Optimal environment-friendly economic restructuring: the United States–China cooperation case study

Abstract

This paper discusses a model for the restructuring of national economies for the purpose of achieving optimal growth under conditions of decreased energy consumption and greenhouse gas emissions. The discussion combines input–output and factorial-decomposition models, and applies projected gradient and factor analysis to find the optimal structural changes that serve all three goals. A comparative analysis of the economies of the United States and China, including opportunities for cooperative restructuring, serves as a case study.

Appendix

In this section, we briefly describe the mathematical tools used to construct the model presented in this paper.

1.1 Factorial decomposition

The technique of factorial decomposition may be traced back to the works of Divisia (1925), Laspeyres (1871), Meerovich (1974), Paasche (1874), Sheremet (1979), and Sheremet et al. (1971).

Laspeyres (1871) and Paasche (1874) introduced a method of estimating the impacts of the factors on the change in the value of the resultant economic indicator. They assumed that a resultant indicator z is a product of the ordered set of factorial indicators x ₁ , x ₂ , …, x _n :

$$ z = x_{1} x_{2} \ldots x_{n} $$

(21)

and suggested decomposing the change in the value of z by the factors x _i by changing the factors in the order of their lower indexes:

$$ \begin{aligned} & \Delta z = \Delta z\left[ {x_{1} } \right] + \Delta z\left[ {x_{2} } \right] + \cdots + \Delta z\left[ {x_{n} } \right], \\ & \Delta z\left[ {x_{i} } \right] = (x_{11} x_{21} \ldots x_{i1} x_{{\left( {i + 1} \right)0}} , \ldots ,x_{n0} ) - (x_{11} x_{21} \ldots ,x_{i1} x_{{\left( {i + 1} \right)0}} , \ldots ,x_{n0} ) \\ \end{aligned} $$

(22)

where ∆z and ∆z[x _i ] stand for the change in the resultant indicator z and its components caused by the change in the factorial indicators x _i , respectively, and x _i0 and x _i1 are the values of the indicator x _i in the base and current periods, correspondingly. As follows from the formula (22), at each step only one factor x _i changes. This approach was further developed in the Index Number Theory (see, for example, Diewert and Nakamura 1993).

Divisia (1925) assumed continuous change in the factorial indicators in time: x _i  = x _i (t), thus avoiding the necessity of ordering them. The following factorial decomposition was obtained:

$$ \Delta z = \varSigma \Delta z[x_{i} ] = {\varvec{\Sigma}}\smallint x_{1} x_{2} \ldots x_{i - 1} x_{i} ' x_{i + 1} \ldots x_{n} dt, $$

(23)

where the symbol Σ stands for the summation, the ∫ symbol stands for the integration, and the factors x _i in the integrand of the formula (23) are substituted term wise for their derivatives x ^’ _i . Integration is done with respect to the model time t over the interval [0, 1]. Each additive term in the formula (23) is attributed to a factor x _i , correspondingly:

$$ \Delta x\left[ {x_{i} } \right] \, = \smallint x_{1} x_{2} \ldots x_{i - 1} x_{i} ' x_{i + 1} \ldots x_{n} dt \, \quad i \, = \, 1, \ldots ,n $$

(24)

Factor models similar to (21) have been studied intensively by Sheremet (1979) and his school of economic analysis. The work of Sheremet et al. (1971) extended the Divisia formula (23) from a product of factorial indicators to arbitrary continuously differential functions. Assuming

$$ z = f\left( x \right) = f\left( {x_{1} , \ldots ,x_{n} } \right) $$

(25)

the authors obtained the factorial decomposition as

$$ \varDelta z = z_{1} - z_{0} = \int\limits_{L} {dz} = \int\limits_{L} {f_{1}^{'} dx_{1} } + \int\limits_{L} {f_{2}^{'} dx_{2} + \cdots } + \int\limits_{L} {f_{n}^{'} dx_{n} } $$

(26)

where

$$ \varDelta z[x_{i} ] = \int\limits_{L} {f_{i}^{'} dx_{i} } = \int\limits_{{t_{0} }}^{{t_{1} }} {f_{i}^{'} } x_{i}^{'} \,dt, $$

(27)

f _i ’ is a partial derivative of the function f(x ₁,…,x _n ) with respect to the i-th argument, and x _i ’ = dx _i /dt.

