A.1. The model

Our model focuses solely on the implications for optimal consumption and production behavior within each period. The advantage of this static approach is that the first-order conditions for the stand-in household and the stand-in firm are given only by observed current variables and we do not have to take a stand on the exact nature of intertemporal opportunities available to households and firms (i.e., the appropriate interest rates for borrowing and lending). In what follows, subscript t, which indicates time period, is omitted in each variable.

The model has two sectors of activity—the nonservices sector (T) and the services sector (S). The nonservices sector includes agriculture and manufacturing. The production function in each sector is assumed to be Cobb-Douglas with constant returns to scale. The static approach allows all variables to change in each period without any exceptions. Then, capital income shares of two sectors, θ<sup>T</sup> and θS, are also assumed to change in each period. The output of services can be used for consumption (CS) and investment (IS). The output of the nonservices sector can be disaggregated into consumption (CT), investment (IT), and net exports (NEXT). The shares of investments and net exports in each sector are exogenously determined in this model. The production structures and their market clearings in the product markets are as follows:

$$\begin{aligned} Y\_S &= A\_S K\_S^{\theta\_S} L\_S^{1-\theta\_S} = \mathbb{C}\_S + I\_S \\ Y\_T &= A\_T K\_T^{\theta\_T} L\_T^{1-\theta\_T} = \mathbb{C}\_T + I\_T + \text{NEX}\_T \end{aligned} \tag{5}$$

where Yi, Ai, Ki, Li are the value added, total factor productivity (TFP), capital stock, employment in i ¼ T, S, respectively. All resources for production (Ki, Li) are fully used, as shown:

$$\begin{aligned} K\_S + K\_T &= K\\ L\_S + L\_T &= L \end{aligned} \tag{6}$$

Pi is the price of sector i, while R and W denote rental rates of capital and employment, respectively, both expressed in nominal currency. The term di (di ≥ 0) denotes a sector-specific

The Declining Labor Income Shares Revisited: Intersectoral Production Linkage in Global Value Chains

<sup>¼</sup> <sup>1</sup> � <sup>θ</sup><sup>T</sup> θT

KT LT

1 þ dS 1 þ dT

<sup>θ</sup><sup>S</sup> with the assumption di <sup>¼</sup> 0. The relative price PS=PT should be the

(9)

http://dx.doi.org/10.5772/intechopen.81316

α<sup>T</sup> þ ð Þ 1 � θLS=θ<sup>N</sup> . Then, α<sup>T</sup> is obtained as

(10)

37

distortion.

basic sectoral data.

<sup>θ</sup><sup>T</sup> <sup>=</sup>ASð Þ KS=LS

(βT), <sup>β</sup><sup>T</sup> is obtained as <sup>β</sup><sup>T</sup> <sup>¼</sup> <sup>θ</sup>LT=θN<sup>∗</sup>

ð Þ 1 þ dS =ð Þ 1 þ dT in Eq. (10) as follows:

ð Þ KT=LT

Dividing these two equations by each other gives:

1 � θ<sup>S</sup> θS

> PS PT

KS LS 

<sup>¼</sup> <sup>1</sup> � <sup>θ</sup><sup>T</sup> 1 � θ<sup>S</sup>

From the second equation in Eq. (8), the implications for relative prices can be derived as:

AT AS k θT T k θS S

In the above equation, kS ¼ KS=LS and kT ¼ KT=LT (the rates of capital deepening for two sectors). The labor income shares of two sectors, 1 � θ<sup>S</sup> and 1 � θT, and the rates of capital deepening for two sectors, kS and kT, are calculated analytically using the model and some

From Eq. (10) in the model, relative labor income share, 1ð Þ � θ<sup>T</sup> =ð Þ 1 � θ<sup>S</sup> , can be calculated by dividing relative price PS=PT by relative labor productivity ð Þ YT=LT =ð Þ¼ YS=LS AT

With macroeconomic labor income share (θN), the ratio of the nonservices sector GDP to total GDP (αT) and the ratio of nonservices sector labor compensation to total labor compensation

