Bilateral multifactor CES general equilibrium with state-replicating Armington elasticities

We measure elasticity of substitution between foreign and domestic commodities by two-point calibration, such that the Armington aggregator can replicate the two temporally distant observations of market shares and prices. Along with the sectoral multifactor CES elasticities which we estimate by regression using a set of disaggregated linked input–output observations, we integrate domestic production of two countries, namely, Japan and the Republic of Korea, with bilateral trade models and construct a bilateral general equilibrium model. Finally, we make an assessment of a tariff elimination scheme between the two countries.


Introduction
Specifically, we evaluate the compound price of each factor input w C i , in terms of domestic and foreign factor input prices (w D i , w F i ) that are observable, via CES aggregation whose macroelasticity replicates the observed domestic-foreign market shares. We then calibrate the microelasticity using w F i and the partner country's domestic price w D0 i in order that the observed partner-ROW market shares are replicated. In this way and based upon 2000-2005 linked input-output tables for Japan and Korea, we construct a multisectoral (395 for Japan and 350 for Korea) general equilibrium model with endogenized bilateral trades, in contrast to the previous studies with limited variety of industrial sectors.
The remainder of the paper is organized as follows. In the next section, we introduce the basics of the two-point calibration of the CES elasticity parameters, i.e., macro and microArmington elasticities, and the multifactor CES elasticity estimation by regression. In Section 3, we apply these protocols using linked inputoutput tables for Japan and for Korea and the UN Comtrade database. In Sect. 4, we integrate domestic and trade models to construct a bilateral general equilibrium model for the welfare analysis of trade liberalization. Section 5 provides concluding remarks.

The model 2.1 Macroaggregator
Assume that foreign and domestic commodities are, to a certain extent, substitutes with a constant elasticity of substitution (CES). Then, a composite product price in a country (whose index i is omitted) can be evaluated by a CES aggregator of foreign and domestic commodity prices as follows: Fig. 1 Nested structure of macro and microArmington elasticities. A foreign commodity price is given by aggregating the partner country's and the rest of the world's (ROW's) commodity prices. The compound commodity price is given by aggregating the domestic and foreign commodities' prices. Finally, the domestic price is given by a multifactor CES aggregator (i.e., unit cost function) Asia-Pac J Reg Sci (2018) 2:431-452 433 where w C is the composite price of a commodity in the concerned country, w F is the price of the imported foreign product (including tariff), and w D is the price of the domestic commodity. Here, the share parameter a 2 ð0; 1Þ and the macroArmington elasticity e are subject to estimation. According to Shephard's lemma, we can obtain the cost share by taking derivatives as follows: where s D and s F denote the market shares of the domestic and imported commodities, respectively. One may verify that s D þ s F ¼ 1 by taking (1) (2), the identities must hold at the reference state and the identities must hold at the current state. By virtue of (3) and (4), e can be solved (two-state calibrated) as follows: where D is the difference operator, i.e., current value minus reference value. 1 In addition, we may solve for the share parameter a as follows: In this way, we obtain the macroArmington aggregator (1) that replicates both the reference and current states specified by (3) and (4), respectively. We also note that the compound price w C will be evaluated assuming (1) and thus it is shown in brackets in Fig. 1.

Microaggregator
Let us indicate the partner country by P and the ROW by R. Assume that the aggregated foreign import product price w F (whose commodity index i is omitted) can be expressed as a CES aggregator function of price of commodity imported from the partner country w P and that from the ROW w R , as follows: where b 2 ð0; 1Þ is the share parameter and g is the microArmington elasticity, both of which are subject to estimation. Note that w R must be evaluated assuming (7) with the calibrated parameters, while w F and w P are statistically observable. 2 Hence, the parameters are calibrated according to the two-state observation of the partner country's market share within the commodity's fraction of imports, i.e., ðs P The following identities must hold at the reference state, according to Shephard's lemma applied to (7): Likewise, the following identities must hold at the current state: By virtue of (8 left) and (9 left), g can be solved (two-state calibrated) as follows: In addition, we may solve for b as follows: Hence, we have the microArmington aggregator (7) that replicates both the reference and current states. Also note that w R will be evaluated by (7): The in-bound price of the product imported from the partner country w P is evaluated by the domestic price at the partner country w D0 and the barrier factor l under the currency exchange factor m. The barrier factor l captures various factors such as insurance, freight, miscellaneous tax, and tariff factors. For further convenience, we may decompose l into the tariff factor 1 þ s, where s represents the tariff rate, and other factors which we denote by q, as follows: As we monitor m and l for the two states, w P can be evaluated accordingly, that is