Formula (26) may be rewritten in the vector form as

$$ \varDelta \varvec{z} = \int\limits_{L} {\nabla \varvec{z}^{T} d\varvec{X}} , $$

(28)

where ∆z is a row vector of factorial decomposition with components ∆z[x _i ], vector

$$ \nabla z = \; < f_{1}^{'} , \ldots ,f_{n}^{'} > $$

(29)

is the gradient column vector of the function f(x ₁ , …, x _n ), upper index T stands for the transposition, and d X is a diagonal matrix with elements dx ₁ , dx ₂ ,…,dx _n .

Meerovich (1974) demonstrated, however, that when structural change is analyzed, the Laspeyres-Paasche approach may lead to results that contradict economic common sense. An example borrowed from that publication may be found in Vaninsky (2014a). Vaninsky and Meerovich (1978) revealed further that the same effect is observed when the Divisia approach is used. That publication also revealed, at the empirical level, how this contradiction may be overcome by using the projected gradient. An axiomatic theory was developed in Vaninsky (1983, 1987). Its brief description may be found in Maital and Vaninsky (2000). This paper applies that theory to the factorial decomposition of energy consumption and CO₂ emissions.

1.2 Projected gradient

In the discussion above, we used the projected-gradient method to find a vector of optimal restructuring. It is known, see, for example, Kaplan (1993), that for a continuously differentiable function of several variables f (x), the gradient vector

$$ \nabla \varvec{f} = \, < f_{1}^{'} , \ldots ,f_{n}^{'} > , $$

(30)

where f ^’ _i is the partial derivative with respect to the i-th argument, provides both the direction and the magnitude of steepest increase in the value of the function. The antigradient vector points in the opposite direction and determines the magnitude of steepest decrease. Optimization algorithms use this property by making steps in the direction of the gradient or antigradient for maximization or minimization, respectively (see, for example, Polak 1997). However, if the variables x = < x ₁ ,…,x _n  > were bound by the equations of interconnections—i.e., in the problems of constrained optimization—the endpoint of the gradient vector might leave the surface defined by these equations. In that case, the gradient vector becomes unfeasible. To avoid the infeasibility, Rosen (1960, 1961) suggested projecting the gradient vector on a plane tangent to the surface defined by the equations of interconnection. This projected-gradient method was developed further in Bertsekas (1976), Calamai and More (1987), Grana Drummond and Iusem (2004), Goldstein (1964), and Levitin and Polyak (1966), among others. In applications, one advantage of the projected-gradient method is the ability to set up optimization problems in terms of a complete set of convenient variables, rather than their contraction to the subset of functionally independent variables.

For the purposes of our discussion here, we use a particular property of the projected gradient, proved in Maital and Vaninsky (1999). That property indicates that the projected gradient provides both the direction and the magnitude of steepest increase in the value of the function limited to the surface of the interconnection of the variables. We use this theorem to find the optimal direction in the presence of the equations of the interconnections between the arguments of the objective function. It is known, for example, from Albert (1972), that if the arguments of a function f(x) are interconnected by a system of equations Φ ( x ) = 0, then the projected gradient may be found as

$$ Proj_{H} \nabla \varvec{f} = \nabla \varvec{f}^{T} \left( {\varvec{I} - \varvec{HH}^{ + } } \right), $$

(31)

where

$$ \varvec{H} = \left( {h_{ik} } \right) = \left( {\partial \varPhi_{k} /\partial x_{i} } \right), \, i = 1, \ldots ,p; \, k = 1, \ldots ,q, $$

(32)

p is the number of variables, q is the number of the equations of the interconnection, I is the identity matrix, H is the Jacobian matrix, and H ⁺ is the general inverse of the matrix H. If the columns of the matrix H are linearly independent, then

$$ \varvec{H}^{ + } = \, \left( {\varvec{H}^{{\varvec{T}}} \varvec{H}} \right)^{ - 1} \varvec{H}^{{\varvec{T}}} $$

(33)

(see Kaplan 1993, for details).

