Substitution elasticities and country size

The analysis of the concept of elasticity of substitution. The model that explains the dependence of the elasticity of substitution on the size of the country for which the elasticity is calculated. Use regression analysis to test this relationship.

Рубрика Экономика и экономическая теория
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Язык английский
Дата добавления 23.09.2018
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Substitution elasticities and country size


elasticity substitution regression

To begin with, it would be useful to specify the object of this research, which is the elasticity of substitution. In this study the concept of the substitution elasticity would be discussed and analyzed. But the main focus will be made on the relationship between elasticity of substitution and a country size.

In order to explain the idea of the research and the main problem, the term of elasticity of substitution should be described. Therefore, we should initially explain the meaning behind elasticity itself. In economics, elasticity shows how one economic variable changes with respect to the change in another variable, where the former one is dependent on the latter. For convenience these changes are usually presented in a percentage form so that variables which may be expressed in different units still have equal changes. The easiest but still one of the most important concepts of elasticity used in economics is price elasticity of demand. It can be measured as the percentage change in quantity demanded for a specific good in response to the percentage change in the price for that good (M. Parkin, M. Powell and K. Matthews, 2002). Elasticity of substitution is a more complex concept firstly introduced by John Hicks in 1932 and it uses a function of not one, but two parameters. Throughout the century many economists introduced their own vision of the elasticity of substitution, which could be derived for two factors and two prices, or two factors and one price or one factor and one price (D. Stern, 2009). But usually, the production or utility function with two production factors or two goods as their parameters is considered. For our analysis it is convenient to show the definition of the substitution elasticity on a formula for two goods used in the Armington model (Armington, 1969):


where and are demands for a good produced in countries and consumed in a country , is the substitution elasticity coefficient. From the formula it follows that elasticity of substitution is the percentage change in a ratio of demands for two goods in relation to the percentage change in the respective prices of these goods. To provide intuition over the concept it is reasonable to show the dependence among the relative expenditure on and which we denote as and the substitution elasticity. Relative expenditure can be expressed in a formula:


The dependence of relative consumption on the substitution elasticity can be seen from the equation:


For , an increase in relative price is dominated by the diminution of the relative quantity demanded which leads to the decrease in relative expenditure. Thus, the goods are considered to be the gross substitutes. Moreover, it can be said that the higher is coefficient for two goods, the more substitutable one of them is for another, meaning that one would more readily choose one good over another. Following the same logic when , goods are considered gross compliments. And if , goods are independent.

Substitution elasticity has been used and calculated in many studies since the introduction of the concept. There were many researches dedicated to the complex analysis and estimation of substitution elasticities, two of the most recent ones were prepared by Christian Broda, Joshua Greenfield and David Weinstein, who modeled how trade enhances growth by creating and importing new varieties of goods. The methodology used for calculation of elasticities coefficient relies on the methods developed in Feenstra (1994). These studies will be mentioned in more detail later in the literature review section. In practice, elasticity is used to forecast trade flows between countries. One of the first international trade models was derived by Paul Armington in 1969, while nowadays the are many more international trade models such as Ricardo-Roy (RR) model (A. Costinot, J. Voge, 2015) which is a new vision of the Ricardian model derived almost 200 years ago.

What is important, the methodology of the estimation of coefficients is based on an implicit assumption that a country for which coefficients are calculated are considered a dot, i.e. all consumers and trade are concentrated in one place in a country. However, if we consider a model where estimation of substitution elasticities takes into account the size of the country, it would become evident that they are higher than those calculated by Broda and Weinstein. The logic behind goes as follows: in a large country which has at least two regions, substitution elasticities may be calculated for these particular regions instead of the aggregation on the country level. The residents of these regions will choose the imported goods based on their preferences. If in any region there is high demand for one particular good and low demand for any other good from a group of similar items, elasticity of substitution is high and vice versa. But if these regions are not separated, on the country level the demand for both goods is high leading to the lower estimates of the substitution elasticity, since people prefer different items. From the following it can be concluded that the larger is the country, the more separate regions' demands will be collapsed into the one, and the higher would be the difference in the coefficients estimated based on the assumption of the “dot-country” and on the assumption of the separate regions.

