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A caution about considering possible changes in $$A_T$$ and $$G_T$$ when the composition of waste for treatment changes
The input-output relationships of waste treatment processes are often closely linked to the chemical properties of the treated waste, particularly in incineration processes. The amount of recoverable heat, and thus the potential heat supply for external uses, including power generation, depends on the heat value of the waste. This heat value is strongly influenced by the waste's composition. Therefoer, any change in the composition of waste can significantly impact $$A_T$$ and $$G_T$$.

To address this aspect of waste treatment, especially in incineration, Nakamura and Kondo recommended using engineering information about the relevant treatment processes. They suggest solving the entire model iteratively, which consists of the WIO model and a systems engineering model that incorporates the engineering information.

Alternatively, Tisserant et al proposed addressing this issue by distinguishing each waste by its treatment processes. They suggest transforming the rectangular waste flow matrix ($$n_w \times n_T$$ not into an $$n_T \times n_T$$ matrix as done by Nakamura and Kondo, but into an $$n_T n_W \times n_T n_W$$ matrix. The details of each column element were obtained based on the literature.

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A dynamic situation of economic growth that is depicted by a linear forward difference equation with constant coefficients:

where $$A$$ is a square nonnegative matrix. A balanced-growth solution of $$x(t)$$ is defined as such a nontrivial solution that the proportions $$x_1(t):x_2(t):..:x_n(t)$$ of the components of $$x(t)$$ remain constant over time. It therefore takes the form:

$$ x(t) = \lambda^t x $$

$$\lambda$$ is a constant and an eigenvalue of $$A$$ in the sense that

$$ \lambda x = A x, x \geq 0 $$

Since $$A$$ is a positive matrix, its Frobenius root $$\lambda(A)$$is positive, and hence for a large $$t$$:

$$ x(t) = \lambda(A)^t x $$

The balanced-growth solution in ($$) lead to the eigen value problem:

$$ \lambda x = A x,  x \geq 0 $$

The Leontef IO model with capital formation endogenized
The IO model discussed above is static because it does not describe the evolution of the economy over time: it does not include different time periods. Dynamic Leontief models are obtained by endogenizing the formation of capital stock over time. Denote by $$y^I$$the vector of capital formation, with $$y^I_i$$ its $$i$$th element, and by $$I_{ij}(t)$$ the amount of capital good $$i$$ (for example, a blade) used in sector $$j$$ ( for example, wind power generation), for investment at time $$t$$. We then have

$$ y^I_i(t) = \sum_j I_{ij}(t) $$

We assume that it takes one year for investment in plant and equipment to become productive capacity. Denoting by $$K_{ij}(t)$$ the stock of $$i$$ at the beginning of time $$t$$, and by $$\delta \in (0,1]$$ the rate of depreciation, we then have:

Here, $$\delta_{ij}K_{ij}(t)$$ refers to the amount of capital stock that is used up in year $$t$$. Denote by $$\bar{x}_j(t)$$ the productive capacity in $$t$$, and assume the following proportionalty between $$K_{ij}(t)$$ and $$\bar{x}_j(t)$$:

The matrix $$B=[b_{ij}]$$ is called the capital coefficient matrix. From ($$) and ($$), we obtain the following expression for $$y^I$$:

$$ y^I(t) = B\bar{x}(t+1) + (\delta - I)\bar{x}(t) $$

Assuming that the productive capacity is always fully utilized, we obtain the following expression for ($$) with endogenized capital formation:

$$ x(t) = Ax(t)+Bx(t+1)+ (\delta-I)Bx(t) + y^o(t), $$

where $$y^o$$ stands for the items of final demand other than $$y^I$$.

Rearranged, we have

$$ \begin{align} Bx(t+1) &= (I-A + (I-\delta)B)x(t) - y^o(t)\\ &= (I - \bar{A} + B)x(t) - y^o(t) \end{align} $$

wehere $$\bar{A}=A + \delta B$$.

If $$B$$ is non-singular, this model could be solved for $$x(t+1)$$ for given $$x(t)$$ and $$y^o(t)$$:

$$ x(t+1) = [I + B^{-1}(I- \bar{A})]x(t) - B^{-1}y^o(t) $$

This is the Leontief dynamic forward-looking model

A caveat to this model is that $$B$$ will, in general, be singular, and the above formulation cannot be obtained. This is because some products, such as energy items, are not used as capital goods, and the corresponding rows of the matrix $$B$$ will be zeros. This fact has prompted some researchers to consolidate the sectors until the non-singularity of $$B$$ is achieved, at the cost of sector resolution. Apart from this feature, many studies have found that the outcomes obtained for this forward-looking model invariably lead to unrealistic and widely fluctuating results that lack economic interpretation. This has resulted in a gradual decline in interest in the model after the 1970s, although there is a recent increase in interest within the context of disaster analysis.

Efficient path of capital formation
Notwithstanding the above property, the dynamic model had been the subject of intensive research in the 1960s to 1970s within the context of the multi-sector growth model. Of particular interest is the case of a closed model, where $$y^C(t)$$ is endogenized as

$$ \check{y}(t) = c v x(t) $$

where $$v$$ stands for the ratio of value added and $$c$$ for the consumption basket of househjods. We then have

$$ \begin{align} x(t+1) &= [I + B^{-1}(I-\check{A})]x(t)\\ &= D x(t) \end{align} $$

where $$\check{A}=\bar{A}+cv$$. The general solution to this system is

$$ x(t) = \sum_{i=1}^{n}c_i \lambda_i^t v_i $$

where $$\lambda_i$$, are the eigenvalues, $$v_i$$ are the corresponding eigenvectors, and $$c_i$$ are constants determined by the initial condition $$x(0)$$.

If $$D$$ is a positive matrix, it has a Frobenis root, $$\lambda_1$$, the largest characteristic value that is positive and real, with its characteristic vector also positive. Because $$\lambda_1$$ is the largest characteristic value and positive, the term involving it will become dominant as $$t$$ becomes large.

$$ x(t) \approx c_1 \lambda_1^t v_1 $$