A Multilateral Exchange Rate Model

Jacques R. Artus; Rudolf Rhomberg

doi:10.5089/9781451969313.024.A002

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A Multilateral Exchange Rate Model

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Jacques R. Artus

Jacques R. Artus
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Rudolf Rhomberg

Rudolf Rhomberg
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Publication Date:: 01 Jan 1973

eISBN:: 9781451969313

Language:: English

Keywords:: SP; capital movement; price elasticity; fund chairman; point of view; budget balance; capital flow; Capital flows; Personal income; Budget planning and preparation; Commercial banks; South America; Central America; Africa; North America; Middle East; Caribbean; volume terms; expenditure elasticity; price elasticities of supply; factor price; price identity; Exchange rate adjustments; Currencies; Demand elasticity; Trade balance

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The Analysis of the effects of changes in exchange rates on foreign trade flows requires that account be taken of the simultaneous interaction among prices, incomes, and spending of the countries whose exchange rates have been altered and of the trading partners of these countries. The complexity of the problem increases when simultaneous changes in the exchange rates of several closely associated countries are examined. The model described below was developed to facilitate the assessment of the trade effects of such changes that would occur after an adjustment period of two to three years. Thus, the model is not directly relevant to estimating either short-run effects (within the first year after the exchange rate changes) or ultimate long-run effects (when all commodity and factor prices may be considered perfectly flexible, and labor and capital perfectly mobile).

Abstract

Main Features of the Model¹

As used in some recent applications, the model identifies 15 countries or groups of countries: Austria, Belgium-Luxembourg, Canada, Denmark, France, Germany, Italy, Japan, the Netherlands, Norway, Sweden, Switzerland, the United Kingdom, the United States, and the rest of the world; and five groups of goods (shown with the Standard International Trade Classification (SITC) number): food, beverages, and tobacco (SITC 0–1), crude materials (SITC 2 and 4), mineral fuels (SITC 3), manufactures (SITC 5–9),² and a class of nontraded goods.³ Furthermore, each good is assumed to be differentiated in use according to the country of production. A good produced by a particular country is named a product. For example, French and Canadian agricultural commodities are the same good but two different products. Hence, the model distinguishes 75 (15 × 5) different products, each supplied by only one country.

For each traded product there is a single supply function and a set of demand functions, one for each country of the model. The world demand for such a product is related to each country’s total money expenditure on the good and to the prices of all products. World demand here includes domestic demand, and the explanatory variables include domestic money expenditure and the domestic price, along with foreign prices and expenditures. For a nontraded product, there is only a domestic demand. The supply of each product is a function of prices in the supplying country. The model is closed by imposing two sets of constraints: (1) for each product, supply and demand must be equal; and (2) for each country, the total real output is exogenously determined.

Demand Equations

The demand equations are based on the theoretical framework developed by Armington for products distinguished by place of production.⁴ A distinction, however, is made between three goods for final demand (food, beverages, and tobacco (SITC 0–1), manufactures (SITC 5–9), and a class of nontraded goods) and two intermediate goods (raw materials (SITC 2 and 4) and mineral fuels (SITC 3)).⁵The proportionate change in demand in geographic market k for any final product ij (i.e., for final good ι supplied by country j), expressed in the numeraire currency, is represented by the sum of the effects of changes (a) in money expenditure on all goods for final demand in market k, (b) in any of the prices of products of class i supplied by the 15 competing countries i, and (c) in any of the prices of final products outside the ith class:

\begin{matrix} \begin{matrix} {\overset{*}{X}}_{i j}^{k} = {\overset{*}{P}}_{i j} + ϵ_{i}^{k} {\overset{*}{D}}_{k} + \underset{l}{Σ} η_{i j / i l}^{k} ({\overset{*}{P}}_{i l} - {\overset{*}{T}}_{k}) + \underset{n \neq i}{\begin{matrix} Σ \end{matrix}} η_{i / n}^{k} [\underset{l}{Σ} S_{n l}^{k} ({\overset{*}{P}}_{n l} - {\overset{*}{T}}_{k})] \end{matrix} & (1) \end{matrix}

where

i, n = 1, 2, and 3 (goods for final demand),

j, k, l = 1,…,15 (countries),

and where

${\overset{*}{X}}_{i j}^{k}$ = proportionate change in demand (in value terms) for product ij in market k in the numeraire currency;
${\overset{*}{P}}_{i j}$ = proportionate change in the price of the i th good supplied by country j expressed in the numeraire currency;
${\overset{*}{T}}_{k}$ = proportionate change in the exchange rate of country k’s currency in terms of the numeraire currency;
${\overset{*}{D}}_{k}$ proportionate change in the money expenditure on all goods for final demand in country k in local currency;
$ϵ_{i}^{k}$ = expenditure elasticity of country k’s demand for the ith good;
$η_{i j / i l}^{k}$ = price elasticity of demand (in volume terms) in market k for the ith good produced by country j with respect to the price of the ith good produced by country l.
$η_{i / n}^{k}$ = price elasticity of demand (in volume terms) in market k for the ith good in general, irrespective of the source of supply, with respect to the price of the nth good in general;
$S_{n l}^{k} = X_{n l}^{k} / \underset{m}{Σ} X_{n m}^{k},$ where $X_{n m}^{k}$ is the value of country m’s exports of

good n to market k in the base year; that is, $S_{n l}^{k}$ is the share of exports of good n by country l in market k.

Similarly, the proportionate change in demand for an intermediate product is expressed as:

\begin{matrix} {\overset{*}{\begin{matrix} X \end{matrix}}}_{i j}^{k} = {\overset{*}{P}}_{i j} + ϵ_{i}^{k} {\overset{*}{I}}_{k} + \underset{l}{Σ} η_{i j / i l}^{k} ({\overset{*}{P}}_{i j} - {\overset{*}{T}}_{k}) + \underset{n \neq i}{Σ} η_{i / n}^{k} [\underset{l}{Σ} S_{n l}^{k} ({\overset{*}{P}}_{n l} - {\overset{*}{T}}_{k})] & (2) \end{matrix}

where

i, n = 4 and 5 (intermediate goods),

j, k, l = 1,…,15 (countires),

and where ${\overset{*}{I}}_{k}$ is the proportionate change in the total money expenditure on the two intermediate goods in country k.

Equation (1) and (2) rest on the following assumptions: (1) the marginal rates of substitution between any two products of the same kind (i.e., belonging to the same good) are independent of the quantities demanded of the products of other kinds,⁶ and (2) the index functions—utility functions for final goods and production functions for intermediate goods—governing choices among various products of the same kind are linear and homogeneous.

The proportionate change in world demand for product ij $({\overset{*}{X}}_{i j})$ can be expressed as a weighted average of the proportionate changes of demands in various markets, with the importance of each market in total exports of good i by country j taken as weights:

\begin{matrix} \overset{*}{X_{i j}} = \underset{k}{Σ} {W_{i j}}^{k} \overset{* k}{X_{i j}} & (3) \end{matrix}

where $W_{i j}^{k} = X_{i j}^{k} / \underset{k}{Σ} X_{i j}^{k},$ and

$X_{i j}^{k}$ = value of country j’s exports of good i to market k in the base year.

