Back Matter
Author: Mr. Joe Crowley

APPENDIX

Mathematical Derivations of Results

This appendix derives the impact on the endogenous variables of changes in the exogenous variables, under alternative assumptions about capital mobility

A1. Perfect capital mobility

Combining equations 9, 13 and 14 we get an expression for inflation:

pt=b1(1λ)+b2+c2+λc1(αt1)b1(1λ+λδ)+b2+c2+λc1(βt1)et(16)+δλb1b1(1λ+λδ)+b2+c2+λc1βpt1+λc1b1(1λ+λδ)+b2+c2+λc1(βt1)rt*+λb1(1λ+λδ)+b2+c2+λc1(βt1)t

When δ=0 and either expectations are static (αt = βt = 0) or the stabilization program is fully credible (αt = βt), the coefficient of the exchange rate crawl is equal to 1 and changes in the exchange rate crawl are translated fully into changes in the inflation rate. As wage indexation becomes more backward-looking (as δ increases), however, the coefficient on the exchange rate crawl decreases, so to achieve a given reduction in inflation a greater reduction in the crawl is needed. The coefficient of pt−1 becomes greater, so the persistence of inflation is increased. Inflation is stabilized with respect to changes in the world interest rate and with respect to supply and demand shocks. The effects of increasing δ are diminished somewhat as β increases.

In a steady state (where et = et−1 = pt−1) with no external shocks. the change in inflation caused by changes in expectations as a result of an announcement in the absence of other policy measures or shocks is equal to the steady state rate of crawl of the exchange rate times the change (caused by the announcement) in the sum of the coefficients on the exchange rate and on past inflation. More simply, let the sum of the coefficients on the exchange rate and on past inflation be equal to a coefficient Zt. Then:

Zt=b1(1λ+δλ)+b2+c2+λc1(αt1)b1(1λ+δλ)+b2+c2+λc1(βt1)(17)

An announcement in a steady state will cause inflation to change by: (ZtZt−1) et.

As seen in equation (17), in a steady state (pt = et = pt−1) a fully credible announcement of a price stabilization program (αt = βt) in the absence of shocks has no effect on inflation since Z is always equal to one. When the program lacks full credibility (αt > βt), however, inflation will increase, since Zt−1 equals one but Zt is greater than one.4 If wages are at all backwardly indexed (δ > 0) the coefficient of the exchange rate crawl is less than one and inflation will include a component that depends on past inflation which is unaffected by the price stabilization program. Assuming that the public is aware of this fact, βt will be smaller than αt as the public will expect inflation to fall by less than the reduction in the rate of crawl of the exchange rate.5 Thus, when backward-looking wage indexation prevails, any announcement of a price stabilization policy will be inflationary.

To solve for real income growth we combine 3, 5 and 15:

yt=A1etB1pt1C1r*t+D1vt+F1μt(18)

where,

A1=δb1c2+λc1(βt1)b1(1λ+λδ)+b2+c2+λc1(βt1)+c1(αβ)b1(1λ+δλ)+b2b1(1λ+δλ)+b2+c2+λc1(βt1)
B1=δb1c2+λc1(βt1)b1(1λ+δλ)+b2+c2+λc1(βt1)
C1=c1b1(1λ+δλ)+b2b1(1λ+δλ)+b2+c2+λc1(βt1)
D1=C1/c1
F1=1D1=c2+λc1(βt1)b1(1λ+δλ)+b2+c2+λc1(βt1)

A1 will increase, so greater backward-looking indexation means that exchange rate policy, which as mentioned above will have less effect on inflation, will have more effect on income growth. This means that backward-looking wage indexation causes policies that reduce inflation by a given amount to have a more recessionary effect. B1 and C1 will increase, so demand shocks and changes in world interest rate growth will have greater effects on income, and D1 will decrease, so supply shocks will have smaller effects on income.

The change in income growth caused by changed expectations resulting from an announcement in the absence of other policy measures or shocks is therefore equal to the steady state rate of crawl of the exchange rate (= the steady state inflation rate) times the change in A1 - B1 that is caused by the announcement. More simply, let:

Zt=A1B1=c1(αtβt)b1(1λ+δλ)+b2b1(1λ+δλ)+b2+c2+λc1(βt1)(19)

An announcement in a steady state will cause income growth to change by: (Zt - Zt−1) et.

