A Method for Calculating Export Supply and Import Demand Elasticities

Contributor Notes

Author’s E-Mail Address: STokarick@imf.org

Trade elasticities are often needed in applied country work for various purposes and this paper describes a method for estimating import demand and export supply elasticities withoutusing econometrics. The paper reports empirical estimates of these elasticities for a large number of low, middle, and upper income countries. One task for which trade elasticities are needed is in developing exchange rate assessments and this paper shows how the estimated elasticities can be used for this purpose.


Trade elasticities are often needed in applied country work for various purposes and this paper describes a method for estimating import demand and export supply elasticities withoutusing econometrics. The paper reports empirical estimates of these elasticities for a large number of low, middle, and upper income countries. One task for which trade elasticities are needed is in developing exchange rate assessments and this paper shows how the estimated elasticities can be used for this purpose.

I. Introduction

This paper sets out the details of a methodology that can be used to calculate export supply and import demand elasticities without using econometrics. There is a vast literature (see Stern, Francis, and Schmacher (1976) and Khan and Goldstein (1985) for surveys) that contains empirical estimates of trade elasticities, but the magnitude of the estimates varies widely, and in some instances, the signs of the estimates are contrary to theory. The methodology presented below uses a well-accepted model of international trade, together with a comprehensive dataset, to calculate elasticity values.

Export supply and import demand elasticities can be calculated by using some results from production theory. For example, it is well-known that the derivative of an economy’s GDP function with respect to an output price gives the general-equilibrium supply function, using Hotelling’s lemma. The demand function for inputs can be obtained in a similar fashion and this approach has been used by Kee et al. (2008) and others to estimate import demand elasticities. More generally, there is a large literature in international trade which uses the GDP function approach to estimate elasticities (see for example Kohli (1991)). In this approach, the demand for imports arises from the demand for imported intermediate inputs. Even if one assumes that there is a demand for imports for consumption, this approach is still valid if one assumes that imports for consumption arise in the production sector, since they have to be combined with wholesaling and retail services before they are consumed. In this sense, even imports for consumption can be thought of as an intermediate input.

II. The Model

A. Assumptions

The methodology uses a standard general equilibrium model from international trade theory, as described in Jones (1965), Dixit and Norman (1980) and Woodland (1982). The model assumes that an economy produces three goods (i) a good which is exported, denoted by (E); (ii) a good which competes with imports, denoted by (M); and (iii) a nontraded good (N). It is assumed that there is no local demand for the exportable good. Each of these three goods is produced using labor (L) which is mobile across sectors; a factor specific to each sector (K); and imported intermediate inputs (I). Since labor is assumed to be mobile across sectors, it must earn the same wage regardless of where it is employed. The return to the specific factor in each sector will, of course, differ. The price of imported intermediates is assumed to be exogenous. The output prices of all three goods are treated as parameters.

It is assumed that the output of each good is produced under constant returns to scale and zero profits. Therefore, the following conditions must hold:


where pE is the domestic price of exports, pM is the domestic price of imports (inclusive of any tariff), pN is the price of the nontraded good, aij is the amount of factor i (i= labor, capital, imported inputs) used per unit of good j, w is the wage rate, rj is the return to capital in sector j, and pI is the exogenously given price of imported intermediate inputs.

The primary factors of production—labor and capital—are assumed to be fully employed:


where Kj is the amount of capital used in sector j, L is the endowment of labor in the economy, and Xj is output of good j.

Equations (1) through (3) reflect the assumption that the price of each good must equal per unit cost. That is, per-unit labor costs, plus per-unit capital costs, plus the per-unit cost of imported inputs must equal the output price of each good. This zero-profit condition implicitly assumes perfect competition. A subsequent section discusses how imperfect competition could be introduced. Equations (4) through (6) represent the assumption that capital is sector specific, reflecting a short to medium-run focus. Equations (7) requires that the labor market clear: the amount of labor used in each sector must equal the economy-wide endowment.

