Editing Input–output model (section)

===The Leontief 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 <math>y^I</math>the vector of capital formation, with <math>y^I_i</math> its <math>i</math>th element, and by  <math>I_{ij}(t)</math> the amount of capital good <math>i</math> (for example, a blade) used in sector <math>j</math> ( for example, wind power generation), for investment at time <math>t</math>. We then have 

<math>
y^I_i(t) = \sum_j I_{ij}(t)
</math>

We assume that it takes one year for investment in plant and equipment to become productive capacity. Denoting by <math>K_{ij}(t)</math> the stock of <math>i</math> at the beginning of time <math>t</math>, and by <math>\delta \in (0,1]</math> the rate of depreciation, we then have:
{{NumBlk|::|
<math>
K_{ij}(t+1) = I_{ij}(t) + (1-\delta_{ij})K_{ij}(t)
</math>
|{{EquationRef|2}}}}
Here, <math>\delta_{ij}K_{ij}(t)</math> refers to the amount of capital stock that is used up in year <math>t</math>. 
Denote by <math>\bar{x}_j(t)</math> the productive capacity in <math>t</math>, and assume the following proportionalty between <math>K_{ij}(t)</math> and <math>\bar{x}_j(t)</math>:

{{NumBlk|::|
<math>
K_{ij}(t) = b_{ij}\bar{x}_j(t)
</math>
|{{EquationRef|3}}}}
The matrix <math>B=[b_{ij}]</math> is called the capital coefficient matrix. 
From ({{EquationNote|2}}) and ({{EquationNote|3}}), we obtain the following expression for <math>y^I</math>: 

<math>
y^I(t) = B\bar{x}(t+1) + (\delta - I)\bar{x}(t)
</math>

Assuming that the productive capacity is always fully utilized, we obtain the following expression for ({{EquationNote|1}}) with endogenized capital formation:

<math>
x(t) = Ax(t)+Bx(t+1)+ (\delta-I)Bx(t) + y^o(t), 
</math>

where <math>y^o</math> stands for the items of final demand other than <math>y^I</math>.

Rearranged, we have

<math>
\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}
</math>

wehere <math>\bar{A}=A + \delta B</math>. 

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

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

This is the '''Leontief dynamic forward-looking model'''<ref>Wassily Leontief, Dynamic Analysis, Ch.3. In: W. Leontief et al.(eds.) Studies in the Structure of the American Economy. 1953, New York, Oxford University Press, 53–90.</ref>

A caveat to this model is that <math>B</math> 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 <math>B</math> will be zeros. 
This fact has prompted some researchers to consolidate the sectors until the non-singularity of <math>B</math> is achieved, at the cost of sector resolution.<ref>{{Cite journal |last=Jorgenson |first=Dale W. |date=February 1961 |title=Stability of a Dynamic Input-Output System |url=https://doi.org/10.2307/2295708 |journal=The Review of Economic Studies |volume=28 |issue=2 |pages=105–116 |doi=10.2307/2295708 |jstor=2295708 |issn=0034-6527|url-access=subscription }}</ref><ref>{{Cite journal |last=Tsukui |first=Jinkichi |date=1968 |title=Application of a Turnpike Theorem to Planning for Efficient Accumulation: An Example for Japan |url=https://www.jstor.org/stable/1909611 |journal=Econometrica |volume=36 |issue=1 |pages=172–186 |doi=10.2307/1909611 |jstor=1909611 |issn=0012-9682|url-access=subscription }}</ref> 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.<ref>Dorfman, Robert, Paul Anthony Samuelson, and Robert M. Solow. ''Linear programming and economic analysis''. RAND Corporation, 1958. Chapter 11. </ref><ref>Jinkichi Tsukui, (1961) On a Theorem of Relative Stability. International Economic Review, 2, 229–230.</ref><ref>{{Cite journal |last=Bródy |first=A. |date=January 1995 |title=Truncation and Spectrum of the Dynamic Inverse |url=http://www.tandfonline.com/doi/full/10.1080/09535319500000022 |journal=Economic Systems Research |language=en |volume=7 |issue=3 |pages=235–248 |doi=10.1080/09535319500000022 |issn=0953-5314|url-access=subscription }}</ref> 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.<ref>{{Cite journal |last1=Steenge |first1=Albert E. |last2=Reyes |first2=Rachel C. |date=2020-10-01 |title=Return of the capital coefficients matrix |url=https://www.tandfonline.com/doi/full/10.1080/09535314.2020.1731682 |journal=Economic Systems Research |language=en |volume=32 |issue=4 |pages=439–450 |doi=10.1080/09535314.2020.1731682 |issn=0953-5314}}</ref>