- Email: [email protected]
Income elasticity and international income differences
Accepted Manuscript Income elasticity and international income differences Junmin Liao, Wei Wang
PII: DOI: Reference:
S0165-1765(18)30192-7 https://doi.org/10.1016/j.econlet.2018.05.019 ECOLET 8054
To appear in:
Economics Letters
Received date : 27 April 2018 Revised date : 13 May 2018 Accepted date : 17 May 2018 Please cite this article as: Liao J., Wang W., Income elasticity and international income differences. Economics Letters (2018), https://doi.org/10.1016/j.econlet.2018.05.019 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Hightlights:
Empirical evidence implies the relative income elasticity is constant
Prevailing Stone-Geary preferences generate diminishing relative income elasticities
Non-homothetic CES preferences do a better job fitting cross-country data
*Title Page
Income Elasticity and International Income Di↵erences Junmin Liaoa , Wei Wangb,⇤ a Center b School
for Economic Development Research, Wuhan University, Wuhan, Hubei 430072, China of International Trade and Economics, University of International Business and Economics, 10 Huixin East Street, Beijing 100029, China
Abstract The constant relative income elasticity generated by non-homothetic CES preferences is consistent with cross-country empirical evidence. We show that nonhomothetic CES preferences fit empirical patterns better than Stone-Geary preferences in a standard framework in the international income di↵erences literature. Keywords: Income elasticity; Income di↵erences; Non-homothetic CES; Stone-Geary CES; Agriculture JEL Classification: O1; O4
⇤ Corresponding
author Email addresses: [email protected] (Junmin Liao), [email protected] (Wei Wang)
Preprint submitted to Economics Letters
May 13, 2018
*Manuscript Click here to view linked References
1. Introduction
5
10
15
20
25
30
Agricultural productivity is extremely low in poor countries compared to rich ones. Yet, poor countries devote most of their labor to agriculture. Agriculture is thus crucial for understanding international income di↵erences(Caselli, 2005; Restuccia et al., 2008; Gollin et al., 2014). The relative income elasticity of demand between agriculture and non-agriculture (non-homotheticity) is important in accounting for these cross-country di↵erences.1 Stone-Geary preferences are prevailing in macro development literature (Restuccia et al., 2008; Lagakos & Waugh, 2013). The relative income elasticity of Stone-Geary preferences vanishes quickly as income grows. Empirical evidence, however, suggests that the relative income elasticity is stable at all income levels. Figure 1 shows the correlations between the logarithm of the relative sectoral consumption and the logarithm of the aggregate real GDP, controlling the prices. Constant slope of the fitted line indicates a constant relative income elasticity between the agricultural and the non-agricultural goods.2 The diminishing relative income elasticity generated by Stone-Geary preferences causes a problem for quantitative analyses. In particular, quantitative models in macro development literature are usually calibrated to match the data of a benchmark economy (mostly the U.S.). In addition, the relative income elasticity parameter of Stone-Geary preferences is assumed to be the same across countries. Hence, the relative income elasticity is lower (higher) in the model than its empirical counterpart for countries that are richer (poorer) than the benchmark economy. As a consequence, the agricultural productivity and employment share of richer (poorer) countries are exaggerated (attenuated) in the model than what is observed in the data. In contrast, the relative income elasticity generated by non-homothetic CES preferences (Comin et al., 2017) is constant at all income levels, which is consistent with Figure 1.3 In the quantitative framework of Restuccia et al. (2008), we show that the non-homothetic CES preference does a better job fitting the exact empirical facts that the Stone-Geary preference is meant to match. The remainder of this paper is organized as follows. Section 2 shows how, in theory, the relative income elasticity changes with income under each of the 1 We follow Comin et al. (2017) to define the income elasticity of demand in each sector as the ratio between the logarithm of the relative sectoral consumption and the logarithm of the aggregate real consumption, holding prices constant. 2 We get data of 43 countries from GGDC 10-Sector Database (Timmer et al., 2015). The ten sectors are aggregated into an agricultural sector and a non-agricultural sector. We use the annual time series of nominal and real value-added data. Relative prices are calculated from the nominal value. We get the residual log GDP by controlling the price based on the following regression: log Ytc = c + 1 log pcat + 2 log pcnt + ⌫tc , where c denotes country and t denotes time. Y denotes the GDP. pa and pn denote the price of agricultural and nonagricultural goods, respectively. is the country fixed e↵ect and ⌫ is the error term. We calculate the residual log value-added share in a similar manner. The adjusted R2 = 0.73. For the micro level evidence, see Comin et al. (2017). 3 They explore the role of non-homothetic CES preferences in the standard framework of structural change.