As seen in equation 19, a fully credible announcement of a price stabilization program (αt = βt) will have no direct effect on income growth since Z will always be equal to zero, but if the announcement is not fully credible (αt > βt), it will have an expansionary effect since Zt−1 will be zero but Zt will be greater than zero. Credible announcements will stabilize income with respect to changes in the world interest rate and demand shocks, but will destabilize it with respect to supply shocks. Increasing the backwardness of indexation will increase the expansionary effect of announcements that lack full credibility.

Combining 14 and 15 yields the growth of the real wage:

wtpt=δ(b1(1λ)+b2+c2+λc1(αt1)b1(1λ+λδ)+b2+c2+λc1(βt1))et(20)+δ(b1(1λ)+b2+c2+λc1(βt1)b1(1λ+δλ)+b2+c2+λc1(βt1))pt1+δλc1b1(1λ+λδ)+b2+c2+λc1(βt1)rt*δλb1(1λ+λδ)+b2+c2+λc1(βt1)t

As can be seen from equation (20), the coefficient of the previous period’s inflation rate increases as δ increases, so backward-looking indexation increases the inertia of real wage growth. Greater backward indexation also leads to a greater impact of the exchange rate crawl, interest rate changes, and supply and demand shocks, and hence it is destabilizing to real wage growth in all ways.

The change in real wage growth caused by changed expectations resulting from an announcement in the absence of other policy measures or shocks is therefore equal to the steady state rate of crawl of the exchange rate (= the steady state inflation rate) times the change (caused by the announcement) in the sum of the coefficients of the exchange rate crawl and past inflation. More simply, let Zt = the coefficient of the exchange rate crawl plus the coefficient of past inflation. Then:

Zt=δλc1(βtαt)b1(1λ+λδ)+b2+c2+λc1(βt1)(21)

An announcement in a steady state will cause wage growth to change by (Zt - Zt-1) et.

As can be seen in equation (21), an announcement of a price stabilization program will have no effect on real wages in the absence of other policy measures or shocks either if it is fully credible (αt = βt), or if there is no backward-looking wage indexation, meaning in either case that Z is always equal to zero. If the policy is not fully credible and there is backward-looking wage indexation, however, real wages will decline (since Zt < 0), and the decline will be greater as δ is larger. Announcements with any degree of credibility (βt > 0) will reduce the impact of world interest rate changes and shocks; the reduction will be more substantial if δ is larger.

Combining 3, 4, 5, 13 and 15 yields the growth of the current account balance:

xt=A2etB2Pt1+C2rt*D2vt+F2μt(22)

where,

A2=(d2d1(c2+λc1(βt1)))(δb1c1(αt(βt1)))b1(1λ+δλ)+b2+c2+λc1(βt1)d1c1(αtβt)
B2=δb1d2d1(c2+λc1(βt1))b1(1λ+δλ)+b2+c2+λc1(βt1)
C2=d1c1b1(1λ+δλ)+b2+d2/d1b1(1λ+δλ)+b2+c2+λc1(βt1)
D2=C2/c1
F2=C2d1=d2d1(λc1(βt1)+c2)b1(1λ+δλ)+b2+c2+λc1(βt1)

As δ increases, in other words when indexation becomes more backward-looking, and assuming that d2/d1< c2 + λc1(βt−1) 6 the coefficients will be affected as follows:

A2 will increase, so exchange rate-based price stabilization programs will have more negative effects on the growth of the current account. Since increasing δ also diminishes the effect of the exchange rate crawl on the inflation rate, it means that reducing inflation by a given amount entails a greater worsening of the current account and therefore increases the likelihood that the price stabilization effort will be unsustainable because of reserve losses. B2 and D2 will decrease, so supply and demand shocks will have less impact on the current account, and C2 will decrease, lessening the impact of world interest rate fluctuations. Thus, backward-looking indexation will stabilize the current account in the absence of inflation reduction programs.

