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Economic Review Federal Reserve Bank of Dallas May 1988  1  Currency Substitution and the International Transmission of Economic Disturbances  W. Michael Cox and Michael Parkin This article isolates one possible source of economic interdependence-currency substitution- and analyzes its implications for the international transmission of monetary disturbances, the output inflation trade-off, and the real exchange rate. We use a rational expectations equilibrium model for a world economy with flexible exchange rates, currency substitution, and incomplete current information. Among other results, we show that in this environment, monetary shocks in one country can generate an exchange rate effect and output and price effects in all countries. 13  A Study of the Relationship Between Economic Growth and Inequality: The Case of Mexico Joseph H. Hasla~ Thomas B. Fomby, and D. }. Slottje  This article uses data for Mexico to test whether changes in the growth of an economy's per capita real gross domestic product lead or lag changes in inequality in the distribution of household expenditures. The findings indicate that increases in economic growth tend to precede decreases in the level of inequality. This result suggests that policy efforts directed toward stimulating economic activity may well give rise to a more equal distribution. The evidence, however, does not indicate that increases in inequality precede changes in growth. Consequently, policy efforts toward redistribution need not retard economic growth.  This publication was digitized and made available by the Federal Reserve Bank of Dallas' Historical Library (  Economic Review Federal Reserve Bank of Dallas May 1988 President and Chief Exe<:utive Officer  Robert H. Boykin First Vice President and Chief Operating Officer  William H. Wallace Senior Vice President and Director of Research  Harvey Rosenblum Vice President and Associate Director of Research  Gerald P. O'Driscoll, Jr. Assistant Vice President and Assistant Director of Research  W. Michael Cox Assistant Vice President and Senior Economist  leroy O. laney  Economists Naiionalllnternational  John K. Hill Robert T. Clair Joseph H. Haslag Cara S. lown Kenneth J. Robinson Regional/Energy  Stephen P. A. Brown William C. Gruben William T. long 11\ Hilary H. Smith Keith R. Phillips  Editorial  Virginia M. Rogers Elizabeth R. Turpin Graphics and Typesetting  Graphic Arts Department  The Economic Review is published by the Federal Reserve Bank of Dallas and will be issued six times in 1988 (January, March, May, July, September, and November). The views expressed are those of the authors and do not necessarily reflect the positions of the Federal Reserve Bank of Dallas or the Federal Reserve System. Subscriptions are available free of charge. Please send requests for single·copy and multiple-copy subscriptions, back issues, and address changes to the Public Affairs Department, Federal Reserve Bank of Dallas, Station K, Dallas, Texas 75222 (2 14/651-6289). Articles may be reprinted on the condition that the source is credited and that the Research Department is provided with a copy of the publication containing the reprinted material  Currency Substitution and the International Transmission of Economic Disturbances W. Michael Cox  Michael Parkin  Assistant Vice President and Assistant Director of Research  Professor of Economics  Federal Reserve Bank of Dallas  Department of Economics University of Western Ontario  During the era of fixed exchange rates, it was a generally held presumption that flexible rates would deliver a better macroeconomic performance. Specifically, it was predicted that the widespread adoption of flexible rates would reduce the international transmission of disturbances and permit countries to pursue more independent macroeconomic policies. Experience with flexible rates since 1973, however, has led to a modification of views concerning their properties and the degree of insulation that they afford. Little perceptible change has occurred in the degree of crosscountry correlation in real economic activity in the flexible rate era, though inflation performances have diverged markedly. Further, the output-inflation trade-off (Phillips curve) that appeared to be remarkably stable during the fixed exchange rate era of the 1950s and 1960s has become "unstable" in the flexible rate period. Finally, much greater deviations from purchasing power parity appear to have occurred than have been experienced under fixed rates. Many potential factors, of course, could account for the experience of the past fifteen years. 1 This article is not designed to outline or review the many possible factors at  work in generating exchange rate movements and transmitting economic disturbances across national borders. Rather, we focus on one particular aspect of international economic relations- currency substitution- in order to analyze its implications for the international transmission of monetary disturbances, for the output-inflation trade-off, and for the real exchange rate. The motivation for this work is our feeling that as compared to a system of fixed exchange rates, a flexible rate system provides a much greater opportunity for profit from participating in international currency markets-thus enhancing the substitution aspect of currencies internationally and providing a channel for the international transmission of disturbances not significantly found under fixed rates. 2 Although the explicit focus of this paper is on currency substitution, a broader interpretation may be placed on the study and its analysis-that of portfolio diversification across securities of varying currency denomination. In the model developed in this article, money is the only asset issued in each country, and currency substitution is therefore equivalent to international portfolio diversification. The work of  Economic Review - May 1988  Lance Girton and Don Roper explicitly examines the distinction between substitution involving currency and that involving securities. 3 Because that distinction is not central to the present study, this paper abstracts from it. Several other important studies of the role of information in international finance include that by Reuven Glick and Clas Wihlborg, who examine the purchase of information and the potential for a free rider problem to arise when agents use information contained in freely available market prices. 4 Earlier, Kent Kimbrough studied the effects of monetary policy in an open economy, and particularly the information content of the exchange rate under alternative exchange rate regimes. s Further, Richard Harris and Douglas D. Purvis studied the exchange market efficiency in the face of incomplete information. 6 The contribution of the present study differs in two important respects from the earlier ones cited above. First, we deal directly with isolating the one phenomenon of currency substitution in the face of incomplete information. Thus, our analysis has been stripped of other possible effects. This has enabled us to take a sharper view of the role of this phenomenon than have the reports of earlier studies. Second, though our focus on one particular channel is narrow, we have been able to study a wider range of variables than have some of the earlier studies. Like earlier studies, our article here derives propositions about the behavior of both the nominal and the real exchange rate. Also derived, however, are results on the behavior of prices and output-on the effects of incomplete information and currency substitution on the division of nominal disturbances between output and the price level. Finally, the results of this article have differed from those of other articles in being derived for the world (two-country) economy and not just a small open economy. In the light of the many possible theoretical approaches that could be used to study the phenomenon of interest, the choice of the model for this study was dictated by a selection of the simplest possible assumptions. In recent years, it has become fashionable to specify models in terms of their "deep structure." This deep structure can be defined as an environment generating movements in exogenous variables, the information-generating process, and preferences and technologies. 7 Given the deep structure approach, decision rules are derived, usually as the solution of a stochastic dynamic programming problem. Such models enable important advances to be made in understanding aggregate fluctuations in the context of a closed economy.B An alternative approach that for many purposes is less satisfactory-but for the present study is appropriate-would be to specify a model in terms of mar2  ket demand and supply functions rather than the primitives of the endowments, preferences, and technologies that underlie them. This approach, which has been adopted here, has been the one most commonly used in the earlier literature on exchange rate determination, currency substitution, and the role of incomplete information in the transmission of economic disturbances. SpeCifically, the model developed in this study is a version of the rational expectations equilibrium approach to macroeconomics initially proposed by Robert E. Lucas, Jr., and Robert J. Barro.9 The difference between the present model and earlier work in this area is that our study explicitly represents the entire world economy (viewed as two countries) and abstracts from it all features of the world that obscure any effects of the one channel of international transmission that is being isolated. Using the model helps to determine the equilibrium levels of output and prices in both economies, as well as the equilibrium exchange rate. Because of the strong assumptions made about the asset and information structure, there are some problems that the model cannot address. Central to these are the determination of interest rates and capital and trade account imbalances. Because these matters have been studied extensively, it is not the goal of this study to add to that body of knowledge. 10 Simultaneously analyzing the intertemporal substitution that drives capital and trade accounts and the information asymmetries that drive the results would involve cumbersome procedures while not appearing to shed any light on the issues at hand. This article is divided into four sections in addressing the study'S issues. In the first section, the abstractions employed are examined and t,he structure of the model is set out. The second section derives the solutions for the exchange rate, prices, and output in the two economies and examines the properties of those solutions in a world of full current information. The third section examines the solutions under incomplete current information and analyzes the properties of the output-inflation trade-off and the behavior of international relative prices. The final section provides a summary and presents the conclusions. The central conclusion reached is that currency substitution with incomplete current information is sufficient for monetary shocks in one country to generate an exchange rate effect and output and price effects in all countries. The slope of the output-inflation trade-off (the response to a domestic monetary shock) depends on the relative variability of monetary disturbances in the domestic and foreign economies. Further, foreign monetary shocks lead to shifts in the domestic output-inflation trade-off. In general, international relative prices are not invariant to purely monetary Federal Reserve Bank of Dallas  ered. Information on the money supplies---domestic and foreign-is available with a discrete, one-period time lag.  shocks, and the size of those effects depends on the degree of currency substitution. Abstractions and structure of the model  The term "currency substitution" will be used to signify portfolio diversification across domestic and foreign currencies. In order to focus attention specifically on the role of currency substitution (so defined) in the international transmission of disturbances, the study has eliminated the other aspects of international economic activity. Specifically assumed are the following: 1.  There is no international trade in commodities. Individuals reside in and buy and sell commodities in only one country in each period of time. This setup is similar to the Phelps "island" parable, which was also employed by Lucas, except that each of the two markets (home and foreign) issues its own currency, and there is a global exchange market in the two currencies. 