2
Residual Log Real Value Added Share
2
1
0
-1
-2 -2
-1
0 1 Residual Log Real GDP
2
Figure 1: Partial Correlations of Relative Value Added Share and GDP Sources: GGDC 10-Sector Database (Timmer et al., 2015). The red line is the linear regression line. The shaded blue region represents the 95% confidence interval.
two preferences. Section 3 conducts quantitative analyses. Section 4 concludes. 2. Income Elasticities 35
40
In this section, we characterize the relative income elasticity for both StoneGeary CES preferences and non-homothetic CES preferences. Consider a static choice problem of a consumer whose income is E. She chooses the consumption of the agricultural good ca and non-agricultural good cn to maximize her utility. Let the non-agricultural good be the numeraire. Denote the price of the agricultural good by pa . The budget constraint is pa ca + cn = E. An index of real income measuring consumer utility, C, is defined explicitly (implicitly) in the Stone-Geary (non-homothetic) CES preference below. 2.1. Stone-Geary CES Preferences The index C is defined explicitly by the following CES aggregator: h C = !a (ca
45
a ¯)
✏
1 ✏
✏
+ !n c n ✏
1
i✏ ✏1
,
(1)
where ✏ > 0 and ✏ 6= 1. a ¯ > 0 is the subsistence parameter. !i > 0 is a strictly positive weight parameter for good i 2 {a, n}. The relative income elasticity can be obtained by solving the related utility maximization problem:
3
@log
⇣
ca cn
@logC
⌘
=
1+
✓
◆✏
!a pa
1
C a ¯
,
1
50
where = !a✏ p1a ✏ + !n✏ ✏ 1 .4 Two implications are in order. First, the relative income elasticity vanishes as C grows. Second, the subsistence a ¯ controls the magnitude of the income elasticity. In particular, all else equal, a larger (smaller) subsistence parameter implies a larger (smaller) relative income elasticity. 2.2. Non-Homothetic CES Preferences With non-homothetic CES preferences, the index C is defined by the following aggregator: !a C
55
µa ✏ ✏
✏
1
c a ✏ + !n C
µn ✏ ✏
✏
1
cn ✏ = 1,
(2)
where µi > 0 controls the income elasticity of demand for good i 2 {a, n}. By solving the related optimization problem, the relative income elasticity can be obtained: ⇣ ⌘ @log ccna = µa µn , @logC which is a constant.5 This feature is consistent with the cross-country evidence described in Section 1.
60
3. Quantitative Analysis
65
In this section, we use the model of Restuccia et al. (2008) to illustrate that the non-homothetic CES preference does a better job fitting the exact empirical facts that the Stone-Geary preference is meant to match. We calibrate common parameters (across countries) to match the data from our benchmark economy, the U.S.6 3.1. Model
70
Consider a two-sector economy in which each sector produces one good: the agricultural good a and the non-agricultural good n. The preference of the representative household can be (1) or (2). The households supply total labor N inelastically. Let Yi and Li be the output and the labor input in sector i 2 {a, n}. The production of the agricultural good uses land input Z and a non-agricultural intermediate input X: Ya = X ↵ Z 1 4 See 5 See 6 See
(ALa )
1 ↵
,
Appendix A.3 for the details. Appendix A.4 for the details. Appendix B.1 for details of the data we use in our quantitative analysis.
