A monetary commitment can survive when people believe in it and fail when they expect it to fail—even though the underlying policy preferences and economic fundamentals are unchanged. Paul De Grauwe’s model makes the mechanism precise: expectations change the economic cost of defending the commitment.
In my March 2023 post on the fragility of an incomplete monetary union, I used Mathematica to introduce this model and its different exchange-rate configurations. This new article develops the explanation much further. We derive the optimal inflation rule, calculate the losses associated with four policy situations, reconstruct eight figures, and use numerical exercises to understand why the same fundamentals can support different outcomes.
I taught this material to first-year master’s students in Strasbourg. That teaching experience shapes the presentation here: I explain what each equation means before using it, show the intermediate algebra, and connect each figure to the policy decision it represents. The aim is to make the reasoning accessible to a reader encountering the model for the first time.
The starting point is Professor Paul De Grauwe’s lecture notes and Chapter 5 of De Grauwe (2020). I retain his equations and notation. Additional algebra explains the intermediate steps, and a separate exercise illustrates the sovereign-debt mechanism of an incomplete monetary union. I would like to thank Professor De Grauwe again for sharing the lecture notes that made the original teaching exercise possible.
The argument belongs to a broader literature. Kydland and Prescott (1977) explain why a policy that looks desirable in advance may cease to be optimal when the moment to implement it arrives. Barro and Gordon (1983) apply this logic to discretionary monetary policy and the inflation bias. The possibility that expectations themselves affect the incentive to abandon a currency commitment is central to Obstfeld (1996). De Grauwe connects this reasoning to the fragility of a monetary union in which national governments issue debt in a currency they do not individually control.
The sequence to keep in mind. First derive the best policy for given expectations. Then impose consistency between expectations and policy. Finally, compare defending and abandoning the peg while holding expectations fixed within each comparison.
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:
| Symbol | Economic meaning |
|---|---|
| \(U\), \(U_N\) | Actual unemployment and the natural unemployment rate. |
| \(\pi\), \(\pi^e\) | Actual inflation and inflation expected when nominal commitments are set. The superscript \(e\) means “expected.” |
| \(a>0\) | The unemployment response to an inflation surprise. |
| \(\varepsilon\) | An unexpected disturbance; a positive value raises unemployment in this equation. |
| \(\bar U\) | The authorities’ unemployment target. The bar identifies a target, rather than an average calculated from data. |
| \(\beta\geq0\) | The weight on the squared unemployment-target gap, relative to an inflation weight normalized to one. |
| \(k\) | The gap between natural and target unemployment, introduced below. |
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.
Mathematical appendix: A1: the substitution into the loss function; A2: reading Figures 1 and 2.
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.
Mathematical appendix: A3: every differentiation and rearrangement step, with the curvature diagram.
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.
A useful limiting case. If the unemployment target equals the natural rate, \(k=0\), the no-shock inflation bias vanishes. It also vanishes when \(\beta=0\). The bias depends on an incentive to use monetary policy to achieve an unemployment target below the natural rate.
Mathematical appendix: A4: the origin of the discretionary equilibrium and Figure 3.
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.
Mathematical appendix: A5: the shock coefficients and their economic meaning.
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.
Mathematical appendix: A5: the Germany–Italy comparisons, including the numerical increase in Italian inflation.
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
| Private expectations | Maintain the peg | Devalue |
|---|---|---|
| \(\pi^e=0\) | \(L_{0,\pi^e=0}=\beta k^2\) | \(L_{\mathrm{cheat}}+C=\dfrac{\beta k^2}{1+\beta a^2}+C\) |
| \(\pi^e=\beta a k\) | \(L_{\mathrm{stab}}=\beta k^2(1+\beta a^2)^2\) | \(L_{\mathrm{DIS}}+C=\beta k^2(1+\beta a^2)+C\) |
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.
Mathematical appendix: A6: the four loss derivations and the meanings of DIS, cheat, and stab.
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.
Mathematical appendix: A7: the gain ratio and the graphical markers in Figures 6 and 7.
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.
Mathematical appendix: A9: two self-consistent outcomes at the same preference weight.
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:
| Range | Incentive comparison | Outcome |
|---|---|---|
| \(\beta<\beta_1\) | \(C_0\) exceeds both gains | Unique maintained-peg equilibrium |
| \(\beta_1<\beta<\beta_2\) | Temptation is below \(C_0\), defense loss difference above it | Two pure equilibria |
| \(\beta>\beta_2\) | \(C_0\) is below both gains | Unique devaluation equilibrium |
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.
Mathematical appendix: A8: the threshold calculations; A10: why the steep curve has no vertical asymptote.