11 Thus, each country is a Phelps ian island with its own currency and a flexible exchange rate. The assumption of no international trade in commodities does not, however, rule out buying in one country in one period and selling in another country in the following period. The only asset, however, that may be used as a store of wealth between the two periods is money, and either domestic or foreign money may be used for that purpose.  2.  There is a discrete, one-period time lag in the flow of price information from one country to the other. This assumption is intended to represent a situation in which markets in goods, services, and factors of production are less centralized than is the market in foreign exchange.  3.  Sufficient information is available to ensure that prior to observing any current prices, countries have the same distribution of prices when measured in terms of one of the currencies.  4.  No interest-bearing assets exist.  s.  There are two countries- the home country and a foreign country (the rest of the world). Each country issues its own money, which serves both as the unit of account and as the medium of exchange for all transactions within the country.  6. There is a free worldwide market in money, and the exchange rate between the two monies is determined by market forces so as to ensure that the quantities of the two monies issued by the separate monetary authorities are held willingly. Exchange controls are not considEconomic Review - May 1988  7.  Information on the exchange rate is continuously available to a!1 agents at zero cost.  8.  Each currency enters the economies via transfer payments to individuals who are currently holding units of that currency.  It should be emphasized that the assumptions listed above have been selected for the specific purposes of the present analysis and are not intended, therefore, to reflect absolute realism. Rather, the goal for this study was to isolate the one phenomenon of primary interest-currency substitution-and subject it to a microscopic study. Some objections may be raised, however, that the assumptions concerning the timing of the arrival of information are more appropriate to the steamship era than to that of the global village that we now inhabit. Nevertheless, it can be seen that the assumptions under which this study was developed are indeed relevant to today's world when it is recognized that the composite commodity is playing the role not only of conventional goods and services but also of factors of production. Because of the aggregation across all markets, the price level should be taken as representing an average of both commodity and factor prices. Also, though various monetary aggregates are observed and reported with high frequency and reasonable accuracy, no presumption is made that the monetary aggregate that actually generates observed behavior is recorded or reported at all. The model for this study contains an aggregate called money, but what the real world counterpart to that aggregate actually is remains a matter of speculation and uncertainty. To the extent that the assumptions in this article are unreasonable, they may err on the side of giving agents information that agents in the actual economy probably never have-that is, knowledge of the true value of the past money supply. The only test of the reasonableness of any assumptions such as these presented here lies in the predictions that they yield. Therefore, a model is developed that incorporates the study'S assumptions and generates some results based on them. The model is presented in terms of the series of equations that describes supply and demand in markets for commodities and money and also sets out the equilibrium conditions in those markets.12 Output supply in the domestic economy is given by  3  where Y= P= M= z= E= I  =  logarithm of real output, logarithm of the price level, logarithm of nominal money balances, stochastic disturbance to supply, expectations operator, and set of information available to agents in the domestic economy.  (Subscripts denote time, and superscripts denote supply,) The aggregate supply function contains three terms, the first of which is an intertemporal substitution effect. The parameter y' measures the strength of that effect. The second term is a real balance effect, which has been specified only in terms of domestic real balances. This is in keeping with the goal of isolating the effects of currency substitution and limited information per se. If foreign currency real balances were placed in the domestic output supply equation, this would provide a direct channel whereby foreign monetary policy could affect domestic economic activity. Since the present study seeks to cut out that effect, the real balance effect has been defined in terms of domestic real balances only. The third term in the supply equation is a random disturbance, more about the properties of which will be discussed later. Like aggregate supply, the aggregate demand for goods and services in the domestic economy also depends on intertemporal substitution, real balances, and a random disturbance. That is,  d  5  Y ,1'/ > O. Since equilibrium prevails in the domestic goods market, the actual quantity traded, y, is equal to the quantity demanded and the quantity supplied, or  (3)  d  YI  5  = YI = YI '  Using equations 1 and 2, together with the equilibrium condition 3, gives  (4) where 4  In words, the logarithm of the price level is a weighted average of the logarithm of the money supply and the expected future price level together with a random disturbance. The weights on the money supply and the expected future price level add to one, and the weight on the money supply is higher the smaller the intertemporal substitution effect and the larger the real balance effect. The foreign economy is assumed to be identical to the domestic economy in terms of the underlying intertemporal substitution and real balance parameters. Further, it is assumed that the real random disturbance, z, sums to zero across the world economy. Also, a positive real shock to the home economy is a negative real shock to the foreign economy. With these assumptions, the supply and demand functions for goods and the equilibrium condition in the foreign economy can be manipulated to yield an expression for the foreign price level, as follows:  It should be noted here that the foreign price level is determined in exactly the same way that the domestic price level is, but it depends on the foreign money supply and the expected future foreign price level. Also, a random disturbance, z, which increases the domestic price level, decreases the foreign price level. This last feature arises from the study's assumption that real shocks are zero for the world as a whole. It would not be difficult to relax this assumption, and for some purposes, it might be desirable to do so. In a more general model, two other margins of substitution in the supply and demand functions would be added. One margin would be between current domestic goods and future foreign goods and the other between current domestic goods and future foreign money. The first of these margins-international intertemporal substitution-is absent from the study's model because to include it would be inconsistent with the assumption made that sufficient information existed to guarantee ex ante purchasing power parity. This assumption implies that when converted at the expected exchange rate, the two intertemporal substitution terms-that between current domestic goods and future domestic goods and that between current domestic goods and future foreign goods- are identical. Thus, although this intertemporal substitution term does not explicitly appear in equations 1 and 2, it could be regarded as being Federal Reserve Bank of Dallas  subsumed under the intertemporal substitution effect already present in these equations. It should be emphasized that the real balance term in the domestic supply and demand functions uses the entire world stock of currency of the domestic economy rather than only that part which is held by residents in the home country. This specification is appropriate, given the assumption that domestic currency enters the economy through transfer payments to individuals who are currently holding units of domestic currency. Random relative shifts in the holdings of domestic currency across countries are thus, in effect, treated as entering through the random disturbance terms, z, as in Barro,13 The preceding statements could be repeated for the foreign economy. Specifying the supply and demand functions in this way has the considerable virtue that it avoids introducing a direct wealth effect from foreign currency to domestic output (and domestic currency to foreign output), and it leaves the analysis examining exclusively the incomplete information-the currency substitution transmission channel. 14 For the two monetary assets in the world, M and M', demand functions exist for each. By Walras' Law, however, only (n - 1) of the n markets in the world economy need be considered in order to determine the general equilibrium. Accordingly, this study chose to focus only on the market for M (domestic money) and ignore that for M (foreign money). Equilibrium in the market for domestic money, M, is given by  Mt = (1 - <I»[Pt - A(E[Xt +1 1IrJ - Xtl]  (6)  + <I>[P; + Xt -  A(E[Xt +1 1I;] - Xt)] ,  where  x = logarithm of the E:ixchange rate between t~e two monies, M and M (units of M per unit of M ), <I>  = the weight in a geometric index of domestic money  held domestically (weighted by 1 - <I> ), and held in the rest of the world (weighted by <I> ), and A = semielasticity of the demand for M with respect to the expected rate of change of the exchange rate. The supply of currency, M, which appears on the left-hand side of equation 6, is assumed to be determined exogenously. The right-hand side of equation 6 is the world demand for the domestic currency. The demand arises from the demand of the home country (given a geometric weight of 1 - <1» and the demand in the foreign country (given a geometric weight of <1». Interest will center on the implications of alternative values of the parameter A. If A took Economic Review - May 1988  the value zero, no currency substitution would occur, and with A infinity, the two currencies would be perfect substitutes. The parameter <I> is also of interest. If <I> were equal to zero, then all the home currency would be held in the home country (no substitutability of M for M) This could be thought of as a situation of full and effective exchange control in which foreign agents were barred from holding the currency of the home country. The money supply in each country has been assumed to be generated by a random walk and to be given by  (7)  .= .  (8)  Mt  Mt- 1  .  + J1t ,  where M t - 1 and M;-1  = known previous period values of the  supplies of domestic and foreign money, respectively, and J1t and J1t = random variable~ with ~~ro means and variances, (1 and (1 • •  They are distributed independently of each other. The only other random variable in the analysis is z. It, too, has a zero mean and constant variance, ... 2 , and is distributed independently of J1 and J1'.15 Two additional propositions are needed to complete the model. It should be noted that because of the assumption that individuals reside in and buy and sell commodities in only one country during each period of time, it is not possible to call on purchasing power parity to equate Pt with (P; + Xt) • However, the assumption of ex ante purchasing power parity is explicitly embodied by specifying that-viewed from the perspective of each country-purchasing power parity is expected to hold at time (t+ 1). In other words,  (9)  E(Pt+1 1/t ) = E(P;+11/t )  (10)  E(Pt+1 1/;)  + E(Xt+1 1/t)  and  = E(P;+11/;) + E(Xt +1 1/;).  