4
Parameter
Calibrated Value
Target
✏ !a !n a ¯
0.9223 0.0048 0.9952 663.99 0.7000 0.4000
Estimated from the U.S. data Estimated from the U.S. data Normalization Agricultural employment share of the U.S. Income share of labor in agriculture Income share of intermediate inputs
↵
Table 1: Calibration for Stone-Geary CES Preferences
75
where 0 < < 1, 0 < ↵ < 1, is the sector-specific productivity in agriculture, and A is the economy-wide productivity. The production technology of the non-agricultural good is linear: Yn = ALn .
80
85
90
95
We assume that all markets are competitive and we let the non-agricultural good be the numeraire. Let pa , w, and ⇡ denote the prices of the agricultural good, labor and the intermediate input, respectively. Due to the friction in the intermediate input market, assume that ⇡ 1. Due to the friction in the labor market, assume that wa = (1 ✓) wn , where 0 < ✓ < 1. 3.2. Stone-Geary CES Calibration. Country-specific parameters A, , ✓, ⇡ and Z/N are calibrated from the data of the corresponding countries.7 All other parameters are common across countries and they are calibrated to match the U.S. data. We choose the same production side parameters and ↵ as in Restuccia et al. (2008). Parameters in the preference are calibrated as follows. We first normalize the weight parameters such that !a + !n = 1. We use the annual data of the U.S. to estimate ✏ and !a based on the first-order condition.8 Then a ¯ is calibrated to match the U.S. agricultural employment share in 1985.9 Notice that !a governs the agricultural employment share in the long run. Hence, our calibration strategy is consistent with Restuccia et al. (2008). Calibrated values are summarized in Table 1. Model Fit. Figure 2 plots the simulated agricultural employment share La /N and agricultural labor productivity Ya /La against their counterparts in the data. The left panel shows the absolute change of agricultural employment share and the right panel shows the relative change (in log scale). On the one hand, the quantitative predictions for rich countries match the data well. On the other 7 ⇡ is assumed to be unity in the U.S. These values are the same as those in Restuccia et al. (2008). 8 See Appendix B.2 for details. 9 The cross-country data from Restuccia et al. (2008) are in 1985.
5
Data Stone-Geary CES
Data Stone-Geary CES Agricultural Employment Share (log-scale)
Agricultural Employment Share
1.00
0.75
0.50
0.25
1.0
0.1
0.00 100
1000 10000 Agricultural Output per Worker (log-scale)
100
1000 10000 Agricultural Output per Worker (log-scale)
Figure 2: Model Fit: Stone-Geary CES Preferences
100
105
hand, since the model predicts significantly larger income elasticities for poor countries, the simulated agricultural employment share of poor countries are significantly larger than what is observed in the data. In particular, for the poorest countries, the relative income elasticity is so large that devoting all labor to agriculture is not enough to meet the demand of the agricultural good in the model.10 In the next subsection, we show that the model with the nonhomothetic CES preferences matches our cross-country data better than the Stone-Geary CES preferences. 3.3. Non-homothetic CES
110
115
Calibration. The only di↵erence in the calibration strategy between the nonhomothetic and the Stone-Geary cases is the identification of the preference parameters. We first use the time series data of the U.S. to estimate ✏, µa and µn based on the equilibrium conditions.11 Without loss of generality, we normalize !a to be 1.12 !n is calibrated to match the agricultural employment share of the U.S in 1985. We summarize the calibrated values in Table 2.13 Model Fit. The simulated agricultural employment share La /N and agricultural labor productivity Ya /La are plotted against their empirical counterparts in Figure 3. In general, the quantitative simulations match the data very well. The reason is that, even though the parameters are estimated from the U.S. data, the relative income elasticity is similar across all countries. 10 We use 100% agricultural employment share to indicate these countries in the graph. Equilibrium does not exist for them. 11 See Appendix B.2 for details. 12 See Appendix B.3 for the justification of this normalization. 13 When µ < ✏, the non-homothetic preference is not globally monotonically increasing or a quasi-concave, but it can be transformed to a locally valid preference. In that case, we search
6
Parameter
Calibrated Value.