Note that when j = l, $η_{i j / i l}^{k}$ is the direct price elasticity of demand in market k for the product ij. When j ≠ 1 $η_{i j / i l}^{k}$ is the cross price elasticity of demand in market k for product ij with respect to the price of the product il. In the special case where i represents a nontradable good, the trade weight $W_{i j}^{k}$ equals zero is k ≠ j, and unity if k = j; the elasticity coefficient $η_{i j / i j}^{j}$ equals the price elasticity of country j’s demand for the (nontradable) good; and, $η_{i j / i l}^{k} = 0,$ if k ≠ j or l ≠ j, or both.

The number of demand price elasticities present in the model is too large to allow a direct econometric estimation of these parameters. However, they can be expressed in terms of a fairly limited number of basic parameters if a few simplifying assumptions are added.

First, the parameters $η_{i / n}^{k}$ , representing the elasticities of demand in volume terms for good i with respect to the price of good n in market k, can be expressed as a function of the corresponding income compensated price elasticities, $λ_{i / n}^{k},$ and the expenditure elasticities, ϵ_i^k, for good i by employing the Slutsky equations:

\begin{matrix} {\begin{matrix} η_{i} \end{matrix}}^{k} / n = {λ_{i}}^{k} / n - {S_{n}}^{k} {ϵ_{i}}^{k}; i, n = 1, 2, and 3 (goods for final demand) & (4) \end{matrix}

and

\begin{matrix} η_{i}^{k} / n = λ_{i}^{k} / n - {S_{n}}^{k} {ϵ_{i}}^{k}; i, n = 4 and 5 (intermediate goods) & (5) \end{matrix}

where S_n^k = share of spending on the nth good in the total money spending on all final goods (or intermediate goods) if n refers to a final good (or intermediate good).

If the various goods are clearly distinguished by the kinds of want or need that they serve, the income-compensated price elasticities $λ_{i}^{k} / n,$ including the income-compensated own-price elasticity $λ_{i}^{k} / n,$ are likely to be quite small. These parameters will be assumed to be equal to zero for purposes of simplification. In this instance, all the price elasticities among goods can be derived from the share matrices and from the expenditure elasticities.

Similarly, the elasticity (in volume terms), $η_{i j / i l,}^{k}$ of demand for product ij with respect to the price of product il can be expressed as a function of the corresponding income-compensated price elasticity, ${\overset{- k}{η}}_{i j / i l},$ and the expenditure elasticity, $ϵ_{i j}^{k}$ for product ij,

\begin{matrix} η_{i j / i l}^{k} = {\bar{η}}_{i j / i l}^{k} - S_{i l}^{k} S_{i}^{k} ϵ_{i j}^{k}, & (6) \end{matrix}

where the term $S_{i l}^{k}$ $S_{i}^{k}$ represents the share of product ij in the total demand in market k.

The basic assumption that the index functions governing choices among various products of the same kind are linear and homogeneous implies that $ϵ_{i j}^{k}$ is equal to $ϵ_{i}^{k} .$ The parameters ${\bar{η}}_{i j / i l}^{k}$ can be expressed in terms of a few basic parameters by further specifying the form of the index functions involved. It is shown in the Appendix that the adoption of CRESH (constant ratios of elasticities of substitution homogeneous) index functions—utility functions for goods for final demand and production functions for intermediate goods—yields the following relations:

\begin{matrix} \bar{η} k_{i j / i j} = \frac{a_{i l}^{k} X_{i l}^{k}}{\underset{m}{Σ} a_{i m}^{k} X_{i m}^{k}} . a_{i j}^{k}, l \neq j, & (7) \end{matrix}

\begin{matrix} \bar{η} \underset{i j / i l}{k} = - (1 - \frac{a_{i j}^{k} X_{i j}^{k}}{\underset{m}{Σ} a_{i m}^{k} X_{i m}^{k}}) a_{i j}^{k} & (8) \end{matrix}

where $a_{i j}^{k}$ = a parameter measuring the substitutability of the product ij’ for all other products of the same kind in market k.

In some calculations with the model, the substitutability of a given product for all other products of the same kind has been assumed to be the same in all the respective foreign markets; that is, $a_{i j}^{k}$ _ij, if k ≠ j. The substitutability in the home market, $a_{i j}^{k},$ could be different from the one in foreign markets. The advantage of this assumption is that it allows the values of the a_ij and $a_{i j}^{i}$ parameters to be derived from the estimated values of the import and export price elasticities corresponding to the various goods in the different countries.

The income-compensated price elasticity of k’s demand for all imports of good i, η_i^k, can be written as an import-weighted average of cross elasticities (with sign reversed) of k’s demand for imports from particular countries with respect to the price of good i produced at home. That is, employing equation (7) to express ${\bar{η}}_{i j / i k,}^{k}$

\begin{array}{l} {\bar{η}}_{j}^{k} = - \underset{j \neq k}{Σ} \frac{X_{i j}^{k}}{\underset{m \neq k}{Σ} X_{i m}^{k}} {\bar{η}}_{i j / i k}^{k} \\ = - (\underset{j \neq k}{Σ} \frac{X_{i j}^{k}}{\underset{m \neq k}{Σ} X_{i m}^{k}} a_{i j}) (\frac{a_{i j}^{k} X_{i j}^{k}}{a_{i k}^{k} X_{i k}^{k} + \underset{m \neq k}{Σ} a_{i j} X_{i m}^{k}}) . & (9) \end{array}

Similarly, the compensated elasticity of the foreign demand for country j’s exports of good i η_ij can be written as a function of the a_ij and $a_{i k}^{k}$ parameters by employing equation (8) aggregated over foreign markets:

\begin{matrix} {\bar{η}}_{i j} = - a_{i j} (1 - \underset{k \neq j}{Σ} \frac{X_{i j}^{k}}{\underset{m \neq j}{Σ} X_{i m}^{k}} . \frac{a_{i j} X_{i j}^{k}}{a_{i k}^{k} X_{i k}^{k} + \underset{m \neq k}{Σ} a_{i j} X_{i m}^{k}}) . & (10) \end{matrix}

The set of 120 equations composed of equations (9) and (10) for the four internationally traded goods and the 15 countries or groups of countries considered in the model implicitly defines the 120 parameters a_ij and $a_{i j}^{j}$ as a function of the trade flows and the compensated import and export price elasticities. In practice, the system of equations was solved for the a_ij and $a_{i j}^{j}$ parameters by employing the Gauss-Seidel⁷ iterative procedure after rewriting equations (9) and (10), respectively, as

\begin{matrix} a_{i k}^{k} = - \bar{η} {}_{i}k / (\underset{j \neq k}{Σ} \frac{X_{i j}^{k}}{\underset{m \neq k}{Σ} X_{i m}^{k}} a_{i j}) (\frac{X_{i k}^{k}}{a_{i k}^{k} X_{i k}^{k} + \underset{m \neq k}{Σ} a_{i j} X_{i m}^{k}}), & (11) \end{matrix}

\begin{matrix} a_{i j} = - \bar{η} i j / (1 - \underset{k \neq j}{Σ} \frac{X_{i j}^{k}}{\underset{m \neq j}{Σ} X_{i k}^{k}} . \frac{a_{i j} X_{i j}^{k}}{a_{i k}^{k} X_{i k}^{k} + \underset{m \neq k}{Σ} a_{i j} X_{i m}^{k}}) . & (12) \end{matrix}

Experience showed that convergence to a solution was not always reached. Broadly speaking, convergence was obtained only when the export price elasticities for a given good were larger than the average of the import price elasticities for the same good. This condition reflects the fact that for any given good, a country has normally a much larger share of the market in its home market than in foreign markets. Most available econometric estimates of import and export price elasticities are consistent with this condition.