It should be noted that there are two explicit conditions here. Purchasing power is expected to hold, given the information available in the domestic economy, It, in equation 9, and also, viewed from the perspective of the foreign economy (given the information set n, purchaSing power is expected to hold at period (t + 1). These two equations are an explicit statement of assumption 3. This completes the specification of the model, and it is now possible to analyze the implications of the assumptions that it embodies. In the next section, the analysis will be 5  based on the assumption that full current information is available, and in the succeeding section, the assumption will be made that current information about the innovations in domestic and foreign money and the real disturbance, z, is not available. Model analysis based on full current information The purpose of this section is to show that the model exhibits the standard properties of monetary neutrality under full current information. Using equations 7 and 8 in equations 4 and 5 yields the following expressions for the domestic and the foreign price level:  (11)  Pt = a(Mt- 1 +  (12)  P; = a(M;_1  + (1  + J.l;) + (1  Xt  (18)  1  + J.l;)  - (1 - 2<1»zt - (1 - <1>)(1 - a - A.)E(Pt+1 1/t) - (1 - <1»A.E(p;+11/a - <1>(1 - a  + A.)E(P;+11/;)  + <1>A.E(pt+1 1/;) . These three equations provide solutions for the domestic and the foreign price level and the foreign exchange rate, given the information on the previous period's (known) quantities of domestic and foreign money, the current innovation in domestic and foreign money, and the current value of the real shock, together with domestic expectations of domestic and foreign prices in the next period and foreign expectations of domestic and foreign prices in the next period. Since the expectations of P'+l and P;+l are rational, they will depend (linearly) on the same variables that determine actual prices, so that the solution to this system may be written as follows:  .  .  = 1t31Mt-1 + 1t32Mt-1 + + + 1t3SZt·  To establish the values for the 1ti;, equations 14 and 15 are led one period, and expectations are taken-conditional on the information available at time t (full current information in the present solution), with the results substituted into equations 11, 12, and 13. Equating the coefficients between equations 11 and 13 and equations 14 and 16 gives the solutions for the values for the 1ti;' This yields the following values for prices and the exchange rate:  - a)E(P;+11/;) -  = <1> + A. [1 - (1 - <1» a] (Mt- 1 + - <1>a(M;_1  (16) Xt  (17)  Zt .  + 1t22M;_1 + + 1t24J.l; + 1t2SZt,  and  - a)E(Pt+1 1/t) + Zt and  Using these two results, together with equations 9 and 10, in equation 6 and solving for the exchange rate gives the following:  (13)  (15) P; = 1t21Mt-1  .= M.-  .  t 1 + - Zt, and  Pt  •  (19) Xt  •  = Mt- 1 + - Mt- 1 - +  [<1> - (1 - <1»] <1> + A. Zt·  These results look very familiar. Each of the prices, P, and P; , is proportional to the own country's money stock, is unaffected by foreign money, and depends, in a natural way, on the real disturbance. The exchange rate is proportional to the ratios of the two monies (difference in logarithms). If the two countries are completely symmetrical, the parameter <1> will equal one-half, and z, the real shock, will have no effect on the exchange rate. To the extent that <1> departs from one-half, real shocks will influence the exchange rate. The values of output in the two countries will be as follows:  -l + tis  (20)  Yc =  (21)  Yt  Y + '1  d Zt  +  l  + tid  Y + tI  z:, and  -l + tis  = - y + tI  These results also look familiar. Under full current information, the model generates predictions that display complete monetary neutrality. The prices and the exchange rate are proportional to money, and real output is independent of money. The next discussion will turn to the more interesting task of examining the solution to the model under incomplete current information. Model analysis based on incomplete current information For the analysis of the model in this section, it can now be assumed that agents do not have complete current infor-  6  Federal Reserve Bank of Dallas  mation. Specifically, there is a discrete, one-period time lag in the arrival of commodity price information across countries. Also, no one has information about current actual monetary innovations and the real disturbance. Individuals in each country are assumed to have full knowledge of the local (own-country) price level and of the foreign exchange rate. The assumptions here on the arrival time of information imply that in the home country, Pt and Xt are observed at date t but that P; is not observed. In the foreign country, P; and Xt are observed, but Pt is not observed. It should be noted from equation 14 that the observation in the domestic economy of .the price level is the equivalent to an observation of (22)  vt  wt  .  .=  vt  7t23llt  .  + 7t241lt + 7t2SZt .  The task of agents in the domestic economy is to extract information from the observation of Vt and W t in order to make the best inference possible about the current values of the domestic and foreign monetary shocks and the real shock. A similar task has to be performed by agents in the foreign economy. To calculate the expected values of the price level at period (t + 1), equations 14 and 15 are led one period, and the conditional expectations of each are calculated for the two information sets (domestic and foreign). These calculations yield the following: (25)  E(pt+1 1/a =  7tl1[Mt_1  + E(llt lIt)]  + 7t12[M;_l + E(Il; 1m, (26)  E(p;+ll/t) =  7t21[Mt-1  + E(llt lIt)]  + 7t22[M;_l + E(Il; 1m, (27)  E(pt+1 1/;) =  7tll[Mt-1  + E(llt II;)]  + 7t12[M;_1 + E(Il; II;)] , and Economic Review - May 1988  7t21[Mt- 1  + E(llt II;)]  + 7t22 [M;_1 + E(Il;l/t)]· The actual values of the money stock in the domestic and foreign economies at time (t -1) are known, but the current innovations have to be inferred from current observations. The rational procedure for achieving this is to compute the least squares' projections from the observed prices and the exchange rate onto domestic and foreign money. Computing these projections gives the rational expectations of the domestic and foreign money stock, conditioned on domestic information and foreign information, as follows: (29)  E(llt lIt)  (30)  • E(llt lIt)  (31)  •• E(Ilt lIt)  (32)  • E(llt lIt)  = 7t331lt + 7t34/4 + 7t3SZt .  In the foreign economy, the exchange rate is observed so that the weighted average, w, in equation 23, is also observed there, along with foreign price. The foreign price observation in the foreign economy amounts to an observation of (24)  E(p;+ll/;) =  .  = 7t13llt + 7t14/4 + 7tlSZt .  Further, the observation of the exchange rate, equation 16, is equivalent to an observation of (23)  (28)  011  021  012  022  0;1'  0;1  0;2'  0;2  = -;rV t + -;r-Wt, 13 33 = -;rVt + -;r-Wt , 14 34 = -;rvt + -;r-wt , and 24 34 = -;rvt + -;r-Wt . 23 33  Equation 29 gives the domestic expectation of the domestic money stock, equation 30 is the domestic expectation of the foreign money stock, equation 31 is the foreign expectation of the foreign money stock, and equation 32 is the foreign expectation of the domestic money stock. The parameters Oij and O;j are set out in the Appendix. The expectations of domestic and foreign monetary disturbances (as viewed from the domestic and foreign information sets) calculated in equations 29 through 32 may be substituted into equations 25 through 28 to obtain the rational expectations of the price levels one period hence, conditioned on the information available in each of the two economies. These solutions may then be substituted into equations 11 through 13. Comparing equations 11 through 13 with equations 14 through 16, term by term, enables the 7tii parameters to be determined. As is well known, closedform solutions for the values for the 7tii are generally not available since the equations determining them are heavily nonlinear. It is possible, nevertheless, to display some of the relationships that obtain among the parameters and also to examine their values under certain limiting conditions. First to be considered are the following effects of domestic monetary innovations on domestic prices and foreign monetary innovations on foreign prices- own-money on ownprice effects: 7  (33)  7t13  = rx + (1 - rx)(011 + 021 ) and  (34)  7t24  = rx + (1 - rx)(0;1 + 0;1).  The two terms are equivalent and may be discussed together. Own prices will respond to own money by a weight of rx plus (1 - rx) multiplied by the amount of information pulled from the domestic price level and the exchange rate, respectively, concerning the own-country's money stock. The lowest possible values for (0" + O2, ) and for (0;, + 0;, ) are zero. This would occur as the monetary variances became infinitely large relative to the variance of the real shock. In that case, the effects of own money on prices would be rx. The upper limit to (0" + O2,), or equivalently (0;, + 0;,), is unity. This would occur when monetary disturbances were small, relative to real disturbances. In that case, own money would have a unit effect on own prices. If the variance of domestic money to foreign money approached zero, 7tH would approach rx, and 7t24 would approach unity (with a symmetrical proposition for the reverse ratios of the variances). The effects of own-country monetary disturbances on own-country output can be calculated. Such a calculation is performed and set out for the domestic economy only in equation 35, as follows: (35)  Yt  = y1 [ '1 dYs -  Yd '1 s](1 -  7t13  )Jlt .  An equivalent proposition (with 7t24 replacing 1r13 and Jl; replacing Jlt) holds for the foreign economy. Making the standard presumption that the first term in square brackets is positive, the effects of own monetary shocks on own output are positive. The output-inflation trade-off can be calculated (assuming zero foreign and real disturbances) by using equation 35 along with the own-country priceresponse coefficient, 7t H . The resulting output-inflation trade-off (Phillips curve) is as follows:  pression for 7tH that as the variance of domestic to foreign money approaches zero, the coefficient (1 - 7t 13 )/7t13 approaches yl'1, and in the foreign country, the value of (1 - 7t 24 )/7t 24 approaches zero. Conversely, as the ratio of the variance of foreign money to domestic money innovations approaches zero, the value of (1 - 7t,3)/7t13 approaches zero, and in the foreign country, the value of (1 - 7t2J/7t24 approaches yl'1. Thus, the slope of the domestic output-inflation trade-off, as a response to domestic monetary disturbances, depends crucially upon the amount of monetary noise generated in the rest of the world. The above result is sufficient to demonstrate the nonindependence of domestic economic performance from foreign monetary behavior. It is a result, however, concerning the degree of responsiveness of the domestic economy to a domestic shock.'6 It is now useful to examine whether or not foreign disturbances directly impact upon the domestic economy. To establish this, it is necessary to examine the 1rij coefficients relating foreign money to domestic prices and, equivalently, domestic money to foreign prices. These coefficients are as follows:  (37)  (38) Since it has already been established that 7tH and 7t24 are nonzero, it follows that 7t'4 and 7t23 are nonzero if 7t34/7t33 is also nonzero. This can be established most readily by considering the limiting case where the monetary variances approach zero. Setting (12 and (1'2 equal to zero in the Oij equations yields the following expression for 7t34 17t33 :  (39)  2  U , U  (36) where  Evidently, the slope of the output-inflation trade-off depends on the amount of information that the domestic agents obtain from observing their own-price level and also from observing the foreign exchange rate. The variance of foreign monetary policy will influence the slope of the domestic trade-off. It is readily verified by examining the ex8  7t34  lim '2  -+0  7t33  -  <Drx  = 1 - (1 - <D)rx .  