Daily Welfare Gains from Trade

Daily Welfare Gains from Trade Hasan Toprak Hakan Yilmazkuday y INCOMPLETE Abstract Using daily price quantity data on imported locally produced agricultural products, this paper estimates the elasticity of substitution between home foreign goods (i.e., the Armington elasticity), the elasticity of substitution across goods, the elasticity of substitution across varieties of goods using estimation methodologies that are robust to any simultaneity bias. The corresponding daily welfare gains from trade are rst shown to depend on the home consumption share, as well as the Armington elasticity, then decomposed (over time) into the e ects due to the nominal exchange rate, consumer price index, timevarying preferences toward foreign goods foreign source prices. The results show that dem shocks (captured by preferences) macroeconomic shocks (captured by in ation) contribute most to the daily welfare gains. JEL Classi cation: F2, F4 Key Words: Armington Elasticity; Daily Data; Turkish Imports The authors would like to thank XXX for their helpful comments suggestions. The usual disclaimer applies. y Department of Economics, Florida International University, Miami, FL 3399, USA; e-mail: hyilmazk@ u.edu

. Introduction This paper investigates the daily welfare gains from trade using daily price quantity data on aggricultural products. The results show that dem shocks (captured by preferences) macroeconomic shocks (captured by in ation) contribute most to the variance decomposition of daily welfare gains. 2. Economic Environment An economy with a nite number of goods, each with a nite number of varieties, is modeled. Individuals consume all varieties of all goods, while retailers are specialized in the sale of a particular variety of a particular good. In the model, generally speaking, X j s;t (i) sts for variable X where j represents the good, i represents the variety of good j, s represents the location of production (either home or foreign), t represents the time period; certain subscripts/supercript drop as X j s;t (i) is aggregated over the corresponding dimensions. 2.. Individuals Individuals maximize nested CES utility consisting of home foreign products at time t: 0 C t @ X s2fh;f g s;t (C s;t ) A where is the (Armington) elasticity of substitution between home H foreign F products. C s;t for s 2 fh; F g is further given by: C s;t X j! " " j " s;t C j " " s;t 2

where C j H;t Cj F;t represent good j produced in home foreign countries, respectively; " is the elasticity of substitution across goods. C j s;t for s 2 fh; F g is further given by: C j s;t X i j s;t (i) C j s;t (i)! where C j H;t (i) Cj F;t (i) represent variety i of good j produced in home foreign countries, respectively; is the elasticity of substitution across varieties. The dem shifter j s;t (i) is further assumed to follow a rom shock according to: j s;t (i) = j s;t (i) exp v j; s;t (i) (2.) where v j; s;t (i) is an i.i.d. shock with zero mean variance 2. For s 2 fh; F g, the optimal allocation of any given expenditure yields the following dem functions: C j s;t (i) = j s;t (i) C j s;t = j s;t P j s;t (i) P j s;t P s;t P j s;t! C j s;t (2.2)! " C s;t (2.3) Ps;t C s;t = s;t C t (2.4) P t where P j s;t (i), P j s;t, P s;t, P t are the corresponding prices per units of C j s;t (i), C j s;t, C s;t, C t, respectively, for s 2 fh; F g. It is implied that prices are connected to each other through the following expressions for s 2 fh; F g: P j s;t X i j s;t (i) P j s;t (i)! (2.5) P s;t X j j s;t! " P j " s;t (2.6) 3

0 P t @ X s;t (P s;t ) A (2.7) s2fh;f g 2.2. Retailers Retailer selling variety i of good j produced in s 2 fh; F g maximizes the following pro t: maxy j P j s;t (i) s;t (i) P j s;t (i) Z j s;t (i) (2.8) subject to Y j s;t (i) = C j j H;t (i), where Ys;t (i) is the quantity sold Z j s;t (i) is the marginal cost of production (under the assumption of constant returns to scale) that follows a rom walk in log-linear terms according to: Z j s;t (i) = Z j s;t (i) exp v j;z s;t (i) (2.9) where v j;z s;t (i) is an i.i.d. shock with zero mean variance 2 Z. The pro t maximization problem results in: P j s;t (i) = Z j s;t (i) (2.0) where represents gross markups. 3. Welfare Gains from Trade Given income P t C t = P s2fh;f g P s;tc s;t, we attempt to measure the welfare costs of autarky in percentage terms (i.e., W GT t ) by which the aggregate price index P t would have to adjust to keep the consumer utility the same between the current openness to trade a hypothetical autarky: exp (W GT t ) = P t A P t = C t C A t where superscript A sts for autarky. 4