Thus, with the help of formulas (4), (5), (6), (11), and (12), all the structural demand parameters can be derived from a set of basic parameters composed of (1) the elasticities of expenditure for the various goods, (2) the import and export price elasticities, and (3) the values of the trade flows and share matrices. Values of foreign trade flows are readily available in Organization for Economic Cooperation and Development, Statistics of Foreign Trade, Series B. Values of domestic trade flows, $X_{i j}^{j},$ which included the value of the nontraded good, can be obtained from national accounts statistics. A priori information or a direct application of econometric methods may be employed to estimate import and export price elasticities and expenditure elasticities for goods.

The parameters ϵ_i^k are assumed to be equal to unity for intermediate goods. Thus, the demand equation for an intermediate product (2) can be rewritten as follows by employing equation (5), with $λ_{i / n}^{k}$ equal to zero, and equation (6),

\begin{matrix} {\overset{*}{X}}_{i j}^{k} = {\overset{*}{P}}_{i j} + [{\overset{*}{I}}_{k} - \underset{n}{Σ} {S_{n}}^{k} (\underset{l}{Σ} S_{n l}^{k} ({\overset{*}{P}}_{n l} - {\overset{*}{T}}_{k})] + \underset{}{\underset{l}{Σ} {\bar{η}}_{i j / i l}^{k} ({\overset{*}{P}}_{i l} - {\overset{*}{T}}_{k}),} & (13) \end{matrix}

with i, n = 4 and 5 (intermediate goods), and where S_n^k is the share of good n in the total spending on intermediate goods.

The second term in equation (13) is simply the proportionate change in the deflated expenditure on all intermediate goods. This variable was assumed to be equal to the proportionate change in the total real output of the corresponding country, ${\overset{*}{O}}_{k,}$ so that equation (13) can be rewritten as

\begin{matrix} \overset{*}{X} k_{i j} = \overset{*}{P_{i j}} + {\overset{*}{o}}_{k} + \underset{l}{Σ} {\bar{η}}_{i j / i l}^{k} (\overset{*}{P_{i l}} - \overset{*}{T_{k}}), & (14) \end{matrix}

with i, n = 4 and 5 (intermediate goods).

The model does not constrain the expenditure elasticities, ϵ_i^k for goods for final demand to be equal to unity; however, these elasticities were taken as being equal to unity in most recent calculations. The demand equations (1) and (14) and the relations (3), (4), (6), (7), (8), (11), and (12) depict the demand side of the model.

Supply Equations and Price Identities

The supply equations are introduced into the model with variables expressed as proportionate changes in value terms, and all elasticities are accordingly defined in value terms (the value elasticity equals the volume elasticity plus unity), that is:

\begin{matrix} \overset{*}{q_{i j}} = \underset{n}{Σ} α_{i j / n j} (\overset{*}{P_{n i}} - \overset{*}{s_{n j}}) + \overset{*}{s_{i j}} + \overset{*}{T_{j,}} & (15) \end{matrix}

where ${\overset{*}{q}}_{i j}$ = proportionate change in value terms in the supply of product ij expressed in the numeraire currency;

α_ij/nj price elasticity of supply (in value terms) of product ij with respect to the price of product nj ;

${\overset{*}{p}}_{n j}$ = proportionate change in the price of product nj expressed in local currency;

${\overset{*}{s}}_{n j}$ = proportionate shift in the supply schedule for product nj owing to changes in the price of the factors of production.

The supply equations represent quantitatively the possibility of transferring resources from the production of one good to that of another in response to price incentives. They also allow for the possibility of an increase in total resource utilization. The aggregate supply elasticity is represented by the algebraic sum over n of the coefficients α_ij/nj, while the transferability of resources in and out of sector ij is represented by the absolute size ofα_ij/ij, or by the absolute size of the sum of the remaining coefficients. These elasticities are defined for a given

set of prices for the factors of production.

The shift factors ${\overset{*}{s}}_{n j}$ are given exogenously and can be viewed as proportionate vertical shifts in the supply function. These shifts reflect changes in prices that are not causally linked with changes in output. They serve to express the shift in the supply of product ij which results from the changes in the prices of imported products used as inputs in the production process and from changes in wages, and indirect taxes induced by the effect of the change in the exchange rate on the cost of living.

The shift factors are specified as follows:

\begin{matrix} \overset{*}{s} = a_{j} b_{i j} [\underset{n}{Σ} \underset{l}{Σ} {S C L}_{n l}^{j} (\overset{*}{P_{n l}} - \overset{*}{T_{j}})] + \underset{n}{Σ} \underset{l \neq j}{Σ} {S P C}_{n l}^{i j} (\overset{*}{P_{n l}} - \overset{*}{T_{j}}), & (16) \end{matrix}

where a_j = proportion of the change in the cost of living in country j that is reflected in money wages and indirect taxes during the period of time considered;

bij = share of the wage and tax bill in the total cost of production in the ijth industry;

$S C L_{n l}^{j}$ share of product nl in the cost of living index in country j ;

$S P C_{n l}^{i j}$ share of product nl (as an input) in the total cost of production in the ij th industry.

The share parameters b_ij, $S C L_{n l}^{j}$ , and $S P C_{n l}^{i j}$ are obtained from input/ output matrices.

The change in the price of product ij expressed in the numeraire currency is defined as the sum of the change in the local currency price plus the change in the exchange rate.

\begin{matrix} \begin{matrix} \overset{*}{P_{i j}} = \overset{*}{p_{i j}} + \overset{*}{T_{j}} & (17) \end{matrix} \end{matrix}

Market Equilibrium Equations

The market equilibrium equations stipulate that the ex post (proportionate) changes in the supply of and demand for each product must be equal:

\begin{matrix} \overset{*}{X_{i j}} = \overset{*}{q_{i j}} & (18) \end{matrix}

Constraint on Total Real Production

The model consisting of these equations expressing demand, supply, and market equilibrium conditions could be closed by considering the change in each country’s total nominal spending as a policy variable. With exchange rates and levels of total nominal spending given exogenously, the system of equations could then be solved for each country’s foreign trade balance⁸ and real output. However, the total nominal spending of a country is not normally considered an important target variable per se and any value assumed a priori would be arbitrary.

The present model employs an alternative approach. The real value of the output of a country is considered to be the important target variable, in part because it implies a particular level of unemployment. Therefore, the model is employed to derive for each country the changes in nominal spending and the trade balance that correspond to the chosen (proportionate) changes in the levels of output and the foreign exchange rates for all countries.

In recent applications, the model was employed to analyze the implications of one or several changes in exchange rates under the assumption that the chosen changes in the total output of each country was nil. This assumption permits isolation of the effects of a change in exchange rates from the effects of output changes. This allows the estimation of the effects of given changes in exchange rates on the trade balances corresponding to the target levels of output. The real output constraint is formulated as follows:

\begin{matrix} {\overset{*}{O}}_{j} = \frac{\underset{i}{Σ} q_{i j} \overset{*}{q_{i j}}}{\underset{i}{Σ} q_{i j}} - \frac{\underset{i}{Σ} q_{i j} \overset{*}{P_{i j}}}{\underset{i}{Σ} q_{i j}} = 0, & (19) \end{matrix}

where ${\overset{*}{O}}_{j}$ = proportionate change in the total real output of country j, and

q_ij = value of the production of product ij in the base year.