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.
| Exchange-rate model | Sovereign-debt setting | |
|---|---|---|
| Commitment | Maintain the peg | Repay public debt |
| Pessimistic expectation | Devaluation and inflation | Default |
| Effect on the decision | Higher unemployment cost of zero inflation | More difficult refinancing and costlier adjustment |
| Adverse action | Devalue | Default |
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
Work through the model in Mathematica. Download the Mathematica notebook (V3, .nb). It includes De Grauwe’s 48 equations, eight reproducible figures, worked exercises, and two interactive explorations of exchange-rate credibility and sovereign financing stress. Save C5_DeGrauwe_Mathematica_V3.nb in a folder on your computer and open it in Mathematica, so that the exported figures can be saved beside it.
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:
| Mathematical symbol | Notebook code |
|---|---|
| \(\pi\), \(\pi^e\) | p, pe |
| \(U_N\), \(\bar U\) | UN, UB |
| \(L_{\mathrm{DIS}}\), \(L_{\mathrm{cheat}}\), \(L_{\mathrm{stab}}\) | LDIS, Lcheat, Lstab |
| \(L_{0,\pi^e=0}\) | L0 |
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 eight numbered figures in the main 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 \(k\).
Interpret the changes
At the initial values, temptation is \(0.5\) and the defense loss difference is \(2\). Raising \(C\) above \(2\) makes maintaining the peg preferable in both expectation configurations. Lowering \(C\) below \(0.5\) makes devaluation preferable in both. Between those values, the optimal decision depends on expectations.
Both gains are proportional to \(k^2\). With \(a=\beta=C=1\), they are \(k^2/2\) and \(2k^2\). Thus the multiple-equilibrium range is \(\sqrt{1/2}<k<\sqrt2\). A larger target gap eventually makes devaluation attractive even without pessimistic expectations. The gap here is a structural policy ambition, rather than a new unemployment shock.
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 \(S=1.7\).
Worked interpretation
Initially, \(B_U=0.5\) and \(B_E=2\), with the cost \(C=1\) between them. The thresholds are approximately \(0.7071\) and \(1.4142\), so \(S=1\) lies in the multiple-equilibrium interval.
At \(\ell=0\), the two benefit curves coincide. For \(S=1\), both benefits equal \(0.5\), below the default cost. The expectation-induced difference disappears in this idealized experiment.
At \(S=1.7\), however, the common default benefit becomes \(0.5\times1.7^2=1.445\), exceeding \(C=1\). Default remains attractive even without financing stress. The backstop removes the liquidity amplification represented by \(\ell\); it does not erase the underlying solvency shock.
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.
Mathematical appendix. Derivations and economic intuition
This appendix supplies the intermediate calculations behind the main article and explains how to read the figures. It retains Professor De Grauwe’s equations and notation, while separating a conditional policy choice from an equilibrium in which policy confirms private expectations. Each derivation is followed by its economic interpretation.
The maintained assumptions are \(a>0\), \(\beta\geq0\), \(U_N>0\), and \(\bar U=\lambda U_N\) with \(\lambda<1\), so \(k=U_N-\bar U>0\). The abandonment cost is positive. Unless an unexpected shock is explicitly introduced, the peg comparisons set \(\varepsilon=0\). Strictly positive gain ratios require \(\beta>0\).
For the policy rule, begin with A3. For the four losses, go to A6. For why the same economy can support two outcomes, read A9. The calculations connect directly to the Mathematica notebook described in Section 13.
A1. From the Phillips curve to the loss function
The calculations retain De Grauwe’s notation. Actual unemployment is \(U\), natural unemployment is \(U_N\), and the authority’s target is \(\bar U\). The bar denotes a policy target. Actual inflation is \(\pi\), whereas \(\pi^e\) is the inflation rate expected when private nominal commitments are set. Begin with the three relationships in Section 2:
Subtract the target from actual unemployment, replacing each by its expression:
The two terms containing \(U_N\) combine because \(U_N-\lambda U_N=(1-\lambda)U_N\):
Substitute this entire gap into the second term of the loss function:
The abbreviation \(k=(1-\lambda)U_N=U_N-\bar U\) produces the equivalent expression:
Economic intuition. The unemployment-target gap has three components: the gap \(k\) already present at natural unemployment, the effect of an inflation surprise, and the shock. With \(U_N>0\) and \(\lambda<1\), the authority desires unemployment below its natural rate. Even when \(U=U_N\), its unemployment objective is therefore unmet. This distinction explains why policy can have an inflationary incentive in an economy experiencing no unemployment shock.
The notes also give \(Y=Y_n+\theta(\pi-\pi^e)+\varepsilon\). In that output equation, a positive disturbance raises output. In the unemployment equation used here, it raises unemployment. Both source equations are retained as written; their shock conventions mean that the same signed disturbance cannot be transferred between them without an explicit mapping.