Using the relationship between C s;t C t, the corresponding consumption shares of home versus foreign goods are given by: Ps;t P s;t C s;t = s;t P t C t (3.) P t P A s;tc A s;t = s;t P A s;t P A t! P A t C A t (3.2) which can be combined to get an expression for exp (W GT t ) as follows: Ps;t C s;t exp (W GT t ) = P t C t where we have used P A H;t CA H;t = P A t C A t (since the only expenditure is on home goods in the case of autarky) the assumption that the marginal cost of producton for each variety i of each good j produced at home Z j H;t (i) is the same across current openness to trade the hypothetical autarky Z j H;t (i) = Zj;A H;t (i). Taking the log of both sides results in an expression for the welfare costs of autarky in percentage terms as follows W GT t = log X H;t (3.3) where X H;t represents the home share of consumption given by: X H;t = P H;tC H;t P t C t = P j t;h Cj t;h P s2fh;f g P s;tc s;t Therefore, welfare gains from trade in percentage terms W GT t is a function of the Armington elasticity the home share of consumption X H;t. 5

3.. Components of Welfare Gains from Trade We can put more structure on the marginal cost of producton for each variety i of each good j coming from abroad as follows: Z j F;t (i) = Zj F;t (i) E t where Z j F;t (i) is the price charged by the foreign supplier in the form of foreign currency (including trade costs) E t is the nominal exchange rate (de ned as the units of home currency to exchange for one unit of foreign currency). It is implied by Equation 2.0 that the retail price of the same variety P j F;t (i) is given by: which further implies for other price indices that: P j F;t (i) = Z j F;t (i) E t (3.4) P j F;t E t X i j F;t (i) Zj F;t (i)! (3.5) P F;t E t P F;t (3.6) where 0 PF;t = @ X j j F;t 0 @ X i j F;t (i) Zj F;t (i)! " A A " represents the foreign price index in terms of the foreign currency. Therefore, for any given Z j F;t (i), foreign price index of P F;t is directly a ected by the nominal exchange rate E t. It is implied for the foreign share of consumption X F;t that: X F;t = P F;tC F;t P t C t = (E t ) F;t P F;t (3.7) which also changes with the nominal exchange rate E t. Using Equation 3.3, we can also write: P t W GT t = log ( X F;t) (3.8) 6

which can be approximated for small values of X F;t as follows (using log ( x) x): W GT t = W GT 0 t R t X (E t ) P F;t F;t = F;t P t (3.9) where R t represents to adjustment of W GT t due to the approximation. Taking log of both sides results in: log W GT t = ( ) log E t + ( ) log P t + log F;t (3.0) log ( ) + ( ) log P F;t + log R t where we know all the variables except for the latent variable of R t, which can be extracted. In terms of economic intuition, this expression reveals that any depreciation of the home currency (i.e., an increase in E t ) results in lower welfare gains from trade (where the magnitude of the reduction is determined by the armington/trade elasticity of ), while any increase in consumer price index P t or time-varying preferences toward foreign goods F;t results in higher welfare gains from trade, although, from the broader perspective, all the changes in W GT t are due to the changes in home expenditure share of X H;t (as shown in Equation 3.3). 3.2. Variance Decomposition of Welfare Gains from Trade We can further have a variance decomposition analysis in order to investigate the contribution of each component on log W GT t by taking the covariance of both sides in Equation 3.0 with respect 7