This equation stipulates that the proportionate change in the real total output of country j, which is defined as the difference between the proportionate changes in the value of country j’s total output and its output deflator, must be equal to zero.

It is important to note that the present model does not contain any behavioral function relating changes in nominal spending to changes in nominal income (i.e., changes in the nominal value of output). The model indicates for each country the level of money spending that is consistent with a particular set of foreign exchange rates and real output constraints for all countries. But it does not give any guidance on the question of how this level of money spending can be reached. Hence, it does not determine by how much the nominal spending of a country will have to be changed by use of monetary or fiscal policies to close the gap between the change in nominal spending that is necessary to meet its real output constraint and the change in nominal spending that results automatically from the assumed change in one or several of the exchange rates.

Summary of the Model

The model is composed of the following 390 equations in 390 endogenous variables:

Endogenous variables	Equations: type and identification number in text	Number of variables and equations of each type
Change in world demand $({\overset{*}{X}}_{i j})$	Demand (3)	5 × 15 = 75
Change in supply $({\overset{*}{q}}_{i j})$	Supply (15)	5 × 15 = 75
Price change in numeraire currency $({\overset{*}{P}}_{i j})$	Identity (17)	5 × 15 = 75
Local currency price change $({\overset{*}{P}}_{i j})$	Market equilibrium (18)	5 × 15 = 75
Shift factors $({\overset{*}{s}}_{i j})$	Shift equation (16)	5 × 15 = 75
Change in total nominal spending $({\overset{*}{D}}_{j})$	Output-constraint (19)	15
		390

Endogenous variables	Equations: type and identification number in text	Number of variables and equations of each type
Change in world demand $({\overset{*}{X}}_{i j})$	Demand (3)	5 × 15 = 75
Change in supply $({\overset{*}{q}}_{i j})$	Supply (15)	5 × 15 = 75
Price change in numeraire currency $({\overset{*}{P}}_{i j})$	Identity (17)	5 × 15 = 75
Local currency price change $({\overset{*}{P}}_{i j})$	Market equilibrium (18)	5 × 15 = 75
Shift factors $({\overset{*}{s}}_{i j})$	Shift equation (16)	5 × 15 = 75
Change in total nominal spending $({\overset{*}{D}}_{j})$	Output-constraint (19)	15
		390

The exogenous variables are the original trade flows $X_{i j}^{k}$ and the changes in exchange rates ${\overset{*}{T}}_{j} .$

The main parameters are the expenditure elasticities for the final goods, the import and export price elasticities, the price elasticities of supply, and the proportion of the change in the cost of living, which is automatically reflected in a change in wages and taxes. The latter proportion has been taken to be 0.75. The values of the expenditure elasticities for final goods have been taken to be 1.0. The values of the import and export price elasticities and supply elasticities as used in some past calculations are given in Tables 1, 2, and 3. All these parameters refer to trade effects after a period of adjustment of two to three years. The other parameters are the trade matrices and the input-output matrices.

^{Table 1.}

Price Elasticities of Demand for Import of Internationally Traded Goods in Volume Terms (η_i^k)

Source: These parameters are to a large extent chosen so as to be consistent with a number of empirical studies completed recently—among others, Grant B. Taplin, “Updated Import Equations for the Expanded World Trade Model” (unpublished, International Monetary Fund, April 19, 1972). However, the choice of elasticities also takes into account various a priori considerations and reflects ultimately the judgment of the authors.

^{Table 1.}

Price Elasticities of Demand for Import of Internationally Traded Goods in Volume Terms (η_i^k)

Country	Food, Beverages, and Tobacco	Crude Materials	Fuels	Manufactures
Austria	–0.1	–0.1	–0.1	–1.0
Belgium-Luxembourg	–0.5	–0.1	–0.1	–1.0
Canada	–1.0	–0.1	–0.1	–2.0
Denmark	–1.0	–0.1	–0.1	–1.5
France	–0.5	–0.1	–0.1	–2.0
Germany	–0.5	–0.1	–0.1	–2.0
Italy	–1.0	–0.1	–0.1	–1.5
Japan	–0.1	–0.1	–0.1	–2.0
Netherlands	–0.1	–0.1	–0.1	–1.0
Norway	–0.5	–0.1	–0.1	–1.0
Sweden	–0.5	–0.1	–0.1	–1.0
Switzerland	–0.1	–0.1	–0.1	–1.0
United Kingdom	–0.5	–0.1	–0.1	–1.0
United States	–1.0	–0.1	–0.1	–2.0
Rest of the World	–0.5	–0.1	–0.1	–1.0

^{Table 1.}

Price Elasticities of Demand for Import of Internationally Traded Goods in Volume Terms (η_i^k)

Country	Food, Beverages, and Tobacco	Crude Materials	Fuels	Manufactures
Austria	–0.1	–0.1	–0.1	–1.0
Belgium-Luxembourg	–0.5	–0.1	–0.1	–1.0
Canada	–1.0	–0.1	–0.1	–2.0
Denmark	–1.0	–0.1	–0.1	–1.5
France	–0.5	–0.1	–0.1	–2.0
Germany	–0.5	–0.1	–0.1	–2.0
Italy	–1.0	–0.1	–0.1	–1.5
Japan	–0.1	–0.1	–0.1	–2.0
Netherlands	–0.1	–0.1	–0.1	–1.0
Norway	–0.5	–0.1	–0.1	–1.0
Sweden	–0.5	–0.1	–0.1	–1.0
Switzerland	–0.1	–0.1	–0.1	–1.0
United Kingdom	–0.5	–0.1	–0.1	–1.0
United States	–1.0	–0.1	–0.1	–2.0
Rest of the World	–0.5	–0.1	–0.1	–1.0

^{Table 2.}

Price Elasticities of Demand for Export of Internationally Traded Goods in Volume Terms (η_ij)

Source: These parameters are based on various empirical studies on export price elasticities or substitution elasticities—among others, H. B. Junz and R. R. Rhomberg, “Price Competitiveness in Export Trade Among Industrial Countries,” American Economic Association, Papers and Proceedings of the Eighty-fifth Annual Meeting (The American Economic Review, Vol. LXIII (May 1973), pp. 412–18. However, here also, the choice of elasticities takes into account various a priori considerations and reflects ultimately the judgment of the authors.