In the factorial models for energy consumption and CO₂ emissions the Eq. (4) is the only one in the system of the equations of interconnections Φ ( x ) = 0. In this case, the Jacobian matrix H in formula (32) becomes a one-column matrix

$$ \varvec{H} = < 1,1, \ldots ,1 >^{T} . $$

(34)

As shown in Vaninsky and Meerovich (1978) and Vaninsky (1983, 1987), the projected gradient on the hyperplane defined by the Eq. (4) for the CO₂ emissions turns out to be (14), and the projected antigradient becomes (15). Similar expressions were obtained for the energy consumption indicator, as given by formula (18).

1.3 Relationship between the coefficient of correlation and cosine of an angle

The following comment is useful for the development of the computational procedure and the interpretation of the results obtained thereby: An ordered set of observations may be viewed from two perspectives. The first is a geometric approach that treats each set as a multidimensional vector. The measure of the closeness of two vectors is a cosine of the angle between them. The second approach considers the sets of observations as the occurrences of two random variables. In this case, the closeness of a pair of sets is measured by a coefficient of correlation. From the vector perspective, a cosine of the angle between two vectors u = < u ₁, u ₂, …, u _n  > and v = <v ₁, v ₂, …, v _n  > is as follows:

$$ cos\,\theta = \frac{{\varvec{u} \cdot \varvec{v}}}{{\left\| \varvec{u} \right\|\,\left\| \varvec{v} \right\|}} = \frac{{u_{1} v_{1} + \cdots + u_{n} v_{n} }}{{\sqrt {u_{1}^{2} + \cdots + u_{n}^{2} } \,\sqrt {v_{1}^{2} + \cdots + v_{n}^{2} } }} $$

(35)

where the symbols “·” and || || stand for the dot product and the norm of a vector, respectively (see Spiegel et al. 2009, for details). On the other hand, if two ordered sets of observations are considered as the occurrences of two random variables, their closeness is measured by a coefficient of correlation r, equal to

$$ r = \frac{{\sum {\left( {u_{i} - \bar{u}} \right)} \left( {v_{i} - \bar{v}} \right)}}{{\sqrt {\sum {\left( {u_{i} - \bar{u}} \right)^{2} \left( {v_{i} - \bar{v}} \right)^{2} } } }}, $$

(36)

where the upper bar symbol “$\bar{\,}$” stands for the average value (see, for example, Spiegel and Stephens 2014). The value of r ² is referred to in statistical literature as a coefficient of determination. It defines a share of the variance of one random variable that may be explained by another random variable. Both measures of closeness—cos θ and r—vary in the same range from −1 to 1, and the greater any of them, the closer the two sets of observations are to each other. In our case, as follows from Eqs. (9), (15), and (18), the average values in formula (36) are equal to zero. This means that, for the purposes of this paper, we can use the geometric or the statistical interpretation interchangeably, as appropriate.

The problem of finding a vector closest to each of the vectors in a set is typical of factor analysis (see Thompson 2004, for details). In this discussion we use a simple version of this technique, with three vectors in a set and one factor vector. Mathematically, we are seeking a vector d as a convex combination of the normalized gradient/antigradients direction vectors

$$ \hat{\varvec{d}}_{GDP} = \varvec{d}_{GDP} /\left\| {\varvec{d}_{GDP} } \right\|,\;\,\hat{\varvec{d}}_{e} = \varvec{d}_{e} /\left\| {\varvec{d}_{e} } \right\|,\;\;\hat{\varvec{d}}_{c} = \varvec{d}_{c} /\left\| {\varvec{d}_{c} } \right\|. $$

(37)

The optimal factor vector d maximizes the sum of squares of the pairwise coefficients of correlation or, in the geometric terms, the sum of squares of the cosines of the angles formed with each gradient or antigradient. Mathematical statement of the problem is given by formulas (19) and (20).