Because A2 contains a factor of (αtβt), a perfectly credible announcement of a price stabilization program made in the absence of other policy measures or shocks will have no effect on current account growth. However if the announcement lacks credibility (α t > βt) there will be a decrease in current account growth in response to capital inflows caused by an expected future real appreciation of the currency. As wage indexation becomes more backward-looking the impact of an announcement will be more negative. Announcements with any degree of credibility (βt > 0) lower the impact of world interest rate fluctuations and demand shocks on current account growth, the impact of supply shocks becomes ambiguous.

The change in the growth of the current account balance in the absence of other policy measures or shocks caused by changed expectations resulting from an announcement is therefore equal to the steady state rate of crawl of the exchange rate (= the steady state inflation rate) times the change in A2 - B2 that is caused by the announcement. More simply, if Zt = A2 - B2, then:

Zt=c1(αtβt)d2d1(b1(1λ+δλ)+b2)b1(1λ+δλ)+b2+c2+λc1(βt1)(23)

An announcement in a steady state will cause current account balance growth to change by (Zt - Zt−1) et. If the announcement is fully credible there will be no change in the current account since Zt will always be zero.

A2. Imperfect capital mobility

The assumption of equal international rates of return based on perfect capital mobility may not be accurate for many countries (see Reinhart and Reinhart (1995)), so an alternative scenario of zero capital mobility is presented. To model this assumption, international interest rates are assumed to have no effect on the economy; equation 7 is removed from the analysis and equation 1 is no longer a rule for endogenously determining money supply growth, but rather a condition according to which the growth of the money supply, which is now an exogenous policy instrument, influences the interest rate. Price expectations still follow the same rule:

Etpt=(1βt)pt,hence:it+rt+(1βt)pt

αt has no effect on economic variables in an imperfect capital mobility framework. The impact of an announcement on variables is calculated from a steady state (e.g. that mt = et = mt-1 = et-1 = pt-1) and assuming that there are no shocks (e.g. μt = vt = 0).

Combining equations 1, 2, 3, 5 and 6 yields the following expression for inflation:

pt=D31(A3mt+B3et+b1δpt1+C3vtμt)(24)

where:

A3=c1a+c1
B3=ac2λ(a+c1)+b1(1λλ)+b2λ
C3=aa+c1
D3=ac2/λ+c1(1+a(βt1))a+c1+b1(1λλ+δ)+b2λ

The response of inflation to increases in δ is qualitatively unaffected by the change in the assumption about capital mobility. Increasing δ increases the persistence of inflation, and reduces the effect of money supply growth, the exchange rate crawl and shocks. An announcement of a price stabilization program in a steady state and in the absence of other policy measures or shocks will reduce inflation and reduce the impact of shocks (since it will lower β and there fore lower D3, the only coefficient that is not constant), but both of these benefits are mitigated if δ increases (since increasing δ dilutes the effects of increasing β).

Solving for inflation also yields an expression for income growth:

yt=A4mtB4etC4pt1+D4vtE4μt(25)

where:

A4=c1a+c1(γχ+γ)
B4=c1(1+a(βt1))(a+c1)(γχ+γ)δb1(χχ+γ),may be<0 for large δ.
C4=δb1(χχ+γ)
D4=1(aa+c1)(χχ+γ)
E4=χχ+γ

and:

χ =ac2+c1λ(1+a(βt1))a+c1
γ =b1(1λ+δλ)+b2

Similar to the case of perfect capital mobility, increasing δ increases the impact of money growth on income growth by decreasing γ. Since it also reduces the effect of money supply growth on inflation, a given reduction in inflation achieved with a monetary stabilization program will have a greater recessionary impact. It will reduce the effect of supply shocks and will increase the persistence of inflation and the impact of demand shocks. Increasing δ will also reduce the absolute value of the coefficient of the exchange rate crawl, but when δ is large compared with the other parameters this coefficient may be either positive or negative so it is not clear whether this will increase or decrease the impact of exchange rate policy on income.7 Hence, depending on parameter values, increasing δ may stabilize or destabilize income during exchange-rate-based stabilization programs.

In the absence of other policy measures or shocks, increasing β has an overall recessionary effect on income which is equal to:

yβt=(A4B4C4)βtmt=(ac1a+c1)βt(γχ+γ)mt(26)

Higher δ will increase the recessionary effect of announcements. Increasing βt will also reduce the impact of demand shocks, but will increase the impact of past inflation and supply shocks.