Turning back to the usage of the elasticity coefficient in the real world the relevance of that study becomes evident. Since there are many models which use elasticity coefficient which up to this point have been calculated without taking the country sizes into account, it could be the case that the trade was modeled with an error corresponding to the incorrectness of the elasticity coefficient.

Thus, the hypothesis of our research is may be formulated as follows: there is a negative correlation between elasticities of substitution and country sizes.

And the goal of this paper is to check whether there is a relationship between country sizes and the corresponding elasticities of substitution.

1. Literature review

As we focus the research on the elasticities of substitution for imported goods, it is important to review prior publications dedicated to their calculation and analysis. The two main studies which will be reviewed in detail are performed by C. Broda and D. Weinstein (2004) and C. Broda, J. Greenfield and D. Weinstein (2006). In the former one economists investigate how the import of new varieties influences the national welfare of the United States, while in the latter economists study the impact of new import varieties for approximately 4000 markets in 73 countries. It was found that in the USA gains from varieties in imports account for approximately 2.8 percent growth of GDP. The results of the latter study showed that in the typical country, new imported varieties contribute to 15 percent of its productivity growth, while in developing countries these effects are larger compared to the developed countries. There the median impact of new imported varieties equals a quarter of national productivity growth.

During the research mentioned economists had to calculate substitution elasticities for the imported goods. As both studies are similar in their logic only the second one will me described below.

The whole study could be separated in two stages. The first stage is focused on the “level effect,” i.e. the productivity gain arising from the import of new and better varieties, while the second stage focuses on the “growth effect”, i.e. the impact that the global rise in imported varieties in the last 10 years has on future output growth. Thereby, the first stage was dedicated to gathering data on imports and analyzing it. After the analysis of 6-digit bilateral flows (from the COMTRADE database) over the period 1994-2003 it was found that in developing countries almost all growth in imports to GDP ratio was brought with new import varieties. And in developed countries the import of new varieties accounted for a half of the growth in imports to GDP ratio. Thus, it became evident that new varieties indeed have an influence on the productivity growth all over the world. In the next step gains from varieties were calculated by applying constant elasticity of substitution specification, which enabled economists to keep track of only two factors. Those were elasticities of substitution between different varieties of a good and changes in the shares of expenditures among the disappearing, remaining and new goods. To calculate substitution elasticities the method described in Feenstra (1994) was implemented, namely, C. Broda, J. Greenfield and D. Weinstein marked each variable with a country index i so that elasticities were estimated for each good and importing country. Then they estimated the system of import demand and export supply equations:



where (i.e., differencing across two different varieties of a given i-g pair), i denotes the importer country, g a 4-digit good, and v (for variety) a particular variety of good g, and are taste or quality shocks to variety v of good g in country i and are shocks to the supply of the same variety.

After certain manipulations with the formulas of import supply and demand it is possible to calculate the elasticities by estimating coefficients from the regression and then substituting them the formula for elasticities of the varieties of imported goods.

Elasticities were then used in the equation of GDP growth which can be decomposed into change in productivity, change in number of goods and production factors. From the formula of the change in productivity it follows that the less substitutable are new varieties with existing ones the larger is the impact of an increase in the number of intermediate or capital goods. Thus, estimation of elasticity of substitution was needed to capture change in the GDP in each of the 73 countries.

Besides the global aim of the study there were found many useful results anв conclusions about the elasticities. While assessing the reasonability of obtained substitution elasticities estimates, economists check for stability by examining whether they fall as the number of varieties rising. To do so elasticities were calculated again using the same data but divided into two time periods. Even though there was a 40 percent increase in the varieties over the period, the median elasticity of substitution was stayed the same (3,6). Another find states that elasticities seem to be unaffected by introduction of the new varieties to the market.

Another interesting feature from the study was that elasticities of substitution were referred to as the import demand elasticities, but the reason why they were considered the same was not discussed. However, the same assumption was later used in the study of the import demand elasticities by Alberto Felettigh and Stefano Federico (2010). The state that the price elasticity on imported good for the country A is the same as the substitution elasticity between varieties of that good in the country A. In order to compare import demand elasticities between the markets of the destination for Italian exports and the destination markets of the other main euro area countries' exports Felettigh and Federico use estimates provided by Broda and Weinstein, but calculate 4 of them related to Italy, France, Germany and Spain using 3 more methods. This is due to the fact that already estimated elasticities vary from 1 to 16808. The extremely high values caused a problem in estimation of weighted export elasticity (weighted average of the import demand in mentioned countries. Also, according to the Mohler (2009) there is no big differences in economic sense between elasticities from the range of 20-30 and the range of 100-1000. Thus, values larger 30 may be simply cut down to 30. Using different methods, such as the one described in Broda and Weinstein but with more relevant data and the method described in Feensta (1994). They conclude that Italy has a lower weighted export elasticity than 3 of its biggest trade partners.