Target
✏ !a !n µa µn
0.6100 1.0000 1503.7 0.2537 0.9529 0.7000 0.4000
Estimated from the U.S. data Normalization Agricultural employment share of the U.S. Estimated from the U.S. data Estimated from the U.S. data Income share of labor in agriculture Income share of intermediate inputs
↵
Table 2: Calibration for Non-homothetic CES Preferences
1.00
Data Non-homothetic CES
Data Non-homothetic CES Agricultural Employment Share (log-scale)
Agricultural Employment Share (log-scale)
1.0
0.75
0.50
0.25
0.1
0.00 100
1000 10000 Agricultural Output per Worker (log-scale)
100
1000 10000 Agricultural Output per Worker (log-scale)
Figure 3: Model Fit: Non-homothetic CES Preferences
7
4. Conclusion
120
This paper illustrates that the non-homothetic CES preferences are consistent with cross-country empirical evidence and quantitatively perform better than the prevailing Stone-Geary preferences. Acknowledgements
125
We are grateful to the associate editor and the anonymous referee for helpful suggestions. We acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 71703017, 71703116), Beijing National Natural Science Foundation (Grant No. 9184031), and Center for Economic Development Research of Wuhan University. Appendix: Model and Quantitative Details Supplementary material related to this article can be found online.
130
References Caselli, F. (2005). Accounting for Cross-Country Income Di↵erences. Handbook of Economic Growth, 1 , 679–741. Comin, D., Lashkari, D., & Mestieri, M. (2017). Structural Change with Longrun Income and Price E↵ects. Working Paper , .
135
Gollin, D., Lagakos, D., & Waugh, M. (2014). Agricultural productivity di↵erences across countries. American Economic Review , 104 , 165–170. Lagakos, D., & Waugh, M. (2013). Selection, Agriculture and Cross-Country Productivity Di↵erences. American Economic Review , 103 , 948–980.
140
Restuccia, D., Yang, D. T., & Zhu, X. (2008). Agriculture and aggregate productivity: A quantitative cross-country analysis. Journal of Monetary Economics, 55 , 234–250. Timmer, M., de Vries, G., & de Vries, K. (2015). Patterns of Structural Change in Developing Countries. In J. Weiss, & M. Tribe (Eds.), Routledge Handbook of Industry and Development (pp. 65–83). Routledge.
for the optimal allocation numerically.
8
PII: DOI: Reference:
S0165-1765(18)30192-7 https://doi.org/10.1016/j.econlet.2018.05.019 ECOLET 8054
To appear in:
Economics Letters
Received date : 27 April 2018 Revised date : 13 May 2018 Accepted date : 17 May 2018 Please cite this article as: Liao J., Wang W., Income elasticity and international income differences. Economics Letters (2018), https://doi.org/10.1016/j.econlet.2018.05.019 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Hightlights:
Empirical evidence implies the relative income elasticity is constant
Prevailing Stone-Geary preferences generate diminishing relative income elasticities
Non-homothetic CES preferences do a better job fitting cross-country data
*Title Page
Income Elasticity and International Income Di↵erences Junmin Liaoa , Wei Wangb,⇤ a Center b School
for Economic Development Research, Wuhan University, Wuhan, Hubei 430072, China of International Trade and Economics, University of International Business and Economics, 10 Huixin East Street, Beijing 100029, China
Abstract The constant relative income elasticity generated by non-homothetic CES preferences is consistent with cross-country empirical evidence. We show that nonhomothetic CES preferences fit empirical patterns better than Stone-Geary preferences in a standard framework in the international income di↵erences literature. Keywords: Income elasticity; Income di↵erences; Non-homothetic CES; Stone-Geary CES; Agriculture JEL Classification: O1; O4
⇤ Corresponding
author Email addresses: [email protected] (Junmin Liao), [email protected] (Wei Wang)
Preprint submitted to Economics Letters
May 13, 2018
*Manuscript Click here to view linked References
1. Introduction
5
10
15
20
25
30
Agricultural productivity is extremely low in poor countries compared to rich ones. Yet, poor countries devote most of their labor to agriculture. Agriculture is thus crucial for understanding international income di↵erences(Caselli, 2005; Restuccia et al., 2008; Gollin et al., 2014). The relative income elasticity of demand between agriculture and non-agriculture (non-homotheticity) is important in accounting for these cross-country di↵erences.1 Stone-Geary preferences are prevailing in macro development literature (Restuccia et al., 2008; Lagakos & Waugh, 2013). The relative income elasticity of Stone-Geary preferences vanishes quickly as income grows. Empirical evidence, however, suggests that the relative income elasticity is stable at all income levels. Figure 1 shows the correlations between the logarithm of the relative sectoral consumption and the logarithm of