A Simple Model of Recurrent Hyperinflation

Revista Española de Economía, Vol. 15, nº 3, 1998 355-377 A Simple Model of Recurrent Hyperinflation * Jesús Vázquez Universidad del País Vasco. Recibido: enero de 1998 Aceptado: julio de 1998 Abstract The model developed in this paper builds on Cagan's demand for money by considering that a share of government expenditure is spent on servicing foreign debt obligations and fixed in foreign currency. Given this fiscal policy the choice of the depreciation rate implies a passive money supply rule. We show in this model that two types of high-inflation path exist. One type is driven by crawling peg rules of the official exchange rate and is characterized by the saddle-stable path dynamics. The other type characterizes hyperinflationary dynamics. This latter type is a consequence of the global dynamics of the model. The existence of these two types of path can generate recurrent hyperinflation. Moreover, hyperinflation can arise independently of government spending. Key words: Crawling peg rules, government debt, global and local dynamics. JEL Classification: E31, F31. Resumen. Un modelo sencillo de hiperinflaciones recurrentes A partir de la función de demanda de dinero de Cagan, el modelo desarrollado en este artículo considera que una fracción del gasto gubernamental se destina al pago de intereses asociado a la deuda externa y que dicho pago es constante medido en unidades de la moneda extranjera. Dada esta política fiscal, la elección de una tasa de depreciación del tipo de cambio implica una política monetaria pasiva. En este modelo, mostramos que dos tipos de sendas de alta inflación existen. Un tipo de estas sendas se deriva de la adopción de reglas sobre la tasa de depreciación del tipo de cambio oficial, conocidas como crawling peg * I am grateful for helpful conversations and comments from Carlos Borondo, Cruz Ángel Echevarría, María José Gutiérrez, Albert Marcet, two anonymous referees and participants in seminars at ESADE (Barcelona), XX Simposio de Análisis Económico (Universitat Autònoma de Barcelona), 95 Atlantic Economic Conference (Vien) and Universidad de Valladolid on previous drafts. A previous version of this paper was circulated under the title «Government Foreign Debt, Crawling Peg Rules and Hyperinflation: The Peruvian Case». Financial support from Ministerio de Educación and Universidad del País Vasco (Spain) through projects DGICYT PS95-0110 and UPV 035.321-HB067/96 is gratefully acknowledged. 356 Vol. 15, nº 3, 1998 Jesús Vázquez rules. Este tipo de sendas están caracterizadas por una dinámica de punto de silla. El otro tipo de sendas caracterizan dinámicas hiperinflacionarias. Este último tipo es una consecuencia de la dinámica global del modelo. La coexistencia de estos dos tipos de sendas puede generar hiperinflaciones recurrentes. Además, las hiperinflaciones pueden surgir independientemente de variaciones del gasto del gobierno. Palabras clave: reglas crawling peg , deuda pública, dinámica global y local. 1. Introduction Many Latin American economies have suffered recurrent hyperinflationary episodes (e.g. Argentina, Bolivia, Brazil and Peru) over the last decade. The observed evolution of inflation in these economies alternates relatively long periods of moderate and steady inflation with a few short periods of hyperinflation. The evolution of inflation for Peru is shown in Figure 1. Similar inflation patterns for Argentina and Brazil are shown by Mondino, Sturzenegger and Tommasi (1996). Moreover, recurrent hyperinflation is quite remarkable because the peaks of the inflation rate seem to be independent of any strong changes in seigniorage. 1 Hyperinflationary episodes are widely understood to be unstable processes where inflation speeds up and real money balances fall. The conventional view of hyperinflation, as stated clearly by Cagan (1956) and expressed by many authors (for instance, Sachs and Larrain (1993: 742-744)), is that hyperinflation arises when required seignorage exceeds its maximum steady state level and when expectations or real money holdings are adaptive. 