α<sup>T</sup> ¼ ð Þ θ<sup>N</sup> � θLS =ð Þ θLT � θLS where θLT and θLS are the labor income shares of the nonservices and services sectors, respectively. Among the variables in this equation, α<sup>T</sup> ¼ ð Þ θ<sup>N</sup> � θLS = ð Þ θLT � θLS , α<sup>T</sup> and θ<sup>N</sup> are the known data. Then, by solving the system of equations involving two variables, θLT and θLS or α<sup>T</sup> ¼ ð Þ θ<sup>N</sup> � θLS =ð Þ θLT � θLS and θLT=θLS (¼ ð Þ 1 � θ<sup>T</sup> =ð Þ 1 � θ<sup>S</sup> ), the labor income shares of the two sectors, θLT and θLS, are obtained. By substituting these labor income shares into Eq. (9), the relative capital deepening rate ð Þ KT=LT =ð Þ KS=LS can be calculated. The two sectors' employment data, LT and LS, are available and then the capital ratio, KT=KS, across the two sectors can be calculated. By using total capital stock data K ð Þ K ¼ KS þ KT , the capital stock of the sectors, KT and KS, are obtained. At this stage, we can calculate the rates of capital deepening of two sectors, kS ¼ KS=LS and kT ¼ KT=LT, respectively. We calibrate the distortion parameter (di) as follows. As mentioned above, di ¼ 0 is set with the assumption that there is perfect factor mobility across the two sectors, and we factorize the relative price changes into relative labor income share and relative labor productivity changes, referring to Eq. (10). The result is disappointing, especially for the period up to the 1970s in Japan and up to the 1980s in Korea, when the per-capita incomes of these countries were relatively low. The analytically calculated relative labor income share and the relative capital deepening differ greatly from the actual data. Based on the result of this simulation, we assume that the degree of distortion depends on the per-capita income level (x) and redefine

absolute relative price in order to obtain relative labor income share 1ð Þ � θ<sup>T</sup> =ð Þ 1 � θ<sup>S</sup> .

<sup>α</sup><sup>T</sup> <sup>¼</sup> <sup>θ</sup>LS=θN<sup>∗</sup>

We assume the period utility function, u Cð Þ <sup>S</sup>; CT is of the form:

$$\mu(\mathbb{C}\_S, \mathbb{C}\_T) = \left[ \omega^\frac{1}{\nu} \overline{\mathbb{C}\_S^{\frac{\varepsilon-1}{\varepsilon}}} + (1-\omega)^\frac{1}{\nu} \overline{\mathbb{C}\_T^{\frac{\varepsilon-1}{\varepsilon}}} \right]^\frac{\varepsilon}{\varepsilon-1} \tag{7}$$

#### A.2. Optimality conditions for production side

Production side efficiency that is used for deriving factor shares of income and the rate of capital deepening for the two sectors is now derived. There is perfect factor mobility across the two sectors if sector-specific distortions to production factors (capital and employment) are cleared. The first-order conditions for the stand-in firm in sector i are given by:

$$\begin{split} R &= \frac{1}{1+d\_S} P\_S \theta\_S A\_S \left(\frac{K\_S}{L\_S}\right)^{\theta\_S - 1} = \frac{1}{1+d\_T} P\_T \theta\_T A\_T \left(\frac{K\_T}{L\_T}\right)^{\theta\_T - 1} \\\ W &= \frac{1}{1+d\_S} P\_S (1 - \theta\_S) A\_S \left(\frac{K\_S}{L\_S}\right)^{\theta\_S} = \frac{1}{1+d\_T} P\_T (1 - \theta\_T) A\_T \left(\frac{K\_T}{L\_T}\right)^{\theta\_T} \end{split} \tag{8}$$

Pi is the price of sector i, while R and W denote rental rates of capital and employment, respectively, both expressed in nominal currency. The term di (di ≥ 0) denotes a sector-specific distortion.