B. Model Solution

Totally differentiating equations (1) through (7) and putting them in proportional change form gives:


In the above equations, θij is the share of good j’s cost accounted for by factor i, λij is the proportion of the supply of factor i used in industry j, and a “^” denotes proportional change, p^=dpp.As a result of the assumed structure, the following relationships must hold:

1.iθij=1,for each sector j;(15)
2.jλij=1,for each sector i.(16)

Each aij, the factor demands per unit of output, depends on the input prices:


Each aij can also be related to the elasticity of substitution between labor, capital, and imported inputs in each sector j. Assuming that the elasticity of substitution among all three factors is the same, the following relationships hold for each sector j, using the definition of the elasticity of substitution, σj :

σj=a^Kja^Ljw^r^j,or σj(w^r^j)=a^Kja^Lj.(17)


σj=a^Kja^Ijp^Ir^Kj,or σj (p^Ir^Kj)=a^Kja^Ij.(18)


σj=a^Lja^Ijp^Iw^,or σj(p^Iw^)=a^Lja^Ij(19)

Cost minimization requires that:


(17) through (22) can be used to solve for each a^ij, as function of the factor prices, the elasticity of substitution between labor, capital, and imported inputs in each sector, and the relevant cost shares (see Jones (1965)). Using the above relationships, the solutions for each a^ij are:

a^KE=σE(w^r^E)θLE+θIEσE (p^Ir^E)(24)

Equations (23) through (31) show how each factor demand (per unit of output) responds to changes in input prices.

Substituting equations (23)) through (31) into equations (8)) through (14), it is possible to solve for all the endogenous variables (w^, r^E, r^M, r^N, X^E, X^M X^N), as a function of the exogenous variables (L^, K^E K^M K^N, p^E, p^M, p^N, p^I).

Since the objective is to determine values for the export supply elasticity and the import demand elasticity, two relationships are of interest:

1. Export supply elasticity = X^Ep^E, which can be obtained from the equation for the output of the exportable good: X^E=F(L^,K^E,K^M,K^N,p^E,p^M,p^I).The coefficient of the term p^E gives the export supply elasticity, which is:


2. Import demand arises from the demand for imported intermediate inputs. Total demand for imported inputs in the economy (MI) is:

MI=jMIj, where MIj = aIj X j.Therefore,

M^I=λIEM^IE+λIMM^IM+λINM^IN, and M^Ij=a^Ij+X^j.

Using the solutions for a^Ij and X^j, it is possible to solve for M^I as a function of p^I.

Therefore, the import demand elasticity = M^Ip^I, which equals:


C. Sensitivity of Elasticities to Parameter Values

This section discusses how the calculated elasticities are affected by the underlying parameters: (i) the elasticity of substitution between factors (σj); (ii) the cost share of factor i in the production of good j (θij); and (iii) the proportion of the total supply of factor i used in the production of good j (λij).

  • Changes in the elasticities of substitution between factors (σj): Increases in σj will increase the magnitude of both the export supply and import demand elasticity, regardless of the sector. The reason for this is that a higher value for σj makes it easier to alter factor proportions, i.e. factor usage, in each sector. Therefore, regarding export supply, a higher σj will make it easier to increase output and will therefore increase the export supply elasticity. Regarding the demand for imported intermediate inputs, a higher value for σj will make it easier for firms to substitute between labor and imported intermediates. Firms can alter factor usage more easily; therefore the import demand elasticity will be larger.

  • Changes in the factor cost shares, (θij): Under model assumptions, capital is assumed to be fixed by sector, while labor can move freely across sectors. Similarly, firms can freely alter the amounts of imported intermediate inputs they use. Therefore, larger values for (θLj) and (θIj), and thus smaller values for (θKj), will increase the magnitude of the export supply elasticity. The larger is θKj, the more difficult it will be for firms to increase output in response to a price change because capital stocks are fixed by sector.

  • Changes in the distributive shares, (λij): The effect of changes in the distributive shares can either raise or lower the magnitudes of the elasticities.

III. Calculation of Elasticities

Equations (32) and (33) give the elasticities of interest and can be calculated for values of σj, θij, and λij. This section explains how values for these parameters can be obtained.

Data on θij, and λij for 87 countries/regions are available from the Global Trade Analysis Project (GTAP) database for the years 1997, and 2001. Data are available for 2004 for 113 countries/regions. The countries/regions in the database include a mix of developed (24) and developing countries (63). This database contains information on value added by sector, as well as its components (primary inputs), since it is based on country input-output tables. The GTAP database is described in Dimaranan and McDougal (2006).

Developed Countries: Australia, New Zealand, China, Hong Kong, Japan, Korea, Canada, United States, United Kingdom, Austria, Belgium, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, Netherlands, Portugal, Spain, Sweden, Switzerland.