2 However, the conventional view fails to explain the existence of recurrent hyperinflation which is not caused by strong movements in the level of seigniorage. Recently, Nicolini and Marcet (1996) and Mondino et al. (1996) have developed hyperinflationary models which accounts for the observations of recurrent hyperinflation. The key feature in Nicolini and Marcet's model is that agents are endowed with a type of bounded rationality whereas the relevant assumption in Mondino et al.'s model is the introduction of financial adaptation to new financial products when those assets start to compete with old depreciating domestic money. This paper develops a perfect foresight inflationary finance model which provides an alternative explanation of the recurrent hyperinflation phenomena observed during the 1980's. The model proposed in this paper is a slight variation on the model considered in the traditional inflationary finance literature (for instance, Evans and Yarrow (1981), Bruno (1989) and Bruno and Fischer (1990)). The model builds on Cagan's demand for money, but, in addition, the model assumes, first, that a portion of government spending is devoted to the 1. See Bruno et al. (1988) and (1991) for details. Nicolini and Marcet (1996) give a neat exposition of the stylized facts during recurrent hyperinflation. 2. See Vázquez (1998) for an alternative view of hyperinflation and for a review of different attempts made in the literature to characterize the explosive evolution during hyperinflationary episodes. A Simple Model of Recurrent Hyperinflation Vol. 15, nº 3, 1998 357 service of foreign debt and fixed in foreign currency. Second, the existence of crawling peg rules for the exchange rate, which are supported by rationing the foreign exchange market. This rationing typically results in the creation of a black foreign exchange market. We believe that these additional features, which are common in many Latin American countries suffering endemic high-inflation processes, are much relevant to understand recurrent hyperinflation. 3 Given a level of government spending, our model exhibits two types of highinflation dynamics. One type is characterized by the local dynamics around the unique steady state of the model. The steady state is saddle-point stable. Therefore, the dynamics of inflation are described in this case by the unique stable manifold converging to the steady state. The steady state rate of inflation under this type of dynamics is determined by the policy choice of the rate of crawl of the exchange rate. The other type of high-inflation dynamics is characterized by the global dynamics of the model. We identify these dynamics as hyperinflationary dynamics. Along these hyperinflationary paths inflation increases, real 3. The German government also had to pay war reparations after World War I, leading to huge budget deficits and hyperinflation. An important difference between Latin American hyperinflationary episodes and German hyperinflation was that the German government did not ration the foreign exchange market. See Bresciani-Turroni (1937) for more details on how the foreign exchange market worked during German hyperinflation. 1988.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8Monthly Inflation Rate 1989.2 1990.0 1990.8 1991.6 Time Figure 1. Peruvian Hyperinflation. Source: IMF Statistics. 358 Vol. 15, nº 3, 1998 Jesús Vázquez money balances fall and the official depreciation rate is fixed. These hyperinflationary dynamics lead the economy to a hyperinflationary plateau which is globally stable. Nevertheless, the hyperinflationary plateau might not be reached for two reasons. First, the black market premium is increasing rapidly along hyperinflationary paths because the black market depreciation rate increases with inflation whereas the official rate stays constant. Thus the proportion of private exports being smuggled will increase, reducing the net tax transfer from the private sector to the government, which will lead to an increase of the official depreciation rate (or equivalently a big devaluation) in order to reduce the black market premium. This policy can ultimately act as an anchor for inflation in order to drive the economy to the local stable manifold, where inflation, official and black depreciation rates remain close to each other, implying a more stable black market premium which leads to a more stable net tax on exports. Second, since domestic price is a jump-variable, changes in beliefs can drive the economy from a hyperinflationary path to the stable manifold and vice versa, generating recurrent hyperinflation. The remainder of the paper is organized as follows. Section 2 introduces a slight variation on the traditional inflationary finance model which builds on Cagan's demand for money. In Section 3, we show how the existence of both local and global dynamics of inflation helps to understand recurrent hyperinflation. The results are illustrated by the Peruvian hyperinflation of 1988-1990. Finally, Section 4 concludes. 