to log W GT t as follows: var (log W GT t ) = cov (( ) log E t ; log W GT t ) Contribution of Nominal Exchange Rate + cov (( ) log (P t ) ; log W GT t ) Contribution of Consumer Price Index +cov log F;t ; log W GT t + cov ( ) log PF;t ; log W GTt Contribution of Preferences Contribution of Foreign Source Prices + cov (log R t ; log W GT t ) Contribution of the Approximation where var () cov () are variance covariance operators, respectively. 4. Data Estimation Methodology We are interested in estimating the elasticities of, " especially that is necessary for the welfare analysis. Since each parameter corresponds to a di erent aggregation, estimation in achieved in the corresponding aggregation levels for each parameter. Since retailers set prices at the variety level, the estimation of the micro-level elasticity of may be subject to simultaneity bias. Accordingly, for the estimation of, we follow the estimation methodology developed in Feenstra (994) which is robust to the consideration of simultaneity bias (since we observe equilibrium quantities prices in our data set). Once is estimated, it is used to construct upper-tier variables, which are further used to estimate ". It is important to emphasize that since the upper-tier variables are arti cially created aggregates, they are not subject to any simultaneity bias. 4.. Data Daily price quantity data of aggricultural products from Turkey are employed. The data distinguish between home foreign products as well as varieties of each good over time. 8

4.2. Estimation of For s 2 fh; F g, the estimation of is achieved by rst considering the dem side through the log version of Equation 2.2: log C j s;t (i) = log P j s;t (i) + log C j s;t P j s;t + log j s;t (i) (4.) then by taking the di erence across dimensions of varieties i time t, which results in: log g C j s;t (i) = log g P j s;t (i) + j;q s;t (i) (4.2) where log g C j s;t (i) = log C j s;t (i) log C j s;t (i 0 ) = log C j s;t (i) log C j s;t (i 0 ) log C j s;t (i) + log C j s;t (i 0 ) log g P j s;t (i) = log P j s;t (i) log P j s;t (i 0 ) = log P j s;t (i) log P j s;t (i 0 ) log P j s;t (i) + log P j s;t (i 0 ) j;q s;t (i) = v j; s;t (i) v j; s;t (i 0 ) where the last equality is due to Equation 2., is the operator of time di erence, i 0 is any alternative variety of good j (other than variety i). Similarly, the supply side of the economy is considered by the log version of Equation 2.0: log P j s;t (i) = log + log Z j s;t (i) (4.3) which can be rewritten by again taking the di erence across dimensions of varieties i time t as follows: log g P j s;t (i) = j;p 9 s;t (i)

where j;p s;t (i) = v j;z s;t (i) v j;z s;t (i 0 ), which is due to Equation 2.9. Estimation is achieved by using the independent relationship between j;q s;t (i) j;p s;t (i) due to v j; s;t (i) v j;z s;t (i) being i.i.d. shocks. In particular, the independence of j;q s;t (i) j;p s;t (i) is used to obtain: j;q s;t (i) j;p s;t (i) = log g P j s;t (i)log g C j s;t (i) + log g 2 P j s;t (i) (4.4) which corresponds to the following expression: log g 2 P j s;t (i) = log g P j s;t (i)log g C j s;t (i) + j s;t (i) (4.5) where j s;t (i) = j;q s;t (i) j;p s;t (i) =. Since quantities prices are correlated with shocks of v j; s;t (i) v j;z s;t (i), j s;t (i) is correlated with the right h side variable in Equation 4.5. Nevertheless, can still be estimated consistently using instrumental-variable (IV) estimator, where instruments are good--variety xed e ects. The corresponding stard errors are calcualted by the Delta method. 4.3. Estimation of " Once is estimated, it is further used to construct the following expression obtained from Equation 2.2 for s 2 fh; F g: log C js;t (i) P js;t (i) Data = log P j s;t C j s;t Good-Source-Time Fixed E ects + log j s;t (i) Residuals where the left h side is constructed by the price quantity data together with the estimated. The only right h side variable corresponds to good-source-time xed e ects, while the preferences are employed as residuals as in Hillberry et al. (2005) Yilmazkuday (202). The estimation of this expression (by pooling data across s 2 fh; F g) provides estimates of preferences j s;t (i) which are further used, together with estimated, to construct P j s;t s according to Equation 0 (4.6)