A2. Reading the loss curves and the Phillips curves
In Figure 1, each curve holds the loss at a fixed level, denoted here by \(L_c>0\):
For \(\beta>0\), divide by \(L_c\) to recognize an ellipse centered on \((\bar U,0)\):
The figure shows its upper-right part, where \(\pi\geq0\) and \(U\geq\bar U\). At target unemployment, the unemployment-loss term vanishes:
At zero inflation, all loss comes from the unemployment-target gap:
A larger \(L_c\) puts both intercepts farther from the preferred point. At a fixed loss level, raising \(\beta\) leaves the vertical intercept unchanged but moves the horizontal intercept toward \(\bar U\): a smaller unemployment deviation now produces the same loss. The ellipse formulas require \(\beta>0\); if \(\beta=0\), loss depends only on inflation.
Economic intuition. Each arc describes combinations the authority regards as equally costly. Moving outward worsens its objective. The corner labeled \(\bar U\) is a shifted horizontal origin, so it does not mean that target unemployment is literally zero. These curves describe preferences; the Phillips curve determines the feasible combinations.
For an illustration, take \(\bar U=4\) and \(\beta=1\), measuring rates in percentage points. The combination \((U,\pi)=(4,2)\) gives \(L=2^2+(4-4)^2=4\). The combination \((6,0)\) also gives \(L=0^2+(6-4)^2=4\). They lie on the same contour. The combination \((6,2)\) gives \(L=8\) and lies on an outer contour.
To understand Figure 2, set \(\varepsilon=0\) and put inflation on the vertical axis. Starting from \(U=U_N+a(\pi^e-\pi)\), subtract \(U_N\):
Divide by \(a>0\), then isolate actual inflation:
At fixed expectations, this line has slope \(-1/a\). The curve for \(\pi^e=0\) passes through \((U_N,0)\). The curve for \(\pi^e=\pi_1\) passes through \((U_N,\pi_1)\), and is higher by \(\pi_1\) at every unemployment rate.
Economic intuition. Actual inflation rising to \(\pi_1\) while expectations remain zero moves the economy along the lower curve, to \(U=U_N-a\pi_1\). If expected inflation also becomes \(\pi_1\), the relevant curve shifts upward and unemployment returns to \(U_N\) at that actual inflation rate. An inflation surprise affects unemployment; fully anticipated inflation produces no unemployment reduction below the natural rate in this no-shock comparison.
A3. The conditional optimum, the chain rule, and curvature
The authority chooses actual inflation after expectations and the observed shock are given. During differentiation, \(\pi^e\), \(\varepsilon\), \(U_N\), and \(\bar U\) are therefore fixed. In particular, \(d\pi^e/d\pi=0\) and the Phillips curve gives \(dU/d\pi=-a\). Rational-expectations consistency is imposed after solving this policy problem.
Differentiate each term of \(L=\pi^2+\beta(U-\bar U)^2\). The inflation term gives \(2\pi\). The square in the unemployment term gives \(2(U-\bar U)\), and the chain rule multiplies this by \(d(U-\bar U)/d\pi=-a\):
Setting the derivative to zero and dividing by two gives:
Economic intuition. At the optimum, the marginal cost of inflation, \(2\pi\), equals the marginal reduction in unemployment loss, \(2\beta a(U-\bar U)\). The latter combines the remaining target gap, the weight placed on that gap, and the response of unemployment to an inflation surprise.
Now differentiate once more. The derivative of \(2\pi\) is \(2\), while the derivative of \(-2\beta a(U-\bar U)\) is \(-2\beta a(-a)\):
The two minus signs explain why the unemployment component adds positive curvature. With \(\beta\geq0\), the loss is strictly convex in actual inflation, so the first-order condition identifies a unique global minimum for these fixed expectations. A positive second derivative means that the slope of the loss curve rises with inflation: it is negative before the minimum, zero at the minimum, and positive afterward. It does not mean that the loss is increasing everywhere.
This unique conditional minimum is compatible with the multiple equilibria discussed later. Here we hold expectations fixed and solve a continuous choice of inflation. In the peg problem, we also compare maintaining the commitment with paying a fixed cost to abandon it. Different expectations can change that discrete decision. Each expectation configuration may have a well-defined preferred policy, while more than one configuration is confirmed by the policy it induces. These are different questions about uniqueness.
To recover the explicit policy rule in Section 3, source equation 9, substitute the unemployment-target gap into the first-order condition:
Distribute \(\beta a\) across the bracket:
Move the term containing actual inflation to the left and factor:
Division by the positive coefficient \(1+\beta a^2\) gives the conditional optimum:
Replacing \(k\) by \((1-\lambda)U_N\) and separating the numerator produces exactly the three fractions printed in the original derivation:
The compact expression is thus an algebraic rewriting of the source equation. Its no-shock version follows simply by setting \(\varepsilon=0\). The denominator arises because increasing inflation both raises its direct cost and reduces the unemployment gap that initially made additional inflation attractive.