the aggregate real GDP, controlling the prices. Constant slope of the fitted line indicates a constant relative income elasticity between the agricultural and the non-agricultural goods.2 The diminishing relative income elasticity generated by Stone-Geary preferences causes a problem for quantitative analyses. In particular, quantitative models in macro development literature are usually calibrated to match the data of a benchmark economy (mostly the U.S.). In addition, the relative income elasticity parameter of Stone-Geary preferences is assumed to be the same across countries. Hence, the relative income elasticity is lower (higher) in the model than its empirical counterpart for countries that are richer (poorer) than the benchmark economy. As a consequence, the agricultural productivity and employment share of richer (poorer) countries are exaggerated (attenuated) in the model than what is observed in the data. In contrast, the relative income elasticity generated by non-homothetic CES preferences (Comin et al., 2017) is constant at all income levels, which is consistent with Figure 1.3 In the quantitative framework of Restuccia et al. (2008), we show that the non-homothetic CES preference does a better job fitting the exact empirical facts that the Stone-Geary preference is meant to match. The remainder of this paper is organized as follows. Section 2 shows how, in theory, the relative income elasticity changes with income under each of the 1 We follow Comin et al. (2017) to define the income elasticity of demand in each sector as the ratio between the logarithm of the relative sectoral consumption and the logarithm of the aggregate real consumption, holding prices constant. 2 We get data of 43 countries from GGDC 10-Sector Database (Timmer et al., 2015). The ten sectors are aggregated into an agricultural sector and a non-agricultural sector. We use the annual time series of nominal and real value-added data. Relative prices are calculated from the nominal value. We get the residual log GDP by controlling the price based on the following regression: log Ytc = c + 1 log pcat + 2 log pcnt + ⌫tc , where c denotes country and t denotes time. Y denotes the GDP. pa and pn denote the price of agricultural and nonagricultural goods, respectively. is the country fixed e↵ect and ⌫ is the error term. We calculate the residual log value-added share in a similar manner. The adjusted R2 = 0.73. For the micro level evidence, see Comin et al. (2017). 3 They explore the role of non-homothetic CES preferences in the standard framework of structural change.
2
Residual Log Real Value Added Share
2
1
0
-1
-2 -2
-1
0 1 Residual Log Real GDP
2
Figure 1: Partial Correlations of Relative Value Added Share and GDP Sources: GGDC 10-Sector Database (Timmer et al., 2015). The red line is the linear regression line. The shaded blue region represents the 95% confidence interval.
two preferences. Section 3 conducts quantitative analyses. Section 4 concludes. 2. Income Elasticities 35
40
In this section, we characterize the relative income elasticity for both StoneGeary CES preferences and non-homothetic CES preferences. Consider a static choice problem of a consumer whose income is E. She chooses the consumption of the agricultural good ca and non-agricultural good cn to maximize her utility. Let the non-agricultural good be the numeraire. Denote the price of the agricultural good by pa . The budget constraint is pa ca + cn = E. An index of real income measuring consumer utility, C, is defined explicitly (implicitly) in the Stone-Geary (non-homothetic) CES preference below. 2.1. Stone-Geary CES Preferences The index C is defined explicitly by the following CES aggregator: h C = !a (ca
45
a ¯)
✏
1 ✏
✏
+ !n c n ✏
1
i✏ ✏1
,
(1)
where ✏ > 0 and ✏ 6= 1. a ¯ > 0 is the subsistence parameter. !i > 0 is a strictly positive weight parameter for good i 2 {a, n}. The relative income elasticity can be obtained by solving the related utility maximization problem:
3
@log
⇣
ca cn
@logC
⌘
=
1+
✓
◆✏
!a pa
1
C a ¯
,
1
50
where = !a✏ p1a ✏ + !n✏ ✏ 1 .4 Two implications are in order. First, the relative income elasticity vanishes as C grows. Second, the subsistence a ¯ controls the magnitude of the income elasticity. In particular, all else equal, a larger (smaller) subsistence parameter implies a larger (smaller) relative income elasticity. 2.2. Non-Homothetic CES Preferences With non-homothetic CES preferences, the index C is defined by the following aggregator: !a C
55
µa ✏ ✏
✏
1
c a ✏ + !n C
µn ✏ ✏
✏
1
cn ✏ = 1,
(2)
where µi > 0 controls the income elasticity of demand for good i 2 {a, n}. By solving the related optimization problem, the relative income elasticity can be obtained: ⇣ ⌘ @log ccna = µa µn , @logC which is a constant.5 This feature is consistent with the cross-country evidence described in Section 1.