Clearly, the dependence of the exchange rate on Jl and Jl' (relevant for the coefficients 7t34 and 7t 33 ) depends only qualitatively on the magnitudes of the variances. Thus, if it has been established that 7t34 17t33 is nonzero for a particular limiting case, it follows that, in general, this ratio will be nonzero. Thus, 7t'4 and 7t23 (the effects of other countries' money on own country's prices) will generally be nonzero.'7 Of primary interest, of course, is the effect of foreign money on domestic output and, equivalently, of domestic money on foreign output.'8 That effect may be calculated readily using the results on prices. For the domestic Federal Reserve Bank of Dallas  economy, with an equivalent proposition for the foreign economy, the effect is expressed as follows: (40)  Yt  1[ds  Sd]  = - y " y -" y  • 1'·  Since from equation 37, 1t14 will be negative, the effect of an innovation in foreign money- continuing to assume that the first term in square brackets is positive-will be to stimulate domestic output. As in the case of a domestic monetary shock, the size of the coefficient linking the foreign monetary shock to domestic output will depend on relative variances, and in a similar way it will be linked to that already discussed in connection with a domestic monetary disturbance. Finally, in this study, we examine the determination of international relative prices or the real exchange rate. In the model as set up, such a variable is of only limited interest since no trade takes place across national boundaries at a given point in time. Trade takes time, so that agents indulging in international trade buy (sell) in one economy at one point in time and sell (buy) in the other economy at another point in time. Thus, the instantaneous real exchange rate is never actually faced by any agent. Nevertheless, it is a price that will be observed in the time series data, ex post, and it is of some interest to examine its behavior. By definition, the real exchange rate is the number of units of the domestic good per unit of the foreign good. Thus, the value of the real exchange rate, R, can be readily computed by adding together equations 15 and 16 and subtracting equation 14-that is, R = p. + X - P. The effects of any given shock on the real exchange rate (R;l, will be the sum of the coefficients 1t2i and 1t3i , minus the coefficient 1t1i (i = 3,4,5).  It is readily verified that the real exchange rate generally will be influenced by monetary disturbances. The particular expressions, however, for the relevant sums are not very illuminating. It is interesting, nevertheless, to examine what happens to the real exchange rate in response to monetary noise in a world in which such monetary noise is unusual. SpeCifically, consider what happens to the real exchange rate when the monetary variances approach zero but the variance of the real shock is not zero. The calculations in this case are as follows:  (41)  (42)  Economic Review - May 1988  (43)  R5  =-  1t15  + 1t2 5 + 1t35 =  - (1 + 2A.) cI> + A.  A positive domestic monetary disturbance will raise the real exchange rate-that is, raise foreign prices relative to domestic prices- if 1X(1 + A. ) < 1, and it will lower that relative price, otherwise, in equation 41 . Thus, if the degree of currency substitution, measured by A. , is large enough, an unanticipated increase in the home money supply will act so as to appreciate home currency-even in real terms. A positive foreign price disturbance will, in this limiting case, unambiguously raise foreign relative prices, in equation 42, and a positive real shock to the domestic economy will lower the foreign relative price.  Summary and conclusions In this paper, we have analyzed the implications of currency substitution for the international transmission of economic disturbances, for the output-inflation trade-off, .and for the real exchange rate. The key feature of the model is a twocountry, two-currency world wherein the market for foreign exchange is centralized, but the market in goods is, at a given point in time, informationally segmented. Our central conclusions derived from this study are as follows. Under incomplete current information, an unanticipated monetary shock in either country will generate output effects in all countries as long as the currencies are substitutes. The slope of the output-inflation trade-off (the Phillips curve) depends on the relative variability of domestic and foreign monetary shocks, and foreign monetary shocks can be viewed as leading to shifts in this trade-off. Purchasing power parity across countries (as well as equality of inflation expectations and future exchange rate expectations) does not prevail in each period. A very notable feature of the results concerns the domestic effects of foreign disturbances. In recent years, complaints from European countries have frequently been made about the adverse domestic effects of U.S. monetary policy. These complaints have usually been inspired by a Keynesian, rigid-price view of the world. Our model shows here that even in an equilibrium, rational-expectations world, such complaints' have a ground basis. An unexpected tightening of foreign (U.S . in this case) monetary policy lowers domestic (European) output and raises the domestic price level- thus generating stagflation. On the other hand, an unexpected rise in the foreign (U.S.) money supply stimulates domestic (European) output and lowers prices. In this article, we have highlighted the possible role played by currency substitution in obtaining these responses. 9  Although we have generated these results in an abstract environment that ignores many obvious features of actual economies, we consider it significant that some of the key results appear to be highly relevant to our real world experience. There is, thus, good reason to believe that for many developed economies-such as Japan, Germany, Switzerland, and the United States-as well as many developing economies, currency substitution may be an important factor in increasing international interdependence under flexible exchange rates.  McCallum, "Real Business Cycle Models," in Robert J. Barro (ed.), Handbook of Modem Business Cycle Theory (Carnegie Mellon, Forthcoming  1988); and Edward C Prescott, "Theory Ahead of Business Cycle Measurement," Quarterly Review, Federal Reserve Bank of Minneapolis, Fall 1986,9-22.  9. Lucas, Models of Business Cycles; "Expectations and the Neutrality of Money," Journal of Economic Theory 4 (April 1972): 103-24; and "Some International Evidence on Output-Inflation Tradeoffs," American Economic Review 63 (June 1973): 326-34; also see Robert J. Barro, "Rational Expectations and the Role of Monetary Policy," Journal of Monetary Economics 2 (January 1976): 1-32. 10. See, in particular, Frenkel and Razin, Fiscal Policies in the World Economy: An Intertemporal Approach.  1. Probably the most important of these factors is the state of fiscal policy in the major economies. This topic, for example, has been extensively studied and most comprehensively surveyed in Jacob A. Frenkel and Assaf Razin, Fiscal Policies in the World Economy: An Intertemporal Approach (Cambridge, Mass.: MIT Press, 1987).  11 . See Edmund S. Phelps, et ai., Microeconomic Foundations of Employment and Inflation Theory (New York: W . W. Norton and Co., 1970): and Lucas, "Expectations and the Neutrality of Money." 12. The supply and demand setup used here is, we feel, compatible with a deeper underlying structure wherein economic agents maximize the expected utility associated with the real returns from holding a portfolio of currencies with declining marginal utility of income (which together imply risk-averse behavior on the part of agents). In this environment, a system of flexible exchange rates would provide a greater opportunity for profit from speculation in foreign exchange markets than would a system of fixed rates; and hence substitution between the currencies, as measured here by A, would generally be greater. Also in such an environment, currencies that tended to exhibit more risk-due, say, to a higher degree of variance in the underlying monetary disturbances-would tend to be less desirable, ceteris paribus, and thus would carry a lower A.  2.  This point was raised in Marc A. Miles, "Currency Substitution: Some Further Results and Conclusions," Southern Economic Journal 48 (July 1981): 78-86, though not formally demonstrated in a rational expectations setting. As we show here, however, currency substitution alone is not sufficient to permit an international transmission of disturbances when expectations are formed rationally. Also see Leroy O. Laney, Chris D. Radcliffe, and Thomas D. Willett, "Currency Substitution: Comment," Southern Economic Journal SO (April 1984): 1196-1200; and Marc A. Miles, "Currency Substitution: Reply," Southern Economic Journal 50 (April 1984): 1201-1203.  3.  Lance Girton and Don Roper, "Theory and Implications of Currency Substitution," Journal of Money, Credit, and Banking 13 (February 1981): 12-30.  13. Barro, "Rational Expectations and the Role of Monetary Policy_"  4.  Reuven Glick and Clas Wihlborg, 'The Role of Information Acquisition and Financial Markets in International Macroeconomic Adjustment," Journal of International Money and Finance 5 (September 1986): 257-84.  14. The omission of the expected rate of change of the exchange rate from the supply and demand functions for goods has been made in order to achieve the study'S goal of isolating information and currency substitution in the transmission process . Specified were demand-for-money functions consistent with this assumption .  5.  Kent P. Kimbrough, "Aggregate Information and the Role of Monetary Policy in an Open Economy," Journal of Political Economy 92 (April 1984): 268-85; "The Information Content of the Exchange Rate and the Stability of Real Output Under Alternative Exchange-Rate Regimes," Journal of International Money and Finance 2 (April 1983): 27-38; and "ExchangeRate Policy and Monetary Information," Journal of International Money and Finance 2 (December 1983): 333-46.  15. The specification of the money supply process in equations 7 and 8 explicitly abstracts from ongOing trend growth in the money stock. This approach has been taken for the sake of clarity. An extension to the case of ongoing inflation (and at different rates in the two countries) is straightforward and contains no additional insights or results.  6.  Richard G. Harris and Douglas D. Purvis, "Diverse Information and Market Efficiency in a Monetary Model of the Exchange Rate," Economic Journal 91 (December 1981): 829-47.  7. For the purposes of this study, the primary benefit of the deep structure approach would be to make endogenous the currency substitution parameter (defined here as A). Ideally, A would be determined as a solution to some sort of underlying optimization process on the part of economic agents, given the structure of the economy. (See n. 12 for a description of one such deep structure environment that we feel would be compatible with the supply and demand setup employed here.)  6. For surveys of the deep structure approach, see Robert E Lucas, Jr , Models of Business Cycles (Oxford: Basil Blackwell, 1987); Bennett T. 10  16. For a demonstration of these results, see Barro, "Rational Expectations and the Role of Monetary Policy," and "A Capital Market in an Equilibrium Business Cycle Model,' Econometrica 48 (September 1980): 1393-1417; Lucas, "Expectations and the Neutrality of Money," and "An Equilibrium Model of the Business Cycle," Journal of Political Economy 83 (December 1975): 1113-44; R. Lucas and L. Rapping, "Real Wages, Employment, and Inflation," in Phelps, Microeconomic Foundations of Employment and Inflationary Theory, 257-305; and McCallum, "Real Business Cycle Models." 17. Actually, for the limiting case calculated in equation 39, the coefficients e21 and e;l are zero; so that in that particular case, 1t14 and 1t23 are, in fact, zero. However, if we move away from the limiting value, the ratio 1t34/ 1t33 remains nonzero, and the coefficients e21 and e;1 become nonzero  Federal Reserve Bank of Dallas  18. There are also some interesting proportionality propositions concerning the effects of domestic and foreign money on domestic and foreign prices. Evidently, from equation 37, it is clear that 7tu  1i33 =  [IX + (1 -  bance. The second equation provides the equivalent proposition for the foreign economy. Combining both equations from above yields the following:  1X)021]  (1 - 1X)021  Also, from equation 38, it is clear that 7t24  1ij4=  [IX + (1 -  1X)0;1]  (1 -IX)0'21  The first equation above states that relative to their effects on the exchange rate, a domestic monetary disturbance affects the domestic price level by a greater amount than does a foreign monetary distur-  Economic Review - May 1988  This states that the combined effects of domestic money on domestic prices and foreign money on foreign prices, 7tu 7t2'" exceeds the combined effects of domestic money on foreign prices and foreign money on domestic prices, 7t14 7t23. It is possible, however, that one of the cross-country price effects, 7t14 or 7t23- exceeds one of the own-price effects, 7tu or 7t24.  "  Appendix  Values of 0;; and O;j (A.1)  11:13  -  (A.2)  12  2  2  '2  11:13(11:1411:340"  11:14  2  '2  2  8  22  ~  11:34(11:1311:330"  2  (A.6)  ) 2  )]0"  = (11:1311:34 -  2  2 2 '2 11:1411:33) 0" 0"  2  2  2  2 )  ,  11:34(11:2311:330"  11:34  2  2  '2  '  2'2  + 11:2511:35' 2  821 = -,-[11:34(11:230" + 11:25'  )]0"  2 )  ~  2  )]0"  2  2)  + 11:1511:35'  -  (A .7 )  )  2  -  (A.S)  ,  11:24(11:2311:330"  11:23  2  '2  2'2  + 11:2511:35' 2  812 = -,-[11:23(11:340" + 11:35'  )]0"  2 )  ~  2  )]0"  '2  -  11:34 2 2 2 2) = ""T[1I:34(1I: 130" + 11:15'  11:14(11:1311:330"  11:24  2  2  + 11:1511:35'  '  811 = -,-[11:24(11:330" + 11:35' ~  + 11:1 511:35' 2  2  2  + 11:1511:35'  812 = ""T[1I:14(1I:330" + 11:35'  -  (A.S)  '2  11:33(11: 1411:340"  11:33  -  (A.4)  '2  821 = ""T[1I:33(1I: 140" + 11: 1 5' -  (A. 3)  2  811 = ""T[1I:13(1I:340" + 11:35'  (A.9)  ,  11:33(11:2411:340"  11:33  2  '2  '2  + 11:2511:35' 2  822 = -,-[11:33(11:240" + 11:25'  2  2  )]0"  2 )  ~  2  )]0"  '2  '2  -  (A.10)  ~  '  11:23(11:2411:340"  = (11:2311:34 -  + 11:2511:35'  2  2  )]0"  2 2 '2 11:2411:33) 0" 0"  + (11:1311:35 -  2 2 2 11:1511:33) 0" ,  + (11:2311:35 -  2 2 2 71:2571:33) 0" ,  + (71:1471:35 -  2 '2 2 71: 1 571:34) 0" ,  + (71:2471:35 -  2 '2 2 71:2571:34) 0" ,  Federal Reserve Bank of Dallas  A Study of the Relationship Between Economic Growth and Inequality: The Case of Mexico Joseph H. Haslag  Thomas B. Fomby  D. J. Slottje  Economist  Associate Professor of Economics  Assistant Professor of Economics  Federal Reserve Bank of Dallas  Southern Methodist University  Southern Methodist University  Consultant  Federal Reserve Bank of Dallas  In his seminal 1955 article on economic growth and income inequality, Kuznets asked whether inequality in the distribution of income increased or decreased during the course of economic growth.' Kuznets argued that growth of the economy produces countervailing effects on inequality in the distribution of income-with no obvious dominance of either set of forces. Central to Kuznets' analysis was the hypothesis that changes in the level of economic inequality stemmed from changes in the rate of growth. Indeed, much of his paper is devoted to theorizing about how changes in the rate of growth affect economic inequality. In contrast to Kuznets' point of view, Myrdal postulated that increased economic inequality within a country adversely affects its rate of growth. 2 Myrdal contended that economic disincentives and political unrest follow when economic inequality reaches some critical level. The ensuing political and social upheaval disrupts business activity and results in lower rates of growth. The purpose of this article is to investigate the empirical question raised by Kuznets' and Myrdal's alternative hyEconomic Review - May 1988  potheses. That is, the investigation will attempt to identify whether growth "causes" economic inequality or vice versa. 3 The question of presence and direction of causality between growth and economic inequality is applied to Mexico. Unfortunately, detailed income and expenditure data on developing countries are rare; Mexico, however, is one exception. Basmann, Molina, Rodarte, and Slottje have painstakingly gathered the data that make possible the present investigation of the relationship between growth and inequality.4 If hopes are realized, this initial investigation will lead to other detailed data collection efforts so that the results obtained here can be critically evaluated with respect to the experiences of other countries as well. The empirical issue of the relationship between growth and inequality has obvious policy implications for a developing country. Should a country actively promote a more "optimal" distribution of income and expenditures for the purposes of nurturing greater economic growth, or should the distribution of income and expenditures be taken as a by-product of a growing economy? 13  The relationship between growth and inequality  Methods to identify causality in economic time series suggest that the relationship between economic inequality and growth may be characterized as one of four mutually exclusive events. If changes in the rate of growth help to predict changes in economic inequality but the converse does not hold, then the growth rate is said to cause economic inequality unidirectionally.5 In contrast, unidirectional causality may run from economic inequality to the rate of growth. Economic inequality causes the rate of growth when changes in the distribution of income and expenditures help predict changes in the rate of growth, but not vice versa. A third alternative involves bidirectional causality, or "feedback." This outcome is suggested when both changes in the rate of growth and changes in economic inequality are helpful in predicting each other. Finally, when changes in the rate of growth and changes in economic inequality are independent of each other and, thus, neither helps predict the other, it is said that no causality exists. Theories that growth"causes" inequality  As noted earlier, Kuznets' apparent hypothesis is that changes in the rate of growth cause changes in inequality in the distribution of income. In his discussion, however, Kuznets suggested that the sign of the total effect of growth on inequality in the distribution of total income is ambiguous. On the one hand, the "concentration of savings" and "industrialization" are cited as factors increasing inequality. On the other hand, "political forces," "population," the "age composition of industries," and "education" are identified as decreaSing inequality. Savings appear to become more concentrated in the high-income groups as an economy grows. Assuming that assets pay a rate of return higher than the rate paid to human capital, the increasing proportion of non-labor income received by individuals in the high-income group means that total income becomes distributed more unequally. To the extent that industrialization corresponds to a shift in the population from rural areas to urban areas, the degree of inequality in the distribution of income is likely to increase. The distribution of aggregate income may be thought of as the sum of weighted income distributions of rural and urban population subgroups. Kuznets noted that income was probably more equally distributed for the rural population than for the urban population. Consequently, as more people moved into cities, a greater weight was given to the more unequal share of the two components. Hence, inequality in the total income distribution should increase. 14  Oppositely, political interference is postulated to decrease inequality in the distribution of income. Other things being constant, progressive income tax rates result in after-tax income being more equally distributed. Transfer programs are also likely to lead to a more equal distribution of income. Population redistribution between the high-income groups and the low-income groups would likely have an effect on inequality in the distribution of total income. Demographics suggest that the population of the high-income groups has grown less rapidly than the population of lowincome groups. Representing a smaller proportion of the total population means that a smaller weight is used for high-income earners. For example, suppose an economy consists of two individuals-one earns $100 per period, and the other earns $50. In this case, the top earner garners two-thirds of total income. Now suppose that the population of $50 wage earners doubles. Total income has risen to $200, with the person in the high-income category receiving one-half of the total. Taken to the extreme, if the number of individuals in the low-income group grows to, say, 1,000, the high-income individual receives roughly 0.2 percent of total income, and the distribution of total income approximates a perfectly equal distribution. Another factor Kuznets considered important was the age composition of industries, which refers to the introduction of new industries into a growing economy. It is believed that entrepreneurial activities are cyclical. Moreover, it is assumed that differentials in rates of return exist between old industries and new, with higher rates going to the newer enterprises. Assuming that the high-income groups have invested in older industries and are unable to move their investments immediately into new opportunities, the higher returns associated with younger industries flow to lowincome groups. Consequently, inequality in the distribution of total income decreases. One rationalization for investment in education is anticipated returns to human capital. In other words, labor productivity rises, which means that for a given price of output, labor services are more highly valued. Kuznets suggested that education may play an important role in understanding how growth affects inequality in the distribution of total income. 6 Investment in education is believed to raise the rate of return to human capital. As education becomes more mass-oriented, those in the low-income groups presumably would benefit relative to those in the high-income groups-the idea being that low-income groups are the last to participate in education. A growing economy gives rise to wider participation in education, thus lowering the differential in returns to human capital between the top and bottom income groups. These events lead to reducing inFederal Reserve Bank of Dallas  equality in the distribution of labor income. Because this is the largest component of total income, it follows that total income is also more equally distributed. Supplementing Kuznets' analysis of the benefits of education, Welch argued that education affects the speed with which individuals are able to respond to economic opportunities. 7 He found evidence that more-educated farmers have an advantage over their less-educated counterparts in responding to the dynamics of growth. Chaudhri found evidence in support of Welch's hypothesis for northern India, where education and productivity were positively related. 8 In the context of educational impacts on agriculture, Welch suggested that the resulting decrease in real factor costs due to higher education levels improved the income positions of low-income groups relatively more than those of high-income groups. Hence, inequality in the distribution of income falls. Schultz believed that the large absolute differences in earnings stimulated increased investment in education.9 More education gives rise to smaller differences in absolute earnings between skilled workers and unskilled workers. As competition causes differences in the levels of rates of return to decrease, inequality in the distribution of total income decreases. Education was also argued to have positively affected household efficiency (that is, how close the head of the household is to full production potential). Michael showed that the effiCiency of heads of households was positively related to education levels. 