Completing the square makes the minimum especially visible. Denote this conditional optimum by \(\pi_{\mathrm{opt}}\), with exactly the value derived above. Then:
The second term is fixed during the policy choice. Any deviation from \(\pi_{\mathrm{opt}}\) adds a nonnegative squared term. For \(a=\beta=k=1\) and \(\pi^e=\varepsilon=0\), the calculation becomes:
The minimum is at \(\pi_{\mathrm{opt}}=1/2\), with \(L=1/2\). Below \(1/2\), a little more inflation reduces total loss; above \(1/2\), it increases total loss. This is the best policy conditional on zero expected inflation, rather than the rational-expectations discretionary equilibrium derived next.
A4. Why discretion generates an inflation bias
“DIS” abbreviates discretion. It describes a policy regime in which the authority chooses inflation after expectations have been formed, without a binding commitment to zero inflation. To determine its no-shock equilibrium, start from the conditional policy rule just derived:
With no remaining uncertainty, rational expectations require the chosen rate to equal the anticipated rate. Impose \(\pi=\pi^e\) now:
Multiply through by the denominator and expand:
The terms \(\beta a^2\pi^e\) cancel from both sides:
Substitute actual and expected inflation into the Phillips curve:
Economic intuition. Private agents anticipate the authority’s incentive to lower unemployment toward \(\bar U\). The resulting inflation is fully expected, so it fails to lower unemployment below \(U_N\). Nevertheless, the target gap remains \(U_N-\bar U=k\), and positive inflation remains. This is the inflation bias discussed in Section 4 and by Barro and Gordon (1983).
In Figure 3, each tangency is the best policy for a particular expected inflation rate. The upward-sloping dashed line joins these conditional optima:
For positive inflation, a loss contour has slope \(-\beta(U-\bar U)/\pi\). Equating it to the Phillips-curve slope \(-1/a\) gives this same optimality condition. The dashed line starts at the preferred point \((\bar U,0)\) and has slope \(\beta a\).
The lower tangency, associated with zero expected inflation, involves surprise inflation and unemployment below \(U_N\). It solves the conditional policy problem but does not fulfill those zero-inflation expectations. The no-shock rational-expectations restriction supplies the vertical line \(U=U_N\). Its intersection with the dashed line is the discretionary equilibrium, with inflation \(\beta ak\). The dashed line compares optima for different beliefs; it does not specify a path through time or a process of learning.
Under a credible commitment to zero inflation, both actual and expected inflation instead equal zero, and unemployment is also \(U_N\). The macroeconomic losses are therefore:
Discretion adds inflation loss without improving unemployment relative to that credible commitment. Maintaining the commitment is difficult because, once people expect zero inflation, the authority would prefer the surprise-inflation choice. The later peg comparisons add the separate abandonment cost and ask whether it is sufficient to prevent that deviation. The belief \(\pi^e=\beta ak\) applies when discretionary devaluation is anticipated; it is not an expectation imposed on every policy regime.
A5. Unexpected shocks and the Germany–Italy figures
This subsection expands Section 5 and the explanations of Figure 4 and Figure 5. Two sources of policy disagreement must be distinguished: different preferences and different disturbances.
Why the shock response is added to expected inflation
Suppose the shock has conditional mean zero when expectations are formed, and the authority observes it before choosing policy. Expected inflation is the expectation of the conditional policy rule derived in A3. With the other parameters known and fixed:
Multiplying by the denominator and cancelling the expected-inflation term yields:
Now let the shock occur. Expectations remain at the value set beforehand. Substitute \(\pi^e=\beta ak\) into the policy rule:
The representation as expected inflation plus a shock response uses the expectation condition just derived. It is not obtained by replacing an arbitrary expectation with actual inflation after the shock. Realized inflation differs from its expectation when the unexpected shock is nonzero.
Next substitute the inflation surprise into the Phillips curve:
Economic intuition. An adverse shock initially raises unemployment. The authority accepts some unexpected inflation to offset part of that increase. It stops before fully eliminating the shock because inflation itself is costly. With \(a=\beta=1\), a unit shock produces an inflation surprise of \(1/2\), which offsets half the unemployment disturbance; unemployment still rises by \(1/2\). This temporary stabilization effect is compatible with the absence of a permanent unemployment gain from anticipated inflation.
For fixed \(U_N\), the variance formula follows by squaring the shock loading, not by squaring the shock variance:
A response of one half therefore means a variance ratio of one quarter. Raising \(\beta\) strengthens stabilization of unemployment, but also raises the average inflation bias \(\beta ak\) when \(k>0\). Average inflation and the response to a new shock are different objects.