Multifactor CES aggregator
Production of industry j (index omitted) is assumed to be carried out under a constant returns multifactor CES (constant elasticity of substitution) whose unit cost function can be described in the following form: where k i 2 ð0; 1Þ and r are the share parameter for the ith input and the multifactor CES elasticity of substitution, respectively, while t denotes the productivity level. While w D is observable, w C i depends on (1) via w D and w F , which are statistically observable, and the calibrated parameters a and e.
We note below that r and t can be estimated by regression, for each industrial sector. The cost share of the ith input s i may be represented according to Shephard's lemma by differentiating (14) as follows: By taking the logarithm of both sides, we have Thus, the difference in (16) between two temporally distant states, i.e., reference and current, is given by the following formula: Note, if r and t are estimated by the slope and the intercept of (17), k i will be determined by (15).

Armington elasticities
A set of linked input-output tables includes sectoral transactions in both nominal and real terms. Hence, such a set of tables provides temporally distant observations of cost shares and prices (as indexes) for all factor inputs (and outputs). In this study, we use the 1995-2000-2005 linked input-output tables for both Japan (MIAC 2011) and Korea (BOK 2009), and we chose the year 2000 for reference and 2005 for the current states. To calibrate macroelasticity e j on two-state observations using (5), we standardize all prices at the current state and evaluate the reference state prices by the current-standardized price index (the inflator), which we denote by q. Specifically, we use the following terms for calibrating the parameters: The parameters of the macroaggregator are thus evaluated by the following formulae, based on (5) and (6): To evaluate microelasticities, we need reference and current observations of the partner country and the ROW market shares ðs P \ ; s P [ Þ within the foreign factor inputs. To this end, we use the 6-digit HS trade data of the UN Comtrade database (Comtrade 2017), spanning 6376 goods, converted into the linked input-output sector classification 3 to obtain the market share of the partner country with respect to that of the ROW in two periods (2000 and 2005). Furthermore, to calibrate the parameters of the microaggregators, we need to specify the in-bound prices of the partner country's commodities as noted in (13). That is, we need the inflator q P , while q F is observable in the linked input-output tables: Therefore, we use the exchange rate that properly scales the two countries' price indexes. Specifically, (13) must be replaced by the following identities: (18), we may use current-standardized index numbers for reference currency exchange factor m \ as well as for the reference barrier factor l \ . In this way, we evaluate the in-bound partner country's commodity inflator q P by way of an inflator of the commodity produced inside the partner country q D0 . Then, according to (10) and (11), the parameters of microaggregator are determined by the following equations: In Fig. 2, we display the two-point calibrated macro and microArmington elasticities of inputs (commodities) whose imports from the partner country are present, for Japan (left) and for Korea (right). We use log-absolute values (with base of 10) for both axes, since most of the calibrated elasticities are very large in absolute magnitude. Note that inputs with positive log-absolute elasticity (at the north or at the east of the ridgeline) are entitled as elastic, since input substitution is responsive with respect to price changes. Conversely, inputs with negative logabsolute elasticity (at the south or at the west of the ridgeline) are entitled as inelastic. Concerning the regionality of inputs in domestic production, inelastic substitution implies that foreign-made or partner-made commodities are difficult to be substituted by domestic-made or ROW-made commodities. List of inelastic inputs from macro and microperspectives is provided in Table 4 (appended).