10 The efficiency argument is tied closely to the information set used by the households. Higher education reduces the cost of using an additional piece of information. Consequently, more information is used, which, in turn, allows quicker response to economic opportunities. Theories that inequality" causes" growth  Myrdal's hypothesis is that the level of economic inequality helps predict growth-exactly opposite to the Kuznets position. This direction of causality rests on the political and economic implications of changes in economic inequality. To illustrate, consider one effect of a change in economic inequality on growth. An increase in inequality, for instance, is assumed to induce changes in the behavior of individuals, particularly those in low-income groups. A political argument asserts that relatively poor individuals respond to "perverse" inequality by attempting to overthrow the current system. Poor individuals form coalitions with the intention of replacing the "unfair" distribution with a system that is conducive to more equality. Both the transi-  Economic Review - May 1988  tion to and the implementation of the new system are characterized by displacement of resources from production activities. Consequently, growth of output slows. Economic theory suggests that the relationship between public policy and economic inequality may also induce responses by agents that affect growth. The signs of these cumulative policy effects, however, are ambiguous. On the one hand, suppose free education is offered. Transfer schemes involving payment in kind are assumed to affect those in low-income groups relatively more than those in high-income groups. With the distribution of economic well-being more equal as a result of implementing this policy, and to the extent that education makes workers more productive, the same level of inputs will subsequently give rise to faster growth. On the other hand, policies that increase inequality may have supply-side effects, which induce more productive activity. For instance, lower income tax rates for individuals in the high-income group will result in more inequality. With higher after-tax real wages, more labor is supplied; consequently, output will subsequently grow. In addition, a higher concentration of savings may also explain how a change in inequality affects the rate of growth. To the extent that wealthier individuals have higher average propensities to save, the rise in inequality would likely increase the level of savings. With a higher level of saving, the level of investment should rise. To conclude the argument, more investment leads to a higher capital stock. Other things being equal, output growth will rise in response to a larger quantity of the capital input. Mexico provides a case study for analyzing these competing theories. First, the proximity of Mexico to the Eleventh Federal Reserve District makes the present case study quite pertinent and also timely. For example, the prospects of enhanced growth in Mexico, "caused" by, say, a redistribution of income and expenditures, would have a direct bearing on the issues of illegal immigration in the Southwest and of increased trade between Mexico and the southwestern United States. In contrast, should growth precede (cause) inequality, attempts to redistribute income would appear to be an impote,nt policy with regard to improving the present circumstances between Mexico and the United States, particularly the Eleventh District. Second, as pointed out earlier, from the standpoint of data availability, the study of Mexico rather than other countries was a choice necessitated by having few alternatives. In the following section, the measure of economic inequality used in this study is developed.  15  Table 1 GINI COEFFICIENTS FOR EXPENDITURE DISTRIBUTION IN MEXICO AND THE LEVEL OF PER CAPITA REAL GROSS DOMESTIC PRODUCT  (es)  Per capita real GOP (peDP)  .633 .634 .646 .682  .792 .796 .796 .829  19.7 21.2 20.0 23 .1  .809 .807 .779 .791 .767  .682 .679 .673 .668 .675  .825 .821 .826 .819 .820  27.8 30.8 29.9 30.5 30.4  .668 .667 .661 .659 .640  .778 .786 .792 .785 .764  .681 .670 .656 .652 .640  .829 .824 .803 .802 .790  31.8 32.2 32.6 34.1 36.8  .515 .510 .513 .504 .502  .640 .630 .619 .617 .628  .761 .754 .749 .750 .756  .642 .642 .641 .636 .625  .794 .798 .790 .796 .781  37.8 39.1 40.2 42.0 43 . 1  .372 .370 .366 .362 .361  .501 .493 .493 .485 .477  .617 .612 .596 .590 .595  .709 .710 .708 .708 .721  .621 .624 .617 .610 .606  .786 .784 .772 .766 .770  44.5 44.8 47.0 49.2 50.5  .360 .415 .433 .427  .476 .558 .579 .577  .597 .678 .706 .696  .722 .796 .823 .820  .599 .678 .699 .692  .759 .821 .837 .831  51.5 51.8 51.7 55.3  Year  Total expend itures (e)  Food (e1)  Clothing (G2)  Housing (e3)  Durables (e4)  1951 1952 1953 1954  .375 .380 .383 .411  .501 .512 .518 .548  .621 .630 .624 .670  .718 .721 .734 .788  1955 1956 1957 1958 1959  .420 .417 .409 .404 .409  .562 .562 .550 .541 .545  .689 .680 .669 .660 .653  1960 1961 1962 1963 1964  .408 .402 .398 .388 .386  .546 .547 .542 .533 .522  1965 1966 1967 1968 1969  .383 .378 .377 .372 .370  1970 1971 1972 1973 1974 1975 1976 1977 1978  Medical services  NOTE: The Gini coefficients are from R. L. Basmann, David J. Molina, Mario Rodarte, and Daniel J. Slottje, " Some New Methods of Predicting Changes in Economic Inequality Associated with Trends in Growth and Development," in Issues in Third World Development, ed . Kenneth C. Nobe and Rajan K. Sam path (Boulder, Colo .: Westview Press, 1983), 128. The per capita real GOP values were calculated from International Financial Statistics (international Monetary Fund) data.  16  Federal Reserve Bank of Dallas  A measure of economic inequality Obviously, the choice of a measure of economic inequality is important to the investigation of the issue at hand. In general, economists are concerned with inequality in the distribution of "welfare."11 A common perception is that some relationship exists between inequality in the distribution of total income and inequality in welfare-which may include such components as consumption expenditures, wealth, justice, human dignity, and safety, among other things. Instead of the distribution of total income, this study uses the distribution of total household expenditures as the measure of welfare. 12 We do so for two reasons. 1. Income statistics as traditionally reported by government agencies do not reflect the transfers in kind (such as subsidized housing) received by lower-income groups; thus, they overstate inequality. To the extent that transfers in kind free up money for additional expenditures, lowerincome groups will account for a larger share of expenditures than before. Consequently, transfers in kind (and, hence, economic well-being) may arguably be reflected more accurately in expenditure statistics than in income statistics. In addition, expenditures are direct antecedents of consumer satisfaction. Income, though spendable, must find appropriate outlets before it can be enjoyed. 2. The inequality of "permanent" income is probably a more appropriate measure of economic well-being than the inequality of current income. Permanent income represents long-run earning capacity, not transitory earning power, and therefore determines the long-run well-being of individuals or groups of individuals. Following Friedman's arguments on permanent income and consumption, expenditures and permanent income are proportionately related; thus, the distribution of expenditures should reflect the distribution of permanent income, the more desired measure of economic inequality.1) For Mexico, the Encuesta Nacional de Ingresos gives survey data on the distribution of total household expenditures and the five expenditure components. 14 These items are food, clothing, housing, durables, and medical services. The data used in the analysis here cover 1951 through 1978. The measure of inequality in the distribution of total expenditures and the five components is the Gini coefficient (discussed in the accompanying box).15 The values of the Gini coefficient for total expenditures- denoted G- and for the components-Gl through G5- over the sample period are reported in Table 1, along with the level of per capita real gross domestic product, which is denoted PGOP. The measure of growth used here is the rate of change in PGOP; thus, the differences in the natural logarithms of PGOP, representing percentage changes, are the basis of the analysis . Economic Review - May 1988  Next, we will specify the methodology employed to detect causality between the rate of growth and the inequality in the distribution of total expenditures and the five components. Following the description are the findings from applying the methodology to the Mexican growth-inequality issue. Causality tests Several alternative methods exist to detect causality. Most notable are the tests developed by Granger, Sims, and Pierce and Haugh .16 In this analysis, we will adopt the procedure developed by Pierce and Haugh, using cross-correlations to detect the presence of causality. The primary reason for adopting the Pierce-Haugh approach to causality testing is identified by Schwert, who states, "application of conventional regression techniques to untransformed economic time series variables can often result in 'spurious regressions."'1? In other words, least squares estimates applied to variables that are trending may incorrectly impute a significant statistical relationship (and, thus, causality) between variables when, in fact, none exists. Pierce and Haugh minimize this problem by looking at "innovations" in the levels of time series. When using the method of least squares on the innovations of each time series, the tendency to accept spurious relationships between variables is greatly reduced. The approach specified in Pierce and Haugh identifies the presence and direction of causality by using a two-step procedure. First, the Box-Jenkins technique is used to "prewhiten" each series. The Box-Jenkins method is applied to the growth rate of per capita real gross domestic product (~lnPGOP) and the levels of the Gini coefficients (G through G5).18 Table 2 presents the results from the Box-Jenkins model identification stage. 19 The results indicate that percentage changes in PGOP may be modeled as a first-order autoregressive-AR(l)- process. The Ljung-Box modified Q statistic for the AR(1) process is 5.44 at 12 lags, less than the critical value of 15.99 for a chi-square distribution with 10 degrees of freedom at a 10-percent significance level. Therefore, we conclude that the AR(l) model of percentage changes in PGOP generates "white-noise" residuals and that no systematic variation remains to be explained.2o Overfitting also indicates that the AR(l) is appropriate because additional parameters are not significantly different from zero at the 10-percent level of significance. The Q statistics of the overfitted models are also consistent with white noise. Similarly, the Q statistics and overfitting the Box-Jenkins models for the Gini coefficients suggest that modeling the levels of the Gini coefficients as autoregressive processes is 17  What Is a Gini Coefficient? Each member of a population spends a fraction of the aggregate expenditures in the economy. We are interested in determining how "evenly" these expenditures are spread among individuals in the society and how inequality in the distribution of expenditures changes over time. To achieve both goals, it is necessary to choose a measure of inequality. The measure selected for our analysis is the Gini coefficient ("Gini" for short). Perhaps the best way to understand how the Gini measures inequality in the distribution of expenditures is to provide a brief description of how the coefficient is calculated. A useful framework for illustrating the Gini incorporates a hypothetical distribution of expenditures to serve as a reference case. Changes in the Gini that are related to greater or less inequality in the distribution of expenditures may be inferred from how the actual distribution relates to the hypothetical case. In the hypothetical case, let each individual spend exactly the same amount. Because aggregate expenditures are equally distributed among members of the population, this is referred to as the "perfect equality" case. Indeed, each individual spends the same fraction of aggregate expenditures. With the percentage of total population on the horizontal axis and the percentage of total expenditures on the vertical axis, Figure 1 shows a graphical depiction of the hypothetical expenditure distribution as a straight line beginning at the origin and having slope equal to 1. The point of having any distribution begin at the origin is that zero percent of the population will spend zero percent of aggregate expenditures. Similarly, 100 percent of the population will necessarily spend 100 percent of aggregate expenditures. Points along the perfect equality line indicate the percentage of total expenditures accounted for by some percentage of the population. For instance, point B on the perfect equality line indicates that x percent of the population spends Pe percent of total expenditures. Of course, in the perfect equality case, x equals Pe . Note that by drawing a line perpendicular to the horizontal axis at the value of 100 percent, we form a triangle having the perfect equality line and the horizontal axis as the other two sides. The next step is to plot the actual distribution of the expenditures on the same diagram. Implicitly, individuals are ranked by the sizes of their expenditures. In effect, the percentage of the population becomes a continuum of the smallest to largest spenders. This ranking is consistent with the hypothetical distribution serving as the reference case here, with the smallest and largest spenders being identical. Given the interpretation of the horizontal axis, the plot of the actual distribution-also known as the Lorenz curve-indicates the cumulative percentage of aggregate expenditures accounted for by some percentage of the population. For example, point C indicates that x percent  18  FIGURE 1 Percent of Total Expenditures  100 - - - - - - - - - - - - - - - - - - A  Pe  P"  o  x  100  Percent of Total Population  of the population (corresponding to the smallest x percent of spenders) spends p. percent of total expenditures. Intuitively, the measure of inequality would capture the disparity between the perfect equality case and the actual distribution. Following our intuition, from Figure 1, the Gini coefficient is simply the ratio of the shaded region between lhe actual distribution and the perfect equality line to the area of the triangle under the perfect equality line. Because the shaded region is necessarily smaller than the triangle, the Gini must be between zero and 1. At one extreme, suppose expenditures are equally distributed among individuals in the population. In this case, the lorenz curve would perfectly coinCide with the perfect equality line. Hence, the shaded region would degenerate, and the Gini would be equal to zero. If expenditures were perfectly concentrated, such that one person's pending was equal to total expenditures, than the lorenz curve would be a single point-namely, point A in Figure 1. The shaded region and the area of the triangle under the perfect equality line would be identical, and the Gini would take the value of 1. The two extreme cases just described suggest the interpretation of changes in the Gini coefficient. That is, as the Gini rises (falis), the shaded region increases (decreases) relative to the area of the triangle under the perfect equality line. In other words, an increasing (decreasing) Gini indicates greater (less) inequality in the size distribution of expenditures.  