Real wage growth is given by:

wtpt=δpt1δpt(27)

When indexation is fully forward-looking real wage growth is constant; otherwise it is not. Since announcements reduce inflation, they will increase the real wage, and this increase will be greater when δ is greater. Announcements also increase the impact of shocks on the real wage.

Using 4, 20 and 21, the current account is given by:

xt=A5mt+B5et+C5pt1D5vt+E5μt(28)

where:

A5=(c1a+c1)(γd1+d2χ+γ)
B5=c1(1+a(βt1))(a+c1)(γd1+d2χ+γ)δb1(χd1d2χ+γ)
C5=δb1(χd1d2χ+γ)
D5=d1(aa+c1)(χd1d2χ+γ)
E5=χd1+d2χ+γ

B5 may be negative when δ > 0, depending on parameter values, although this would be inconsistent with the experience of countries that have used exchange rate-based stabilization programs. Assuming B5 is positive, increasing δ will have the following effects:

A5 will increase if and only if d 2/d x < χ. The effect is not terribly important since the coefficient is negative, meaning that monetary stabilization programs are not threatened by outflows of reserves. B5 may increase or decrease, depending on parameter values. The adverse effects of indexation on the response of current account growth to exchange rate based stabilization programs is reduced as capital mobility decreases. C5 increases in absolute value, so the effects of past inflation are greater. The effect on D 5 is uncertain. E 5 decreases, so current account growth is stabilized with respect to supply shocks.

The overall effect on the current account of an announcement that increases βt is that there will be an increase given by:

CAβ=(A5+B5+C5)βtmt=(ac1a+c1)βt(γd1+d2χ+γ)mt(29)

Increasing δ may increase or decrease this effect, depending on parameter values. Announcements reduce the impact of demand shocks, but increase the impact of supply shocks.

If any action were taken in the period of the announcement, the effects of higher δ would generally be the same whether expectations were rational or static. The effects of an announcement would generally be to reduce the impact of all coefficients, with the following exceptions. Under perfect capital mobility the impact of the exchange rate crawl on inflation, income growth and the real wage, and the impacts of past inflation and supply shocks on past inflation will be increased, while under imperfect capital mobility the impact of the exchange rate on income growth and on current account growth and of the world interest rate on current account growth will be uncertain, while the impacts of world interest rates and supply shocks on current account growth will increase.

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1

The author would like to thank Charles Adams, Michael Hadjimichael, Robert Kahn, Carmen Reinhart, Federico Rubli-Kaiser and Alessandro Zanello for helpful comments and suggestions.

2

As capital becomes less mobile, however, the current account growth reaction to an exchange rate-based stabilization will diminish and perhaps even change sign. Also when capital is not mobile, while the income growth reaction to an exchange rate-based stabilization will become more negative, it is a positive reaction when indexation is mostly forward-looking, and therefore some backward-looking wage indexation could stabilize income after an announcement of an exchange rate-based program.

4

The program might also lack credibility in a different way; it could be believed that a recession would cause the authorities to abandon the price stabilization effort and increase the rate of crawl of the exchange rate (α < 0). This possibility is examined later in the section where different scenarios of price stabilization efforts are presented.

5

Not explicitly modeled here is the fact that changing δ will reduce β, in other words increasing the backwardness of indexation will reduce expectations of any reduction in inflation. The exact response of β to δ will depend on how knowledgeable the public is about how the rate of inflation is determined. The limiting case is the one where the public has full information about all coefficients in the model and about the probability density functions of the shocks, in which case β will be between zero and α. If β reacts strongly to changes in δ, the effects of changing δ on the coefficients may be greater than in the case of static expectations.

6

It is most likely true that d2/d1 < c2 + λc1(βt+1-1) since otherwise the indirect effect the growth of the exchange rate has on the growth of the current account by affecting income growth would be larger than the direct effect it has by the changing relative price growth of exports and imports. This would also be contrary to the experience of countries that have used exchange rate-based stabilization programs.

7
Specifically, when
δ > c1(1+a(βt+11))b1(a+c1)(γχ)
the coefficient will be negative.
The Effects of Forward-Versus Backward-Looking Wage Indexationon Price Stabilization Programs
Author: Mr. Joe Crowley