2. The Inflationary Finance Model In this section we develop a model built on Cagan's model, which considers dual foreign exchange markets. There is no uncertainty. The money market equilibrium condition is given by , [1] where ln( m ) denotes the log of real money balances, π is the expected rate of inflation, γ and λ are constant positive parameters. The right-hand side of [1] is a continuous time version of Cagan's demand for money which establishes that the demand for real money balances is inversely related to expected inflation. 4 We assume perfect foresight, that is, actual and expected rates of inflation are equal. Moreover, we assume that a share of government spending is made on the payments associated with service obligations of foreign debt, and fixed in foreign currency. We assume that government spending is entirely financed by issuing domestic money according to the following expression , [2] 4. Appendix 1 provides a microeconomic foundation of Cagan's demand for money. This microeconomic foundation is considered below, first, to characterize the hyperinflationary dynamics and second, to rule out hyperdeflationary paths. mt()()ln γλπt()–= M ˙ed Pg+= A Simple Model of Recurrent Hyperinflation Vol. 15, nº 3, 1998 359 where M is the nominal level of money supply, e is the official exchange rate, d is the government expenditure devoted to the service of foreign debt (fixed in dollars), P is the domestic price level and g is the government spending made on consumption goods. As usual a dot on a variable denotes its time derivative. Both d and g are assumed to be constants to show the ability of the model to generate recurrent hyperinflation even when the level of government expenditure and the service obligations of foreign debt remain constant. With this assumption we do not intend to claim that changes in government spending cannot contribute to explaining hyperinflation in practice. Rather, we are aiming to suggest a mechanism which can amplify the effects of small changes of government spending in inflation. Following Kharas and Pinto (1989) and Pinto (1991) we have in mind the existence of a private export/import sector which is not modeled in this paper for the sake of simplicity. Private residents are expected to surrender all their export receipts, which are nominated in foreign currency, to the Central Bank for domestic currency at the official exchange rate because they are not allowed to hold foreign currency. However, they do so illegally. Therefore, there is currency substitution with residents holding domestic and foreign currencies. In the black market, the (black) foreign exchange rate is f ≥ e , with smuggled exports flowing into this market to obtain (illegally) higher returns. In the official exchange rate market, the official exchange rate e is managed by the government. Rationing in this official market works as follows. A certain amount of surrendered private export receipts are set aside for paying the service of foreign debt. The rest of surrendered export receipts are allocated to the private sector by allowing some types of imports to be made eligible for foreign exchange at the subsidized official exchange rate. Official reserves (fixed in dollars) are then protected and remain constant The rationing in the official market induces a black market. Thus the remainder of private export receipts are smuggled into the black market at rate f . 5 A government may have many incentives for rationing the foreign exchange market during a high-inflation episode. One possibility is that the rationing of the foreign exchange market works as a way of reducing the pressure on depreciating domestic money. Another possibility is that the black market premium generates an implicit net tax transfer from the private export sector to itself. Another possibility is that the government can subsidize some categories of imports by made them eligible for foreign exchange at the highly subsidized official exchange rate. 6 5. In this paper we omit for the sake of simplicity the demand for money by private exporters. Alternatively, we can assume that the total demand for money in the economy is given by [1'] instead of equation [1], in order to capture the idea that demand for money by some agents depends on the black market depreciation rate. Equation [1'] is similar to the money demand suggested by Abel, Dornbusch, Huizinga and Marcus (1979) which reflects that foreign assets and real assets can both be substitutes for domestic money. This money demand specification was tested by Vázquez (1995) using data from the German hyperinflation. As shown in Appendix 2, replacing [1] by [1'] has no effect on the dynamic analysis of the model. 