Federal Reserve Bank of Dallas  Table 2 RESULTS OF BOX-JENKINS UNIVARIATE MODEL IDENTIFICATION PROCESS APPLIED TO MEXICO DATA Equation  1 ....  Estimated relationships  ~ln(PGDPt)  = - .04 + .45 (.03)  ~ln(PGDPt_d  2 ... .  Gt = .11 + 1.1 6 Gt- 1 - .46 Gt- 2 + V,; (.01) (.18) (.19)  3 ....  G1 t = .16 (.01)  +  Ut;  Q = 5.44.  (.18)  + 1.15G1'_1 - .45 G1'_2 + v1 t ; (.18)  Q=1.81 . Q = 2.98.  (.19)  4 . . ..  G2, = .21 + 1.10 G2'_1 - .43 G2'_2 + v2,; (.01) (.20) (.18)  Q = 6.28.  5 ... .  G3, = .22 + 1.08 G3'_1 - .38 G3 t_ 2 + v3,; (.20) (.01 ) (.18)  Q = 8.54.  6 ....  G4, = .14 + .78 G4'_1 + v4,; (.01) (.13)  Q = 6.00.  7 ....  G5, = .21 + .73 G5'_1 + v5 t ; (.01) (.14)  Q = 4 .85.  NOTE: Q indicates the value of the Ljung·Box Q statistic at 12 lags. Figures in parentheses are standard errors  appropriate. As indicated in Table 2, Gini coefficients for total expenditures, food, clothing, and housing may be modeled as second-order autoregressive-AR(2)- processes, although the Ginis for expenditures on durables and medical services are modeled as AR(1) processes. In each case, the Ljung-Box Q statistic is consistent with the supposition of white-noise residuals. The second step in the Pierce-Haugh procedure is to compute cross-correlation coefficients for the innovations in the two series. In the time series literature, innovations refer to those parts of the time series that are not predicted on the basis of the past history of the series. In other words, in using the Box-Jenkins model, the innovations are identical to the residuals . The innovations in the growth rate of per capita real gross domestic product are denoted Ut while the innovations in the Gini coefficients are denoted v, (with V,j, i = 1, ... , 5, corresponding to the innovations in Gi, i = 1, ... , 5). The cross-correlation coefficient is defined as  Causality is suggested if any value of p(k) is Significantly different from zero. For example, the rate of growth causes the Gini if p(k) is significantly different from zero for only some positive values of k. 21 Conversely, when p(k) is staEconomic Review - May 1988  tistically significant for only some negative values of k, the Gini causes the rate of growth. Finally, if statistical significance is found for some k > 0 and k < 0, the causality is said to be bidirectional. The results  Six combinations of Gini coefficients and the rate of growth in per capita real gross domestic product were analyzed. Cross-correlations of the innovations were estimated for each combination. The values of k are defined with respect to the rate of growth. That is, ut in the above equation reflects innovations in the growth rate of per capita real gross domestic product. Results are reported in Charts 1 through 6. In three of the six plots of the cross-correlation function, there is a value of the cross-correlation coefficient that is significantly different from zero. These coefficients are identified by plots that lie outside the 90-percent confidence interval marked for each graph. In Chart 1, for instance, the coefficient at k = 2 exceeds the confidence boundary. In the three cases in which a statistically significant crosscorrelation coefficient is present, the plots indicate that significant values are always associated with positive k'S .22 These results suggest that Kuznets' direction of causation, if not his rationale, is supported. That is, the rate of growth causes (leads) inequality in the distribution of expenditures 19  Chart 1  Cross-Correlations Between Growth Rate of Per Capita Real GOP and Gini Coefficient for Total Household Expenditures VALUE OF CROSS-CORRELATION  .5 .4  .3  .~~~~~~~.~.~!..~~.~.~~~.~.~~~ ..~.?~~.~.~.~~............................................... .  .2  .1 O~~~~~~~-y-L-r-r-r. . . .~,-,-,-~--.-  -.1 -.2  -.3  -.4  90-Percent Confidence Boundary  -.5~~~  -10  __  -8  ~  __- L__- L__  -6  -4  -2  ~  __~__~__~__~__~  0  2  4  6  8  10  LAG LENGTH (k)  Chart 2  Cross-Correlations Between Growth Rate of Per Capita Real GOP and Gini Coefficient for Food Expenditures VALUE OF CROSS-CORRELATION  .5 .4 .3  ~~~~.~.r. ~.~~~..~~.~~~~.~.~~~ ..~.?~~.~~.~~............................................... .  ..  .2 .1 O~~~~~-L-L-L-L-r~. .,-,-~~~~r-~~  -.1  -.2 -.3  - .4  90-Percent Confidence Boundary  __ __- L__ __ -8 -6 -4 -2 0  -.5~~~  -10  ~  ~  ~  __~__~__~__L -__L  2  4  6  8  10  LAG LENGTH (k)  20  Federal Reserve Bank of Dallas  Chart 3  Cross-Correlations Between Growth Rate of Per Capita Real GOP and Gini Coefficient for Clothing Expenditures VALUE OF CROSS-CORRELATION  .5  .4 .3  ,.~~:~~.~~.~~!..~~.~~~~.~.~~~ ..~.?~~.~.~~~............................................... .  .2 .1 O~~~-L-L~~~~~~~~~~-r--~~'-  -.1  -.2  -.3  -.4  90-Percent Confidence Boundary -8  -6  -4  -2  0  2  4  6  8  10  LAG LENGTH (k)  as a measure of economic inequality. The three exceptions occur with inequality in the distribution of clothing, housing, and medical service expenditures. In other words, the lack of significant cross-correlation coefficients is consistent with the independence of the rate of economic growth and the level of inequality in the distribution of clothing, housing, and medical service expenditures. In addition, results reported in the Appendix of this article, as well as some that are not reported, show that the signs of the cross-correlation coefficients are suggestive of the impact of changes in the rate of growth on the distribution of expenditures. For example, with respect to the distribution of total expenditures, the data suggest that a permanent increase of 1 percent in the Mexican growth of per capita real gross domestic product resulted, on average, in a permanent decrease of 0.0064 for the Gini coefficient on total expenditures. Thus, given the present analysis, growth in Mexico directly benefits its poor in that the lower-income groups gain disproportionately relative to the higher-income groups. Conversely, declines in growth disproportionately hurt the poor relative to wealthier groups. Conclusions The studies of Kuznets and Myrdal propose an interesting empirical question for a developing country: Does ecoEconomic Review - May 1988  nomic inequality cause changes in the rate of growth, do changes in the growth rate cause inequality, or both of these? This research effort constitutes a modest first step toward addressing this very important question. Using expenditure and per capita real gross domestic product data for Mexico, we applied the causality testing procedure of Pierce and Haugh and simple transfer functions to glean what must be viewed as very tentative conclusions concerning the Kuznets-Myrdal growth-inequality issue. First, increased growth apparently does promote greater equality in the distribution of economic welfare. That is, the fruits of growth need not be restricted to higher-income groups. The benefits of growth, given the case of Mexico, do seem to "trickle down." Conversely, economic policies of a developing country like Mexico can have a cruel effect on low-income groups because they often bear a disproportionate share of the burden of economic decline. Disgruntlement among lower-income groups is likely to lead to increased pressures to immigrate illegally. Thus, growth in Mexico and increases in attempts at illegal immigration appear to be inexorably linked. Similarly, the argument can be advanced that with slow growth in Mexico, the U.S. exportation of "blue-collar" goods to Mexico is likely to suffer more than the exportation of "luxury" goods. 21  Chart 4  Cross-Correlations Between Growth Rate of Per Capita Real GOP and Gini Coefficient for Housing Expenditures VALUE OF CROSS-CORRELATION  .5 .4 .3  .?~:~~.~~.~. ~!..~~.?~!~.~.~ ~~ ..~.?~.~.~~.~I .............................................. .  .2  .1 O~~---A-L-L~--~~L-~~~-T-r-r-r~,-  -.1 -.2 -.3  ............. .......... ......... .......... ........ ...... ............................ .... .................. .  -.4  90-Percent Confidence Boundary  -.5 ....................---'-........-10 -8 -6  .......-4  .......- - - ' - -.........-2 0 2  .........4  .......- - " - -...... 8 10 6  LAG LENGTH (k)  Chart 5  Cross-Correlations Between Growth Rate of Per Capita Real GOP and Gini Coefficient for Ourables Expenditures VALUE OF CROSS-CORRELATION  .5 .4 .3  .?~:~~.~~.~~!..~?':'.~!~.~.~~~ ..~.?~~.~.~. ~~............................................... .  .2 .1 O~~~~~-A~--L-r-r-r~~~~,-~~  __~  -.1  -.2 -.3 - .4  90-Percent Confidence Boundary  -.5 ..........'-----'-........- .......- ........---'--.........- .........- .......- - " - -...... - 10 -8 -6 -4 -2 0 2 4 6 8 10 LAG LENGTH (k)  22  Federal Reserve Bank of Dallas  Chart 6  Cross-Correlations Between Growth Rate of Per Capita Real GOP and Gini Coefficient for Medical Service Expenditures VALUE OF CROSS-CORRELATION  .5r---------------------------------------------------------------------------------.4 .3  ~~~~~.~~.e.~~ ..~~.~.~~~.~.~~~.~.~~~.~.~.~~............. .. ....... ..........................  ..  .2 .1 O~~~~-L-L~__~~r-r<~,,~~-r-r~~  -.1  -.2  -.3  -.4  gO-Percent Confidence Boundary  -10  -8  -G  -4  -2  0  2  4  G  LAG LENGTH (k)  Another conclusion, very tentative as well, is that direct and swift redistributions of economic welfare among the Mexican citizenry need not guarantee advances in economic growth in Mexico. This conclusion follows because the data do not seem to indicate that changes in the distribution of expenditures precede (cause) changes in the growth rate of per capita real GOP. Given the paucity of the present Mexican data, as well as the data for other developing countries, it is difficult to discern what role public education, retraining, and public health programs, or other possible government initiatives, play in promoting economic growth. If such in-kind transfers were properly accounted for in the data made available to researchers, it is always possible that a bidirectional (feedback) relationship between economic welfare distribution and economic growth could be uncovered. For example, as Kuznets suggested, growth possibly promotes more mass-oriented education, which, in turn, makes previously uneducated workers and households more productive. With more productivity, education then leads to more growth. Whether the Mexican experience as reported here is "typical" of other developing countries is yet to be seen. Should these results turn out to be typical, however, it might be tentatively suggested that more efforts of the governments of developing countries should be focused on Economic Review - May 1988  growth-promoting fiscal, monetary, and trade policies rather than "qUick-fix" policies involving attempts to reshape the distribution of economic welfare to promote growth. Also, to the extent that the present results are typical, a developed country can indirectly promote greater economic equality in less developed countries. For instance, by adopting foreign aid and trade policies that promote growth in the less developed nation, the developed country can reduce the level of inequality in the less developed nation.  