Multifactor CES elasticities
We estimate multifactor CES elasticities for all production sectors according to the regression equation (17). However, in this case, we must measure D ln w C i between current and reference states in advance, using the macroaggregator (1) whose parameters are measured via the two-point calibration method presented previously. The reference and current compound prices evaluated with respect to the price indexes (inflators) used for domestic and foreign commodities are as follows: Using these values, we estimate r via (17). Specifically, 1 À r is estimated by the slope of the following linear regression equation: where s i is the cost share of input i for the concerned industrial sector whose reference and current values are both available in a set of linked input-output tables 4 and u i is the disturbance term. Furthermore, note that growth of productivity, i.e., D ln t, is estimable from the intercept of the regression line, although that analysis is beyond the purpose of this study. We must note that linked input-output tables do not provide price indexes for the primary input (comprising labor and capital), which we aggregate as a single input in this study. To address this, we use the quality-adjusted price indexes of labor and capital compiled by JIP (2015) for Japan and by KIP (2015) for Korea for the corresponding periods to inflate the value-added observed in nominal values. In Fig. 3, we report the estimated multifactor CES elasticities r for all sectors (left) with the corresponding statistical significances (right) for Japan. Figure 4 is the equivalent figure for Korea. Furthermore, we shall note that the average of the estimated elasticities (ignoring statistical significance) is 1.46 for Japan and 1.53 for Korea, and these values are almost identical to those estimated using q D i instead of q C i in regression equation (19) as reported in Kim et al (2017).  Fig. 2 Scatter plot of log-absolute micro and macroArmington elasticities, i.e., log 10 je j j vs log 10 jg j j for Japan (left) and Korea (right). List of inputs with inelastic substitution (with negative log-absolute elasticity) is provided in Table 4 −1 In this section, we construct a bilateral multisectoral general equilibrium model that reflects all measured elasticities for the two countries. Let us first focus on one country's general equilibrium state of multisectoral production. We shall calibrate the share parameters at the current state to examine various policy shifts (such as tariff elimination) on the basis of the current state. As we have previously arranged that all current prices be the basis of price standardization, we may calibrate the share parameters k i at the current state where the productivity is standardized at unity t ¼ 1, according to (15): Here, a i is the current state input-output coefficient (i.e., cost share) of input i for the industry (output) concerned, and thus, P n i¼0 a i ¼ 1. We may express the system of unit cost functions (14) as follows: or more concisely as The model for both countries according to the multifactor CES aggregator (14), the macroaggregator (1), and the microaggregator (7), can be expressed as follows, where J and K indicate Japan and Korea, respectively: Note that we eliminate w C 0 from the multifactor CES aggregator, since it is fixed as constant and w R from the microaggregators as we assume that ROW import prices are invariable (under the small-country assumption).
To close (integrate) the model, we must introduce a weighted converter that connects the foreign sector with the domestic sector classifications in terms of 6digit HS transactions. Specifically, a sector-HS converter z jk that assigns a sectoral commodity j to an HS item k has the following form: where x jk represents the amount of import of HS item k that belongs to sector j. As we represent Japan's sector-HS converter by matrix z J and Korea's sector-HS converter by z K , Korea's 350 sectors can be converted into Japan's 395 sectors by z K z | J , and likewise, Japan's sectors can be converted into Korea's by z J z | K , where | indicates transposition. Thereupon, we introduce the following identities, according to (12): where angle brackets indicate diagonalization. In addition, we know that ml ¼ 1 from (18) at the current state. Hence, we know that the equilibrium solution to the bilateral integrated price system (22)-(25) at the current state is unity for all, i.e.,

Tariff elimination
We first calculate the equilibrium price when all tariffs that levied against the partner country's commodities in both countries were eliminated. For the purpose, we specify the tariff rates levied at the current state, and thus, we used the UNCTAD Trade Analysis Information System (TRAINS 2017) database. Specifically, we used the tariff rates evaluated by way of customs duties-imported values that were converted into ratios and distributed over the linked input-output product classifications. In Fig. 5, we display the estimated tariff rates levied against the partner country's commodities for all sectoral commodities, for both countries. Note that ''Refined sake'' (59.0%) and ''Beef cattle'' (22.5%) were among the higher tariff rate commodities in Japan against Korea, whereas ''Vegetables'' (53.6%) and ''Fruits'' (37.4%) were among the higher tariff rate commodities in Korea against Japan.
Let us now consider what happens if the tariff between the two countries was entirely eliminated over the current state. In that event, the ex ante barrier factor l Ã will equal q instead of qð1 þ sÞ, in regard to (12). 5 Thus, because ml ¼ 1 at the current state according to (18), l Ã must be evaluated as follows: and hence, we must modify (25) when evaluating tariff-eliminated bilateral general equilibrium prices, as follows: Hereafter, let us denote by p the tariff-eliminated bilateral general equilibrium prices. That is More specifically, p is the fix point of the mapping G : R 4ðn J þn K Þ ! R 4ðn J þn K Þ which comprises of the functions (22-24) and (26) i.e., 6 Note that G is a concave and monotone increasing mapping, because CES aggregators H, U, and V are all concave functions, and linear functions (26) are also concave (although not strictly concave). Thus, G becomes a contraction mapping and we may solve (27) for the fixed point by recursive means (see e.g.Kennan 2001; Krasnosel'skiǐ 1964) from arbitrary initial guess such as 1, that is