^{Table 2.}

Price Elasticities of Demand for Export of Internationally Traded Goods in Volume Terms (η_ij)

Country	Food, Beverages, and Tobacco	Crude Materials	Fuels	Manufactures
Austria	–1.0	–0.5	–0.5	–2.5
Belgium-Luxembourg	–1.0	–0.5	–0.5	–3.0
Canada	–1.0	–0.5	–0.5	–1.5
Denmark	–1.0	–0.5	–0.5	–2.0
France	–1.0	–0.5	–0.5	–2.5
Germany	–1.0	–0.5	–0.5	–1.5
Italy	–1.0	–0.5	–0.5	–2.5
Japan	–1.0	–0.5	–0.5	–3.0
Netherlands	–1.0	–0.5	–0.5	–2.5
Norway	–1.0	–0.5	–0.5	–2.5
Sweden	–1.0	–0.5	–0.5	–2.5
Switzerland	–1.0	–0.5	–0.5	–2.5
United Kingdom	–1.0	–0.5	–0.5	–2.0
United States	–1.0	–0.5	–0.5	–2.5
Rest of the World	–1.0	–0.5	–0.1	–2.5

^{Table 2.}

Price Elasticities of Demand for Export of Internationally Traded Goods in Volume Terms (η_ij)

Country	Food, Beverages, and Tobacco	Crude Materials	Fuels	Manufactures
Austria	–1.0	–0.5	–0.5	–2.5
Belgium-Luxembourg	–1.0	–0.5	–0.5	–3.0
Canada	–1.0	–0.5	–0.5	–1.5
Denmark	–1.0	–0.5	–0.5	–2.0
France	–1.0	–0.5	–0.5	–2.5
Germany	–1.0	–0.5	–0.5	–1.5
Italy	–1.0	–0.5	–0.5	–2.5
Japan	–1.0	–0.5	–0.5	–3.0
Netherlands	–1.0	–0.5	–0.5	–2.5
Norway	–1.0	–0.5	–0.5	–2.5
Sweden	–1.0	–0.5	–0.5	–2.5
Switzerland	–1.0	–0.5	–0.5	–2.5
United Kingdom	–1.0	–0.5	–0.5	–2.0
United States	–1.0	–0.5	–0.5	–2.5
Rest of the World	–1.0	–0.5	–0.1	–2.5

^{Table 3.}

Price Elasticities of Supply in Volume Terms for All Countries

^¹For the “Rest of the World,” the direct price elasticity of supply was taken to be equal to 1.0; and the cross elasticities of supply with respect to a change in the price of manufactures and nontraded goods were taken to be equal to—0.2 and—0.3, respectively.

^{Table 3.}

Price Elasticities of Supply in Volume Terms for All Countries

	With Respect to a Change in the Price of
	Food, beverages, and tobacco	Crude materials	Fuels	Manufactures	Non- traded goods
Food, beverages, and tobacco	2.0 ¹	0.0	0.0	–0.3 ¹	–0.7 ¹
Crude materials	0.0	1.0	0.0	–0.3	–0.2
Fuels	0.0	0.0	0.0	0.0	0.0
Manufactures	–0.1	–0.05	0.0	3.0	–1.2
Nontraded goods	–0.1	–0.02	0.0	–0.7	3.0

^{Table 3.}

Price Elasticities of Supply in Volume Terms for All Countries

	With Respect to a Change in the Price of
	Food, beverages, and tobacco	Crude materials	Fuels	Manufactures	Non- traded goods
Food, beverages, and tobacco	2.0 ¹	0.0	0.0	–0.3 ¹	–0.7 ¹
Crude materials	0.0	1.0	0.0	–0.3	–0.2
Fuels	0.0	0.0	0.0	0.0	0.0
Manufactures	–0.1	–0.05	0.0	3.0	–1.2
Nontraded goods	–0.1	–0.02	0.0	–0.7	3.0

Applications of the Model

The model described in this paper can be employed to estimate the effect on the trade balance of each country of a single exchange rate change or of an exchange rate realignment in which several rates change at the same time.

The model can also be used to estimate the exchange rate changes required to achieve a set of trade balance targets. For this purpose, the trade effects of an isolated change by x per cent in the exchange rate of each country or group of countries is first calculated.⁹ This procedure yields a matrix (A) of coefficients a_kj indicating the effect of a change in the exchange rate of country k by x per cent on the trade balance of country j. If linearity problems are ignored, the vector of changes in trade balances (B) caused by a set of changes in exchange rates (T) can be computed as follows:

\begin{matrix} \frac{A T}{x} = B . \end{matrix}

If the vector of changes in trade balances (B) is specified as a set of targets, the changes in exchange rates required to reach these targets are

T = x A^{- 1} B,

where A^-1 is the inverse of the matrix A. Although the effects of changes in exchange rates are in this model not exactly proportionate to the size of the changes in exchange rates, the linearity approximation does not result in significant errors as long as the components of the vector of changes in exchange rates T are of the same order of magnitude as the change x.

To illustrate these applications, the effects of an isolated devaluation by 10 per cent of the currency of each country were calculated from trade data for 1971 and from the parameters listed in Tables 1–3. Effects on the trade balance and on import and export prices of the devaluing country are presented in Table 4. This table also shows the change in final domestic expenditure, in money and in real terms, which is implied by the assumption that demand policies are conducted in such a manner as to keep output constant in spite of the exchange rate changes. The matrix A is presented in Table 5; the entries indicate the calculated response, measured in U. S. dollars, of the trade balance of the country or area in the heading (j in the notation used in the preceding paragraph) to a devaluation by 10 per cent (x) of the currency of the country or area in the stub (k).

^{Table 4.}

Effects of a 10 Per Cent Devaluation on the Foreign and Domestic Trade Flows of the Devaluing Country Under the Assumptions of the Multilateral Exchange Rate Model ¹

^¹For the United States, the figures relate to the effects of a 10 per cent revaluation of all other currencies with respect to the U.S. dollar.

^²Effects on the trade balance are evaluated at the scale of world trade in 1971.

^{Table 4.}

Effects of a 10 Per Cent Devaluation on the Foreign and Domestic Trade Flows of the Devaluing Country Under the Assumptions of the Multilateral Exchange Rate Model ¹

	Effects on			Required Change in Final Domestic Expenditure
	Trade balance (f.o.b.) ^²	Average import price in terms of balance local currency	Average export price in terms of local currency	Required Change in Final Domestic Expenditure
	Trade balance (f.o.b.) ^²		Average export price in terms of local currency	In money terms	In real terms
	Million U.S. dollars	In per cent
Austria	368	9.9	6.5	1.7	–3.6
Belgium-Luxembourg	1,458	9.7	7.2	–1.2	–7.0
Canada	1,532	9.8	6.6	1.8	–3.1
Denmark	309	9.9	6.9	2.1	–3.5
France	3,188	9.5	4.3	0.6	–3.0
Germany	4,096	9.2	4.2	0.4	–3.4
Italy	2,009	9.7	5.1	0.5	–3.3
Japan	3,399	9.7	4.8	1.4	–2.3
Netherlands	1,336	9.7	7.0	0.3	–6.0
Norway	301	9.9	6.7	1.4	–3.9
Sweden	633	9.8	6.4	1.8	–3.4
Switzerland	931	9.9	5.8	–1.1	–5.5
United Kingdom	1,952	9.8	6.0	0.7	–2.9
United States	7,431	8.5	2.3	0.3	–1.0
Rest of the World	5,131	9.0	5.8	0.4	–1.1

^¹For the United States, the figures relate to the effects of a 10 per cent revaluation of all other currencies with respect to the U.S. dollar.

^²Effects on the trade balance are evaluated at the scale of world trade in 1971.