Another study was focused on the understanding of how countries specialize their trade. The diversification of specialization is two-dimensional so that countries decide which goods to produce and how many and to which countries to export (W. Steingress, 2015). Thу paper uses Ricardian model of comparative advantage in order to determine the factors which constitute the specialization preferences of the countries. It was found that the key determinants are substitution elasticity, trade costs and the degree of absolute and comparative advantage. The elasticity is used to estimate the production of the composite good which can be later used either for production of the final goods or another intermediate good. An interesting point is that the elasticity coefficient is not used here in the equation for the sensitivity of trade between two countries, which contradicts to the many other models, such as Armington. This is so due to the assumption that elasticities of substitution are the same within countries so that they do not change relative prices for goods across countries. The main use of the elasticity is, however, in the key equation of the model - the intensive Theil index of imports for a certain country. If the elasticity coefficient us large it helps to enlarge specialization allowing producers in the composite intermediate good sector to substitute expansive and cheap products and focus expenditure in these sectors. To sum up, the higher is the elasticity of substitution the more substitutable are goods which allows countries to focus their expenditure on the low-price products.

In conclusion to the literature review section, it should be pointed out that even for a particular area of studies of the international trade, the elasticity of substitution coefficient is used in completely different models, which indicates its importance for the economy in the modern world. Summarizing, we can proceed to the formation of a theoretical model for this study.

2. The model

2.1 Theoretical model

As it is evident from the literature review, substitution elasticity has been a topic of many studies for almost a century. The main use of that type of elasticities is both in the field of production and manufacturing where the substitutability of the production factors is evaluated (R.K. Anderson, J.R. Moroney, 1993) and in the models of international trade. But we will focus on the import demand elasticities or elasticities of substitution between imported goods.

For our research to achieve the goal, which is to check if there is a negative correlation between sizes of countries and elasticities of substitutions for imported goods we use the most well-known and wide-used Armington model of trade (Armington, 1969), which combines varieties of goods differentiated by the country of their origin and constant elasticity of substitution preferences. The model has several assumptions and specifications which will be described below.

Firstly, goods produced in one country are assumed absolutely homogeneous, while those originated from other parts of the world are imperfect substitutes. Consumers or buyers differentiate goods not only by the type (i.e. fruits and bread), but also by the supplier's area of residence (i.e. fruits from Russia and Georgia). Armington names a variety of good from the specific country a product. This allows similar products to be traded in markets of each country.

Secondly, Armington model uses an assumption of independence between consumers preferences over the products of some particular category and consumption of the products from another category. Thus, the demand function for a separate category of products or, simply, for a good may be distinguished.

The third assumption of the model is of high importance. Elasticities of substitution for each pair of products or varieties of the same good imported from different countries is the same in the market where they are consumed. This assumption allows to present elasticities of substitution for particular category of goods in each country as a single number, without specifying each country-pair.

Based on all of these assumptions the model is formulated. In order to obtain the demand function for the imported goods from each country two maximization problems for a representative consumer should be solved. On the first step consumer solves the problem of allocating costs to the separate categories of goods. Thus, the utility function for representative consumer is of a kind:


and it should be maximized subject to the budget constrained:


where is a composite index of consumption of goods from a group , is a composite index of prices for goods from a group or the average price in the market j, represents expenses of a representative consumer in the country for consumption of goods in that country.

The solution of a problem would be .

During the second step the representative consumer optimizes its purchases of products for each category of goods, according to the share of money for each category found in the first step. A representative consumer now derives his utility from consumption of the products imported from all over the world and it may be represented by a function with a constant elasticity of substitution (CES):


where is the substitution elasticity coefficient between products of the same category of a good, is an exogeneous preference parameter (where ), is the quantity of a good k imported from the country i to the country j, and S is the total number of trade partners ().