60
3. Quantitative Analysis
65
In this section, we use the model of Restuccia et al. (2008) to illustrate that the non-homothetic CES preference does a better job fitting the exact empirical facts that the Stone-Geary preference is meant to match. We calibrate common parameters (across countries) to match the data from our benchmark economy, the U.S.6 3.1. Model
70
Consider a two-sector economy in which each sector produces one good: the agricultural good a and the non-agricultural good n. The preference of the representative household can be (1) or (2). The households supply total labor N inelastically. Let Yi and Li be the output and the labor input in sector i 2 {a, n}. The production of the agricultural good uses land input Z and a non-agricultural intermediate input X: Ya = X ↵ Z 1 4 See 5 See 6 See
(ALa )
1 ↵
,
Appendix A.3 for the details. Appendix A.4 for the details. Appendix B.1 for details of the data we use in our quantitative analysis.
4
Parameter
Calibrated Value
Target
✏ !a !n a ¯
0.9223 0.0048 0.9952 663.99 0.7000 0.4000
Estimated from the U.S. data Estimated from the U.S. data Normalization Agricultural employment share of the U.S. Income share of labor in agriculture Income share of intermediate inputs
↵
Table 1: Calibration for Stone-Geary CES Preferences
75
where 0 < < 1, 0 < ↵ < 1, is the sector-specific productivity in agriculture, and A is the economy-wide productivity. The production technology of the non-agricultural good is linear: Yn = ALn .
80
85
90
95
We assume that all markets are competitive and we let the non-agricultural good be the numeraire. Let pa , w, and ⇡ denote the prices of the agricultural good, labor and the intermediate input, respectively. Due to the friction in the intermediate input market, assume that ⇡ 1. Due to the friction in the labor market, assume that wa = (1 ✓) wn , where 0 < ✓ < 1. 3.2. Stone-Geary CES Calibration. Country-specific parameters A, , ✓, ⇡ and Z/N are calibrated from the data of the corresponding countries.7 All other parameters are common across countries and they are calibrated to match the U.S. data. We choose the same production side parameters and ↵ as in Restuccia et al. (2008). Parameters in the preference are calibrated as follows. We first normalize the weight parameters such that !a + !n = 1. We use the annual data of the U.S. to estimate ✏ and !a based on the first-order condition.8 Then a ¯ is calibrated to match the U.S. agricultural employment share in 1985.9 Notice that !a governs the agricultural employment share in the long run. Hence, our calibration strategy is consistent with Restuccia et al. (2008). Calibrated values are summarized in Table 1. Model Fit. Figure 2 plots the simulated agricultural employment share La /N and agricultural labor productivity Ya /La against their counterparts in the data. The left panel shows the absolute change of agricultural employment share and the right panel shows the relative change (in log scale). On the one hand, the quantitative predictions for rich countries match the data well. On the other 7 ⇡ is assumed to be unity in the U.S. These values are the same as those in Restuccia et al. (2008). 8 See Appendix B.2 for details. 9 The cross-country data from Restuccia et al. (2008) are in 1985.