^{Table 4.}

Effects of a 10 Per Cent Devaluation on the Foreign and Domestic Trade Flows of the Devaluing Country Under the Assumptions of the Multilateral Exchange Rate Model ¹

	Effects on			Required Change in Final Domestic Expenditure
	Trade balance (f.o.b.) ^²	Average import price in terms of balance local currency	Average export price in terms of local currency	Required Change in Final Domestic Expenditure
	Trade balance (f.o.b.) ^²		Average export price in terms of local currency	In money terms	In real terms
	Million U.S. dollars	In per cent
Austria	368	9.9	6.5	1.7	–3.6
Belgium-Luxembourg	1,458	9.7	7.2	–1.2	–7.0
Canada	1,532	9.8	6.6	1.8	–3.1
Denmark	309	9.9	6.9	2.1	–3.5
France	3,188	9.5	4.3	0.6	–3.0
Germany	4,096	9.2	4.2	0.4	–3.4
Italy	2,009	9.7	5.1	0.5	–3.3
Japan	3,399	9.7	4.8	1.4	–2.3
Netherlands	1,336	9.7	7.0	0.3	–6.0
Norway	301	9.9	6.7	1.4	–3.9
Sweden	633	9.8	6.4	1.8	–3.4
Switzerland	931	9.9	5.8	–1.1	–5.5
United Kingdom	1,952	9.8	6.0	0.7	–2.9
United States	7,431	8.5	2.3	0.3	–1.0
Rest of the World	5,131	9.0	5.8	0.4	–1.1

^¹For the United States, the figures relate to the effects of a 10 per cent revaluation of all other currencies with respect to the U.S. dollar.

^²Effects on the trade balance are evaluated at the scale of world trade in 1971.

^{Table 5.}

Effects of a 10 Per Cent Devaluation by the Countries or Groups of Countries in the Heading on the Trade Balances (F.O.B.) of the Countries in the Stub Under the Assumptions of the Multilateral Exchange Rate Model ^1,2

(In millions of U.S. dollars)

^¹Evaluated at the scale of world trade in 1971.

^²For the United States, the figures relate to the effects of a 10 per cent revaluation of all other currencies with respect to the U.S. dollar.

^{Table 5.}

(In millions of U.S. dollars)

	Austria	Belgium- Luxembourg	Canada	Denmark	France	Germany	Italy	Japan	Netherlands	Norway	Sweden	Switzerland	United Kingdom	United States	Rest of the World
Austria	368	–9	–11	–5	–26	22	–20	–26	–10	–5	–10	–20	–18	–168	–62
Belgium-Luxembourg	–20	1,458	–9	–11	–222	–314	–111	–88	–94	–17	–25	–41	–22	–386	–98
Canada	–5	–15	1,532	–4	–40	–56	–37	–108	–22	–6	–13	–13	–53	–983	–177
Denmark	–3	–7	–9	309	–18	–5	–16	–19	–5	–12	–5	–10	–23	–109	–68
France	–33	–195	–66	–35	3,188	–681	–302	–226	–149	–19	–49	–100	–163	–616	–554
Germany	–130	–336	–102	–53	–733	4,096	–374	–322	–256	–62	–135	–293	–248	–305	–747
Italy	–12	–120	–54	–11	–298	–476	2,009	–142	–96	–11	–40	–75	–98	–260	–316
Japan	–24	–99	–121	–30	–268	–421	–180	3,399	–117	–30	–60	–72	–220	–970	–787
Netherlands	–19	–64	10	–25	–220	–263	–116	–82	1,336	–16	–23	–46	–105	–251	–116
Norway	–2	–8	10	–1	–25	–39	–11	–13	–8	301	2	–2	–15	–151	–38
Sweden	–10	–23	–27	–9	–73	–54	–37	–64	–21	–26	633	–21	–32	–110	–126
Switzerland	–12	–22	–20	–6	–74	–60	–10	–83	–24	–10	–22	931	–42	–377	–169
United Kingdom	–28	–98	–21	–1	–198	–291	–126	–230	–95	–11	–41	–60	1,952	–510	–241
United States	–43	–256	–962	–67	–647 -	–1,030	–428	–1,298	–308	–35	–144	–137	–445	7,431	–1,631
Rest of the World	–27	–206	–150	–51	–346	–428	–241	–698	–131	–41	–68	–41	–468	–2,235	5,131

^¹Evaluated at the scale of world trade in 1971.

^²For the United States, the figures relate to the effects of a 10 per cent revaluation of all other currencies with respect to the U.S. dollar.

^{Table 5.}

(In millions of U.S. dollars)

	Austria	Belgium- Luxembourg	Canada	Denmark	France	Germany	Italy	Japan	Netherlands	Norway	Sweden	Switzerland	United Kingdom	United States	Rest of the World
Austria	368	–9	–11	–5	–26	22	–20	–26	–10	–5	–10	–20	–18	–168	–62
Belgium-Luxembourg	–20	1,458	–9	–11	–222	–314	–111	–88	–94	–17	–25	–41	–22	–386	–98
Canada	–5	–15	1,532	–4	–40	–56	–37	–108	–22	–6	–13	–13	–53	–983	–177
Denmark	–3	–7	–9	309	–18	–5	–16	–19	–5	–12	–5	–10	–23	–109	–68
France	–33	–195	–66	–35	3,188	–681	–302	–226	–149	–19	–49	–100	–163	–616	–554
Germany	–130	–336	–102	–53	–733	4,096	–374	–322	–256	–62	–135	–293	–248	–305	–747
Italy	–12	–120	–54	–11	–298	–476	2,009	–142	–96	–11	–40	–75	–98	–260	–316
Japan	–24	–99	–121	–30	–268	–421	–180	3,399	–117	–30	–60	–72	–220	–970	–787
Netherlands	–19	–64	10	–25	–220	–263	–116	–82	1,336	–16	–23	–46	–105	–251	–116
Norway	–2	–8	10	–1	–25	–39	–11	–13	–8	301	2	–2	–15	–151	–38
Sweden	–10	–23	–27	–9	–73	–54	–37	–64	–21	–26	633	–21	–32	–110	–126
Switzerland	–12	–22	–20	–6	–74	–60	–10	–83	–24	–10	–22	931	–42	–377	–169
United Kingdom	–28	–98	–21	–1	–198	–291	–126	–230	–95	–11	–41	–60	1,952	–510	–241
United States	–43	–256	–962	–67	–647 -	–1,030	–428	–1,298	–308	–35	–144	–137	–445	7,431	–1,631
Rest of the World	–27	–206	–150	–51	–346	–428	–241	–698	–131	–41	–68	–41	–468	–2,235	5,131

^¹Evaluated at the scale of world trade in 1971.

^²For the United States, the figures relate to the effects of a 10 per cent revaluation of all other currencies with respect to the U.S. dollar.

In interpreting the estimates in Tables 4 and 5, it is important to bear in mind that the calculation is based on the trade matrix for 1971 and that the growth of world trade and shifts in its geographic distribution and commodity composition since then would affect the results; that the representation of supply factors in the model is somewhat schematic; that many of the parameters of the model are either crudely estimated, guessed, or merely assumed; that it was assumed in the calculation on which Tables 4 and 5 are based that the authorities of all countries alter demand policies in such a way as to keep output unchanged in the face of expansionary or contractionary influences of the exchange rate changes; and, finally, that the trade effects calculated from this model may be expected to materialize in full only after the lapse of some period of time, say, two to three years.