The (3) equation should be maximized subject to the budget constraint:


After solving the maximization problem, the demand equation for a particular good as a function of prices is derived:


where can be seen that consumption of the good (demand) transferred from country i to j; is the total demand for the good k derived from the first optimization problem, is an exogenous preference parameter, is j's composite price index for the good k; is the price of the good k.

The import trade flow, thus, is a function depending on the value of the substitution elasticity in the country j for a particular good, as it is considered to be constant within the varieties of the same good in one market. This implies that knowing all other variables, such as the import demand itself and price for the imported goods, the elasticity coefficient may be calculated.

However, Armington model does not account for the size of the countries. On the contrary, the methodology of the estimation of coefficients is based on an implicit assumption that a country for which coefficients are calculated is considered a dot, i.e. all consumers and trade are concentrated in one place in a country. And all studies which were reviewed during out analysis prove that elasticities of substitution were not associated with the country size in any way. In our hypothesis we assume that the country size is negatively correlated with the value of the substitution elasticity coefficient. To explain that let us introduce two models, where the first one takes into account the size parameter of the country and the second one does not, but which both calculate the coefficient using the same methodology.

For the first model where estimation of substitution elasticities takes into account the size of the country let us consider a country A which may be divided into two regions 1 and 2 (picture 1). The country A has two trade partners B and C, which export a specific good k to the both regions of A. Country be is situated near the region 1 while C is close to the region 2 of the country A. Knowing the trade imports for both regions, elasticities of substitution may be calculated using the equation (5). If in region 1 the demand for a good produced in B is low, but high for the same good produced in C then the elasticity of substitution is relatively high. As consumers prefer only one of the goods and thus, they are highly substitutable, and vice versa. This logic follows from the definition of the substitution elasticity and equation (1). Same can be said for the region 2: if the demand for both products and is high or at least the difference between them is not dramatic, then the elasticity of substitution would be low, as even with the change in prices for the goods, their relative consumption would not change.

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Now let us move on to the second model with a basic assumption that all trade in a country is concentrated in one place. Thus, the only available demand for import from countries B and C is the sum of their regional demands and . From this information the elasticity of substitution may be calculated, however, it might be the case that the real elasticity is high and buyers from the region 1 consumes only the product from B, while consumers from the region 2 only demand the product from C. But from and it can only be seen that both varieties are consumed in the country A. Thus, the estimated elasticity coefficient is relatively low. The situation is depicted on the picture 2.

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Thus, if the country size is not taken into account and the country is considered to be a dot, rather than a composite of several regions, then for big countries the estimated coefficient of substitution elasticity is lower than its value calculated for the distinct region. From the following it can be concluded that the larger is the country, the higher would be the difference in the coefficients estimated in two models.

From this the theorem of inequality between elasticities of substitution may be written:


Where is the elasticity coefficient obtained without consideration of the country size and is the “true” elasticity of substitution coefficient estimated using a new model which takes into account the country size. The obvious implication of the difference between the “true” and the other coefficient not taking into account the size of a country is that trade models give inaccurate forecasts of the trade flows.

The simplest way to create the new model is to use (5) equation in estimation of the substitution elasticity for each region in a country A and then take an average of them considering a share from a total demand:


where is a region from the total number of regions M in a country j, means the elasticity of substitution in a region m, is the consumption of the good k in the region m and is the total consumption of the good k in j.

The suggested model does not necessarily obtain better estimates of , it is only one of the possible variants. Our research, however, on this stage does not focus on the new model estimating elasticity of substitution.

2.2 Empirical model and analysis

To prove the hypothesis and check if there is the relationship between the country size and the elasticity of substitution, elasticities for regions of the countries may be calculated. Unfortunately, it is not possible to obtain data on imports for regions in the countries. That is why the regression analysis is considered as a main tool for achieving the goal of this study. The regression is given in the following way:


where is a substitution elasticity for a category of goods for particular country (since is assumed to be the same for each country pair for a single good, there is only one observation for the country instead of the numbers of elasticities for each product variety), is a constant, Landarea is the size of country and D is a dummy variable for each category of the goods to capture fixed effects.

Other variables were not included in order to estimate the precise effect that the land area has on the elasticity coefficient. Moreover, some parameters such as GDP per capita were found to be insignificant (Broda, Weinstein, 2004).