Figure 4: different preferences
In the stylized Germany–Italy preference comparison, \(a\) and \(k\) are the same but \(\beta_I>\beta_G\). The countries are labels for contrasting policy preferences in the model. The diagram is not an estimate of their actual preferences. Each country’s conditional tangencies lie on:
The Italian ray is steeper. At the no-shock rational-expectations equilibria, each country is at its natural unemployment rate, so:
The horizontal line \(\pi^*\) marks Germany’s reference inflation. The Italian panel illustrates conditional choices for different expectations: zero inflation, Germany’s reference rate, and Italy’s own discretionary rate. The intermediate tangency is especially informative. If Italians expect \(\pi_G\), their unconstrained authority still prefers more inflation than that expectation:
Economic intuition. Announcing Germany’s lower inflation does not remove Italy’s incentive to deviate after expectations are fixed. A credible constraint on policy can lower inflation while leaving unemployment at its natural rate once expectations adjust. Delivering that lower inflation while people still expect Italy’s higher discretionary rate instead creates an unemployment cost. The credibility of the commitment determines which comparison applies.
Figure 5: the same preferences, but a shock in Italy
Preferences are now identical. Germany has no shock, while Italy has \(\varepsilon>0\), with previous expectations held fixed. The Italian Phillips curve shifts upward by \(\varepsilon/a\) when inflation is on the vertical axis. The upward-sloping optimality line does not shift, because preferences and the unemployment target are unchanged. Their new intersection has both higher inflation and higher unemployment.
Let \(\Delta\pi\) and \(\Delta U\) denote changes from Italy’s initial no-shock position. Subtract the initial Phillips curve from the new one, keeping expectations fixed:
Subtract the initial optimality condition from the new one:
Combine these two relationships and collect the inflation terms:
The numerical coordinates in the V3 notebook use \(a=1.2\), \(\beta=0.45\), \(U_N=2.9\), and target unemployment at the diagram’s normalized origin, \(\bar U=0\). The code’s delta = 1.5 is the vertical shift of the Phillips curve. Since that shift equals \(\varepsilon/a\), the unemployment shock is:
The initial inflation rate and Italy’s increase are therefore:
| Situation | Inflation | Unemployment |
|---|---|---|
| Germany, unchanged; Italy before the shock | \(1.566\) | \(2.9\) |
| Italy accommodates part of its shock | \(2.155806\) | \(3.992233\) |
| Italy keeps inflation at the original rate | \(1.566\) | \(2.9+1.8=4.7\) |
Economic intuition. The additional inflation prevents approximately \(4.7-3.992233=0.707767\) units of unemployment, but does not eliminate the shock. Germany has no corresponding disturbance, so its preferred policy does not change. These are illustrative model units, not historical observations or a prediction for the two countries. The figure shows why their desired responses can differ; determining a common central bank’s actual choice would require a joint policy objective.
A6. Deriving the four losses and interpreting their labels
The four situations in Section 8 combine two private expectations with two policy decisions. Throughout this comparison, set the shock to zero, \(\varepsilon=0\), and retain De Grauwe’s definition \(k=U_N-\bar U=(1-\lambda)U_N\). The building blocks are:
The conditional inflation rule comes from the optimal policy equation in Section 3. Set its shock term to zero, substitute \(k=(1-\lambda)U_N\), and factor the remaining two fractions:
This rule applies when the authority is free to choose inflation after expectations have been set. Maintaining the peg instead imposes \(\pi=0\). In either case, the Phillips curve determines unemployment, and the loss function evaluates the resulting inflation and unemployment-target gap.
A6.1. The peg is expected and maintained: \(L_{0,\pi^e=0}\)
Private agents expect zero inflation and the authority delivers it: \(\pi^e=\pi=0\). The Phillips curve gives \(U=U_N\). Subtract the unemployment target:
Substitution into the loss function gives:
Economic interpretation. A credible peg removes inflation loss, but it does not eliminate the gap between natural unemployment and the authority’s more ambitious target. This remaining gap explains why a temptation to create surprise inflation can exist even when the peg works as promised.
A6.2. The peg is expected, but the authority devalues: \(L_{\mathrm{cheat}}\)
Keep expectations fixed at \(\pi^e=0\) and use the conditional best response:
Because actual inflation exceeds expected inflation, unemployment falls:
The loss function requires the gap relative to \(\bar U\), rather than the deviation from \(U_N\). Subtract the target and use \(U_N-\bar U=k\):
Insert inflation and this gap into the two components of loss:
Both terms have the same squared denominator. Square their numerators, add them, and factor:
Cancel one factor \(1+\beta a^2\):
Economic interpretation. The label “cheat” identifies an unexpected departure from the peg. The authority accepts some inflation in exchange for unemployment closer to its target. For positive \(a\), \(\beta\), and \(k\), unemployment lies below \(U_N\) but remains above \(\bar U\): eliminating the entire target gap would require more inflation than the authority finds worthwhile. This surprise outcome is a deviation evaluated at previously fixed expectations; it is not a rational-expectations equilibrium with \(\pi^e=0\).