An additional use of the model concerns the weighting of simultaneous changes in the exchange rates of various countries for the purpose of calculating an indicator of the extent to which the external value of any currency has moved relative to other currencies in general. Such an indicator, referred to hereafter as the effective exchange rate change, can be seen as a weighted average of exchange rate changes against the currency of a country, with the relative magnitude of their effects on its balance taken as weights. More precisely, the effective exchange rate change of a currency can be defined as the change that would induce the same alteration in its trade balance expressed in the numeraire currency as that brought about by a given realignment of all exchange rates. In terms of the present notation, the vector of changes in nominal exchange rates (T) is defined as

E \equiv B' A * x = T' A' A^{*}

where Bʹ is the transpose of the vector of changes in the trade balance B and the matrix A^* is a diagonal matrix that has for diagonal elements the reciprocals of the principal diagonal elements of the matrix A (i.e., a_kk^* = and a_kj^* = 0 if k ≠ j).¹⁰

APPENDIX A Note on the Income-Compensated Price Elasticities of Demand Used in the Multilateral Exchange Rate Model

Paul S. Armington^{^*}

In the paper, “A Theory of Demand for Products Distinguished by Place of Production,” ¹¹ it was assumed that the elasticity of substitution between the “products” of any two supplying countries competing in any geographic market for the i th “good” is a constant and is equal to the elasticity of substitution between any other pair of products competing in the same market. This assumption was introduced by specifying the quantity of the i th good demanded, say Q_i, as a CES (constant elasticity of substitution) index function of the quantities of the corresponding products demanded, Q_im. That is,

\begin{matrix} Q_{i} = {(\underset{m}{Σ} b_{i m} {Q_{i m}}^{- ρ i})}^{- \frac{1}{ρ i} .} & (20) \end{matrix}

On the assumption that buyers minimize the money cost of obtaining any given quantity of Qi (on the basis of prevailing product prices), there exists in the CES case a constant, uniform elasticity of substitution in the ith market, that is,

\begin{matrix} σ_{i} \equiv \frac{1}{1 + ρ_{i}} & (21) \end{matrix}

This substitution elasticity, along with the value shares of each supplier in the ith market, determines all the income-compensated price elasticities (both direct and cross elasticities) of demand for each product Q_ij. By differentiation of the product demand functions that are implied by the CES model, the following elasticities were derived: ¹²

\begin{matrix} \bar{η_{i j}} / i j = - (1 - S_{i j}) σ_{i} = a n y compensated d i r e c t elasticity, and & (22) \end{matrix}

\begin{matrix} {\bar{η}}_{i j} / i l = S_{i l} σ_{i} = any compensated c r o s s elasticity, where & (23) \end{matrix}

S_{i j} \equiv \frac{X_{i j}}{\underset{m}{Σ} X_{i m}} = the market share by value of  the j th product.

It was apparent that use of the CES model would severely restrict the range of variation, from one product to another, in the direct price elasticities of demand. In particular, use of the CES function in obtaining the price parameters of the multilateral exchange rate model would result in direct price elasticities of demand for exports (with respect to any one good) that could differ very little from one exporting country to another. To enhance the flexibility of this model for simulation purposes, a more general index function than the CES was sought—a linear homogeneous function that would allow relaxation of the assumption that the elasticity of substitution is the same as between any two products competing in a market.

Such a more general index function is now available: the CRESH (constant ratios of elasticities of substitution homogeneous) function, whose properties have been explored by Professor Hanoch.¹³ On the assumption that the good Qi is linear homogeneous with respect to the constituent products Q_im, the CRESH function defines the quality Q_i implicitly as follows:

\begin{matrix} \underset{m}{Σ} b_{i m} (Q_{i m} {/ Q_{i})}^{- ρ_{i m}} - 1 = 0 & (24) \end{matrix}

In contrast to the CES case (equation 20), the behavioral parameter p, governing the substitutability of the various products, carries the product subscript m. If this subscript were eliminated (that is, if this parameter were identical with respect to all products in the ith market), then (24) would reduce to (20). In other words, the CES function is a special case of the CRESH function.

Hanoch defines a parameter a_ij that measures the substitutability of the j th product for all other products in the market. Analogous to σ_i in (21), α_ij is defined:

\begin{matrix} a_{i j} \equiv \frac{1}{1 + ρ_{i j}} & (25) \end{matrix}

Derivation and analysis of the product demand functions lead Hanoch to the following formulas for the compensated price elasticities—analogous to equations (22) and (23): ¹⁴

\begin{matrix} {\begin{matrix} \bar{η} \end{matrix}}_{i j / i l} = - (1 - \frac{a_{i j} S_{i j}}{\underset{m}{Σ} a_{i m} S_{i m}}) a_{i j} = - (1 - \frac{a_{i j} X_{i j}}{\underset{m}{Σ} a_{i m} X_{i m}}) a_{i j}, & (26) \end{matrix}

and

\begin{matrix} {\bar{η}}_{i j} / i l = \frac{S_{i l} a_{i j} i l}{\underset{m}{Σ} S_{i m} a_{i m}} = \frac{a_{i l} X_{i l}}{\underset{m}{Σ} a_{i m} X_{i m}} . a_{i j} & (27) \end{matrix}

These two formulas appear in the main text above as equations (7) and (8).

The correspondence between equations (26) and (22), and between (27) and (23), becomes obvious if the complex ratios in (26) and (27) are interpreted as modified market shares. That is, the “modified share” of any product could be defined as

\begin{matrix} S_{i j}^{*} \equiv \frac{a_{i j} X_{i j}}{\underset{m}{Σ} a_{i m} X_{i m}}, & (28) \end{matrix}

whereupon (26) and (27) can be rewritten simply

\begin{matrix} {\bar{η}}_{i j / i j} = - (1 - S_{i j}^{*}) a_{i j}, & (29) \end{matrix}

and

\begin{matrix} {\begin{matrix} \bar{η} \end{matrix}}_{i j / i l} = S_{i l}^{*} a_{i j} . & (30) \end{matrix}

The idea of modifying the trade flows themselves to reflect substitutability is suggestive of an alternative method of deriving formulas (26) and (27)—that is to say, (7) and (8), respectively—without adducing the CRESH function. The CES model could be formally retained if it were assumed that only a certain proportion, C_ij of the value of the corresponding trade flow, X_ij, “effectively competed” in the market at the uniform substitution elasticity σ_i—while the remaining portion, 1–C_ij, did not compete at all (i.e., was subject to a substitution elasticity of zero). Then the compensated direct price elasticity of demand for the j th product could be expressed as a weighted average—with C_ij and 1-C_ij as weights—of the compensated direct price elasticities of demand for the competing and noncompeting elements, respectively. That is, using equation (22), and recognizing that S_ij must correspondingly be defined in terms of “competing” or “noncompeting” trade flows:

\begin{array}{l} {\begin{matrix} \bar{η} \end{matrix}}_{i j / i j} = C_{i j} [- (1 - \frac{C_{i j} X_{i j}}{\underset{m}{Σ} C_{i m} i m}) σ_{i}] + (1 - C_{i}) [- (1 - \frac{(1 - C_{i j}) X_{i j}}{\underset{m}{Σ} (1 - C_{i m}) X_{i m}}) 0] \\ = - (1 - \frac{C_{i j} X_{i j}}{\underset{m}{Σ} C_{i m} X_{i m}}) C_{i j} σ_{i} & (31) \end{array}

Introducing a_ij by definition,

\begin{matrix} a_{i j} \equiv C_{i j} σ_{i,} & (32) \end{matrix}

equation (31) reduces immediately to equation (26). By a similar analysis of the recorded flows into competing and noncompeting elements, the cross elasticity formula (27) can be obtained from equation (23).