Data gathering

For our estimations we use elasticities calculated by Christian Broda, Joshua Greenfield and David Weinstein (2006) for 73 countries. The data is represented in the table where one column is names of countries, another one is the 3-digit HS code, and the third one corresponds to elasticities. 3-digit HS code is based on the international Harmonized Commodity Description and Coding System where, where 6-digit code means a single variety, 4-digit HS code is an aggregation of some level of the 6-digit codes-varieties and the 3-digit HS code is a certain aggregation of the low-level codes. For example, 10 is cereals, 521 is industrial transport equipment and other. Overall, there are 11295 values for 171 goods categories. During the analysis it was found that there were several extremely high values of the elasticity such as 16808,04 or 1560, 588 and almost one thousand high-valued elasticities which if combined yield the mean value of 81. This was still a very large number in contract to the mean of the rest data (3,84). Thus, using Mohler (2009) proposition of economic indifference between any values higher than 30, all coefficients larger than 15 were eliminated, so that there became 10391 observations which still was a great amount of data. The main reason for elimination of the large coefficients was to obtain more desirable results for the study. For the same reason these values were eliminated, but not cut to the certain level, in order not to create any artificial changes to the sample (while there is still 10000 observations).

Data for the country size was obtained from the World bank website for countries in 2017. Only Venezuela did not have recent information about the country size, so that the value was taken from the previous year. We can take any reasonably appropriate year using the knowledge that elasticities do not change over time (Broda, Greenfield, Weinstein, 2006).

Estimation and analysis

Using Stata software, the regression was calculated for 3 data sets:

1) When all values of were deleted from the sample (pic. 4)

2) When all values of were deleted from the sample (pic. 5)

3) When all values of were deleted from the sample (pic. 6)

Running regressions for 3 data sets showed even though was small each time, it was increasing with the more aggressive reduction of the high values from the data set from 0,0448 to 0,068 and 0,0689. Thus, there is a possibility that the determination coefficient would be higher for the smaller. On the other hand, it is reasonable that the determination coefficient is low, as the country size definitely is not the main factor for the value of sigma. Overall, might be low due to the broad distributions of elasticities for each country, where the values might vary from 1 to 200 and higher, and due to the heterogeneity of the countries and categories of goods.

The P-Values, however, are small enough to reject the hypothesis that country size term is insignificant at a 10% significance level in the first case, at 5% significance level in the second case and at any reasonable level in the third case. Thus, the country size coefficient is indeed significant and there is a dependence of the substitution elasticities on the country size.

Moreover, for each of 3 data sets, the coefficient of Landareasqkm term is negative. Also, the fixed effects for most of the categories of goods are also negative for the datasets upper bounded by 10 and 40 (not for 15, as the value of a constant is much smaller in compared to the other datasets models).

Lastly, the linear graph for the data set upper bounded by 10 was made (Picture 7). Here, X-axis is the country size and Y-axis is the value of elasticity. Blue dots represent elasticities by categories of goods for each country. The trend line (in orange color) was added to the graph to show that overall, there is a hint of negative relationship between the country sizes and elasticities of substitution.

The next step was to run a simple regression for a sample of categories of goods:


Here the upper values were not limited, however, for the chosen categories of goods, the highest sigma was only 131. From there it is obvious that no consistent conclusions can be made: is very low for all five categories of goods (less than 1%), while the coefficient for variable corresponding to the country size changes from positive to negative for different varieties. It can be seen from the graphs (pictures 8). It is negative for the goods of a code 80, 150, 550, 600, 740 and 903, but negative for 290 and 480. P-Values are too high for all categories, but for 600 it is around 90%, probably, because there are 2 outliers, as all other sigmas are smaller than 15.

To conclude, simple regression analysis cannot verify that there is a strong dependence between the values of substitution elasticity and country size. However, the negative relationship is observable both as a trend in the graphs and from calculated values of coefficients for land area. For the first three regressions country size proved to be a significant determinant, while in the regressions for each category, the significance was not verified. Thus, it may be said that our hypothesis is partially confirmed. For the topic to be fully covered, further analysis should be performed in a form of the regression analyses using more appropriate data for elasticities and the sample of countries or the elasticities for different regions of countries may be estimated.