Dividing these two equations by each other gives:

A.1. The model

36 Globalization

Our model focuses solely on the implications for optimal consumption and production behavior within each period. The advantage of this static approach is that the first-order conditions for the stand-in household and the stand-in firm are given only by observed current variables and we do not have to take a stand on the exact nature of intertemporal opportunities available to households and firms (i.e., the appropriate interest rates for borrowing and lending). In

The model has two sectors of activity—the nonservices sector (T) and the services sector (S). The nonservices sector includes agriculture and manufacturing. The production function in each sector is assumed to be Cobb-Douglas with constant returns to scale. The static approach allows all variables to change in each period without any exceptions. Then, capital income shares of two sectors, θ<sup>T</sup> and θS, are also assumed to change in each period. The output of services can be used for consumption (CS) and investment (IS). The output of the nonservices sector can be disaggregated into consumption (CT), investment (IT), and net exports (NEXT). The shares of investments and net exports in each sector are exogenously determined in this model. The

what follows, subscript t, which indicates time period, is omitted in each variable.

production structures and their market clearings in the product markets are as follows:

<sup>S</sup> ¼ CS þ IS

where Yi, Ai, Ki, Li are the value added, total factor productivity (TFP), capital stock, employment in i ¼ T, S, respectively. All resources for production (Ki, Li) are fully used, as shown:

KS þ KT ¼ K

εC ε�1 ε

Production side efficiency that is used for deriving factor shares of income and the rate of capital deepening for the two sectors is now derived. There is perfect factor mobility across the two sectors if sector-specific distortions to production factors (capital and employment) are

> <sup>¼</sup> <sup>1</sup> 1 þ dT

> > <sup>¼</sup> <sup>1</sup> 1 þ dT

KS LS � �<sup>θ</sup><sup>S</sup>

<sup>S</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>ω</sup> <sup>1</sup>

h i <sup>ε</sup>

εC ε�1 ε T

PTθTAT

KT LT � �<sup>θ</sup>T�<sup>1</sup>

PTð Þ 1 � θ<sup>T</sup> AT

KT LT � �<sup>θ</sup><sup>T</sup>

<sup>T</sup> ¼ CT þ IT þ NEXT

LS <sup>þ</sup> LT <sup>¼</sup> <sup>L</sup> (6)

ε�1

(5)

(7)

(8)

<sup>S</sup> <sup>L</sup><sup>1</sup>�θ<sup>S</sup>

<sup>T</sup> <sup>L</sup><sup>1</sup>�θ<sup>T</sup>

YS <sup>¼</sup> ASK<sup>θ</sup><sup>S</sup>

YT <sup>¼</sup> ATK<sup>θ</sup><sup>T</sup>

We assume the period utility function, u Cð Þ <sup>S</sup>; CT is of the form:

A.2. Optimality conditions for production side

PSθSAS

<sup>R</sup> <sup>¼</sup> <sup>1</sup> 1 þ dS

<sup>W</sup> <sup>¼</sup> <sup>1</sup> 1 þ dS u Cð Þ¼ <sup>S</sup>;CT <sup>ω</sup><sup>1</sup>

cleared. The first-order conditions for the stand-in firm in sector i are given by:

KS LS � �<sup>θ</sup>S�<sup>1</sup>

PSð Þ 1 � θ<sup>S</sup> AS

$$\frac{1-\theta s}{\theta\_S} \left(\frac{K\_S}{L\_S}\right) = \frac{1-\theta r}{\theta\_T} \left(\frac{K\_T}{L\_T}\right) \tag{9}$$

From the second equation in Eq. (8), the implications for relative prices can be derived as:

$$\frac{P\_S}{P\_T} = \frac{1 - \Theta\_T}{1 - \Theta\_S} \frac{A\_T}{A\_S} \frac{k\_T^{\theta\_T}}{k\_S^{\theta\_S}} \frac{1 + d\_S}{1 + d\_T} \tag{10}$$

In the above equation, kS ¼ KS=LS and kT ¼ KT=LT (the rates of capital deepening for two sectors). The labor income shares of two sectors, 1 � θ<sup>S</sup> and 1 � θT, and the rates of capital deepening for two sectors, kS and kT, are calculated analytically using the model and some basic sectoral data.