A6.3. Devaluation is expected and occurs: \(L_{\mathrm{DIS}}\)
DIS means discretion. It labels the loss at the discretionary equilibrium, where private agents anticipate the authority’s unconstrained inflation choice. To obtain these expectations, impose \(\pi=\pi^e\) in the conditional best response:
The \(\beta a^2\pi^e\) terms cancel, leaving \(\pi^e=\beta ak\). Given this expectation, the authority’s best response is:
Actual inflation equals expected inflation. Consequently:
The loss is:
Economic interpretation. Discretion produces positive inflation without reducing unemployment below its natural rate. The unemployment-target gap remains \(k\), and inflation adds another loss. The difference from \(L_{\mathrm{cheat}}\) is that private agents anticipate policy: the surprise that temporarily improved unemployment in the preceding case is absent.
A6.4. Devaluation is expected, but the authority defends the peg: \(L_{\mathrm{stab}}\)
Keep the pessimistic expectation \(\pi^e=\beta ak\) fixed, but now impose \(\pi=0\). This is a counterfactual policy choice under the same prior beliefs as in A6.3. It does not require those beliefs to remain correct when the authority unexpectedly defends the peg. The label “stab” refers to stabilizing the exchange rate.
Subtracting the target gives:
Inflation loss is zero, so the authority bears only the unemployment component:
Economic interpretation. Delivering inflation below expectations raises unemployment above \(U_N\). The original target gap \(k\) is enlarged by \(\beta a^2k\). The square on \(1+\beta a^2\) appears because the loss function squares the entire enlarged gap. Stabilizing the exchange rate in this situation entails an unemployment cost.
A6.5. Compare decisions while holding expectations fixed
The additional abandonment cost \(C\) belongs in both devaluation cells. It is fixed conditional on abandoning the peg, so it does not change the first-order condition for inflation. It does change whether abandonment is preferable to defence.
| Private expectations | Maintain the peg | Devalue |
|---|---|---|
| \(\pi^e=0\) | \(L_{0,\pi^e=0}=\beta k^2\) | \(L_{\mathrm{cheat}}+C=\dfrac{\beta k^2}{1+\beta a^2}+C\) |
| \(\pi^e=\beta ak\) | \(L_{\mathrm{stab}}=\beta k^2(1+\beta a^2)^2\) | \(L_{\mathrm{DIS}}+C=\beta k^2(1+\beta a^2)+C\) |
Read horizontally within each row. At the moment of choice, expectations have already been set. The authority cannot choose directly between \(L_{\mathrm{cheat}}\) and \(L_{\mathrm{DIS}}\), because those outcomes presuppose different expectations. It chooses the lower total loss within the relevant row. Rational-expectations consistency is then checked for the resulting expectation–policy combination.
For a concrete check, set \(a=\beta=k=1\). A trusted, maintained peg has loss \(1\). Surprise devaluation generates inflation \(1/2\) and a remaining unemployment-target gap \(1/2\): the two squared losses are \(1/4\) each, giving \(L_{\mathrm{cheat}}=1/2\). Under anticipated discretion, inflation is \(1\) and the target gap is \(1\), giving \(L_{\mathrm{DIS}}=1+1=2\). Defending against that expectation sets inflation to zero but enlarges the target gap to \(2\), giving \(L_{\mathrm{stab}}=2^2=4\). These four values exclude \(C\). Notice why natural unemployment does not mean zero unemployment loss: the authority’s target lies below the natural rate.
A7. Temptation, the cost of defence, and the role of confidence
The two gains in Section 9 are reductions in inflation–unemployment loss from abandoning the peg under different expectations. Each must be compared with the additional abandonment cost \(C\).
A7.1. Temptation when the peg is trusted
With \(\pi^e=0\), subtract the optimized surprise-devaluation loss from the maintained-peg loss:
Writing one over the common denominator leaves \(\beta a^2\) in the numerator:
In Figure 6, the maintained peg is the point \((U_N,0)\) on the outer loss contour. Surprise devaluation moves up and left along the Phillips curve to the lower-loss tangency. Inflation rises, but unemployment moves toward its target. The gain measures the net improvement after accounting for both effects.
A7.2. The cost of defence when devaluation is expected
With \(\pi^e=\beta ak\), subtract the discretionary loss from the loss of defending:
In Figure 7, defending produces zero inflation and unemployment above \(U_N\), on the outer contour. Devaluing produces the inner tangency at \((U_N,\beta ak)\). The cost of defence is the extra loss from choosing defence over devaluation under these same pessimistic beliefs. It is the difference \(L_{\mathrm{stab}}-L_{\mathrm{DIS}}\), rather than \(L_{\mathrm{stab}}\) by itself, and is distinct from the abandonment cost \(C\).
Reading the graphical markers carefully. The thick segment and horizontal markers in Figures 6 and 7 schematically compare loss contours. The horizontal bars connect contour intercepts; their endpoints are not both actual unemployment outcomes. In particular, the left endpoint of Figure 7’s bar is not \(U_N\), and Figure 6’s left bar endpoint is not unemployment at the surprise-devaluation tangency. Their lengths should not be read as literal unemployment changes or numerical welfare differences. The loss differences are calculated with the formulas above.