To conclude, the theoretical model of the dependency between country sizes and the substitution elasticities coefficients was formulated on the existing Armington model of trade. The regression analysis showed that the hypothesis is partially confirmed while there is a space for further analysis. We would suggest running regressions using another data and to calculate elasticities for different countries' regions. The model for calculation of the “true” elasticity of substitution should also be developed.

List of References

1. Anderson, J. (1979). A Theoretical Foundation for the Gravity Equation. American Economic Review, 69(1), 106-116.

2. Ando, M., Estevadeordal, A., & Volpe Martincus, C. (2009). Complements or Substitutes? Preferential and Multilateral Trade Liberalization at the Sectoral Level (Working Papers on Regional Economic Integration No. 39). Asian Development Bank. Retrieved from

3. Armington, P. S. (1969). A Theory of Demand for Products Distinguished by Place of Production (Une thйorie de la demande de produits diffйrenciйs d'aprиs leur origine) (Una teorнa de la demanda de productos distinguiйndolos segъn el lugar de producciуn). Staff Papers (International Monetary Fund), 16(1), 159-178.

4. Baumgдrtner, S., Drupp, M. A., & Quaas, M. (2015). Subsistence, Substitutability and Sustainability in Consumption. Environmental and Resource Economics.

5. Bergstrand, J. (1989). The Generalized Gravity Equation, Monopolistic Competition, and the Factor-Proportions Theory in International Trade. The Review of Economics and Statistics, 71(1), 143-153.

6. Broda, C., Greenfield, J., & Weinstein, D. (2006). From Groundnuts to Globalization: A Structural Estimate of Trade and Growth (Working Paper No. 12512). National Bureau of Economic Research. Retrieved from

7. Broda, C., & Weinstein, D. E. (2004). Globalization and the Gains from Variety (Working Paper No. 10314). National Bureau of Economic Research. Retrieved from

8. Constant elasticity of substitution. (2018, April 20). In Wikipedia. Retrieved from

9. Costinot, A., & Vogel, J. (2015). Beyond Ricardo: Assignment Models in International Trade. Annual Review of Economics, 7(1), 31-62.

10. Feenstra, R. C. (1994). New Product Varieties and the Measurement of International Prices. American Economic Review, 84(1), 157-177.

11. Feenstra, R. C., Luck, P. A., Obstfeld, M., & Russ, K. N. (2014). In Search of the Armington Elasticity (Working Paper No. 20063). National Bureau of Economic Research. Retrieved from

12. Felettigh, A., & Federico, S. (2010). Measuring the price elasticity of import demand in the destination markets of Italian export (Temi di discussione (Economic working papers) No. 776). Bank of Italy, Economic Research and International Relations Area. Retrieved from

13. K. Anderson, R., & R. Moroney, J. (1993). Morishima elasticities of substitution with nested production functions. Economics Letters, 42, 159-166.

14. Kee, H. L., Nicita, A., & Olarreaga, M. (2008). Import Demand Elasticities and Trade Distortions. The Review of Economics and Statistics, 90(4), 666-682.

15. Krugman, P. R. (1979). Increasing returns, monopolistic competition, and international trade. Journal of International Economics, 9(4), 469-479.

16. McDaniel, C., & J. Balistreri, E. (2003). A Discussion on Armington Trade Substitution Elasticities. SSRN Electronic Journal.

17. microeconomics-perloff-jeffrey.pdf. (n.d.). Retrieved from

18. Nadezhda, I. (2005). Estimation of own- and cross-price elasticities of disaggregated imported and domestic goods in Russia (EERC Working Paper Series No. 05-13e). EERC Research Network, Russia and CIS. Retrieved from

19. Saito, M. (2004). Armington Elasticities in Intermediate Inputs Trade; A Problem in Using Multilateral Trade Data (IMF Working Papers No. 04/22). International Monetary Fund. Retrieved from

20. Steingress, W. (2015). Specialization Patterns in International Trade (Working papers No. 542). Banque de France. Retrieved from

21. Stern, D. (2009). Elasticities of substitution and complementarity. Journal of Productivity Analysis, 36, 79-89.

22. Григорьевич, П. М. (2012). Оценка эластичностей замещения между и импортной и отечественной продукцией. Региональные проблемы преобразования экономики, (4). Retrieved from

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