6. For instance, Lago (1991: 302) reports that 72 percent of total peruvian imports were given access to official market during Peruvian hyperinflation. Lago claims that this policy was aiming to prompt a quick recovery by boosting import levels. m()ln γλπ–θf ˆ –= 360 Vol. 15, nº 3, 1998 Jesús Vázquez In order to close the model we need to specify how the black market depreciation rate is determined. Following Culbertson (1975) we assume that the black market depreciation rate is simply a linear function of domestic inflation π and the official market depreciation rate . 7 The idea behind this assumption is that a free (black) foreign exchange rate is largely determined by the purchasing power of domestic currency relative to the purchasing power of foreign currency. Since the foreign inflation can be considered nearly constant when analyzing countries involved in hyperinflationary process, the rate of depreciation of the black foreign exchange rate is then mainly determined by the rate of domestic inflation. In Kharas and Pinto (1989) domestic inflation is determined by the production sector, being a weighted average of official market depreciation rate and black market depreciation rate whereas the black depreciation rate is determined by the portfolio balance condition. In contrast, domestic inflation in our model, which builds on Cagan's demand for money, is determined by money creation, and the black market depreciation rate is mainly caused by inflation. In other words, our model follows a quantity theory of money approach whereas Kharas and Pinto (1989) follow a portfolio balance approach. Appendix 3 shows evidence that causality ran from inflation rate to black market depreciation rate during Peruvian hyperinflation. This empirical evidence supports the approach followed in this paper. 3. Hyperinflationary Dynamics In this section, we characterize the two types of hyperinflationary dynamics described by the model. Let us denote by m e . Using this definition we have that m = m e s , [3] where denotes the official real exchange rate. Implicitly we have normalized the level of foreign prices to one. Further, differentiating with respect to time and using [2] we obtain . [4] 7. The black market depreciation rate depends, of course, upon other many factors such as the variation of the level of reserves, changes in the penalty structure associated with black market activities and so on. Culbertson (1975) provides a theoretical framework for assessing the main determinants of black market exchange rates. As in Kharas and Pinto (1989), we are implicitly assuming that there is no net accumulation of reserves or changes in those other factors affecting black market exchange rate. fˆf˙ f ---= ze˙ e --= M e ----- se P ---= meM e -----= m ˙edg s --- mez–+= A Simple Model of Recurrent Hyperinflation Vol. 15, nº 3, 1998 361 Equation [4] can be viewed as a passive monetary rule determined by the choice of d , g and z . The choice of the rate of crawl for the official exchange rate z has been common in Latin America (called «tablitas») as reported by many authors (Kharas and Pinto (1989), Pinto (1991), Lago (1991) among others). The government choice of the rate of crawl for the official exchange rate z can be due to three reasons. First, as mentioned above, management of the exchange rate generates an implicit tax on exports. Second, the government at some point in time could have incentives to reduce the black market premium φ . These incentives appear because for sufficiently high φ the private export receipts being smuggled can increase so much that the tax transfer from the private sector to the government induced by the rationing decreases. Finally, the government can change the subsidizes of the imports eligible for official foreign exchange rate by varying z . From the definition of the official real exchange rate we have that . [5] The money demand function, equation [1], can be written as . Substituting this expression in [5] we obtain . [6] Equations [4] and [6] form a pair of differential equations describing the dynamics of the economy. Figure 2 summarizes these dynamics. Appendix 2 shows that for a given choice of the rate of crawl of the official exchange rate, z , there is a unique steady state which is saddle-point stable. 8 Moreover, Appendix 2 shows that the slope of is greater than the slope of and both slopes are negative as drawn in Figure 2. This model displays two sources of high-inflation dynamics. First, as in Kharas and Pinto's model, based on local dynamic analysis, our model displays a continuum of saddle-point stable equilibria, each of them corresponding to alternative policy choices of z . According to equation [5], an increase of z drives the steady-state inflation rate to a higher level. Eventually for high levels of z the economy will be evolving in a high-inflation scenario. Second, for a given level of z the global dynamics of the system characterize paths asymptotically tending to m e -axis. 9 Along this type of path, characterized by the global dynamics of the model, s goes to zero and m e goes to infinity. Since real money balances by definition are deter8. Appendix 2 shows that the determinant of the Jacobian associated with the linear approximation of the system is , which is always negative. 