2.5. The constructed P j s;t s are further used in the estimation of Equation 2.3 in expenditure log terms: log Ps;tC j j s;t = ( ") log P j s;t + log ((P s;t ) " C s;t ) Data Constructed Variable Source-Time Fixed E ects + log j s;t Residuals where the left h side is calculated by the implications of having a nested CES, P j s;tc j s;t = P i P j s;t (i) C j s;t (i). The estimation is achieved by pooling data across s 2 fh; F g, " is identi ed from the coe cient in front of log P j s;t. (4.7) In a similar way, estimated " is combined with estimated j s;t s (as residuals) in Equation 4.7 to construct P s;t s according to Equation 2.6, which are further used in the estimation of 2.4 in expenditure log terms: log (P s;t C s;t ) = ( ) log P {z s;t } Data Constructed Variable + log (P t ) C t Time Fixed E ects + log s;t Residuals (4.8) where the left h side is again calculated by the implications of having a nested CES, P s;t C s;t = P P j i P j s;t (i) C j s;t (i); in this estimation, data are pooled across pooling data across s 2 fh; F g, is identi ed from the coe cient in front of log P s;t. Since P j s;t s P s;t s are generated using estimated parameters predicted residuals from a prior regression of Equation 4.6, there is a generated regressor problem (Pagan, 984); i.e., the stard errors are invalid. Following Efron Tibshirani (993), we employ bootstrap techniques to obtain stard errors that explicitly take into account the presence of generated regressors. In particular, for each bootstrap b, (i) we resample (with replacement) the bilateral good-level trade values by using the tted values residuals in Equation 4.6, (ii) estimate Equation 4.6 with Note that j s;t (i) s are identi ed only in relative terms due to the restrictions imposed by the regression. Nevertheless, this does not create any problems in our investigation, since such scale e ects are captured by constant terms or other xed e ects in the following log-linear regressions.

the resampled left h side, (iii) use the estimated parameters predicted residuals from this regression to generate bootstrap prices of P jb s;t s P b s;t s, (iv) estimate Equations 4.7 4.8 using P jb s;t s P b s;t s to estimate " (b) (b). We repeat this exercise 25 times compute the bootstrap stard errors of " as follows: S.E. (") = 25 X25 b= (" (b) ") 2! 2 S.E. () = 25 X25 b= ( (b) ) 2! 2 where " are the original coe cients estimated by Equations 4.7 4.8. 5. Empirical Results 5.. Estimation Results The estimates of, " are given in Table. As is evident, the elasticity of substitution across varieties is about.95, which is higher (as expected) than the elasticity of substitution " across goods that is about.9. Although these estimates have importance of their own, we are mostly interested in the Armington elasticity of between home foreign products, which is estimated about 2:07. Table - Estimation Results Elasticity " Coe cient Estimate :95 :9 2:07 Stard Error (0:0) (0:32) Compared to the existing literature,... 2

5.2. Implications for the Welfare Gains from Trade The variance decomposition of the welfare gains from trade is given in Table 2, where preferences contribute most to the variance of welfare gains from trade, followed by CPI. Foreign prices have a smoothing role, while the contribution of the exchange rate the approximation are almost none. Table 2 - Variance Decomposition of Welfare Gains var (log W GT t ) Exchange Rate CPI Preferences Foreign Prices Appr. Levels 3:43 0:0 0:76 2:97 0:37 0:09 Percentage 00% 0:38% 22:07% 86:54% 0:8% 2:58% It is implied that dem shocks (capture by preferences in this paper) are the most e ective factors in the determination of daily welfare gains from trade, followed by the macroeconomic shocks (captured by in ation in this paper). 5.3. Robustness Checks In order to control for potential measurement errors in the data, we repeat our overall investigation after ltering price quantity data by ignoring observations/outliers that are ve stard deviations away from their mean over the sample period. In such a case, the estimation results in Table are replaced with those in Table 3. As is evident, the results are very similar. Table 3 - Estimation Results Ignoring Outliers Elasticity " Coe cient Estimate :88 :20 2:28 Stard Error (0:0) (0:32) 3

After ignoring outliers, the variance decomposition of welfare gains in Table 2 is replaced with the one in Table 4. As is evident, the results are very similar qualitatively, where preferences CPI contribute most to the daily welfare gains from trade (after ignoring the contribution of the approximation). Table 4 - Variance Decomposition of Welfare Gains Ignoring Outliers var (log W GT t ) Exchange Rate CPI Preferences Foreign Prices Appr. Levels 3:72 0:03 0:50 3:02 0:53 0:76 Percentage 00% 0:79% 3:48% 8:28% 4:27% 20:30% 6. Conclusions By using daily aggricultural price quantity data, this paper has shown that the daily welfare gains are mostly derived by dem shocks (captured by preferences) with a contribution of about 80% macroeconomic shocks (captured by in ation) with a contribution of about 20%. 4