A7.3. Why the incentive is stronger when confidence is absent
For positive \(a\), \(\beta\), and \(k\), divide the two gains:
Cancel \(\beta^2a^2k^2\). Dividing by the remaining reciprocal multiplies by another factor \(1+\beta a^2\):
Economic intuition. When the peg is trusted, unemployment under defence equals \(U_N\): the authority’s concern is the original gap \(k\). When devaluation is expected, defence pushes unemployment above \(U_N\), enlarging the target gap to \(k(1+\beta a^2)\). Defending a distrusted peg is consequently more costly than maintaining a trusted one. The quadratic loss magnifies the effect of that larger gap.
The square in the ratio can be understood directly from this gap. For any fixed nonnegative expected inflation rate in the no-shock comparison, the unemployment-target gap under a maintained peg is \(k+a\pi^e\). Substituting the optimal inflation rule into the loss function shows that the gain from devaluing is:
The coefficient is the same in both expectation rows. Changing expectations from zero to \(\beta ak\) multiplies the gap inside the square by \(1+\beta a^2\). It therefore multiplies the gain by \((1+\beta a^2)^2\). The squared ratio follows from applying the same policy optimization to a larger initial unemployment-target gap.
With \(a=\beta=k=1\), temptation is \(1/2\), the cost of defence is \(2\), and their ratio is \(4\). If \(C=1\), confidence makes the gain from abandoning the peg too small to justify its cost; pessimism makes the gain large enough. Expectations change the authority’s decision by changing the unemployment cost of keeping its promise. This provides the economic basis for the multiple-equilibrium region examined next. At \(\beta=0\), both gains are zero and this ratio is undefined; maintaining the peg remains preferable when \(C>0\).
A8. Computing the two thresholds and reading Figure 8
The horizontal axis of Figure 8 measures \(\beta\), the weight attached to unemployment relative to inflation in the loss function. Moving to the right changes the authority’s preferences while holding \(a\), \(k\), and the abandonment cost \(C_0\) fixed. The symbols \(\beta_1\) and \(\beta_2\) identify two critical values of this same preference parameter. They are boundaries of an equilibrium region, rather than the two equilibria themselves.
For the numerical exercise, set \(a=k=C_0=1\). To keep the threshold calculation readable, denote temptation by \(T(\beta)\) and the cost of defence by \(D(\beta)\):
These functions measure reductions in inflation–unemployment loss from devaluing, under different expectations. The authority compares each reduction with the additional abandonment cost, now equal to one.
This distinction is essential when reading the vertical axis. A height of two means that devaluing reduces the authority’s macroeconomic loss by two units relative to maintaining the peg under the stated expectations. It does not mean two percentage points of inflation or unemployment. Paying the separate abandonment cost is worthwhile only if the loss reduction is larger than that cost.
The first threshold: defence becomes too costly under pessimistic expectations
The left intersection occurs when the cost-of-defence curve meets the horizontal line \(C_0=1\). Thus:
Expanding and moving one to the left gives:
This cubic has a unique positive root. Indeed, \(D(\beta)\) starts at zero and increases strictly for positive \(\beta\). A simple numerical bracket makes the calculation transparent:
The crossing lies between these two values. Refining the numerical solution yields:
Economic interpretation. Below this threshold, the authority prefers to maintain the peg even when people expect devaluation. Above it, pessimistic expectations make the unemployment cost of defence large enough for devaluation to become preferable.
The second threshold: devaluation becomes attractive even when the peg is trusted
The right intersection occurs when the temptation curve meets the abandonment cost:
Multiplying both sides by \(1+\beta_2\), which is positive for the admissible values of \(\beta\), gives:
Rearranging and applying the quadratic formula gives:
The negative root is inadmissible because \(\beta\) is a nonnegative preference weight. Therefore:
Economic interpretation. Beyond this threshold, even the gain from surprising people who trusted the peg exceeds the abandonment cost. Confidence can no longer make maintaining the peg the authority’s preferred choice.
The two intersections divide the horizontal axis into three regions. The following values show their meaning; every entry is compared with the same cost \(C_0=1\).
| Preference weight | Temptation | Cost of defence | Policy incentives |
|---|---|---|---|
| \(\beta=0.5\) | \(1/6\simeq0.1667\) | \(0.375\) | Both gains are below one: maintain the peg under either expectation. |
| \(\beta=1\) | \(0.5\) | \(2\) | Temptation is below one and the cost of defence is above it: expectations change the preferred policy. |
| \(\beta=2\) | \(4/3\simeq1.3333\) | \(12\) | Both gains exceed one: devalue under either expectation. |
Exactly at a threshold, the authority is indifferent in the corresponding comparison. At \(\beta=0\), both gains vanish; paying a positive abandonment cost has no benefit, so maintaining the peg is preferable.