Prospective analysis
Since we know by the Shephard's lemma that the factor input can be obtained by differentiating the unit cost function, inputs in physical units per physical unit output for all sectors, or the physical input-output coefficient matrix, can be obtained as the gradient of (21), that is where r 0 H is an n row vector, while rH is an n Â n matrix. For later convenience, let us use the following terms to indicate monetary input-output coefficient matrices for current and posterior (with tariff elimination) states: Note that we set p C 0 ¼ 1 as we take the primary input i ¼ 0, which is not produced industrially, as the numéraire good. In addition, a 0 and A are the current state (observed) value-added and input-output coefficients, respectively.
Below is the commodity balance in monetary terms: where x denotes domestic output, y denotes domestic final demand, e denotes export, and m denotes import, all in column vectors of monetary terms, while Ax represents the intermediate demand.
Here, we may recall that we have obtained the current state foreign share of a commodity s F [ ¼ 1 À a, by the amount of import m (i.e., an element of m whose index is omitted) and the domestic total demand, that is For further convenience, let us define s ¼ s F [ ¼ 1 À a and endogenize import with respect to the domestic total demand as follows: Furthermore, we may recall that the import from the partner country m P can be replicated by the current share of the partner country's commodity s P [ , which we hereafter denote s P for convenience, as follows: Displayed below is the commodity balance of the posterior state: x ¼Ãx þỹ þ e W þẽ P À Á Àm W þm P À Á : The posterior state values are distinguished by tildes. We assume that imports and exports are subject to change due to tariff elimination, except for the exports to the ROW. Notice that imports from the partner and the ROW are assumed to be proportional to the total domestic demands in the following manner: As indicated above, the export against the partner country is determined by the import from the partner's partner country. 7 The import coefficients are determined by (2) and (9) as follows: The posterior value-added (external inputs) total can be evaluated by the import endogenized model in regard to the posterior commodity balance equation (29): 8 where the import coefficient s Ã is specified as follows, according to (30): We assume an economy to maximize its final demand given the external inputs total, and to this end, the compensation of increased exports against the partner country can be spent for whatever commodity demanded. We incorporate such external inputs into the domestic production in such a way that the external inputs (value-added) total is fortified. 9 In particular, we are to find a scalar d of the following problem that maximizes the total ex ante value of the current-proportioned final demand, i.e.,ỹ ¼ hp D iyd, given the ex ante total value added (31), which is limited to the sum of the locally existing primary factor ' ð¼ a 0 xÞ, and the compensation of exports against the partner country, that is Note that the solution of (32) determines the posterior total domestic demands and thus the imports from the partner country which, in turn, determines the compensation of exports against the partner's partner country via (30) that must enter into the constraint of the parter country's problem. In other words, (32) must be solved recursively for both countries under the condition given by the partner country.
Figures 6 and 7 illustrate the increments of maximized current-proportioned final demand, i.e.,ỹ À y and the corresponding redistribution of the external inputs, i.e., a 0x À a 0 x for Japan and Korea, respectively, under the tariff elimination between the two countries. 10 Notice that BJPY stands for billion Japanese yens and BKRW for billion Korean wons. The total effects are summarized in Table 1. The net benefit (in terms of gained final demand Dy) of tariff elimination is 853 BJPY (about 0.17% of the current GDP) for Japan, whereas it is 6309 BKRW (about 0.74% of the current GDP) for Korea. As regards the redistribution of the external inputs, currentproportioned final demand maximization suggests that sectors such as j ¼ 302 (House rent), j ¼ 25 (Fisheries), j ¼ 352 [Medical service (medical corporations, etc.)], and j ¼ 329 (Information services) must be reinforced, and curtailed in sectors such as j ¼ 65 (Other liquors), j ¼ 75 (Woolen fabrics, hemp fabrics and other fabrics), j ¼ 145 (Miscellaneous leather products), and j ¼ 17 (Hogs), for Japan. On the other hand, preferable policy for Korea is to reinforce in sectors such as j ¼ 71 (Other liquors), j ¼ 283 (Wholesale and Retail trade), j ¼ 19 (Pigs), and j ¼ 18 (Beef cattle), and to curtail in sectors such as j ¼ 53 (Raw sugar), j ¼ 26  where naturally, Df JK þ Df KJ ¼ 0. In Table 2, we display the positive entries of the net export from Japan to Korea Df JK . 11 Likewise, Table 3 is the positive entries of the net export from Korea to Japan.
We may notice from these tables that a lot of meat (i.e., Slaughtering and meat processing) will be exported from Korea to Japan, whereas Japan will export fish (i.e., Fisheries, Frozen fish, and shellfish) to Korea, under tariff elimination. Other notable features are that Korea will net-export petrochemical products (e.g., petrochemical aromatic products (except synthetic resin), coal mining, crude petroleum and natural gas, petrochemical basic products, petroleum refinery products (incl. greases), etc.) to Japan, whereas Japan will net-export mechanical and assembling products (e.g., motor vehicle parts and accessories, machinery and equipment for construction and mining, electric audio equipment, rotating electrical equipment, etc.) to Korea.