5
Data Stone-Geary CES
Data Stone-Geary CES Agricultural Employment Share (log-scale)
Agricultural Employment Share
1.00
0.75
0.50
0.25
1.0
0.1
0.00 100
1000 10000 Agricultural Output per Worker (log-scale)
100
1000 10000 Agricultural Output per Worker (log-scale)
Figure 2: Model Fit: Stone-Geary CES Preferences
100
105
hand, since the model predicts significantly larger income elasticities for poor countries, the simulated agricultural employment share of poor countries are significantly larger than what is observed in the data. In particular, for the poorest countries, the relative income elasticity is so large that devoting all labor to agriculture is not enough to meet the demand of the agricultural good in the model.10 In the next subsection, we show that the model with the nonhomothetic CES preferences matches our cross-country data better than the Stone-Geary CES preferences. 3.3. Non-homothetic CES
110
115
Calibration. The only di↵erence in the calibration strategy between the nonhomothetic and the Stone-Geary cases is the identification of the preference parameters. We first use the time series data of the U.S. to estimate ✏, µa and µn based on the equilibrium conditions.11 Without loss of generality, we normalize !a to be 1.12 !n is calibrated to match the agricultural employment share of the U.S in 1985. We summarize the calibrated values in Table 2.13 Model Fit. The simulated agricultural employment share La /N and agricultural labor productivity Ya /La are plotted against their empirical counterparts in Figure 3. In general, the quantitative simulations match the data very well. The reason is that, even though the parameters are estimated from the U.S. data, the relative income elasticity is similar across all countries. 10 We use 100% agricultural employment share to indicate these countries in the graph. Equilibrium does not exist for them. 11 See Appendix B.2 for details. 12 See Appendix B.3 for the justification of this normalization. 13 When µ < ✏, the non-homothetic preference is not globally monotonically increasing or a quasi-concave, but it can be transformed to a locally valid preference. In that case, we search
6
Parameter
Calibrated Value.
Target
✏ !a !n µa µn
0.6100 1.0000 1503.7 0.2537 0.9529 0.7000 0.4000
Estimated from the U.S. data Normalization Agricultural employment share of the U.S. Estimated from the U.S. data Estimated from the U.S. data Income share of labor in agriculture Income share of intermediate inputs
↵
Table 2: Calibration for Non-homothetic CES Preferences
1.00
Data Non-homothetic CES
Data Non-homothetic CES Agricultural Employment Share (log-scale)
Agricultural Employment Share (log-scale)
1.0
0.75
0.50
0.25
0.1
0.00 100
1000 10000 Agricultural Output per Worker (log-scale)
100
1000 10000 Agricultural Output per Worker (log-scale)
Figure 3: Model Fit: Non-homothetic CES Preferences
7
4. Conclusion
120
This paper illustrates that the non-homothetic CES preferences are consistent with cross-country empirical evidence and quantitatively perform better than the prevailing Stone-Geary preferences. Acknowledgements
125
We are grateful to the associate editor and the anonymous referee for helpful suggestions. We acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 71703017, 71703116), Beijing National Natural Science Foundation (Grant No. 9184031), and Center for Economic Development Research of Wuhan University. Appendix: Model and Quantitative Details Supplementary material related to this article can be found online.
130
References Caselli, F. (2005). Accounting for Cross-Country Income Di↵erences. Handbook of Economic Growth, 1 , 679–741. Comin, D., Lashkari, D., & Mestieri, M. (2017). Structural Change with Longrun Income and Price E↵ects. Working Paper , .
135
Gollin, D., Lagakos, D., & Waugh, M. (2014). Agricultural productivity di↵erences across countries. American Economic Review , 104 , 165–170. Lagakos, D., & Waugh, M. (2013). Selection, Agriculture and Cross-Country Productivity Di↵erences. American Economic Review , 103 , 948–980.
140
Restuccia, D., Yang, D. T., & Zhu, X. (2008). Agriculture and aggregate productivity: A quantitative cross-country analysis. Journal of Monetary Economics, 55 , 234–250. Timmer, M., de Vries, G., & de Vries, K. (2015). Patterns of Structural Change in Developing Countries. In J. Weiss, & M. Tribe (Eds.), Routledge Handbook of Industry and Development (pp. 65–83). Routledge.
for the optimal allocation numerically.
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