9. As pointed out by George and Oxley (1991) ignoring global dynamics and focusing on local dynamics could yield misleading conclusions. The analysis of our model provides, as a by-product, an example of the consequences of ignoring global dynamics. s ˙sz π–()= m e ()ln γs()ln– λπ–= s ˙s λ --- λzm e ()ln s()ln γ–++[]= d λm e ---------– m ˙e0= s˙0= 362 Vol. 15, nº 3, 1998 Jesús Vázquez mined by m = mes, what happens with inflation along these paths? The precise characterization of the global dynamics of the system of equations [4] and [6] can only be undertaken numerically.10 Numerical simulations of the global dynamics of the model show that starting at any point located to the left of the saddle-stable manifold inflation reaches a hyperinflationary plateau and real balances m approach zero as shown by Figures 3 and 4, respectively.11 Numerical simulations also provide a picture of the relative size of inflation along hyperinflationary paths compared to the rate of inflation along the stable manifold, i.e, in the vicinity of the saddle-point steady state. More specific, using the parameter values displayed in footnote 10 the monthly inflation rate at the hyperinflationary plateau is 271.2% whereas the inflation rate at the saddle-point stable steady state is 14%. 10. We used the software package SIMGAUSS V.20 and the Graphic software package, both written in GAUSS V3.01, to derive the numerical solutions of the differential equation system and Figure 2, respectively. The parameter values used to carry out the numerical simulations and to draw Figure 2 are the following: λ = 1.33, g = 0.2, d = 0.02, z = 0.14 and γ = 1. Appendix 4 provides an explanation of how the parameter values were chosen. In Appendix 4, we have also performed a sensibility analysis to see whether alternative values for the parameters give reasonable values of the rate of inflation at the hyperinflationary plateau. Notice that the existence of the two types of hyperinflationary dynamics does not depend on the parameter values chosen provided that the unique steady state exists. The computer programs are available from the author upon request. 11. Global dynamics also characterize hyperdeflationary paths. Along this type of path the economy tends to travel along the locus which implies that tends to infinity (see the phase diagram displayed in Figure 2). As shown in Appendix 1 these hyperdeflationary paths are not feasible because they violate the transversality condition of the underlying household optimization problem, equation [A.7]. Figure 2. Phase Diagram. 0 m e * m e s s* s ·= 0 m · e = 0 m ˙e0= mm e s= A Simple Model of Recurrent Hyperinflation Vol. 15, nº 3, 1998 363 Figure 3. Evolution of Simulated Inflation. Figure 4. Evolution of Simulated Real Money Balances. 20 0 Inflation 0.4 0.8 1.2 1.6 2.0 2.4 2.8 040 60 80 100 120 Time 20 Real Money Balances 0.4 040 60 80 100 120 Time 0.8 1.2 1.6 2.0 2.4 370 Vol. 15, nº 3, 1998 Jesús Vázquez where γ = k–λβ. Equation [A.10] is a continuous-time non-stochastic version of Cagan's demand for money, where the semielasticity of the demand for money with respect to inflation, λ, according to [A.5], is the marginal utility of consumption. The dynamics of the inflationary finance model based on Cagan's demand for money under perfect foresight are well known (Evans and Yarrow (1981), Bruno and Fischer (1990)). We summarize their results here. The traditional inflationary finance model assumes that high-powered money is used according to the following expression . [A.11] For the sake of simplicity we drop the index t from now on. The definition of real money balances and equation [A.11], imply that . Substituting [A.10] in this equation we have the following first-order differential equation . [A.12] Figure A.1 shows that dual steady states may co-exist for a given level of government spending whenever this level is not too high. This figure also shows that the high-inflation steady state, m1, is stable and the low-inflation steady state, m2, is unstable. 