A9. Why two equilibria exist between the thresholds—and disappear beyond the second
An equilibrium combines a policy that is optimal given private expectations with expectations that are confirmed by that policy. The two equilibria discussed here exist at the same value of \(\beta\), with the same fundamentals and abandonment cost. They differ in expectations and the resulting policy.
Choose one point between the thresholds in Figure 8 and read vertically. The temptation curve is below \(C_0\), while the cost-of-defence curve is above it:
The two heights on this vertical line refer to two expectation configurations for the same economy. They do not represent a change in preferences or a movement from one threshold to the other.
Confidence and peg survival confirm one another
If private agents expect the peg to survive, then \(\pi^e=0\). Because temptation is below \(C_0\), the authority prefers to maintain zero inflation:
The resulting policy confirms expectations:
Pessimism and devaluation also confirm one another
If private agents expect the discretionary inflation rate, then \(\pi^e=\beta ak\). Maintaining zero inflation would now push unemployment above its natural level. Because the cost of defence exceeds \(C_0\), the authority prefers to devalue:
Its optimal inflation choice then confirms the pessimistic expectation:
No forecasting error is required in either equilibrium. The unemployment cost of defending the peg changes with expectations, so the preferred decision changes too.
The Phillips curve supplies the mechanism. With a trusted peg, zero actual inflation matches zero expected inflation and unemployment equals its natural rate. With a distrusted peg, delivering zero inflation would fall below expectations and raise unemployment. The authority therefore evaluates a different cost of honouring its commitment, although its preferences have not changed.
Take \(a=k=C_0=1\) and hold \(\beta=1\) fixed throughout. Because \(\beta_1<1<\beta_2\), both outcomes are possible. Compare the total losses within each row:
| Private expectation | Maintain the peg | Devalue, including the abandonment cost | Optimal policy |
|---|---|---|---|
| \(\pi^e=0\) | \(L_{0,\pi^e=0}=1\) | \(L_{\mathrm{cheat}}+C_0=0.5+1=1.5\) | Maintain the peg: \(\pi=0\). |
| \(\pi^e=1\) | \(L_{\mathrm{stab}}=4\) | \(L_{\mathrm{DIS}}+C_0=2+1=3\) | Devalue: \(\pi=1\). |
The authority cannot choose between the two expectation rows after expectations have been set. It chooses between the two policies within the row it faces. Both rows nevertheless describe self-consistent outcomes when the authority’s decision is combined with private expectations.
Why the peg equilibrium disappears beyond \(\beta_2\)
To the right of the second threshold, temptation exceeds the abandonment cost:
Rearranging makes the authority’s choice explicit:
Even if people expect zero inflation, the authority prefers to devalue and chooses positive inflation. The proposed expectation is then contradicted:
A trusted, maintained peg therefore fails the equilibrium conditions. By contrast, when people expect \(\pi^e=\beta ak\), the authority still prefers devaluation and chooses \(\pi=\beta ak\). That expectation remains consistent with policy.
Between the thresholds, confidence can sustain the peg. Beyond \(\beta_2\), even complete confidence is insufficient to make maintaining the peg optimal. Below \(\beta_1\), the reverse logic applies: the authority prefers defence even under pessimistic expectations, so an anticipated devaluation is not confirmed. The thresholds delimit the region where expectations can change the preferred decision.
A10. The steep curve has no vertical asymptote at \(\beta_2\)
The dashed vertical lines in Figure 8 project the intersections with \(C_0\) onto the horizontal axis. They mark parameter values. The cost-of-defence curve becomes steep, but it does not become infinite at the second threshold.
With \(a=k=C_0=1\), the function \(D(\beta)=\beta^2(1+\beta)\) is a polynomial and is continuous. Its limit at \(\beta_2\) is therefore obtained by direct substitution:
The threshold equation gives \(1+\beta_2=\beta_2^2\). Substituting this identity yields:
Using the positive root from A8:
Hence the limit from either side is finite:
At the same parameter value, \(T(\beta_2)=C_0=1\). Defence is already considerably more costly than abandonment, but it is not infinitely costly.
The numerical thresholds and this limit belong to the illustrative normalization \(a=k=C_0=1\). Changing those parameters changes the intersections. The economic comparison remains the same: locate where each gain from devaluation first reaches the given abandonment cost.
The two functions diverge only as \(\beta\) itself tends to infinity. Their growth rates differ:
Here, \(\sim\) means that the ratio of the two expressions tends to one. Cubic growth explains why the defence curve rises much faster than temptation. The policy threshold arises from comparing a finite gain with \(C_0\), not from the cost of defence becoming infinite. Once temptation exceeds \(C_0\), even a trusted peg is no longer optimal.
References and related posts
- 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.
- De Grauwe, Paul. 2011. The Governance of a Fragile Eurozone. CEPS Working Document No. 346, May. CEPS publication page.
- 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.
- 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.
- 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.
- 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.”