Developing Countries: Rest of countries in Oceania, Taiwan, Rest of East Asia, Indonesia, Malaysia, Philippines, Singapore, Thailand, Vietnam, Rest of Southeast Asia, Bangladesh, India, Sri Lanka, Rest of South Asia, Mexico, Rest of North America, Columbia, Peru, Venezuela, Andean Pact countries, Argentina, Brazil, Chile, Uruguay, Rest of South America, Central America, Countries of the Caribbean, Countries of the free trade area of the Americas, Rest of European Free Trade Area, Rest of Europe, Albania, Bulgaria, Croatia, Cyprus, Czech Republic, Hungary, Malta, Poland, Romania, Slovak Republic, Slovenia, Estonia, Latvia, Lithuania, Russia, Rest of former Soviet Union, Turkey, Rest of the Middle East, Morocco, Tunisia, Rest of North Africa, Botswana, South Africa, Rest of South African Customs Union, Malawi, Mozambique, Tanzania, Zambia, Zimbabwe, Madagascar, Uganda,

Rest of South African Development Community, Rest of sub-Saharan Africa.

To apply the methodology described above, the following strategy is adopted:

  • The GTAP database contains data on labor, capital, and imported inputs for 57 sectors. Starting from the full database, the 57 sectors in each country were aggregated into 3 sectors per country (exportables, importables, and nontraded) using sectoral data on trade flows. If exports or imports from a sector was 10 percent of value added or less, the sector was classified as nontradeable. A sector was considered exportable if exports exceeded imports and exports exceeded 10 percent of value added. A sector was importable otherwise.

  • The GTAP database contains a value for the elasticity of substitution among factors used in the 57 sectors for each country. The values are taken from various econometric studies. It assumes that these elasticities are the same for all 87 countries and regions. The elasticities are aggregated from the 57 sectors into 3, using data on value-added shares in each sector.

  • Once the sectors were classified into 3 categories, λij and θij were calculated for each country. Then, using the elasticities of substitution, equations (32) and (33) were used to calculate the elasticities.

Import demand and export supply elasticities are calculated using the procedure described above. There are several “types” of elasticities:

  • Both a “short run” and a “long run” import demand and export supply elasticity were calculated. The short-run elasticities correspond to a set of short-run elasticities of substitution among inputs (sigmas), while the long-run elasticities correspond to a long-run set of sigmas. In general, the long-run sigmas are higher in magnitude, compared to the short-run sigmas.

  • Import demand and export supply elasticities are calculated for two sets of assumptions: (i) elasticities are computed with respect to their own price (the standard definition of elasticities); and (ii) including general equilibrium effects, where the latter takes into account changes in both the own price and the price of other traded goods. For example, the own export supply elasticity measures how export supply changes as the price of exports changes, holding all other prices constant. A devaluation for example increases the prices of imported intermediate inputs, which will reduce the export supply response to the extent that exports use imported inputs. The mathematical formulas for the “general equilibrium” elasticities are shown in the appendix.

Elasticity values are reported in the following tables:

Table 1: Import demand elasticities. This table reports estimated import demand elasticities from various studies, as well as from the method described above. The study labelled “World Bank” refers to a study conducted by Kee, Nicita, and Olarreaga (2008). In that study, the authors estimated import demand elasticities, using the GDP function approach, which is similar to the one used in this paper. The study labelled “Senhadji” refers to a study conducted by Senhadji (1998), who estimated import demand elasticities using what might be termed the “traditional” approach: regressing imports on relative prices and real income. GTAP 2001 refers to the data from version 6 of the GTAP database (data for the year 2001), while GTAP 2004 refers to data from version 7 of the GTAP database (data for the year 2004). For each of these two datasets, a set of “short-run”(SR) and “long-run” (LR) elasticities are reported. The short run is defined to be about six months to a year at the most, while long run refers to as many as three years or more. As well, a set of both short-run and long-run elasticities are reported that take into account general equilibrium effects. The columns on the far right of the table show the average elasticities calculated for 2001 and 2004.

Table 1.

Import Demand Elasticities

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Note1: Income group: Economies are divided according to 2007 GNI per capita, calculated using the World BankAtlas method. The groups are: low income, $935 or less; lower middle income, $936 - $3,705; upper middle income, $3,706 - $11,455; and high income, $11,456 or more. Source: World BankNote: The CGER import-demand elasticity (not shown) is calculated keeping import elasticities to -0.92 for all countries except China (Im: -0.67), Malaysia (Im: -1) and Colombia (Im: -1)