0 m · AB m Figure A.1. Phase Diagram for the Degenerated Model. g+m(ln(m)–γ)/λ M ˙t() Pt()gt()= m ˙gmπ–= m ˙gm λ ---- m()γ–()ln[]+= A Simple Model of Recurrent Hyperinflation Vol. 15, nº 3, 1998 371 Appendix 2 The dynamics in our model are characterized by the following pair of first-order differential equations: , [4] . [6] The expression for the locus is given by . The expression for the locus is given by . Equating the last two expressions we obtain the unique steady state of the model: . . The slope of is given by . The slope of is given by . Let us show that , at the steady state. Since the steady state is unique, the last inequality must always hold, that is, the slope of is always greater than the slope of . Since z>0 the last inequality implies that . Using the expressions for the slopes we have . m ˙edg s --- mez–+= s˙s λ --- λzm e ()ln s()ln γ–++[]= m ˙ e 0= m e d z --- g sz ----+= s˙0= meeγλz– s ------------= s*z d --- eγλz–g d ---–= me *deγλz– zeγλz–g– ----------------------= m ˙e0= me ∂ s∂ --------- m ˙e0= g s2z -------–= s˙0= me ∂ s∂ --------- s˙0= me s ------–= me ∂ s∂ --------- s˙0= me ∂ s∂ --------- m ˙e0= < m ˙e0= s˙0= zme ∂ s∂ --------- s˙0= zme ∂ s∂ --------- m ˙e0= < zme s -------- –g s2 ---- –< 372 Vol. 15, nº 3, 1998 Jesús Vázquez At the steady state . Then, the last inequality can be written as , which is true. In order to study the local dynamics of the system formed by equations [4] and [6], we linearize the system around the unique steady state . Formally, . The determinant of the Jacobian matrix J is , to get the last equality we use the fact that at the steady state or equivalently, . Next we consider the demand for money proposed by Abel et al. (1979) to see that the dynamics of the model remain the same. The money market equilibrium condition is . [1'] We assume in the paper that black market depreciation rate is a linear combination of inflation and official depreciation rates, i.e. . Substituting this expression in [1'] we obtain . Rearranging, . Using this expression and after a little algebra we obtain . [6'] Notice that equation [4] is independent of the assumption made on the money demand. Therefore, the dynamics associated with money demand [1'] are completely characterized by [4] and [6']. Notice that the slope of and the phase diagram do not change when considering [1'] instead of [1]. Therefore, the steady state is also unique. zmedg s ---+= d s ---–0< m e * s * ,() m ˙ e s ˙   z–g s 2 ---- – s λme --------- 1 λ ---      m e m e * – s s – *    = J z λ ---– g λmes ------------+ d meλ --------- 0<–== m ˙ e 0= z d m e ------ g sme --------+= m()ln γλπ–θf ˆ –= f ˆαπ 1α–()z+= m()ln γλθα+()π–θ1α–()z–= π1 λθα+ --------------- γm()ln– θ1α–()z–[]= s ˙ sz π–() s λθα+ --------------- λθ+()zm e ()ln s()ln γ–++[]== s ˙ 0= A Simple Model of Recurrent Hyperinflation Vol. 15, nº 3, 1998 373 In order to study the local dynamics of the system formed by equations [4] and [6'], we linearize the system around the unique steady state . Formally, . The determinant of the Jacobian matrix J is , to get the last equality we use the fact that at the steady state or equivalently, . Therefore, the unique steady state is also saddle-point stable using [1'] instead of [1]. Appendix 3 This appendix analyzes the causality between the rate of inflation and the black market depreciation rate by performing Granger causality tests using data from the Peruvian hyperinflation. More specifically, the VAR coefficients from a fivelag, three-variable system formed by inflation, black depreciation and official depreciation rates, i.e. is considered. To carry out this exercise we use the time series reported by Lago (1991) from July 1985 to June 1990. Augmented Dickey-Fuller tests show evidence that all three variables are integrated of order 1, i.e. I(1).17 The VAR sys17. Under the null hypothesis that a particular variable is I(1) the augmented Dickey-Fuller statistic (ADF with drift) takes the following values: ADF(π) =–1.17, ADF(f) =–1.13 and ADF(z) =–3.02 for inflation, black depreciation and official depreciation rates, respectively. Under the null hypothesis that a particular variable is I(2) the ADF with drift takes the following values ADF(π) =–5.13, ADF(f) =–4.62 and ADF(z) =–5.68. Since the critical values for a sample size of 50 reported by Fuller (1976) are τ0.01=–3.58 and τ0.05=–2.93, it can be concluded that inflation and black depreciation rates are I(1) whereas for the official depreciation rate the I(1) hypothesis against the stationary alternative can only be rejected at the 1% significance level. me ** s** ,() m ˙ e s ˙   z–g s 2 ---- – s λθα+()m e -------------------------- 1 λθα+ ---------------      m e m e ** – s s – **    = J z λθα+() -------------------– g λθα+()m e s ----------------------------+ d meλθα+() -------------------------- 0<–== m ˙ e 0= z d m e ------ g