From Eq. (10) in the model, relative labor income share, 1ð Þ � θ<sup>T</sup> =ð Þ 1 � θ<sup>S</sup> , can be calculated by dividing relative price PS=PT by relative labor productivity ð Þ YT=LT =ð Þ¼ YS=LS AT ð Þ KT=LT <sup>θ</sup><sup>T</sup> <sup>=</sup>ASð Þ KS=LS <sup>θ</sup><sup>S</sup> with the assumption di <sup>¼</sup> 0. The relative price PS=PT should be the absolute relative price in order to obtain relative labor income share 1ð Þ � θ<sup>T</sup> =ð Þ 1 � θ<sup>S</sup> .

With macroeconomic labor income share (θN), the ratio of the nonservices sector GDP to total GDP (αT) and the ratio of nonservices sector labor compensation to total labor compensation (βT), <sup>β</sup><sup>T</sup> is obtained as <sup>β</sup><sup>T</sup> <sup>¼</sup> <sup>θ</sup>LT=θN<sup>∗</sup> <sup>α</sup><sup>T</sup> <sup>¼</sup> <sup>θ</sup>LS=θN<sup>∗</sup> α<sup>T</sup> þ ð Þ 1 � θLS=θ<sup>N</sup> . Then, α<sup>T</sup> is obtained as α<sup>T</sup> ¼ ð Þ θ<sup>N</sup> � θLS =ð Þ θLT � θLS where θLT and θLS are the labor income shares of the nonservices and services sectors, respectively. Among the variables in this equation, α<sup>T</sup> ¼ ð Þ θ<sup>N</sup> � θLS = ð Þ θLT � θLS , α<sup>T</sup> and θ<sup>N</sup> are the known data. Then, by solving the system of equations involving two variables, θLT and θLS or α<sup>T</sup> ¼ ð Þ θ<sup>N</sup> � θLS =ð Þ θLT � θLS and θLT=θLS (¼ ð Þ 1 � θ<sup>T</sup> =ð Þ 1 � θ<sup>S</sup> ), the labor income shares of the two sectors, θLT and θLS, are obtained. By substituting these labor income shares into Eq. (9), the relative capital deepening rate ð Þ KT=LT =ð Þ KS=LS can be calculated. The two sectors' employment data, LT and LS, are available and then the capital ratio, KT=KS, across the two sectors can be calculated. By using total capital stock data K ð Þ K ¼ KS þ KT , the capital stock of the sectors, KT and KS, are obtained. At this stage, we can calculate the rates of capital deepening of two sectors, kS ¼ KS=LS and kT ¼ KT=LT, respectively.

We calibrate the distortion parameter (di) as follows. As mentioned above, di ¼ 0 is set with the assumption that there is perfect factor mobility across the two sectors, and we factorize the relative price changes into relative labor income share and relative labor productivity changes, referring to Eq. (10). The result is disappointing, especially for the period up to the 1970s in Japan and up to the 1980s in Korea, when the per-capita incomes of these countries were relatively low. The analytically calculated relative labor income share and the relative capital deepening differ greatly from the actual data. Based on the result of this simulation, we assume that the degree of distortion depends on the per-capita income level (x) and redefine ð Þ 1 þ dS =ð Þ 1 þ dT in Eq. (10) as follows:

$$\frac{1+d\_S}{1+d\_T} = \exp\left(\alpha x + \beta\right) \tag{11}$$

where α assumes a negative value as the degree of the distortion diminishes along economic development. We calibrate values of α and β to dissipate the difference between the simulated results and the real data of relative labor income share and relative capital deepening. The results are α = �0.0001 and β = 0.4. Eq. (11) is called the "implied distortion index" and is applied to all sample countries.