This method of deriving the elasticity formulas now used in the exchange rate model, although less neat theoretically than the alternative of using the CRESH function, nevertheless has some appeal. Such differences as may exist among countries with respect to price elasticities of demand for exports, for example, may well reflect, in the main, differences in the commodity composition of export trade that cannot be adequately dealt with in the framework of a general equilibrium model of trade among many countries. To the extent that the delineation of “goods” for purposes of such a model is inadequate (inadequate, for example, in that one country’s exports in a given SITC category may be much more specialized with respect to buyers’ needs than are another country’s exports in the same SITC category), the remedy might be fairly interpreted as an adjustment to the trade flows themselves. The alternative interpretation—using the CRESH model—is commended by the fact that it clearly links the elements of the partial price elasticities to behavioral parameters that may be said to underline the demand equations; but this interpretation sidesteps the more fundamental problem—which can never be fully resolved—of making an optimal selection of merchandise categories for purposes of the kind of model set forth in this paper.

Mr. Artus, economist in the Special Studies Division of the Research Department, has degrees from the Faculty of Law and Economics in Paris and from the University of California at Berkeley.

Mr. Rhomberg, Assistant Director in the Research Department, is a graduate of the University of Vienna and of Yale University, and has been a member of the faculty of the University of Connecticut and of Yale University. He has contributed chapters to several books on economic subjects and articles -to economic journals.

Some of the work that forms the background of this research was previously published in Staff Papers (see Paul S. Armington, “The Geographic Pattern of Trade and the Effects of Price Changes,” Vol. XVI (1969), pp. 179–201; “Adjustment of Trade Balances: Some Experiments with a Model of Trade Among Many Countries,” Vol. XVII (1970), pp. 488–526; and “A Many-Country Model of Equilibrating Adjustments in Prices and Spending,” published as the appendix to Rudolf R. Rhomberg, “Possible Approaches to a Model of World Trade and Payments,” Vol. XVII (1970), pp. 1–27.

“Manufactures” include miscellaneous goods (SITC 9).

“Nontraded goods” refers to all goods and services that are not internationally traded—for example, houses, domestic means of transportation, administrative services, and health and education services.

Paul S. Armington, “A Theory of Demand for Products Distinguished by Place of Production,” Staff Papers, Vol. XVI (1969), pp. 159–78. The equations were derived from Armington’s article by totally differentiating his market demand functions (6) and his product demand functions (7) and employing the relation that he obtains in footnote 29, page 174, to express changes in “good” prices in terms of changes in product prices.

This particular delineation of final and intermediate goods is quite rough, reflecting the high degree of aggregation employed at this stage of the project.

For an explanation of the independence assumption, see Robert M. Solow, “The Production Function and the Theory of Capital,” The Review of Economic Studies, Vol. XXXVIII (1955–56), pp. 101–108.

See Brice Carnahan, H.A. Luther, and James O. Wilkes, Applied Numerical Methods (New York, 1969), pp. 299–301.

As a solution of the system of equations, we obtain all domestic and foreign trade flows. The foreign trade balance of country j for a particular good is equal to $\underset{l}{Σ} {\hat{X}}_{i j}^{l} - \underset{l}{Σ} {\hat{X}}_{i l}^{j}$ where $\underset{l}{Σ} {\hat{X}}_{i j}^{l}$ (country j’s total output of good i) and $\underset{l}{Σ} {\hat{X}}_{i j}^{j}$ total absorption of good i) are obtained from the solutions of the system of equations and are defined in value terms.

In the present model the U. S. dollar has been chosen as the numeraire currency. In the matrix A the column corresponding to the United States refers to an x per cent revaluation of all other currencies with respect to the U. S. dollar.

Mr. Armington, Assistant Chief of the Current Studies Division of the Research Department, is a graduate of Swarthmore College and the University of California at Berkeley.

The effective exchange rate change for the United States is defined as the uniform percentage change in all other currencies that would have the same effect on the U. S. trade balance as any realignment in question.

See Armington, “A Theory of Demand for Products Distinguished by Place of Production” (cited in footnote 4), p. 167.

Ibid., p. 175.

See Giora Hanoch, “CRESH Production Functions,” Econometrica, Vol. 39 (September 1971), pp. 695–712. In this article, Hanoch interprets the CRESH function as a production function defined on various factors; but the reinterpretation of the relevant logical steps, in terms of merchandise demand rather than factor demand, poses no difficulties.

Ibid., pp. 698 and 699.

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IMF Staff papers: Volume 20 No. 3

Author:

International Monetary Fund. Research Dept.

Volume/Issue:: Volume 1973: Issue 003

Publisher:: International Monetary Fund

ISBN:: 9781451969313

ISSN:: 1020-7635

Pages:: 310

DOI:: https://doi.org/10.5089/9781451969313.024

Keywords
Main Features of the Model1
Demand Equations
Supply Equations and Price Identities
Market Equilibrium Equations
Constraint on Total Real Production
Summary of the Model
Applications of the Model
- APPENDIX A Note on the Income-Compensated Price Elasticities of Demand Used in the Multilateral Exchange Rate Model

^{Table 5.}

(In millions of U.S. dollars)

	Austria	Belgium- Luxembourg	Canada	Denmark	France	Germany	Italy	Japan	Netherlands	Norway	Sweden	Switzerland	United Kingdom	United States	Rest of the World
Austria	368	–9	–11	–5	–26	22	–20	–26	–10	–5	–10	–20	–18	–168	–62
Belgium-Luxembourg	–20	1,458	–9	–11	–222	–314	–111	–88	–94	–17	–25	–41	–22	–386	–98
Canada	–5	–15	1,532	–4	–40	–56	–37	–108	–22	–6	–13	–13	–53	–983	–177
Denmark	–3	–7	–9	309	–18	–5	–16	–19	–5	–12	–5	–10	–23	–109	–68
France	–33	–195	–66	–35	3,188	–681	–302	–226	–149	–19	–49	–100	–163	–616	–554
Germany	–130	–336	–102	–53	–733	4,096	–374	–322	–256	–62	–135	–293	–248	–305	–747
Italy	–12	–120	–54	–11	–298	–476	2,009	–142	–96	–11	–40	–75	–98	–260	–316
Japan	–24	–99	–121	–30	–268	–421	–180	3,399	–117	–30	–60	–72	–220	–970	–787
Netherlands	–19	–64	10	–25	–220	–263	–116	–82	1,336	–16	–23	–46	–105	–251	–116
Norway	–2	–8	10	–1	–25	–39	–11	–13	–8	301	2	–2	–15	–151	–38
Sweden	–10	–23	–27	–9	–73	–54	–37	–64	–21	–26	633	–21	–32	–110	–126
Switzerland	–12	–22	–20	–6	–74	–60	–10	–83	–24	–10	–22	931	–42	–377	–169
United Kingdom	–28	–98	–21	–1	–198	–291	–126	–230	–95	–11	–41	–60	1,952	–510	–241
United States	–43	–256	–962	–67	–647 -	–1,030	–428	–1,298	–308	–35	–144	–137	–445	7,431	–1,631
Rest of the World	–27	–206	–150	–51	–346	–428	–241	–698	–131	–41	–68	–41	–468	–2,235	5,131

^¹Evaluated at the scale of world trade in 1971.

^²For the United States, the figures relate to the effects of a 10 per cent revaluation of all other currencies with respect to the U.S. dollar.

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Abstract

Main Features of the Model1

Demand Equations

Supply Equations and Price Identities

Market Equilibrium Equations

Constraint on Total Real Production

Summary of the Model

Applications of the Model

APPENDIX A Note on the Income-Compensated Price Elasticities of Demand Used in the Multilateral Exchange Rate Model

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Main Features of the Model¹