1.  Simon Kuznets, "Economic Growth and Income Inequality: American Economic Review 45 (March 1955): 1-28.  2.  Gunnar Myrdal, "Equality and Democracy: chap. 16 in Asian Drama: An Inquiry into the Poverty 01 Nations, vol. 2 (New York: Twentieth Century Fund and Random House, Pantheon 800ks, 1968).  3.  In economics, the word "cause" is used very carefully. A universally accepted definition of causality has not been offered and may not be possible because of the nature of statistical inference. It is more accurate to state, for instance, that changes in the rate of growth help to predict changes in economic inequality. For brevity, however, the word "cause" is often substituted .  4 . R. l. Basmann, David J. Molina, Mario Rodarte, and Daniel J. Slottje, "Some New Methods of Predicting Changes in Economic Inequality Associated with Trends in Growth and Development," in Issues in Third World Development, ed . Kenneth C Nobe and Rajan K. Sampath  23  (Boulder, Colo .: Westview Press, 1983), 71-141 . 5.  For a comprehensive list of the possible causal orderings and their defi· nitions, see Balvir Singh and Balbir S. Sahni, "Causality Between Public Expenditure and National Income," Review of Economics and Statistics 66 (November 1984): 630-44.  6.  See Simon Kuznets, Modern Economic Growth: Rate, Structure, and Sp read (New Haven: Yale University Press, 1966). Kuznets implicitly assumed that more widespread education accompanies a growing economy  7.  F. Welch, "Education in Production," Journal of Political Economy 78 (January/February 1970): 35-59.  8.  D. P. Chaudhri, Education, Innovations and Agricultural Development: A Study of North India (1961-72) (London: Croom Helm; New Delhi: Vikas Publishing House, 1979), chap . 3.  9. Theodore W. Schultz, "Optimal Investment in College Instruction: Equity and Efficiency," Jou rnal of Political Economy 80 (May/June 1972, pt. 2): 52-530. 10 Robert T. Michael, The Effect of Education on Efficiency in Consumption, NBER Occasional Paper no . 116 (New York: National Bureau of Economic Research, 1972). 11. See A. B. Atkinson and F. Bourguignon, "The Comparison of MultiDimensioned Distributions of Economic Status," Review of Economic Studies 49 (April 1982): 183-201, for a thorough discussion of the appropriate measure(s) of economic welfare and the multidimensional approach to measuring economic welfare. 12. See Basmann and others, "Some New Methods of Predicting Changes in Economic Inequality," for a detailed discussion of a comprehensive economic inequality measure that adopts a hypothetical joint distribu· tion of components of income and commodity expenditures  13. The permanent income hypothesis originated in Milton Friedman, A Theory of the Consumption Function (Princeton: Princeton University Press for National Bureau of Economic Research, 1957). 14. Data were obtained from various issues of Encuesta Nacional de Ingresos y Gastos de los Hogares, 1977 (Mexico: Secretarfa de Programaci6n y Presupuesto). For a given level of total expenditures, the survey records the number of households Within each subgroup. 15. The Gini coefficient is calculated using a hypothetical joint distribution function to describe the distribution of total expenditures . The functional form chosen is the Beta distribution of the second kind (Beta-II), which is defined in Maurice G. Kendall and Alan Stuart, The Advanced Theory of Statistics, vo l. 1 of 3, Distribution Theory, 3d ed. (London : Charles Griffin & Company, 1969), 150-51 . This hypothetical joint distribution possesses the properties that the marginal density takes the same form as the joint density and that summing the marginal denSity functions provides the joint density. For the study on Mexico, the Gini coefficient may be calculated using the parameters estimated from data obtained from the Encuesta Nacional de Ingresos The parameters are estimated using the generalized method of moments discussed in W . Palin Elderton, Frequency Curves and Correlation, 3d ed . (Cambridge:  24  Cambridge University Press, 1938). See D. J. Slottje, "Relative Price Changes and Inequality in the Size Distribution of Various Components of Income," Journal of Business & Economic Statistics 5 (January 1987): 19·26, for a thorough discussion of the useful properties of the Beta-ll distribution . 16. See C. W . J. Granger, "Investigating Causal Relations by Econometric Models and Cross·spectral Methods," Econometrica 37 (July 1969): 424-38; Christopher A. Sims, "Money, Income, and Causality," American Economic Review 62 (September 1972): 540-52; and David A. Pierce and Larry D. Haugh, "Causality in Temporal Systems: Characterizations and a Survey," Journal of Econometrics 5 (May 1977): 265-93. 17. G William Schwert, "Tests of Causality: The Message in the Innovations," Carnegie·Rochester Conference Series on Public Policy 10 (1979): 79 . 18. Visual inspection of the plots of the various Gini coefficients indicated stationarity in the mean and variance. Also, the autocorrelation function damped sufficiently rapidly to indicate that no transformation of the Gini coefficient series was needed . In contrast, visual inspection of the plot of per capita real GDP indicated non stationarity in the mean and variance, which suggests that the appropriate transformation is differences in natural logarithms (that is, percentage changes). A quickly damping autocorrelation function and plot of the transformed data indicated that stationarity had been achieved . In the inequality literature, the relationship between changes in the level of inequality is often related to both changes in the level of eco· nomic activity and changes in the growth of economic activity. In a sense, our analysis implicitly tests both hypotheses simultaneously. When testing the relationship between the level of inequality and the growth rate of per capita real gross domestic product, as the study here does, we are indirectly testing the relationship between inequality and the level of per capita real gross domestic product as well. An increased growth rate implies an increased level.  19. George E. P. Box and Gwilym M. Jenkins provide a detailed account of their procedure for analyzing time series data in their book entitled Time Series Analysis, Forecasting and Control, rev. ed . (San Francisco: Holden·Day, 1976), chap . 10. 20 See G. M. Ljung and G. E. P. Box, "On a Measure of Lack of Fit in Time Series Models," Biometrika 65, no . 2 (1978): 297-303 . 21 R. Ashley, C. W J Granger, and R. Schmalensee, "Advertising and Aggregate Consumption: An AnalYSis of Causality," Econometrica 48 (July 1980): 1149-67, state the cross-correlation coefficient, p(k), as being asymptotically distributed as an independent normal with mean zero and variance l/n . Therefore, the test statistic for a null hypothesis that the coefficient is zero has a standard deviation equal to l/(n 11 2), where n is the number of observations. The 90-percent confidence interval in the charts is found by multiplying the standard deviation by 1.645 . 22. Note that the cross-correlation function is limited to analyzing values of cross-correlations between - 10 and 10. Small sample size is the primary reason for restricting our reporting to these coefficients. The meaning of a statistically significant coefficient at, say, k = 20 is tenuous because of the small number of degrees of freedom . Consequently, we chose not to report higher orders of the cross-correlation function .  Federal Reserve Bank of Dallas  Appendix  Transfer Function Analysis As a supplement to the causality tests in the foregoing article, this Appendix presents an analysis of the "steadystate" gain in the level of inequality of total expenditures resulting from a permanent change in the growth rate of per capita real gross domestic product. For the analysis, we examine plausible level equations-transfer functionsthat relate the Gini coefficient of total expenditures to lagged percentage changes in per capita real gross domestic product. Let Gt denote the Gini coefficient for total expenditures at time t and Xt denote the percentage change in per capita real gross domestic product. We initially estimated the following transfer function (autoregressive distributed lag): (A.l)  Gt = lXo  +  +  1X1Gt-1  P3 Xt-3  +  +  P1Xt-l  P4 Xt-4  +  +  P2 Xt-2  Il t ·  The fitted equation obtained was (A.2) Gt = .555 + .851 Gt - 1 - .006 Xt _ 1 - .080 Xt- 2 (1 .21) (7.22) (-.19) (-1 .99) - .027 Xt _ 3 + .025 Xt - 4 . ( -.72) (.69) R2  = .79;  R2  = .73; Q = 4.43 (Pr = .97).  The figures in parentheses under the coefficient estimates are asymptotic t statistics; R2 and R2 are, respectively, the unadjusted and the adjusted coefficients of determination; and Q is the Ljung-Box chi-square statistic at lag 12, with the accompanying probability level in the parentheses. The observed value of Q supports the contention of white-noise residuals, so higher-order lags on G and X seem unwarranted. Because the second lag of X is the only significant coefficient in (A. 2), we fit the following parsimonious transfer function:  Economic Review - May 1988  (A. 3)  Gt = .053 + .857 Gt _, - .091 Xt(1.28) (8.01) ( -3.05) R2  = .78;  R2  2,  = .76; Q = 1.89 (Pr = 1.00).  where the reported statistics are as defined for (A.2). Given the increase in R2, the highly Significant coefficients, and the plausibility of white-noise residuals, (A.3) is an improvement over the initial specification, (A.2). Other simplifications of (A.l) obtained by deleting insignificant variables invariably lead back to (A.3). Thus, we choose (A.3) as our "final form" specification. Given (A.3), changes in the growth rate of per capita real gross domestic product precede changes in the Gini coefficient for total expenditures by two years. This sequence corroborates the results obtained from the crosscorrelation function of the innovations of the two series. Also, given this transfer function, the steady-state "gain" is computed to be P2/(1 - IX,) = -.64. For example, suppose the Mexican annual growth rate of per capita real gross domestic product is initially zero percent for a sustained period of time. Then, if by some means the growth rate is increased permanently to 0.01 (that is, 1 percent), the cumulative change in the Gini coefficient for total expenditures that is implied by the transfer function (A.3) is [-.09/(1 - .86)] (.01) = -.0064. To put this calculation in perspective, the range of the Gini coefficient for total expenditures in the present data is (.433 .360 = .073). A permanent increase of 1 percent in Mexico's annual growth rate results in a permanent decrease in the Gini (toward more equality of expenditures) that is equivalent to 9 percent of the previous range of the Gini coefficient. In contrast, a permanent decrease of 1 percent in Mexico's annual growth rate leads to an increase in the Gini coefficient (toward greater inequality).  25  ANNOUNCEMENT Fall Economic Conference "The Southwest and the International Economy" The Federal Reserve Bank of Dallas is pleased to announce an upcoming conference this October. The conference will examine the increasing importance of the global economy to the Southwest. Topics will include: Trade Policy and the Southwest Economy, The Value of the Dollar, Internationalization of the Southwest Economy, Foreign Investment in the Southwest, and the Outlook for the Southwest. Speakers will include prominent business leaders from the Southwest and the nation. If you would like to receive information regarding this important event, please fill out the information card and return it to: Federal Reserve Bank of Dallas Public Affairs Department Station K Dallas, Texas 75222 Attn : Fall Conference  Please add me to the mailing list to receive information about the Fall Conference. Name: _______________________________________ Title: Company: ____________________________________ Address: _____________________________________ ___________________Zip _______________ Business phone ----'____-'-_________________________  Economic Review - May 1988  27