Concluding remarks
The highlight of this study may, perhaps, be the discovery of a way to calibrate the parameters of a two-input CES aggregator in order that the aggregator completely replicates the observed two temporally distant shares of inputs in both monetary and physical terms. The elasticity parameters, i.e., the Armington elasticities that we obtained by way of this approach (i.e., two-point calibration), were found to be much larger than those observed in the previous studies based upon time series, implying almost complete substitutability between foreign and domestic commodities, which should not be too surprising. We then used the Armington aggregator functions to uncover the composite price index for each commodity, which is key for modeling production activities comprising many factor inputs, including imported commodities, for each industrial sector.
As we are concerned with multisectoral production functions of multiple (more than two) factor inputs, we estimated multifactor CES production elasticities by linearly regressing the growth of commodity-wise cost shares against the relative growths of factor prices. We used published statistics, namely, linked input-output tables and the UN Comtrade database, to measure all the concerned elasticities (i.e., multifactor CES production, and micro and macroArmington elasticities) for both Japan and Korea. The two multisectoral general equilibrium models for Japan and Korea were integrated by the bilateral trading models which reflect the trade barriers between the two countries.
Since the models presume constant returns in all activities and thus interact entirely in terms of unit costs and prices, we were able to simulate the bilateral general equilibrium consequences of eliminating tariffs between the two countries, without (physically) quantitative consideration. The consequential social benefits and costs of tariff elimination were estimated by the amount of linear final demand that can potentially be enhanced under the projected structure for a given total primary input. The result implies positive effects (in terms of total net benefit) for both countries, while considerable structural change is expected to be inevitable. The framework we introduced in this paper can further be advanced from several perspectives.
The framework of the study will benefit considerably if we can endogenize variables that are still exogenous, such as labor, capital services, and exchange rates. The model becomes dynamicalized if capital services are endogenized, while in that event, the model of consumer behavior needs more sophistication. Finally, we shall note, as one may be aware, that trilateralization of trades, which we leave for future investigation, can become rather intricate. Table 4 Qualitative evaluation of macro and microArmington elasticities. Regarding Fig. 2, an input is entitled as inelastic if it is positioned at the south or the west, and as elastic if it is positioned at the north or the east of the ridgeline. Note that inputs that are elastic in both stages (macro and micro) and those whose imports from the partner country are absent, are eliminated from the list