fulfilling market expectations. This provides an empirical application of the argument; the numerical illustrations in the present notebook are theoretical examples rather than estimates from that study.

13. How the Mathematica notebook implements the argument

The notebook follows the same sequence as the economics. The displayed equations retain De Grauwe’s notation, while a few plain code names make symbolic assignments easier to read:

The first calculations encode the Phillips curve, the target, and the loss function. The ordinary multiplication signs in the code make the economic products explicit:

(* Run after the notebook's ClearAll and assumptions cell. *)
U = UN + a*(pe - p) + \[Epsilon];
UB = \[Lambda]*UN;
L = p^2 + \[Beta]*(U - UB)^2;
firstOrder = D[L, p];
pOptimal = FullSimplify[
  p /. First[Solve[firstOrder == 0, p]], assumptions];

D differentiates the loss. Solve returns a rule for the inflation variable, and /. substitutes that rule into p. FullSimplify uses the parameter assumptions to express the result compactly. The algebra still corresponds to the chain rule and rearrangement shown above.

Rational expectations then impose consistency on this conditional solution:

peRE = FullSimplify[
  pe /. First[Solve[pe == (pOptimal /. \[Epsilon] -> 0), pe]],
  assumptions];

The replacement \(\varepsilon=0\) selects the no-shock case. The equation inside Solve equates expected inflation to the authority’s best response. It is imposed at this stage because expectations must be treated as fixed during the earlier optimization.

For the graphs, Plot draws the Phillips and incentive curves, ParametricPlot draws loss contours, and Graphics supplies labels, arrows, and guides. For a fixed positive loss level and \(\beta>0\), the contour parameterization is:

Substitution verifies the contour: the two loss terms become \(L\sin^2t\) and \(L\cos^2t\), which sum to \(L\). Restricting \(t\) to the first quadrant produces the portions drawn in Figure 1. This ties the picture directly to the loss function.

The evaluated V3 notebook reports 15 checks passed out of 15. These cover the four loss formulas, the two loss differences, and the graphical objects. The figures in this article are the actual Mathematica exports accompanying that evaluated notebook.

To reproduce the calculations, save and open the notebook, then choose Evaluation → Evaluate Notebook. Its export section writes all eight figures in PDF, SVG, and 300 dpi PNG formats to a DeGrauwe_Figures folder beside the notebook. The final interactive panels also have buttons to export the graph at the selected parameter values.

14. Two interactive exercises to finish the argument

Open the final section of the accompanying Mathematica notebook (V3) to use the two interactive Manipulate panels. Predict a result first, change one control at a time, and then connect the change on screen to a particular equation. The first panel uses the exchange-rate losses derived above. The second introduces an explicitly separate illustration of sovereign-debt fragility.

14.1 Expectations and the credibility of a peg

Begin at \(a=\beta=k=C=1\). The panel displays the four total losses and identifies the two-equilibrium region. Increase \(C\), then restore its initial value and decrease it. Next, restore \(C=1\) and increase the target gap \(k\).

14.2 Financing stress and a liquidity backstop

The second exercise illustrates the sovereign-debt discussion in De Grauwe (2020) and the benefit-cost framework in the appendix to De Grauwe (2011). It uses the following auxiliary quadratic functions:

These are teaching assumptions, not further equations obtained from the Phillips-curve derivation. Here \(S\) denotes an adverse solvency shock, distinct from the exchange-rate notation used earlier. \(B_U\) is the benefit of default when default is unexpected; \(B_E\) is the benefit when it is expected and financing stress raises that benefit. The parameter \(\ell\geq0\) controls the stress effect. In this exercise, \(C\) denotes the cost of default.

The difference between the two benefit curves is \(\ell S^2\). When the default cost lies between them, repayment is preferred with favorable financing conditions, but default is preferred under stress. The two threshold shocks follow by setting each benefit equal to \(C\):

Start with \(b=1/2\), \(\ell=1.5\), \(C=1\), and \(S=1\). Then set financing stress to zero to represent an ideal liquidity backstop. Finally, increase the solvency shock to \(S=1.7\).

15. What the equations reveal about monetary fragility

The central result rests on a sequence of choices. Private expectations affect unemployment under a peg, or financing conditions in a sovereign-debt setting. Those conditions change the authority’s incentive to honor its commitment. In the intermediate region, a favorable expectation makes the commitment worth maintaining, while an adverse expectation makes abandoning it optimal.

Fundamentals still matter. They determine whether the economy is in a region where expectations can select between outcomes. The model also shows why announcing a policy is insufficient: credibility requires that carrying it out remain optimal after private decisions have been made.

Working through the equations and reproducing the figures makes this reasoning inspectable. Every curve has an economic meaning, every loss corresponds to a specified expectation and action, and every threshold comes from a comparison of explicit alternatives. That is the value of combining De Grauwe’s exposition with Mathematica: the reader can move from an institutional argument to the calculations that support it, and then change the assumptions to see which conclusions survive.

References and related posts

1. Barro, Robert J., and David B. Gordon. 1983. “A Positive Theory of Monetary Policy in a Natural Rate Model.” Journal of Political Economy 91(4): 589–610. https://doi.org/10.1086/261167.
2. De Grauwe, Paul. 2011. The Governance of a Fragile Eurozone. CEPS Working Document No. 346, May. CEPS publication page.
3. De Grauwe, Paul. 2020. Economics of Monetary Union. 13th edition. Oxford University Press. Chapter 5, “The Fragility of Incomplete Monetary Unions.” ISBN 9780198849544. Publisher’s page. The equations and eight-figure sequence follow Professor De Grauwe’s accompanying lecture notes.
4. De Grauwe, Paul, and Yuemei Ji. 2013. “Self-fulfilling Crises in the Eurozone: An Empirical Test.” Journal of International Money and Finance 34: 15–36. https://doi.org/10.1016/j.jimonfin.2012.11.003. CAMA working-paper version.
5. Kydland, Finn E., and Edward C. Prescott. 1977. “Rules Rather than Discretion: The Inconsistency of Optimal Plans.” Journal of Political Economy 85(3): 473–491. https://doi.org/10.1086/260580.
6. Obstfeld, Maurice. 1996. “Models of Currency Crises with Self-Fulfilling Features.” European Economic Review 40(3–5): 1037–1047. https://doi.org/10.1016/0014-2921(95)00111-5.

Earlier EconMacro post: Saadaoui, Jamel. March 4, 2023. “The fragility of an incomplete monetary union with Mathematica.”