sme --------+= πtα0αkπtk– k1= 5 ∑β k f tk– k1= 5 ∑ρ k z tk ,– k 1= 5 ∑ +++= f t γ 0 γ k π tk– k1= 5 ∑δ k f tk– k1= 5 ∑k k z tk– , k 1= 5 ∑ +++= z t τ 0 τ k π tk– k1= 5 ∑φ k f tk– k1= 5 ∑λ k z tk– , k 1= 5 ∑ +++= 374 Vol. 15, nº 3, 1998 Jesús Vázquez tem is estimated using an unrestricted least square estimator (see Lütkepohl (1991: 369)). Then two null hypothesis are tested: β k = 0 and γ k = 0 with k = 1,2,…,5. Given that the time series are I (1) we use the following Wald statistic (see Lütkepohl (1991: 378-379) , where T = 52 is the sample size, is the unconstrained least square estimator which is a ( pr 2 +3)x1 vector constructed by stacking the coefficients of the VAR system as follows 18 . is a consistent estimator of the variance-covariance matrix of . To calculate the null linear hypothesis has to be written in matrix form as . For the statistic takes the value 7.912. For the statistic takes the value 30.545. Since χ 0.05 (5) = 11.070, it can be concluded that the data do not reject the possibility that the black depreciation rate does not cause the rate of inflation, but the data do reject the possibility that inflation does not cause the black depreciation rate. 19 In summary, the empirical evidence from Peruvian hyperinflation supports the quantity theory of money approach followed in this paper, that is, causality runs from inflation to black depreciation rate. 18. p , r and N denote the number of lags, the dimension of the VAR system and the number of linear restrictions imposed by the null hypothesis, respectively, in this application, we have p = 5 r = 3 and N = 5. 19. Table A.1 shows that the results of the Granger causality tests are robust to the choice of VAR order. In particular, we have checked that the results of the Granger causality tests given above hold for the following choices of p = 2,3,4,5,6,7. The reason why we have performed the tests for alternative VAR orders is that the usual criteria for VAR order selection are not conclusive in this particular case. λωTCα ˜c–()'CΣ ˜ C'() 1– Cα ˜ c–()χ 2 N()→= α ˜ α ˜α 0 γ 0 τ 0 α 1 γ 1 τ 1 β 1 δ 1 φ 1 ρ 1 k 1 λ 1 α 2 γ 2 …ρ 5 k 5 λ 5 ,,,,,,,,,,,,,,,,,()= Σ ˜α ˜ λ ω Cα ˜ c= H 0 : βk0= λω H0: γk0= λω Table A.1. Granger causality tests for alternative VAR orders. p λω for βk = 0 λω for γk = 0 1 2.2715 1.8342 2 2.6329 7.1890 3 2.8104 29.6645 4 4.8163 30.3171 5 7.9116 30.5446 6 8.2256 79.1872 7 9.7147 74.7034 A Simple Model of Recurrent Hyperinflation Vol. 15, nº 3, 1998 375 Appendix 4 Taking into account the empirical evidence from Peruvian hyperinflation reported by Lago (1991), this appendix explains how the parameter values used in numerical simulations were chosen. Since per capita endowment (y) is distributed in our model between private consumption (x) and government consumption, normalizing y to one, we assume that the government expenditure is 20% of the endowment. This value is close to the average of government spending as a percentage of GDP during Peruvian hyperinflation as reported by Lago (1991, Table 9.8). Therefore, in our simple model that figure implies that domestic private consumption as a percentage of GDP during Peruvian hyperinflation was around 75% and the trade balance surplus (exports minus imports) was 5% of GDP (see Lago (1991 Table 9.6 and 9.8)). The utility of consumption is assumed to be logarithmic, so we have that . The value reported by Lago (1991) for debt service payments on public foreign debt as percentage of GDP is around 2%.20 z is chosen equal to 0.14. This value is roughly the value taken by inflation during the initial periods of Peruvian hyperinflation from January to July 1988 (see Lago (1991: 304)). Since the steady state values of inflation and depreciation rates of the official and black markets are all the same, by choosing that value for z we are conjecturing that the Peruvian economy was close to the steady state during that period. We next perform a sensibility analysis to see whether alternative values for the parameters give reasonable values of the rate of inflation at the hyperinflationary plateau. As discussed in Section 3, the rate of inflation at the plateau does not depend on the values chosen for d and z. Moreover, γ is a scale parameter measuring units of the money demand. Therefore, the sensibility analysis, which is relevant, is to analyze how alternative values for g and λ affect the rate of inflation at the hyperinflationary plateau. Table A.2 displays the results. The values of the rate of inflation at the plateau are of reasonable size when comparing with the peaks of inflation observed during hyperinflationary episodes. 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